Quantitative Carrier Transport in Quantum Dot Photovoltaic ...

Quantitative Carrier Transport in Quantum Dot Photovoltaic

Solar Cells from Novel Photocarrier Radiometry and Lock-in

Carrierography

by

Lilei Hu

A thesis submitted in conformity with the requirements

for the degree of Doctor of Philosophy

Department of Mechanical and Industrial Engineering

University of Toronto

© Copyright by Lilei Hu, 2017

Toronto, Canada

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Quantitative Carrier Transport in Quantum Dot Photovoltaic

Solar Cells from Novel Photocarrier Radiometry and Lock-in

Carrierography

Lilei Hu

Doctor of Philosophy

Department of Mechanical and Industrial Engineering

University of Toronto

2017

Abstract

Colloidal quantum dots (CQDs) are promising candidates for fabricating large-scale, low-cost,

flexible, and lightweight photovoltaic solar cells. However, their power conversion efficiency

is still insufficient for commercial applications, partly and significantly, due to the not-well-

understood carrier transport mechanisms and the lack of effective characterization techniques.

Addressing these issues, carrier transport kinetics in CQD systems were studied to develop

high-frequency dynamic testing and/or large-area quantitative imaging techniques:

photocarrier radiometry (PCR), and homodyne (HoLIC) and heterodyne (HeLIC) lock-in

carrierographies.

Based on the discrete carrier hopping transport in CQDs, various carrier drift-diffusion

current-voltage (J-V) analytical models and new concepts including the imbalanced carrier

mobilities, reversed Schottky barrier, and double-diode model were developed to

quantitatively interpret carrier transport and J-V characteristics in CQD solar cells. The further

quantitative study of carrier mobility, CQD bandgap energy, phonon-assisted carrier transport,

and open-circuit voltage deficit revealed CQD solar cell efficiency optimization strategies.

Applying these energy transport mechanisms, for the first time, an analytical PCR signal

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generation model for CQD systems was developed from a novel trap-state-mediated carrier

hopping transport theory. Therefore, multiple carrier transport parameters including carrier

hopping lifetime, diffusivity, and diffusion length were extracted to investigate carrier

transport dependencies on temperature, quantum dot size, surface-passivation ligands, and

carrier hopping activation energies. As an imaging extension of PCR, using a heterodyne

method to overcome the limitations of camera frame rate and exposure time of even the state-

of-the-art InGaAs cameras, the first camera-based HeLIC theoretical model for ultrahigh-

frequency (up to 270 kHz) imaging of CQD solar cells was achieved. Therefore, quantitative

imaging of carrier lifetime, diffusivity, and diffusion and drift lengths of CQD solar cells was

accomplished to explore the influences of carrier transport and contact/CQD interface on CQD

solar cells. Also, low-frequency HoLIC large-area imaging evaluated the sample homogeneity

and quality, reflecting preliminary carrier lifetime distribution.

The combination of the novel carrier discrete hopping transport mechanism, J-V models,

PCR, and the lock-in carrierography techniques (HoLIC and HeLIC) shows great potential for

quantitative carrier transport study of CQD solar cells and for fast, all-optical, contactless,

large-area, and nondestructive characterization of commercial photovoltaic materials and

devices.

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Acknowledgements

I am sincerely and deeply indebted to Prof. Andreas Mandelis, my supervisor, inspiration,

and mentor, for his generosity of mind, spirit, and guidance during the course of my Ph.D.

studies. His proficiency, intelligence, and high efficiency ensure the progress of my Ph.D.

research project while these inspirations and impacts on me will benefit me through my whole

life. I would also like to extend my deep appreciation of the freedom that Prof. Mandelis

offered. There is nowhere for me to find a better supervisor than Prof. Mandelis.

I would like to thank all my Ph.D. committee members, Prof. Olivera Kesler and Prof.

Edward H. Sargent, for their constructive feedbacks and encouragement. I would like to thank

Prof. Daniele Fournier and Prof. Axel Guenther for agreeing to act as the external and internal

examiners, respectively.

I would like to thank all my colleagues at the Center for Advanced Diffusion Wave and

Photoacoustic Technologies (CADIPT) for their discussions and friendship. I would especially

like to mention Dr. Xinxin Guo, Dr. Qiming Sun, Dr. Alexander Melnikov, Lixian Liu, Huiting

Huan, Pantea Tavakolian, Dr. Edem Dovlo, Sean Choi, Dr. Wei Wang, Yingcong Zhang, Dr.

Bahman Lashkari, and Dr. Koneswaran Sivagurunathan for their help. Particularly, I would

like to thank Dr. Sun for the very useful and valuable discussions and Dr. Guo for her kind

encouragement and suggestions for my study.

Additionally, I would like to thank Prof. Sargent for inviting me to present my research in

his group meeting and for his new constructive criticism and guidance throughout. I also would

like to thank Prof. Sargent’s team members, Dr. Xinzheng Lan, Dr. Zhenyu Yang, Mengxia

Liu, Grant Walters, Dr. Oleksandr Voznyy, Olivier Ouellette, and Dr. Sjoerd Hoogland for

their kindly collaboration. Dr. Xinzheng Lan, Dr. Zhenyu Yang, and Mengxia Liu are generous

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with their time, materials, devices, insightful comments, and deep knowledge of a variety of

fabrication issues.

Last, but never the least, I would like to express my immense gratitude to my parents, sister,

and grandparents for their support and belief in me.

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Contents

Abstract ................................................................................................................................... ii

Acknowledgements ................................................................................................................ iv

Contents .................................................................................................................................. vi

List of Tables ........................................................................................................................... x

List of Figures ........................................................................................................................ xi

1 Preamble ............................................................................................................................... 1

1.1 The Imperative of Solar Cells ......................................................................................... 1

1.2 The Imperative of Nondestructive Testing of Photovoltaics .......................................... 3

1.3 Objectives of This Work ................................................................................................. 4

1.4 Thesis Outline ................................................................................................................. 7

2 Introduction to Quantum Dots and Colloidal Quantum Dot Solar Cells ..................... 10

2.1 Synthesis of Colloidal Quantum Dots ........................................................................... 10

2.2 Electrical Properties of Colloidal Quantum Dots ......................................................... 13

2.2.1 Effects of Interdot Distance and QD Disorder ....................................................... 14

2.2.2 Effects of Temperature ........................................................................................... 16

2.2.3 Effects of Quantum Dot Size and Polydispersity ................................................... 19

2.3 Colloidal Quantum Dot Solar Cells .............................................................................. 20

2.4 Conclusions ................................................................................................................... 24

3 Non-destructive Testing (NDT) Techniques for Carrier Transport in Quantum Dot

Materials and Solar Cells ..................................................................................................... 26

3.1 Literature Review and Classification ............................................................................ 26

3.2 Traditional Methodologies for CQD Carrier Transport Characterization .................... 30

3.2.1 Short-Circuit Current/Open-Circuit Voltage Decay (SCCD/OCVD) .................... 30

3.2.2 Photoconductance Decay (PCD) ............................................................................ 33

3.2.3 Time-resolved PL (TRPL, transient PL) ................................................................ 35

3.2.4 Carrier Diffusion Length Measurements ................................................................ 38

3.3 Photocarrier Carrier Radiometry (PCR) ....................................................................... 42

3.3.1 Photocarrier Radiometry Instrumentation .............................................................. 42

3.3.2 Theory of Lock-in Amplifier Signal Computation................................................. 44

3.3.3 General Theory of Photocarrier Radiometry .......................................................... 47

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3.4 Homodyne (HoLIC) and Heterodyne (HeLIC) Lock-in Carrierography ...................... 52

3.4.1 Instrumentation and Signal Processing Techniques used in HoLIC and HeLIC

Imaging ............................................................................................................................ 52

3.4.2 Requirements for HeLIC Response to Laser Excitation: Non-linear

Photoluminescence Processes ......................................................................................... 57

3.5 Comparison of Different Techniques and Advantages of PCR and HeLIC ................. 60

3.6 Conclusions ................................................................................................................... 65

4 Quantitative Carrier Transport Study through Current-voltage Characteristics ...... 66

4.1 Introduction ................................................................................................................... 66

4.2 Derivation of Current-voltage Model from Hopping and Discrete Carrier Transport .. 69

4.2.1 Carrier Hopping Diffusivity and Mobility in Quantum Dot Systems .................... 69

4.2.2 Current Density J(x) across CQD Solar Cells ........................................................ 75

4.3 Imbalanced Charge Carrier Mobility and Schottky Junction Induced Anomalous J-V

Characteristics of CQD Solar Cells .................................................................................... 78

4.3.1 CQD Solar Cell Fabrication and Current-voltage Characterization ....................... 79

4.3.2 Double-diode-equivalent Hopping Transport Model ............................................. 82

4.3.3 Origins of Anomalous Current-voltage Curves...................................................... 84

4.3.4 Open-circuit Voltage Origin of CQD Solar Cells .................................................. 89

4.3.5 Temperature-dependent Carrier Hopping Transport and CQD Solar Cell

Performance..................................................................................................................... 93

4.4 Conclusions ................................................................................................................. 101

5 Colloidal Quantum Dot Solar Cell Efficiency Optimization: Impact of Hopping

Mobility, Bandgap Energy, and Electrode-semiconductor Interfaces .......................... 102

5.1 Introduction ................................................................................................................. 102

5.2 Derivation of Carrier Hopping Drift-diffusion J-V Model for CQD Solar Cells........ 104

5.3 Experimental CQD Solar Cell Efficiency Optimization ............................................. 108

5.4 Non-constant Photocurrent in CQD Solar Cells ......................................................... 110

5.5 Impact of Hopping Mobility and Bandgap Energy ..................................................... 118

5.6 Impact of Electrode-semiconductor Interfaces Using Homodyne Lock-in

Carrierography .................................................................................................................. 128

5.7 Conclusions ................................................................................................................. 135

6 Photocarrier Radiometry Study of Quantitative Carrier Transport in CQD Thin

Films ..................................................................................................................................... 138

6.1 Introduction ................................................................................................................. 138

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6.2 PCR Theory for CQDs: Trap-state-mediated Carrier Transport Model ..................... 140

6.3 CQD Thin Film Homogeneity and Optical Properties ............................................... 145

6.4 Carrier Transport Kinetics in Various CQD Thin Films ............................................ 148

6.4.1 Temperature-dependent Carrier Transport Kinetics ............................................. 148

6.4.2 Carrier Hopping Activation Energy and Exciton Binding Energy....................... 153

6.4.3 Ligand- and Size-dependent Carrier Transport Kinetics ...................................... 158

6.5 Fitting Uniqueness and Reliability – Parameter Extraction from PCR ...................... 162

6.6 Conclusions ................................................................................................................. 166

7 Carrier Recombination Mechanism, Energy Band Structure, and Inhomogeneity-

affected Carrier Transport in Perovskite Shelled PbS CQD Thin Films Using PCR and

HoLIC .................................................................................................................................. 168

7.1 Introduction ................................................................................................................. 168

7.2 Experimental Details and CQD Thin Film Synthesis ................................................. 169

7.3 Charge Carrier Recombination Mechanism for PbS CQDs: Nonlinear Response ..... 170

7.4 Energy Band Structure ................................................................................................ 176

7.4.1 Photoluminescence of CQD Thin Films .............................................................. 177

7.4.2 PCR Photothermal Spectra of CQD Thin Films .................................................. 178

7.5 Large-area Imaging and Carrier Transport of CQD Thin Films ................................. 181

7.5.1 Qualitative Large-area Imaging ............................................................................ 181

7.5.2 PCR Characterization of Carrier Transport Parameters ....................................... 183

7.6 Conclusions ................................................................................................................. 188

8 Heterodyne and Homodyne Lock-in Carrierography Imaging of Carrier Transport in

CQD Solar Cells .................................................................................................................. 189

8.1 Introduction ................................................................................................................. 189

8.2 Theories of Homodyne and Heterodyne Lock-in Carrierography .............................. 191

8.3 Carrier Transport Theory of CQD Solar Cells under Modulated Photoexcitation ..... 193

8.4 Quantitative Colloidal Quantum Dot Solar Cell Imaging ........................................... 197

8.4.1 Device Fabrication and Characterization Details ................................................. 197

8.4.2 Quantitative HeLIC Imaging of Carrier Transport in CQD Solar Cells .............. 197

8.5 Further HeLIC Carrier Lifetime Imaging of CQD Solar Cells ................................... 205

8.5.1 HeLIC Imaging at Various Frequencies ............................................................... 205

8.5.2 HeLIC Lifetime Imaging of CQD Solar Cells with/without Plasma Etching ...... 207

8.5.3 HeLIC Lifetime Imaging for Interface Effects on CQD Solar Cells ................... 209

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8.6 Conclusions ................................................................................................................. 210

9 Conclusions and Outlook ................................................................................................ 212

9.1 Conclusions ................................................................................................................. 212

9.1.1 Advances in Carrier Transport and J-V Mechanisms of CQD Systems............... 212

9.1.2 Advances in Ultrahigh-frequency Diagnostics of Carrier Transport ................... 213

9.1.3 Advances in Quantitative Large-area Ultrahigh-frequency Imaging ................... 214

9.2 Outlook ....................................................................................................................... 215

References ............................................................................................................................ 217

Chapter 1 ....................................................................................................................... 217

Chapter 2 ....................................................................................................................... 218

Chapter 3 ....................................................................................................................... 221

Chapter 4 ....................................................................................................................... 226

Chapter 5 ....................................................................................................................... 231

Chapter 6 ....................................................................................................................... 236

Chapter 7 ....................................................................................................................... 242

Chapter 8 ....................................................................................................................... 245

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List of Tables

Table 3.1: Common optoelectronic parameters of colloidal quantum dot materials and

devices from various characterization techniques…............................................................. 26

Table 4.1: Summary of best-fitted parameters using Eq. (4.29)………………………….....91

Table 4.2: Summary of best-fitted parameters…..................................................................100

Table 5.1: Parameters used for the CQD solar cell simulations…………….………….......123

Table 5.2: Optical counterparts of CQD solar cell electrical parameters, obtained through

best-fitting of the experimental data in Figs.5.11(b) and (d) to Eq. (5.25)………….......….133

Table 6.1: Best-fitted parameters for PbS-MAPbI3 CQD thin films at different

temperatures………………………………………………………………………………...150

Table 6.2: Activation energies at different temperatures for PbS CQD thin films passivated

with various ligands…...........................................................................................................155

Table 6.3: Summary of the best-fitted parameters for CQD thin films surface passivated with

various ligands. These parameters were evaluated for 100 K measurement………….....…160


various ligands. These parameters were evaluated for 300 K measurement…………….…160

Table 7.1: Summary of best-fitted parameters using Eq. (6.18) in Sect. 6.2…………........184

Table 8.1: Summary of the parameters used for heterodyne lock-in carrierography best-fits to

Eq. (8.17)………………………………………………….. ………………………………202

xi

List of Figures

1.1 (a) Global net electricity generation from renewable energy source from 2012 to 2040.

The unit is trillion kilowatthours. It should be noted that the other generation includes waste,

biomass, and tide/wave/ocean. (b) Total jobs of 8.1 million created by renewable energy by

the end of 2015. Moreover, 60% of these solar energy jobs were created in China…………2

1.2 Structure of the research discussed in this thesis. NDT = nondestructive testing; HoLIC =

homodyne lock-in carrierography; HoLIC = heterodyne lock-in carrierography; PCR =

photocarrier radiometry; CQD = colloidal quantum dots; CQDSC = colloidal quantum dot

solar cell….................................................................................................................................5

2.1 (a) Schematic of La Mer and Dinegar model for the synthesis of monodisperse CQDs. (b)

Representation of the apparatus employed for CQD

synthesis……………….………………………………………………………..…..…..……12

2.2: Transmission electron spectroscopy (TEM) imaging of CdSe QDs: low-resolution (a)

and high-resolution (b) and (c) ……………………………………………………………...13

2.3 Different carrier transport mechanisms for QD thin film systems: (a) bulk crystal-like

Bloch state carrier transport; (b) carrier tunneling transport from one dot to another without

phonon assistance; (c) carrier transport with over-the-barrier activation energy mechanism;

and (d) hopping transport with phonon-assistance……………………………...........……...14

2.4 Schematic of charge carrier transport within colloidal quantum dot array. Energy states

including trap states and Fermi levels are represented by solid and dashed lines, respectively.

Nearest-neighbor hopping (I) and variable range hopping (I+II) occur through carriers

transport within quantized states (long solid lines, 1Se and 1Sh for electrons and holes,

respectively) and surface trap states (short solid lines). The variation in quantum dot creates

energy and spatial disorders that weaken interdot coupling effects and disturb carrier

transport………………………………………………………………………………….…..17

2.5 Current-voltage characteristics at different temperatures for PbSe CQD thin films

vacuum-annealed at 473 K (a) and 523 K(b).The low-right insets in both figures show the

Arrhenius plots of conductivity G same as 𝜎 in Eq.(2.1), and the up-left insets depict the

TEM images of PbSe CQD arrays after vacuum annealing……….........................……...…18

2.6 Experimental and modeled electron (a) and hole (b) mobility of PbSe QDs, at a different

interdot distance, as a function of QD diameter…………………..……………...……....….19

2.7: Configuration of different types of CQD solar cells, Schottky (a, b), heterojunction (c,

d), and CQD sensitized solar cells (e, f). The top row (a, c, and e) illustrates the device

structure and the bottom row (b, d, and f) depict the energy band structure with carrier

transport mechanism displayed. Only the lowest energy state levels are shown (i.e., 1S and

1P) for simplification...............................................................................................................21

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2.8: Schematic (a) and energy band diagram (b) of the planar depleted CQD heterojunction

solar cell at short circuit. The energy diagram is plotted along the A-A’ cross-section.

Correspondingly, the schematic (c) and energy diagram (d, a cross-section along B-B’) for

bulk heterojunction. The vertical ZnO nanowires were grown using solution-processed

hydrothermal methods to produce ordered nanowire array within the PbS CQD thin film....23

3.1: A representative of open-circuit voltage decay curve for CQD solar cells, the linear best

fit (dashed red line) is used for the determination of recombination-determined lifetime…..31

3.2: A typical spectrum of the time-resolved PL for PbS CQDs. The inset illustrates the

exciton transport with diffusion and dissociation processes…................................................35

3.3: The mechanism for quasi-static PL imaging (a). (b) The mechanism for low-frequency

modulated PL imaging…………………………………………………………………….....37

3.4: Schematic PL quenching method for diffusion length measurements. An acceptor layer

of CQDs with smaller bandgap should be pre-coated.……………………………………....40

3.5: Schematic of transient photoluminescence study of carrier diffusion length, diffusivity,

and lifetime. At x = L the boundary condition can be either with or without quenching…...40

3.6: Schematic of experimental instrumental setup for photocarrier radiometry…………....43

3.7: (a) Energy diagram of an n-type semiconductor with the illumination of photoexcitation,

and radiative and nonradiative recombination. Defects related states are also depicted to

carry radiative and non-radiative recombination………………………………………….....49

3.8: Schematic of one-dimensional Si wafer where an emission photon distribution is yielded

following laser excitation and carrier-wave generation. (a) A representative semiconductor

slab with thickness dz, centered at z. (b) Reflection photons from backing support material.

(c) Emissive IR photons from backing support materials at temperature Tb. ∆𝑁(𝑧, 𝜔) represents the depth- and frequency- dependent carrer-diffusion-wave, and L is the thickness

of the Si wafer. Other parameters can be found in the text. R1,2,b(λ) are reflectivity of the

front surface, back surface, and the backing support material. It should be noted that the

backing material is used to support the wafer but not necessary to be in contact with the

sample………………………………………………………………………………….….....50

3.9: Experimental setup for homodyne (HoLIC) and heterodyne (HeLIC) lock-in

carrierography………………..................................................................................................53

3.10: Schematic of oversampling (a) and undersampling (b) signal processing methods. For

sampling, 16 samples are taken per one cycle (waveform), and one circle (waveform) is

skipped for undersampling……………………………………….……………………….....54

3.11: Schematic of camera-based HeLIC imaging using an undersampling method (a) and

modulation laser frequency mixing mechanism for HeLIC imaging……………….…….....55

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3.12: The nonlinear dependence of DC (a) and HeLIC (b) signals on photoexcitation laser

power density for CQD solar cells with a typical structure: Au/PbS-EDT/PbS-

PbX2(AA)/ZnO/ITO as discussed in detail in Sect.7.3 ………………………………….......58

4.1: (a) A virtual volume element dV = Adx, resembling a QD, illustrates the discrete

hopping transport of excitons and charge carriers. (b) Schematic of the discrete particle flux

into and out of three adjacent virtual planes. All planes have an area of A across the thickness

direction of a CQD solar cell…………………………………………………………….…..71

4.2: Schematic of one CQD solar cell energy band structure…..............................................75

4.3: Schematic of the double-layer CQD solar cell with the structure: ITO/ZnO/PbS-TBAI

QD/PbS-EDT QD/Au……………………………………………………………………......81

4.4: Device energy band diagram under illumination. PbS-EDT acts as an electron blocking

layer and a Schottky barrier is formed for holes, thus prevent their extraction to the Au

anode….……………...……………………………………………………………………....81

4.5: Equivalent electric circuit of a double-diode model, consisting of a heterojunction diode

between ZnO and PbS-QD layers and a Schottky diode between PbS-EDT and Au………..81

4.6: I-V characteristic curves of a CQD solar cell measured at 300K (a), 250K (b), 230K (c),

200K (d), 150K (e), and 100K (f)……………………………………. ………….……….....85

4.7: (a) Current-voltage curves at various µs, while other parameters are kept constant, and

(b) solar cell FF as a function of µs……………………………………...………………..…86

4.8: (a) Current-voltage curves at various Ds, while other parameters are kept constant, and

(b) solar cell FF as a function of Ds…………………………………………...…..…………88

4.9: (a) Figure of the measured open-circuit voltage (Voc) and short-circuit current (Isc) as a

function of temperature. Equation (4.29) was used for the best-fitting of Voc. (b) Voc at

various temperatures. (c) The ratio 𝛥𝑉𝑜𝑐𝑟𝑎𝑑 / 𝛥𝑉𝑜𝑐

𝑛𝑜𝑛 as a function of temperature…………...92

4.10: Arrhenius plots of (a) the ratio Th(T)/Dh(T) and (b) the ratio Ts(T)/Ds(T). The mobilities

and diffusivities were calculated and fitted for the PbS-TBAI and the PbS-EDT interface,

respectively………………………………………………………………………………......94

4.11: (a) The CQD solar cell FFs measured at various temperatures. (b) Maximum power of

as-studied CQD solar cell measured at various temperatures…………….……………...….95

5.1: Schematic of the as-fabricated CQD solar cell sandwich structure. PbX2 and AA

represent lead halide and ammonium acetate, respectively, acting as exchange-ligands for

PbS CQDs…………………………………………………………………………………..103

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5.2: Schematic of the CQD solar cell energy band structure. PbX2 and AA represent lead

halide and ammonium acetate, respectively, acting as exchange-ligands for PbS CQDs...107

5.3: (a) Experimental data and theoretical best-fits of current density vs. voltage under

illumination and in the dark; (b) The dark current density in (a) amplified. Comparison

between Jdark - Jsc, and Jdark, as well as Jillu, as a function of voltage, is also shown in (a).

Equations (5.9), (5.11) and (5.20) were used for the best-fits of the J-V characteristics. The

best-fitted Jph at Va = 0 (representing Jsc) is 24.9 mA and 7.9×10-7 mA under illumination and

in the dark, respectively …………………………………………………………………....112

5.4: Simulated photocurrent density Jph (a) using Eq.(5.16) without Jph,diff, and (b) using Eqs.

(5.16) and (5.19) with Jph,diff at different effective carrier hopping mobilities; and (c) (Jillu -

Jdark)/𝐽𝑝ℎ𝑚𝑎𝑥 as a function of the external voltage at various effective mobilities using

Eq.(5.20)................................................................................................................................116

5.5: (a) Simulated carrier-mobility-dependent J-V characteristics; (b) open-circuit voltage

Voc, and short-circuit current density Jsc; (c) fill factor FF; and (d) power conversion

efficiency PCE (d). The CQD thin film bandgap used in 1.32 eV same as our experimentally

optimized bandgap for the CQD solar cell in Fig. 5.3. The CQD solar cell carrier hopping

mobility was estimated from our previous study [20]. Equations (5.9), (5.11) and (5.20) were

used for the simulations …………………………………………………………………....119

5.6: Theoretical simulations of CQD solar cell electrical parameters: (a) Voc and Jsc, (b) PCE,

and (c) FF as functions of CQD bandgap energy (Eg) for five different carrier hopping

mobilities. The maximum photocurrent 𝐽𝑝ℎ𝑚𝑎𝑥 is the same as Jsc at the mobility of 0.1 cm2/Vs.

The illumination intensity used for the simulation is AM1.5 spectrum at 1 sun intensity.

Equations (5.9), (5.11) and (5.20) were used for the simulations.……….………………...124

5.7: (a) Simulated CQD solar cell (a) Voc and Jsc, as well as FF and PEC (b), as functions of

the illumination intensity. Equations (5.9), (5.11) and (5.20) were used for the

simulations.............................................................................................................................127

5.8: (a) Experimental J-V characteristics; and (b) output power curves as a function of

photovoltage. Continuous lines are best fits to J-V characteristics and output power using

Eqs. (5.9), (5.11) and (5.20)…..............................................................................................129

5.9: (a) A photograph of a CQD solar cell sample; and (b) its LIC image at open circuit after

the cell was flipped over. The excitation laser was frequency-modulated at 10 Hz at a mean

intensity of 1 sun. The eight Au-coated thin film electrodes on the top in (a) are electrical

contacts while dark brown regions are without Au contact layers. Both regions have an

energy structure as shown in Fig. 5.2. The Au electrode circumscribed with a dashed

rectangle in (a) and also shown in the flipped over orientation in (b) is further studied in Figs.

5.10 and 5.11.……………………………………………………………………….…...…130

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5.10: HoLIC images of the circumscribed CQD solar cell Au electrode in Fig. 5.9(a) at open-

circuit 0.64 V (a), 0.60 V (b), 0.56 V (c), 0.35 V (d), 0.20 V (e), and short-circuit (f). The

excitation laser was frequency-modulated at 10 Hz at a mean intensity of 1 sun.................132

5.11: LIC of the circumscribed CQD solar cell Au electrode in Fig. 5.9(a): (a) LIC (Voc) -

LIC(Vsc); (b) [LIC (Voc) - LIC (V)] vs. V; (c) [LIC (Voc) - LIC(VPM)]VPM ; and (d) [LIC(Voc) -

LIC(V)]V vs. V characteristics. The excitation laser was frequency-modulated at 10 Hz at a

mean intensity of 1 sun. (b) and (d) are best-fitted to Eq. (5.25). Points A, B, C, and the

dashed rectangle region are shown in (a) and (c). It should be noted that values calculated for

the dashed rectangle region are based on averaging the LIC amplitudes over all pixels in this

region.………………………………………………………………………………….…...134

5.12: Open-circuit voltage Voc LIC contour mapping of the circumscribed CQD solar cell Au

electrode in Fig. 5.9.……………………………………………………..……. ……….….135

6.1: (a) Schematic of carrier hopping transport in PbS CQD thin films embedded in a

surface-passivation ligand matrix when excited by a frequency-modulated laser source. (b)

Illustration of carrier generation, dissociation, hopping transport, and trapping processes in a

CQD assembly. Se and Sh are the ground states for electrons and holes, respectively. Ea,1 and

Ea,2 are the activation energies associated with exciton binding energy (Eb) and trap-mediated

transition process, respectively. Eg and Eg, opt are, respectively, the electronic and optical

band gap energy………………………………………………………………………….....140

6.2: Photoluminescence (PL) spectra of four PbS CQD thin films surface passivated with

MAPbI3, EDT, and TBAI…………………………..……………………………….…...…146

6.3: Homodyne lock-in carrierography images of PbS-MAPbI3 (a), PbS-MAPbI3-B (b), PbS-

EDT (c), and PbS-TBAI (d) measured at 10 Hz. Note all the samples were placed on an

aluminum platform for imaging………………………….……………...…………...…….147

6.4: PCR amplitude (a) and phase (b) of MAPbI3-passivated CQD thin films (PbS-MAPbI3)

measured at various frequencies ranging from 10 Hz to 100 kHz and temperatures between

100 K and 300 K……………………………...………………………………………….....149

6.5: Best-fitted hopping diffusivity Dh (a), and Arrhenius plot of Dh for the extraction of the

carrier hopping transport activation energy (b) of the MAPbI3-passivated (PbS-MAPbI3)

CQD thin film. For the same sample, (c)-(e) are the best-fitted effective exciton lifetime 𝜏𝐸,

carrier trapping rate RT , and Arrhenius plot of thermal emission rate 𝑒𝑖, respectively. (f)

Carrier hopping diffusion length Lh calculated from the best-fitted 𝜏𝐸 and Dh values……..151

6.6: Temperature scans of the PCR amplitude for different ligands passivated PbS CQD thin

films. The continuous lines are the best fits to each set of data using Eq. (6.20)………......153

6.7:100 K PCR amplitudes (a) and phases (b) of CQD thin films passivated with four

different ligands, and the best fits to each curve using Eq. (6.18)………….………………159

xvi

6.8: The determinant of diffusivity Dh (a) and effective carrier lifetime 𝜏𝐸 (b) in the PCR

phase channel. Diamonds indicate frequencies at which linear dependence occurs; no such

linear dependencies were found for the amplitude channel of all parameters. (c) and (d) are

the sensitivity coefficients of 𝜏𝐸 in the amplitude and phase channel, respectively. Besides

the measured parameters in this figure, other parameters were also treated similarly to yield

the best-fitted values for all samples as shown in Tables 6.1, 6.3, and 6.4…………….…..163

7.1: Schematic of PbS QDs in MAPbI3 matrixes, i.e. MAPbI3-passivated PbS QDs. Due to

the slight lattice mismatch, chemical bonds can be satisfied at the interfaces between QD and

MAPbI3, and the unsatisfied bonds act as surface trap states, denoted by red circles……...170

7.2: Near-band-edge photoluminescence via variable radiative and nonradiative transitions.

(a) Free-exciton recombination, (b) and (c) recombination of donor (D)- and acceptor (A)-

bound excitons (DX, AX), (d) donor-acceptor pair recombination (DA), (e) recombination of

a free electron with a neutral acceptor (eA), (f) recombination of a free hole with a neutral

donor (hD)…………………………………………………………….. …………………..171

7.3: PCR amplitude vs. excitation power at three different temperatures for sample A (a) and

sample B (b), at 10 kHz laser modulation frequency………………………...…………….174

7.4: (a) Photoluminescence (PL) spectra of MAPbI3-passivated PbS (MAPbI3−PbS) thin

films (samples A and B) spin-coated on glass substrates. (b) PL spectra of MAPbI3−PbS thin

films fabricated through the same process as that of samples A and B but with different QD

sizes……………………………………………………………………………………...…177

7.5: Photocarrier radiometry (PCR) photothermal spectra of MAPbI3-passivated PbS

(MAPbI3-PbS) thin films spin-coated on glass substrates, samples A (a) and B (b). (c)

Arrhenius plots of the PCR phase troughs, I, II, and III, as shown in (a), and best-fitted to Eq.

(7.9) for the extraction of activation energies for each sub-bandgap trap level…………...180

7.6: (a) Schematic of PbS QDs in MAPbI3 matrixes, i.e. MAPbI3-passivated PbS QDs. Due

to the slight lattice mismatch, chemical bonds can be satisfied at the interfaces between QD

and MAPbI3, and the unsatisfied bonds act as surface trap states, denoted by red circles (a);

(b) energy band structure (assumed n-type) of a PbS- MAPbI3 nanolayer, sample A, featuring

shallow and deep level trap states. Excitons are excited in the right QD and diffuse through

nearest-neighbor-hopping (NNH) to the next QD, during which process the coupling strength

between two QDs dissociate excitons into free charge carriers. Carriers may experience

radiative recombination or captured by different types of trap states, where non-radiative

recombination or de-trapping may occur………………………………..……….…………181

7.7: Photos of MAPbI3 -PbS thin films, (a) sample A and (b) sample B. 1 kHz homodyne LIC

amplitude images of MAPbI3- PbS thin films, (c) sample A and (d) sample B. 20 kHz

heterodyne LIC amplitude images of MAPbI3-PbS thin films, (e) sample A and (f) sample B.

Note the very different signal strength scales associated with the two samples…………...182

xvii

7.8: Phase diagram of PCR frequency scans in three different regions 1-3 as shown in Fig.7.7

(c)-(d) and the best-fits of experimental data to Eq. (6.18) in Sect.6.2………………….…184

7.9: PCR phase dependence on time over 25 minutes, the duration of a PCR frequency scan.

Sample A at 100 kHz laser modulation frequency…………………………………………187

8.1: Schematic of CQD solar cell sandwich structure (a), and the corresponding band energy

structure (b) also showing the illumination depth profile, the photocarrier density wave

distribution and the intrinsic and external electric fields……………………………..…….194

8.2: (a) A photograph of the CQD solar cell sample under study, and (b) the corresponding

HoLIC image of this solar cell. The dashed-rectangle-circumscribed solar cell A is selected

for further studies as shown in Figs. 8.3 and 8.4. The HoLIC characterization was carried out

at 10 Hz. It should be noted that for carrierographic imaging, the sample was flipped over

with the top Au contact on the bottom, resulting in mirror image positions being assumed in

(a) and (b) by the dashed rectangles and inscribed solar cells……………………….……..198

8.3: Current-density-voltage characteristic of the CQD solar cell shown circumscribed by a

dashed rectangle in Fig.8.2 (a) …..........................................................................................199

8.4: Frequency-dependent PCR phase spectra of the solar cell electrode units A, B, and area

C without Au contact (Figs. 8.2) at 200 K. Equation (8.15) is used for the best fitting of each

curve. The characterization spot area of the single-detector based PCR is the same as the area

of the circular Au contact tip.…............................................................................................200

8.5: High-frequency HeLIC images at 1 kHz (a) and 100 kHz (b) for the CQD solar cell

shown in Fig. 8.2.…...............................................................................................................200

8.6: The frequency-dependent average HeLIC image amplitudes of the CQD solar cell shown

in Fig. 8.2. The HeLIC images in Fig.8.5 are also included.………………….....................201

8.7: (a) 400 Hz HeLIC image of the CQD solar cell region E, Fig. 8.2, and its carrier lifetime

τ image (b) at 200 K. (c) For comparison, the carrier lifetime image of the same electrode E

at 293 K. (d)-(f) are images of carrier diffusivity, diffusion, and drift lengths, respectively, at

temperature 200 K.................................................................................................................203

8.8: HeLIC images of a CQD solar cell at different modulation frequencies 1 kHz (a), 10 kHz

(b), 50 kHz (c), and 100 kHz (d) as labeled. …………………………………………….....206

8.9: 270 kHz HeLIC images of the same CQD solar cell shown in Fig. 8.8………….........206

8.10: (a) Frequency-dependent HeLIC image average amplitude for regions C and D of the

CQD solar cell shown in Fig. 8.8(a) without plasma etching; and (b) lifetime imaging of the

same CQD solar cell. (c) Furthermore, bar-plotted lifetime statistical distribution for the

above CQD solar cell without plasma etching and another CQD solar cell of the same type

except with 15 s plasma etching………………………………………...………...……..…207

xviii

8.11: (a) Lifetime image of two adjacent CQD solar cell units A and D which reveals the

device homogeneity and influence of electrode contacts on carrier hopping transport, and (b)

the barplot of carrier hopping lifetime image in (a). It should be noted that the top Au

contacts as shown in Fig. 8.2 (a) were on the bottom through flipping the sample over for all

the HeLIC imaging…………………………………………………………………………210

1

Chapter 1

Preamble

1.1 The Imperative of Solar Cells

The importance of solar cells arises from its capabilities in solving the global problems of

the energy crisis and greenhouse gas-induced climate change. Originating from scarce fossil

fuels and emitted carbon dioxide from combusting these fossil fuels, the energy crisis and

climate change are two significant and exigent worldwide problems that humanity needs to

solve in the next several decades. Historically, global economic growth significantly relies on

the consumption of fossil fuels including coal, gas, and oil [1]. In the long term, these sources

of energy are scarce in storage, threatening human beings with energy poverty, especially in

developing countries [1]. However, due to their high energy density, fossil fuels remain the

main energy source driving the world economy while the energy crisis and climate change can

be solved with the application of renewable energies such as wind, solar, biomass, and

geothermal energies. Although nuclear energy does not emit carbon dioxide, they are non-

renewable energy resources and the nuclear waste is radioactive, which can be hazardous to

human health for thousands of years. Germany has announced an end to all nuclear energy

production by 2022 and committed to replacing them with renewable energy. Compared with

other types of renewable energy, solar energy is abundant, renewable, compatible with the

environment, and can be installed wherever there is sunlight. Solar energy is, therefore, a more

suitable renewable energy source for solving the global problems of the energy crisis and

climate change.

The solar photovoltaic (PV) market has seen remarkable growth with a five-year (2012-

2016) average growth rate of about 22% [2]. The growth rate in 2015 was as high as 28%. At

CHAPTER 1. 2

the end of 2015, about 22 countries had enough solar PV electricity capacity to meet more than

1% of their electricity demands; for example, Italy at 7.8%, Greece at 6.5%, and Germany at

6.4% [2]. According to the U.S. energy information administration [as shown in Fig.1.1(a)],

renewable energy is projected to increase to more than 10 TWh in 2040, in which year the total

worldwide electricity generation is estimated at ca.35 TWh [3]. Among all the renewable

sources, solar electricity generation is the fastest-growing with an 8.3% growth rate per year.

Additionally, as shown in Fig.1.1(b), the rapidly growing solar PV market increased

employment by 5 % in 2015, totaling 8.1 million jobs in the renewable energy market [2].

Among these, solar PVs have created 2.772 million jobs worldwide, and 60% of which are in

China.

Figure 1.1: (a) Global net electricity generation from renewable energy sources from 2012 to

2040. The unit is trillion kilowatthours. It should be noted that the other generation includes

waste, biomass, and tide/wave/ocean. Adapted from ref. [3]. (b) Total jobs of 8.1 million were

created in the market of renewable energy by the end of 2015. Moreover, 60% of these solar

energy jobs were created in China. Adapted after [2].

Due to the limited roof area, normal residential installations of solar panels are insufficient

for the electricity consumption of traditional living houses. Strategies of combining solar PV,

solar thermal, and mechanical system optimization have been attempted. Comparatively,

solution-processed solar cells (such as colloidal quantum dot solar cells) are more promising

due to their capability of spray coating and the relatively transparent property that can be

CHAPTER 1. 3

deposited on walls and windows, therefore, generating higher energy outputs. The novel

solution-processed colloidal quantum dot solar cells have high potential for the realization of

large-area, flexible, light-weight, and roll-to-roll processed solar cells. These properties can

further decrease solar PV electricity cost, and their large-scale fabrication feasibility facilitates

Swanson’s law while also being suitable for the markets of automobiles and smart zero-energy

buildings.

1.2 The Imperative of Nondestructive Testing of Photovoltaics

The semiconductor associated industries have been growing rapidly for decades, while new

materials, fabrication techniques, device architectures, and new concepts of optoelectronics

and photovoltaics are still emerging. The rapid growth of these semiconductor industries

significantly relies on the increasing abundant knowledge of semiconductor materials and

devices, which relies on various testing and characterization techniques [4, 5]. Testing and

characterization techniques have played and will continue to play essential roles in

semiconductor industries with their applications ranging from raw material fabrication to

device design and manufacturing. With regards to photovoltaics, to guarantee high-quality and

high-stability photovoltaic products, the various solar cell fabrication processes require precise,

non-destructive, and fast monitoring at all fabrication stages. Hence, fast, in-line, non-contact,

and non-destructive characterization techniques are becoming more and more urgent for large-

scale solar cell manufacturing and utility-scale installations as one bad solar panel is fatal to

the entire PV system.

Following Swanson’s law, to decrease solar PV electricity cost, large-area PV devices

prevail nowadays. Therefore, large-area imaging characterization techniques are garnering

significant interest either in in-line industrial photovoltaic manufacturing or off-line

CHAPTER 1. 4

photovoltaic solar panel maintenance. However, advanced knowledge is established on small-

spot rather than large-area based characterization techniques such as electrical current-voltage

characteristics and photoluminescence techniques. Hence, non-destructive and large-area

imaging techniques are in demand for product quality and stability monitoring, and for the

characterization of critical semiconductor electrical and optical properties including carrier life,

mobility, diffusivity, doping concentration, and defect and trap states. In addition to the need

for large-area characterization, due to the fragile nature of new-generation solar cells including

organic, colloidal quantum dot, and polymer solar cells, full optical and non-destructive

techniques are also essential for photovoltaics. Characterization techniques that need to contact

with photovoltaic materials and devices pose potential damages, even to Si-based solar cells.

Therefore, full optical non-destructive techniques have been developed through static

photoluminescence [6, 7] and frequency modulated quasi-steady-state techniques [8] to extract

various solar cell parameters. Although conventional electrical parameters including open-

circuit voltage, resistance, and short-circuit current have been investigated, carrier transport

dynamic properties such as carrier lifetime, diffusivity, mobility, and diffusion length have not

been comprehensively studied. These parameters, however, are indispensable for solar cell

efficiency optimization, and particularly for the understanding of fundamental working

principles of new-generation solar cells. Therefore, nondestructive testing techniques or

photovoltaics are imperative and urgent.

1.3 Objectives of This Work

Figure.1.2 shows the structure of this work that consists of three essential components: the

theoretical studies of carrier hopping transport and current-voltage characteristics in CQD

systems, large-area and/or high-frequency all-optical NDTs (HoLIC, HeLIC, and PCR), and

CHAPTER 1. 5

CQD thin film materials and solar cells. This thesis focuses on the intersection of either

two/three of the above components and starts with the study and development of theoretical

carrier transport and electrical current-voltage models, which are the basis for the theoretical

derivation of signal generation analytical models for NDTs used for characterizing CQD

materials and solar cells in this thesis. The understanding of carrier hopping transport behavior

and NDT signal generation mechanisms plays key roles in interpreting electrical and optical

properties of CQD materials and devices for solar cell efficiency optimization. The results of

CQD solar cell optimization provide feedbacks to theoretical understanding and models,

therefore, forming a complete and self-consistent system.

Figure 1.2: Structure of the research discussed in this thesis. NDT = nondestructive testing;

HoLIC = homodyne lock-in carrierography; HoLIC = heterodyne lock-in carrierography; PCR

= photocarrier radiometry; CQD = colloidal quantum dot; CQDSC = colloidal quantum dot

solar cell.

As shown in Fig. 1.2, the final objective of this research is to improve CQD solar cell

efficiency and two approaches were attempted: first, better understanding of carrier transport

and recombination dynamics, and J-V characteristics in CQD solar cells; second, dynamic

high-frequency PCR and HeLIC study of carrier transport behaviors in CQD materials and

CHAPTER 1. 6

solar cells, as well as the large-area imaging of CQD systems using HoLIC and HeLIC for

homogeneity and quality monitoring. Therefore, the sub-objectives generated within this

project can be summarized as follows

a. Understand how excitons and charge carriers transport and recombine in the emerging

cutting-edge CQD thin films and solar cells, and how they determine the current-voltage

characteristics of these high-efficiency CQD solar cells [9-11].

Solution-processed CQD thin films and solar cells are different from traditional Si

counterparts with respects to confined carriers, strong exciton binding strength, significant

spatial and energy disorder associated discrete energy states, and the more than three orders of

magnitude lower in carrier mobility. These have led to different experimental electrical and

optical properties. To further increase the efficiencies of CQD solar cells, carrier transport and

recombination dynamics, and J-V behaviors in CQD systems should be better understood.

b. Develop dynamic NDT (i.e. HeLIC, PCR, and HoLIC) signal generation mechanisms

and theoretical models for the characterization of CQD thin films and solar cells [12-17].

HoLIC and HeLIC are frequency-modulated large-area imaging techniques that show

enormous potential in industrial inline roll-to-roll solar cell quality monitoring, while PCR and

HeLIC can perform high-frequency characterization of high-rate carrier transport behaviors.

The development of dynamic NDT (i.e. HeLIC, PCR, and HoLIC) signal generation analytical

models for CQD systems are useful for basic physical studies of energy transport and loss

mechanisms in CQD materials and devices and for industrial quality inspection of

photovoltaics.

Attempts for the first sub-objective are discussed in Chapters 4 and 5, while the second

sub-objective is accentuated in Chapters 6-8.

CHAPTER 1. 7

1.4 Thesis Outline

Chapter 2 reviews the fundamentals of QDs and CQD solar cells, starting from the

solution-processed CQD synthesis methods, which are followed by the discussion of CQD

electrical properties and their dependence on interdot distance, temperature, spatial and energy

disorder, QD size, and QD size polydispersity. This chapter ends with the review of CQD solar

cell architectures, working principles, and efficiency-limiting factors.

Chapter 3 first presents a brief overview of current conventional NDTs that have been used

widely in characterizing carrier transport parameters of QD materials and devices. These

conventional techniques include short-circuit current/open-circuit voltage decay,

photoconductance decay, transient photoluminescence, and various diffusion length

characterization methodologies. The differences and connections between these techniques

and the novel PCR and HeLIC are discussed based on the discussion of the instrumentation

and signal generation theories of these two new techniques. Furthermore, nonlinear

photoluminescence response to laser excitation serving as a prerequisite for HeLIC is

experimentally demonstrated and theoretically analyzed through quantitative models. The

advantages of PCR and HeLIC techniques are also addressed.

Chapter 4 is concerned with the discrete carrier hopping transport behavior and the

associated carrier drift-diffusion current-voltage characteristics in CQD solar cells [9, 10].

Novel definitions of carrier hopping diffusivity and mobility are presented from the description

of discrete carrier hopping nature in CQD thin films. Drift-diffusion current-voltage models

are detailed and act as the basis for the development of a double-diode electrical model to

interpret anomalous current-voltage behaviors observed in CQD solar cells, as these behaviors

were found to reduce CQD solar cell efficiency significantly. The mechanism of open-circuit

CHAPTER 1. 8

dissipation at interfaces was quantitatively analyzed, while the phonon-assisted carrier

hopping transport was revealed through the obtained carrier transport dynamics.

Chapter 5 gives an account of a new CQD solar cell efficiency optimization study [11]

based on the theoretical findings in Chapter 4. Common senses for CQD efficiency

optimization such as constant photocurrent, and high mobility and low energy bandgaps

leading to high efficiency were examined to be invalid. The improved current-voltage model

explored the effects of carrier hopping mobility, bandgap energy, and illumination intensity on

CQD solar cell efficiency. Large-area quantitative imaging of open-circuit voltage and carrier

collection efficiency was addressed to show CQD thin film/metal contact interface influence

on CQD solar cell

Chapter 6 focuses on PCR study of phonon-assisted carrier hopping transport dynamics in

CQD thin films [12, 14]. Combing the novel trap-state-mediated carrier transport model, an

analytical expression for PCR signal generation was developed for CQD thin films. The

influence of temperature, dot-size, and interdot linking ligand on carrier transport kinetics was

studied in various CQD thin films. Additionally, the trap-state-mediated carrier hopping

activation energies are also discussed.

Chapter 7 studies the trap-state-mediated exciton and charge carrier transport in

methylammonium lead triiodide (MAPbI3) perovskite-passivated PbS CQD thin films using

PCR [13]. Carrier recombination mechanisms in CQDs were quantitatively described.

Furthermore, the shallow and deep level trap states were characterized through the PCR

photothermal spectra and static photoluminescence, respectively, which result in the

construction of a complete band energy structure of the CQD ensemble. The interpretation of

CHAPTER 1. 9

large-area HoLIC and HeLIC images of CQD thin films for preliminary carrier lifetime and

homogeneity was discussed.

Chapter 8 reports the large-area carrier transport parameter imaging of CQD thin films and

solar cells using HeLIC [15, 16, 17]. The heterodyne signal processing principles of HeLIC

are developed. Furthermore, the novel theoretical models for HeLIC and HoLIC were derived

and were combined with the excess carrier density wave in CQD solar cells to yield the final

demodulated analytical expressions for both PCR and HeLIC. The applications of HoLIC are

principally for material and device homogeneity and quality characterization. Using HeLIC,

quantitative imaging of carrier hopping lifetime, diffusivity, and drift and diffusion lengths

was achieved, further exploring the physical carrier transport dynamics and interface influence

on CQD solar cell efficiency.

Chapter 9 summarizes key contributions and findings and emphasizes future potential

investigations as well as possible applications of the yielded knowledge in this thesis.

10

Chapter 2

Introduction to Quantum Dots and Colloidal Quantum Dot Solar

Cells

Due to the confined particle motion in three spatial dimensions, the unique optical and

electrical properties of QDs make them promising candidates in fabricating low-cost, large-

area, lightweight, flexible, high-efficiency photovoltaic solar cells. Therefore, this chapter will

introduce QDs and CQD solar cells from solution-based CQD fabrication methodologies to

discussing effects of ligand-determined interdot distance, temperature, spatial and energy

disorders, dot size, and dot size polydispersities on QD electrical properties. Lastly, a review

of CQD solar cells is focused on their fundamentals, classification, working principles, and

efficiency-limiting factors.

2.1 Synthesis of Colloidal Quantum Dots

Two main approaches have been developed for synthesizing QDs in the last several

decades: (1) solution-processed colloidal chemistry methods, and (2) lithographic growth

which includes the subsequent processing techniques such as various deposition and etching

methodologies. The colloidal chemistry fabrication starts with the rapid injection of

semiconductor precursors into a hot and vigorously stirred organic solvent, which contains

organic molecules with long chains and can coordinate with the precipitated CQD particles on

the surface. Through proper surface engineering using various ligands, these water-soluble

CQDs are suitable for different optoelectronic and photovoltaic applications. In comparison,

the lithographic growth of QDs is more time-consuming and expensive. Moreover, QDs

fabricated through this technique are more easily contaminated during the fabrication process.

CHAPTER 2. 11

The contamination can introduce various material defect states, a high degree of dot size

polydispersity, and poor interface quality. There are two types of QD epitaxy growth

techniques: the vapor phase epitaxy (VPE) and the liquid phase epitaxy (LPE). LPE is

commonly used for fabricating semiconductor materials on the micro-scale but infrequently

used in QD fabrications. The metalorganic vapor phase epitaxy (MOVPE) using a

metalorganic medium, and the molecule beam epitaxy (MBE) through the Stranski-Krastanov

growth mode are the two main techniques that have been widely used for QD synthesis. For

example, MBE has been widely implemented in investigating single-photon sources and

quantum computation. When compared with lithography growth, although epitaxy growth can

produce QDs with relatively higher quality, it is uncommonly used for large-scale QD

fabrications.

With respect to QD quality and size polydispersity, QD synthesis through the pyrolysis of

metalorganic precursors is the most successful nanoparticle preparation method. A detailed

review of such techniques has been reported by Wang et al. [1]. As shown in Fig.2.1, the QD

preparation mechanism generally can be understood through La Mer and Dinegar’s model in

the way that precursor nucleation occurs through the rapid injection of QD precursors into a

coordinating organic solvent [2]. In the organic solvent, semiconductor precursors thermally

decompose into monomers at a temperature ranging from 120 oC to 360 oC, a process to

increase the monomer concentration in the solvent. When the monomer concentration

surpasses the nucleation threshold concentration, nucleation processes begin and nanoparticles

grow quickly through absorbing monomers from the solution-phase. New nuclei can no longer

be formed once the monomer concentration is smaller than the critical nucleation threshold

concentration, which keeps a constant population of CQDs while the dot size continues to grow

CHAPTER 2. 12

through absorbing more monomers in the solution. This process will continue until the

monomers are depleted. Due to the depletion of monomers in the solution, the CQD growth

process evolves to the Ostwald stage, where smaller CQDs dissolve into monomers because

of their higher surface energy. The dissolved monomers contribute to the further growth of

large CQDs. In other words, the concentration of CQDs in the solution reduces with time,

while the dot size increases, as shown in Fig.2.1. However, La Mer and Dinegar’s model for

CQD synthesis is a simplified mechanism without considering the concurrence of

semiconductor precursor nucleation and the CQD growth. Furthermore, ligands in the solution

may additionally influence the nucleation process.

Figure 2.1: (a) Schematic of the La Mer and Dinegar’s model for the synthesis of

monodispersed CQDs. (b) Representation of the apparatus employed for CQD synthesis.

Adapted from ref. [2].

CHAPTER 2. 13

Figure 2.2: Transmission electron spectroscopy (TEM) imaging of CdSe QDs: low-resolution

(a) and high-resolution (b) and (c). Adapted from ref. [3]. \

In addition, for a better control of the CQD synthesis process, slow temperature ramping

can be used to trigger the precursor supersaturation and nucleation. Furthermore, proper

temperature control can also be implemented to avoid additional nucleation processes. As

shown in Fig.2.1, the CQD size can be feasibly adjusted through changing the hydrothermal

fabrication time. This technique generally can fabricate CQDs with a dot size distribution of <

7 %, which can further be reduced to less than 5 % through various purification methods. As

an example, for CQD dimensions, the transmission electron spectroscopy images of CdSe

CQDs at high- and low-resolution are shown in Fig.2.2.

2.2 Electrical Properties of Colloidal Quantum Dots

Transport of free charge carriers and excitons is of great interest to better understand

fundamental carrier transport dynamics and energy loss mechanisms in QD systems. Both the

QD dimensions and the surrounding environment have a significant influence on electrical

properties of QD systems. Moreover, QD electrical properties play decisive roles in QD-based

photovoltaic devices. Therefore, this section discusses the dependencies of temperature,

interdot distance, dot size, QD size polydispersity, and spatial and energy disorder on carrier

transport in CQD thin films.

CHAPTER 2. 14

2.2.1 Effects of Interdot Distance and QD Disorder

Figure 2.3: Different carrier transport mechanisms for QD ensembles: (a) bulk crystal-like

Bloch state carrier transport; (b) carrier tunneling transport from one dot to another without

phonon assistance; (c) carrier transport with over-the-barrier activation energy mechanism;

and (d) hopping transport with phonon-assistance. Adapted from ref. [4].

Depending on the QD spatial and energy state disorder, there are four potential carrier

transport mechanisms in CQD ensembles as shown in Fig.2.3 [4]. Generally, at room

temperatures, without specifying the quantum dot materials and quantitative characterization

techniques, all four transport mechanisms are possible. First, as shown in Fig.2.3 (a), bulk

crystal-like Bloch states will be formed when the inter-dot distance is sufficiently small with

a quasi-monodispersed dot size distribution in the whole QD matrix. In this case, strong

coupling strength is formed and leads to a short-range continuous energy band in the QD thin

films. Crystal-like Bloch states are not common in real-world QD systems due to the inevitable

dot size polydispersity, which originates even from today’s state-of-the-art techniques as

discussed in Sect.2.1. For crystal-like Bloch states with extended energy bands, due to the

strong interdot coupling strength, excitons dissociate into free electrons and holes immediately

CHAPTER 2. 15

upon generation. Second, Fig.2.3 (b), with the increase of interdot distance or dot size

polydispersity, the tunneling mechanism starts to dominate the carrier transport behavior in

QD thin films. In this situation, charge carriers can transport from one QD to its neighbors

without phonon-assistance as the interdot coupling strength is still sufficiently strong. Third,

Fig. 2.3 (c), the over-the-barrier activation mechanism is dominant when the energy barrier

between two QDs are low enough that charge carriers and excitons can be thermally excited

to transport over the energy barriers. In other words, for CQD systems in this regime, carriers

can transport freely over the energy barrier from one QD to another.

More importantly, fourth, Fig.2.3 (d), phonon-assisted hopping is the most extensively

observed mechanism in QD systems [5-11]. Depending on the interdot distance, coupling

strength, temperature, and QD dimensions, charge carriers can hop from one dot to its

neighboring dots with the assistance of one or multiple phonons. As the population of phonons

is temperature-dependent, phonon-assisted hopping transport is also temperature-dependent.

This temperature-dependent property leads to increased mobility and conductivity with the

increase of temperatures, a contrasting phenomenon to bulk semiconductors, in which carrier

mobility and conductivity decrease with temperatures because of the enhanced carrier

scattering probability with phonons at high temperatures. Therefore, the interdot coupling

strength that increases with reduced ligand-determined interdot distances has a significant

influence on carrier transport behavior in QD ensembles. Through enhancing interdot coupling

strength, it was found that the electrical properties of a PbSe QD ensemble evolve from the

Coulomb blockade dominated insulating regime to a hopping conduction featured

semiconductor regime [5, 12]. Furthermore. high interdot coupling strength can also assist

CHAPTER 2. 16

exciton dissociation into free electron and hole charge carriers during carrier tunneling or

hopping processes [13, 14].

2.2.2 Effects of Temperature

Carrier transport is also temperature-dependent. With the decrease of temperature, carrier

transport mechanism can evolve from the nearest-neighbor-hopping (NNH) to the Efros-

Shklovskii-variable-range-hopping (ES-VRH). The threshold temperature has been reported

as ca. 200 K by refs. [6, 8, 10], while the further investigation found that the threshold

temperatures could be affected by the QD size [8]. NNH and ES-VRH exhibit the same

temperature-dependent influence on solar cell current density and conductivity through

phonon-assisted carrier hopping transport dynamics. In other words, at high temperatures, the

high population of phonons facilitates carrier hopping transport mobility [5, 9]. The

dependence of conductivity on temperature for hopping conduction takes the general form:

𝜎 = 𝜎0𝑒𝑥𝑝[−(𝑇∗/𝑇)𝑧] (2.1)

where 𝜎0 is the conductivity pre-exponential factor, 𝑇∗ is a fitting parameter with a unit of

Kelvin, and z is a parameter associated with different hopping transport mechanisms.

Specifically, as shown in Fig.2.4, z = 1 is for the NNH carrier transport within CQDs, while z

= 0.5 is for the ES-VRH. Moreover, for Mott variable-range-hopping (M-VRH), z is equal to

0.25 in a three-dimensional transport model and is equal to 0.33 for a two-dimensional carrier

transport model.

At low temperatures, due to strong spatial and energy disorders, ES-VRH is the dominant

carrier transport mechanism for CQD systems with localized states. Carriers with an initial

state energy Ei can be thermally activated and hop to a nearby energy state with an energy of

Ef. The hopping probability is determined by the energy difference between these states, i.e.

CHAPTER 2. 17

ΔE = Ei - Ef, and by the hopping distance. In other words, a large state energy difference with

short hopping distance facilitates the ES -VRH hopping process, as shown by the carrier

hopping paths I and II in Fig.2.4. However, the NNH tends to take place at high temperatures,

such as the hopping path I shown in Fig. 2.4, in which carriers hop from a 1Se state in CQD 1

to another 1Se state or a trap state in CQD 2. Moreover, carriers can also hop between trap

states such as the one from CQD 3 to CQD 2, Fig. 2.4. Furthermore, the conduction mechanism

can transfer from ES-VRH to M-VRH if the state energy difference ΔE is equal to the Coulomb

gap energy δ. For weakly coupled QDs, the Coulomb gap energy can be approximated to δ

≈2Ec, in which Ec is the charge energy. The charge energy is the energy required to add or

remove a charge carrier from a QD. For a spherical QD, the charge energy can be obtained

through

𝐸𝑐 =𝑒2

4𝜋𝜀𝑟 (2.2)

where 𝑟 the radius of the QD is, 𝑒 is the elementary charge, and 휀 is the material permittivity.

Figure 2.4: Schematic of charge carrier transport within a colloidal quantum dot array. Energy

states including trap states and Fermi levels are represented by solid and dashed lines,

respectively. Nearest-neighbor hopping (I) and variable range hopping (I+II) occur through

carrier transport within quantized states (long solid lines, 1Se and 1Sh for electrons and holes,

respectively) and surface trap states (short solid lines). The variation in quantum dot creates

energy and spatial disorders that weaken interdot coupling effects and disturb carrier transport.


CHAPTER 2. 18

Figure 2.5: Current-voltage characteristics at different temperatures for PbSe CQD thin films

vacuum-annealed at 473 K (a) and 523 K (b). The lower right insets in both figures show the

Arrhenius plots of the conductivity G which is proportional to 𝜎 in Eq. (2.1). The upper left

insets illustrate the TEM images of PbSe CQD arrays after vacuum annealing. Adapted from

[5].

Romero et al. [5] used the abovementioned model in Fig. 2.4 to study the carrier transport

behavior in PbSe CQDs, which are surface-capped with oleic acid. It was found that for CQD

thin films annealed in vacuum at a lower temperature (373 K), the Coulomb blockade was the

dominant influential factor of the carrier transport and led to an insulating conductivity

property in PbSe CQD thin films. The insulating behavior also originates from the higher

charge energy 𝐸𝑐 of 36 meV than the thermal energy of ~32.5 meV at 373 K. In contrast, under

high annealing temperatures (such as 473 K), the interdot distance was found to be obviously

reduced, leading to a conductive electrical property. As shown in Fig.2.5, the fitting of

conductivity to Eq.(2.1) reveals a 𝑧 value of 0.95~1.05 and 0.48-0.55 for high and low

temperatures, respectively, which indicates that the NNH carrier hopping process dominates

at high temperatures while ES-VRH dominates at low temperatures.

CHAPTER 2. 19

2.2.3 Effects of Quantum Dot Size and Polydispersity

In QD thin films, different charge carriers (electrons and holes) exhibit different transport

dependencies on the QD size. Lee et al. [15] and Liu et al. [7] found that both electron and

hole mobilities increased by 1-2 orders of magnitude with the growth of the PbSe QD size.

Specifically, carrier mobility is generally found to increase with dot size, while further

investigation showed that electron mobility decreased when the quantum dot size was further

increased [Fig. 2.6 (a)], which led to an optimized mobility peaked at a QD diameter of ca. 6

nm. In contrast, hole mobility shows a monotonic increase with the QD dot size as shown in

Fig. 2.6 (b). The increase of carrier mobility with QD diameter can be attributed to the reduced

activation energy in large QDs, while the decrease of electron mobility for further increased

QD size can be ascribed to the weakened electronic coupling strength amongst large QDs as

discussed by Lee et al. [15]. Furthermore, both electron and hole mobilities drop exponentially

with the increase of ligand length, consistent with the framework of the hopping/tunneling

transport mechanism, i.e. short interdot distance results in higher coupling strength and

narrower energy barrier width.

Figure 2.6: Experimental and modeled electron (a) and hole (b) mobilities of PbSe CQDs of

different diameters. Dependence on CQD interdot distances is also illuminated. Adapted from

[15].

CHAPTER 2. 20

In addition, the QD size polydispersity has been found to significantly deteriorate the

interdot coupling strength, therefore, playing key roles in determining charge carrier transport

in CQD ensembles. Within CQD ensembles, due to the dot-size-dependent quantum

confinement effects, CQDs with varied sizes correspond to different bandgap energies, hence,

the size polydispersity results in a bandgap spectrum. Meanwhile, large CQDs with small

energy bandgap can act as trap states for carrier transport. For CQDs with a given/fixed density

of surface trap states, Liu et al. [7] discovered that carrier mobility was independent of CQD

size polydispersity. This finding was also confirmed by Zhitomirsky et al. [16] by showing

that the CQD polydispersity had a negligible impact on photovoltaic device performance with

a fixed concentration of surface trap states. However, through decreasing surface traps to a

very low level, further studies elucidated that an improved photovoltaic device performance is

achievable when CQD size polydispersity can be successfully suppressed [16].

2.3 Colloidal Quantum Dot Solar Cells

Colloidal quantum dot solar cells have attracted considerable attention due to their much

higher theoretical solar energy to electricity conversion efficiency of ~ 65% than the Shockley-

Queisser limit for Si solar cells (33%) [17, 18-20]. This is due to the tunable CQD energy

bandgap through effective dot-size control, and due to solution-oriented fabrication processes,

which are suitable for fabricating low-cost, flexible, light-weight, and large-area photovoltaic

solar cells. CQD-sensitized solar cells and CQD heterojunction solar cells are the two

prevailing CQD solar cell architectures of intense research interest [21]. Although CQD solar

cells have been fabricated with various architectures, typical CQD solar cells have four main

components: a transparent conduction electrode (indium tin oxide and fluorine tin oxide), a

metal oxide semiconductor thin film with a thickness from tens of nanometers to several

CHAPTER 2. 21

hundred nanometers, light absorbing QD thin films of several hundred nanometers or thinner

(depending on the carrier diffusion length), and a metal electrode such as gold (Au). However,

the first generation CQD solar cells are based on Schottky diodes as shown in Fig.2.7 (a) and

(b) [21].

Figure 2.7: Configuration of different types of CQD solar cells, Schottky (a, b), heterojunction

(c, d), and CQD-sensitized solar cells (e, f). The top row (a, c, and e) illustrates the device

structures and the bottom row (b, d, and f) depicts the energy band structures with carrier

transport mechanisms. It should be noted that, for simplification, only the lowest energy state

levels are shown (i.e., 1S and 1P). Adapted from ref. [21].

Formed at the electrode/CQD thin film interface, the Schottky diode induces a depletion

region with an intrinsic electric field to drift electrons and holes to the Al (anode) and ITO

(cathode) electrodes, respectively. CQD thin films are light absorption layers and generate

excitons upon photoexcitation. The photoexcited excitons diffuse along the exciton density

gradient from the ITO layer, where the excitation light impinges. Meanwhile, the interdot

coupling strength can dissociate these electron-hole pairs (i.e. excitons) into free charge

CHAPTER 2. 22

carriers. Furthermore, excitons can also dissociate at the CQD/Al interfaces. However, this

process delimitates the device efficiency as hole charge carriers need to travel all the way back

to the ITO electrode. In addition, the interface-induced carrier recombinations will be

simultaneously enhanced. Figure 2.7 (b) shows that the dissociated electrons diffuse in the

quasi-neutral region and drift in the depletion region to reach the Al electrode. Generally, open-

circuit voltage, Voc is a function of the energy difference between the quasi-fermi levels of

electrons 𝐸𝐹,𝑛 and holes 𝐸𝐹,𝑝 in the form:

𝑉𝑜𝑐 =𝐸𝐹,𝑛−𝐸𝐹,𝑝

𝑞 (2.3)

With the application of metal electrodes, the quasi-fermi levels can be approximated by the

work functions of the corresponding metals, which can also be seen in Fig.2.7. Schottky-diode-

based CQD solar cells endure low fill factors (FF) and Voc for a given short-circuit current

density Jsc. In addition, as shown in Fig.2.7 (b), hole charge carriers can also be injected into

the electron extraction electrode due to the low energy barriers formed at the Schottky diode,

which results in enhanced carrier recombinations and decreased CQD solar cell efficiency. In

comparison, CQD sensitized solar cells consist of a CQD sensitized photoelectrode [titanium

dioxide, TiO2, Figs.2.7 (e) and (f)] and a counter electrode, which are separated by a liquid

electrolyte. In CQD-sensitized solar cells, CQDs act as the light absorbing layers. Due to the

thinness of the CQD layers, solar cells of this type tolerate low Jsc, however, enhanced fill

factor FF and Voc can be obtained. Heterojunction CQD solar cells [Figs.2.7 (c) and (d)] follow

similar carrier extraction mechanisms to Schottky diode based CQD solar cells, except for

additional exciton dissociation sources that arise from the heterojunction interfaces. Metal

oxide materials such as TiO2 and ZnO in nanostructures are often used as an n-type

semiconductor component of the heterojunction pn junction structure. The depleted

CHAPTER 2. 23

heterojunction structure can simultaneously maximize FF, Voc, and Jsc. Therefore,

heterojunction structures have been reported to yield various CQD solar cells with high

efficiencies, although different QD materials or electrodes were used [21-23].

Figure 2.8: Structure schematic (a) and energy band diagram (b) of the planar depleted CQD

heterojunction solar cells at short circuit. The energy diagram is plotted along the A-A’ cross-

section. Correspondingly, the structure schematic (c) and energy diagram (d, a cross-section

along B-B’) for bulk heterojunction is also illuminated. The vertical ZnO nanowires were

grown using solution-processed hydrothermal methods to produce ordered nanowire array

within the PbS CQD thin film. Adapted from ref. [22].

Typically, there are two types of heterojunction CQD solar cells: planar depleted

heterojunction and bulk heterojunction [22, 23]. Figure 2.8 shows the device structures and

working principles of these two categories of heterojunction CQD solar cells. As shown in

Figs.2.7 (e) and (f), planar heterojunction solar cells have the typical structures as discussed

above. Figures 2.8 (a) and (b) also exhibit the structure of planar heterojunction CQD solar

cells, which consist of ZnO nanoparticles and PbS CQDs. The depletion region is formed and

centered at the ZnO/CQD thin film interfaces and extends into the CQD thin films. To harvest

more charge carriers for high Jsc, high-efficiency CQD solar cells need to absorb more light,

which depends directly on the thickness of the CQD light absorber. However, as carriers need

CHAPTER 2. 24

to travel across the entire thin film to be extracted by the respective electrodes, the thickness

of CQD thin films is limited by the exciton or free charge carrier diffusion lengths (generally,

in several hundred nanometers). Thereby, the thinness of light absorber CQD layers, which is

limited by carrier diffusion lengths restricts the further enhancement of short-circuit current

density, Jsc.

To increase Jsc, bulk heterojunction CQD solar cells are designed with the application of

pillars [24] or nanowire arrays [22, 25-30], which are interpenetrated into CQD thin films.

These pillars and nanowires are generally nanostructured metal oxide such as ZnO and TiO2

as shown in Fig. 2.8(c). One of the major advantages of bulk heterojunction CQD solar cells

is their ability to extend the depleted regions to several micrometers, which leads to improved

charge separation and collection efficiency. Using bulk heterojunction CQD solar cells,

evidential improvements of Jsc have been reported as high as 30 mA/cm2 [25, 26, 30]. However,

the overall CQD solar cell efficiency has not progressed substantially due to the loss of Voc

from enhanced carrier recombination, which is additionally augmented through increased

CQD/metal oxide interfacial trap states. Bulk heterojunction CQD solar cells still have great

potential for CQD solar cell performance optimization with respect to the increased Jsc when

CQD and interface quality have been significantly improved in the future.

2.4 Conclusions

This chapter discusses solution-based CQD fabrication processes, which can produce high-

quality CQDs with low dot size polydispersity. High-efficiency solar cells have been achieved

using these solution-processed CQDs. Ligand-exchanges play essential roles in strengthening

CQD interdot coupling, leading to extended mini-bands in CQD ensembles. The temperature-

dependent phonon-assisted hopping transport in CQD ensembles is the dominant carrier

CHAPTER 2. 25

transport mechanism, which has been experimentally demonstrated. Furthermore, large dots

facilitate carrier hopping transport with the exception that electron mobility decreases with the

further increased quantum dot size. CQD size polydispersity acts as trap states to trap carriers,

and should be eliminated or reduced for high-performance solar cell fabrication. The

heterojunction structure can effectively improve CQD solar cell efficiency through

simultaneously increasing Jsc and Voc. The bulk-structured heterojunction can significantly

improve Jsc, but reduced Voc was reported due to enhanced carrier recombinations at

CQD/metal oxide interfaces. The improvements of CQD quality through various methods are

still the main efforts applied by researchers to boost CQD solar cell efficiency.

26

Chapter 3

Non-destructive Testing (NDT) Techniques for Carrier Transport

in Quantum Dot Materials and Solar Cells

3.1 Literature Review and Classification

Charge carrier transport properties such as carrier mobility, diffusion length, doping

density, carrier lifetime, and trap density are essential parameters for CQD solar cell efficiency

optimization. Including transient photoluminescence, short-circuit/open-circuit voltage decay,

and photoconductance decay, many techniques have been developed to measure these carrier

transport properties. This thesis aims to develop novel high-frequency and/or large-area non-

destructive techniques through the derivation of new theoretical models to quantitatively probe

carrier transport dynamics in CQD materials and solar cells. For comparison with our new

techniques (PCR, HoLIC, and HeLIC), several widely-used testing techniques (although not

all of them are NDTs) for carrier transport parameter characterization of CQD systems are

reviewed in this chapter.

Table 3.1: Carrier transport parameters of colloidal quantum dot materials and solar cells

measured by various characterization techniques.

Property Testing method Related

parameters

NDT Devices

or

substrates

References

Carrier

mobility

CELIV J, time No Both [1-3]

TOF J, time No Both [3, 4]

Transient

photovoltage

Voc No Device [2, 5]

FET Id, Vg No Device [2, 6, 7]

CHAPTER 3. 27

CELIV = carrier extraction by linearly increasing voltage, TOF = time of flight, FET = filed

effect transistor, PCR = photocarrier radiometry, PL = photoluminescence.

Table 3.1 summarizes these key carrier transport parameters that are measured by various

commonly used techniques. For the sake of comparison, both NDTs and non-NDTs have been

reviewed and tabulated in Table 3.1. These techniques can be classified into the following

three categories [13]:

1. Steady-state techniques that require contact with samples.

Steady-state techniques are the most widely used and well-developed techniques, including

the surface photovoltage method for minority carrier lifetime characterization [14, 15], two-

point and four-point probe for resistivity measurements [16, 17], secondary ion mass

spectroscopy for ultra-shallow dopant profile characterization [18-21], and Rutherford

backscattering spectrometry (RBS) for measuring dopant/carrier distribution, structure and

composition of materials [22, 23]. Although widely used in characterizing semiconductor

Diffusion

length

PL

quenching

PL intensity,

layer thickness

Yes Substrate [8]

Voc transient decay Voc No Device [4]

PCR Modulated PL Yes Both [9]

Doping

Density

Capacitance-

voltage

Capacitance No Device [2]

Carrier

lifetime

Voc transient decay Voc No [3]

Transient PL PL Yes Both [2, 10]

PCR Modulated PL Yes Both [11, 12]

Trap

density

Voc transient decay Voc, Jsc No Device [4]

Thermal

admittance

spectroscopy

Capacitance,

frequency of ac

signal

Yes Device [4]

CHAPTER 3. 28

materials and devices, a dominant disadvantage of these techniques is the incapability of

measuring optoelectronic dynamic properties due to the steady-state nature. Furthermore,

many of these techniques are expensive, such as the second beam spectroscopy and RBS,

limiting their applications in industrial photovoltaic manufacturing and maintenance.

Additionally, the direct or indirect contact with samples poses potential damages to samples.

At last, the scanning point-by-point mapping method for most small-spot testing based

techniques to get large-area imaging is time-consuming, not suitable for in-line manufacturing

inspection.

2. Transient and quasi-steady-state contact techniques.

Quasi-steady-state (or frequency-modulated measurements) at high modulation

frequencies can detect high-rate dynamic properties of photovoltaic materials and devices with

high signal-to-noise ratio. Including quasi-steady-state photoluminescence (QSSPL) [24, 25]

and microwave photoconductance decay (μ-PCD, a golden standard for carrier lifetime time

measurement), these contactless quasi-steady-state techniques can be used for measuring

effective lifetime of minority charge carriers [26]. Furthermore, modulated photovoltage [27,

28] and electroluminescence (EL) [27] methods were also reported recently for solar cell

quality inspection. These techniques can detect multiple carrier transport dynamic parameters

including carrier lifetime, diffusion length, and surface recombination velocity through fitting

experimental data to theoretical models. However, the requirements for contacting with

samples limit their further applications as discussed for techniques in category 1.

3. Quasi-steady-state (Frequency-modulated) contactless techniques.

Reviewing the development of characterization techniques for semiconductors, the

properties of being all-optical, fast, contactless, nondestructive, and capable of dynamic testing

CHAPTER 3. 29

of high-rate carrier transport behaviors will be the features of future techniques. Up to today,

this type of characterization techniques include frequency-modulated PL for carrier lifetime

and surface recombination characterization [29], coupled photocurrent and photothermal

reflectance techniques that are sensitive to carrier diffusivity and lifetime [30, 31], modulated

free carrier absorption that are capable of measuring multiple carrier transport parameters [32-

34], modulated photothermal reflection (PDT) for detecting photothermal and electro-thermal

responses) [35, 36], photothermal radiometry (PTR) which is capable of measuring carrier

lifetime and surface recombination velocities [37], and photocarrier radiometry (PCR) that can

also qualitatively measure carrier lifetime and surface recombination velocities with higher

accuracy than PTR methods. These emerging techniques have been widely used in

characterizing a wide scope of materials and devices with remarkable success.

In the following sections, conventional characterization techniques including short-circuit

current/open-circuit voltage decay (SCCD/OCVD) — techniques that have been extensively

used in characterizing CQD solar cells although they are not NDTs, static and transient PL,

photoconductance decay (PCD), and various methods which are developed for carrier

diffusion length measurements are reviewed. Our new technique PCR is one of the essential

quasi-steady-state or frequency-modulated contactless NDTs for carrier transport parameter

testing, therefore, the general features of PCR are addressed in this chapter and compared with

the abovementioned techniques elaborated upon. The principles of this technique specific for

CQD thin films will be detailed in Chapter 6. A major disadvantage of the above-mentioned

techniques is the small-spot based characterization, which limits their applications in

characterizing large-area photovoltaic materials and devices. The solution to this limitation

involves the application of the our novel camera-based large-area dynamic imaging techniques,

CHAPTER 3. 30

HoLIC and HeLIC (modulated at ultrahigh frequencies), which are the imaging evolutions of

the PCR technique and are also discussed with the discussion of instrumentation and signal

processing principles, while their particular theory and models for CQD solar cells are

developed in detail in Chapter 8.

3.2 Traditional Methodologies for CQD Carrier Transport

Characterization

3.2.1 Short-Circuit Current/Open-Circuit Voltage Decay

(SCCD/OCVD)

SCCD and OCVD are two valuable techniques for measuring carrier lifetimes in CQD

solar cells [3]. These techniques probe pn junction voltage and short circuit current decay after

the photoexcitation of electron-hole pairs to measure carrier recombination lifetimes [38, 39].

Here, these two techniques are addressed in comparison to the PCR technique for CQD lifetime

measurements. Unlike most techniques that can characterize only one carrier transport

parameter, the combination of SCCD and OCVD can measure both the carrier lifetime, τ𝑟 and

the back-surface recombination velocity, sr, of a solar cell. The theoretical model is derived

from the analysis of minority carrier diffusion within a pn junction, and the solar cell back-

surface is treated through boundary conditions. The differential equation for minority carrier

concentration in the solar cell base can be expressed by

𝜕∆𝑛(𝑥,𝑡)

𝜕𝑡= 𝐷

𝜕2∆𝑛(𝑥,𝑡)

𝜕𝑥2−∆𝑛(𝑥,𝑡)

𝜏𝑟+ 𝐺(𝑥, 𝑡) (3.1)

in which ∆𝑛(𝑥, 𝑡) is the excess minority carrier density, 𝐷 is the diffusivity, and 𝐺(𝑥, 𝑡) is the

generation rate which equals zero after the photoexcitation is turned off. The solution to Eq.

(3.1) is subject to the following boundary equations [39].

1

∆𝑛(𝑥,𝑡)

𝜕∆𝑛(𝑥,𝑡)

𝜕𝑥= −

𝑠𝑟

𝐷𝑛 𝑓𝑜𝑟 𝑥 = 𝑑 (3.2)

CHAPTER 3. 31

with

∆𝑛(0, 𝑡) = 0 (3.3)

for short-circuit current method, and

𝜕∆𝑛(𝑥,𝑡)

𝜕𝑥= 0 𝑓𝑜𝑟 𝑥 = 0 (3.4)

for the open-circuit voltage method. The term d is the device thickness. When considering the

above boundary conditions, the solution to Eq. (3.1) exhibits an exponential short-circuit

current and open-circuit voltage decay profile with time. The decay behavior has a time

constant that is determined by the time-dependent excess carrier density.

Figure 3.1: A representative of open-circuit voltage decay curve for CQD solar cells, the linear

best fit (dashed red line) is used for the determination of recombination-determined carrier

lifetime. Adapted from ref. [3].

For the open-circuit voltage decay method, generally, researchers calculate the minority

carrier lifetime through [3, 40]

CHAPTER 3. 32

𝜏𝑟 = −𝐹𝑘𝑇/𝑞

𝑑𝑉(𝑡)/𝑑𝑡 (3.5)

with 𝑘 the Boltzmann constant, T the absolute temperature, q the elementary charge, and F a

constant, ranging between 1 at low carrier injection levels, and 2 at high carrier injection levels.

Therefore, using the open-circuit voltage decay curve, the carrier lifetime can be resolved.

Figure 3.1 shows a representative curve of Voc decay and the best fit to Eq. (3.5) for the

extraction of the minority carrier lifetime. The minority carrier lifetimes extracted using this

method for PbS CQD solar cells in a Schottky architecture range from 1 ms at low intensities

to 10 µs at high intensities.

The disadvantages of SCCD and OCVD techniques can be summarized as follows: first,

these techniques need to be in contact with solar cell devices. Measurements through contact

with samples are time-consuming and can present damages to photovoltaic samples. Second,

SCCD and OCVD are based on the theoretical model as presented in Eqs. (3.1) and (3.5),

therefore, it is evident that only effective carrier lifetimes can be measured and these

techniques cannot distinguish bulk and surface lifetimes. Although the back surface

recombination velocity is introduced through boundary conditions, the front surface

recombination velocity is forsaken. Third, these techniques cannot detect depth-resolved

carrier transport parameters as the signals of SCCD and OCVD are from the device’s overall

short-circuit current and open-circuit voltage, which are contributed by carriers at different

depths. Fourth, these techniques can only be used for complete solar cells, while not applicable

to semiconductor wafers or incomplete photovoltaic devices, which have no short-circuit

currents and open-circuit voltages.

CHAPTER 3. 33

3.2.2 Photoconductance Decay (PCD)

PCD was developed in 1955 for semiconductor lifetime characterization [17] and it has

evolved to be a golden standard for minority carrier lifetime measurements. Based on the

excess minority carrier decay which is directly associated with carrier lifetime, this technique

can measure effective carrier lifetime with high accuracy and reliability. In PCD, electron-hole

pairs are generated through the pulse photoexcitation, and their time-dependent concentration

decay is monitored with respect to the time following the cessation of the photoexcitation.

During the measurements, sample are in contact with PCD but it can also be made contactless

if microwave is used in reflection and transmission modes, i.e. the μ-PCD technique [41, 42].

For μ-PCD, photoconductivity is monitored through microwave reflection or transmission.

The theory of PCD starts with the expression for conductivity σ,

𝜎 = 𝑞(𝜇𝑛𝑛 + 𝜇𝑝𝑝) (3.6)

in which 𝑞 is the charge element, and n and p are the electron and hole concentrations,

respectively. The term 𝜇𝑛,𝑝 represents the electron (n) or hole (p) mobility, respectively.

Furthermore, n = n0+Δn and p=p0+Δp (n0 and p0 are electron and hole concentrations at

equilibrium, respectively); considering identical electron and hole mobilities, at low injection

levels, i.e. the excess carrier concentration is much lower than the carrier concentration at

equilibrium. For low trapping conditions, i.e. Δn = Δp, the measurements of conductance

change correspond to the measurements of excess carrier changes, which are given by

𝛥𝑛 =𝛥𝜎

𝑞(𝜇𝑛+𝜇𝑝) (3.7)

CHAPTER 3. 34

Therefore, assuming constant mobility, the measurement of 𝛥𝑛 can be carried through

measuring 𝛥𝜎. For the calculation of carrier lifetimes, the dependence of carrier concentration

decay on time is determined by the carrier lifetime τ through [43]

𝛥𝑛(𝑡) = 𝛥𝑛(0)exp (−𝑡

𝜏) (3.8)

PCD measures an effective minority carrier lifetime, 𝜏𝑒𝑓𝑓 and cannot distinguish bulk lifetime,

𝜏𝐵 and the surface lifetime, τs, in other words, 𝜏 in Eq. (3.8) is an effective carrier lifetime

𝜏𝑒𝑓𝑓 and can be expressed by

1

𝜏𝑒𝑓𝑓=

1

𝜏𝐵+

1

𝜏𝑠 (3.9)

Therefore, if either bulk or surface lifetime is of interest, the other carrier lifetime must be

already known.

However, PCD also has many disadvantages [13]. First, PCD has a relatively very complex

instrumentation system, for example, the conventional μ-PCD in the contactless model has

both the photoexcitation and microwave conductance testing systems. In comparison, most of

other carrier lifetime characterization techniques only require a photoexcitation system.

Second, as discussed above, the use of Eqs. (3.8) and (3.9), which depict the definition of

effective minority carrier lifetime, indicates that PCD can only measure effective lifetimes that

originate from the overall effects of the bulk lifetime, surface lifetime, diffusivity, and surface

recombination velocities. Specifically, μ-PCD cannot distinguish these parameters, although

PCD through contact with samples has this capability, thereby compromising the properties of

being all-optical, contactless, and nonconductive. Third, the signals collected by PCD are the

depth integration of the overall signals along the sample thickness, which imply that PCD is

unable to characterize sample properties at different depths. In other words, these trap states

can influence the detected integration signals no matter how deep the trap states lie as long as

CHAPTER 3. 35

carriers exist within the trap state regions. Therefore, although μ-PCD can construct carrier

lifetime images, it cannot perform depth-resolved characterization. This is an essential

drawback or limitation for the characterization of p-n junction based devices.

3.2.3 Time-resolved PL (TRPL, transient PL)

Photoluminescence techniques detect only radiative recombination induced photoemission.

For example, depending on the specific semiconductor energy bandgap, the center

photoemission wavelength is 1.2 µm for Si semiconductors. Therefore, InGaAs is the most

commonly used material for PL detectors. PL based techniques can be further divided into

steady-state and time-resolved PL (TRPL). The former can characterize semiconductor

material optical and electrical properties including energy bandgap and trap states. In

comparison, TRPL is capable of measuring carrier lifetimes, back and front recombination

velocities, and diffusivity with proper theoretical models.

Figure 3.2: A typical spectrum of the time-resolved PL for PbS CQDs. The inset illustrates the

exciton transport with diffusion and dissociation processes. Adapted from ref. [45].

CHAPTER 3. 36

With regards to the TRPL spectra at different decay time ranges, a PL vs. time curve may

have different slopes that correspond to different carrier recombination mechanisms. Therefore,

different TRPL theoretical models can be applied within different decay time ranges for the

extraction of carrier lifetime and other carrier transport parameters, which correspond to

different carrier transport mechanisms [29, 44]. As an example, a typical spectrum of TRPL is

shown in Fig. 3.2. Upon the generation of excitons in PbS CQDs, these excitons diffuse

through hopping or tunneling and dissociate into free electrons and holes as described in the

inset of Fig.3.2. Figure 3.2 reveals that there are two different carrier decay mechanisms of

excitons and charge carriers in CQD thin films, i.e. the fast PL emission decay component that

corresponds to the exciton dissociation process and the slow exponential decay component

originated from free charge carrier trapping in CQD surface states [45].

The excess carrier decay in TRPL is also described by Eq. (3.8). The PL intensity is the

depth integration of excess carrier density along the sample thickness, i.e.

∅𝑃𝐿 = 𝐾 ∫ ∆𝑛(𝑥, 𝑡)𝑑𝑥𝑑

0 (3.10)

where ∅𝑃𝐿 is the PL intensity, 𝑑 is the sample thickness, and 𝐾 is a constant. Therefore, the

minority carrier lifetime can be extracted from fitting the exponent PL intensity decay profile

to a single-exponential equation [46]:

∅𝑃𝐿~𝑒𝑥𝑝(−𝑡/𝜏𝑃𝐿) (3.11)

Sometimes a multi-exponential decay model is used for better fitting and describing carrier

decay mechanisms. The analytical model, however, can be very complicated depending on the

carrier transport parameters that need to be extracted from a PL decay spectrum [44].

According to the above discussion, transient PL measures the effective minority carrier

lifetime. For example, when self-absorption is considered for extra electron-hole-pair

CHAPTER 3. 37

excitation (the influence of self-absorption is more significant for direct bandgap

semiconductors such as PbS), the PL lifetime, therefore, is defined by [47]

1

𝜏𝑃𝐿=

1

𝜏𝑛𝑜𝑛−𝑟𝑎𝑑+

1

𝜏𝑆+

1

𝛾𝜏𝑟𝑎𝑑 (3.12)

where the terms 𝜏𝑛𝑜𝑛−𝑟𝑎𝑑, 𝜏𝑆, and 𝛾𝜏𝑟𝑎𝑑 are the nonradiative, surface, and radiative lifetimes,

respectively, and the term 𝛾 denotes the photon recycling factor. However, the effect of self-

absorption is not substantial for indirect bandgap semiconductors.

Figure 3.3: The mechanism for quasi-static PL imaging (a). Adapted from ref. [48]. (b) The

mechanism for low-frequency modulated PL imaging. Adapted from ref. [49].

In recent years, InGaAs camera based PL imaging of semiconductor materials and devices

is emerging with increasing frequency. However, being limited by the relatively low camera

frame rate and exposure time, PL imaging cannot be constructed at high modulation

frequencies, which, however, are essential for the characterization of high-rate carrier transport

behaviors. In other words, as most PL imaging techniques are performed at a steady state [24,

48] or low modulation frequencies [49], PL imaging cannot detect high-rate carrier transport

behaviors. Figure 3.3(a) depicts the schematic of PL imaging in a quasi-steady-state model,

where four images are taken at a low-frequency-modulated square wave excitation. The

CHAPTER 3. 38

effective minority carrier lifetime can be measured through fitting each pixel of these images

to a time domain theoretical model. Similarly, Fig.3.3(b) presents the mechanism for PL

imaging at low modulation frequencies. The low-frequency-modulated PL imaging can also

yield effective carrier lifetime images from proper frequency domain theoretical models.

Transient PL has many advantages, for example, the applied near-infrared InGaAs camera

or single detector does not need cooling, which eases the requirements for the testing

environment and improves measurement accuracy. In addition, transient PL directly measures

carrier radiative recombination without influence from thermal emission. Therefore, the

theoretical computation processes can be simplified. Currently, with the increase of the

InGaAs camera frame rate, relatively high-rate carrier transport property can be characterized

but still not high enough for low carrier lifetime CQDs. However, transient PL still has many

disadvantages as summarized below: first, without the application of a lock-in amplifier,

transient PL has a low signal-to-noise ratio (SNR), which means an extremely low signal if the

carrier lifetime is very small. In addition, system calibrations are required [24, 25]. Second,

the PL signal is a depth integration of carrier radiative recombination along the sample

thickness, which indicates that carrier transport properties at different depths cannot be

distinguished, limiting its applications in photovoltaic device characterization. Third, although

transient PL can measure multiple carrier transport parameters, this technique is still

constrained by lower camera frame rate and the requirements for high exposure time.

3.2.4 Carrier Diffusion Length Measurements

Diffusion length is the distance that photoexcited excitons or free charge carriers can

diffuse through before they recombine radiatively or non-radiatively. Given a low-level

injection approximation, the diffusion length is the square root of the product of the carrier

CHAPTER 3. 39

diffusivity and the lifetime. To measure carrier diffusion lengths in CQD thin films,

Zhitomirsky et al. [8] applied the QD bandgap tunability to develop a PL quenching method.

Specifically, in their 1D method, as shown in Fig.3.4 (a), the CQD thin film of interest is

denoted as a donor, and correspondingly, an acceptor CQD layer with smaller energy bandgap

is pre-coated atop the donor CQD layer. Laser-based photoexcitation is introduced from the

donor side, from where excitons are generated and diffuse within the donor to the acceptor

CQD layer. At steady state, the carrier concentration 𝑛(𝑥) can be approximated by

𝐷𝑑2𝑛(𝑥)

𝑑𝑥2−𝑛(𝑥)

𝜏= 𝑔(𝑥) (3.13)

in which 𝑔(𝑥) is the generation rate at x, 𝐷 is the diffusivity, and 𝜏 is the carrier lifetime.

Solving Eq. (3.13), the expression for 𝑛(𝑥) can be obtained with proper boundary conditions.

The photoluminescence intensity is proportional to 𝑛(𝑥), a theoretical fundamental that is also

applied in PCR. The PL flux is detected from the acceptor layer at different thin film

thicknesses, and the diffusion length of the donor CQD thin film can be obtained by fitting the

experimental data to the following exponential equation [8]:

𝑃𝐿𝐹𝑙𝑢𝑥 =1

𝐿𝐷(𝐴𝑒

−𝑑

𝐿𝐷 − 𝐵𝑒𝑑

𝐿𝐷) + 𝛼𝐶𝑒−𝛼𝑑 (3.14)

where 𝑃𝐿𝐹𝑙𝑢𝑥 is the PL flux from the acceptor thin film, 𝐿𝐷 is the carrier diffusion length, 𝑑 is

the thin film thickness, 𝛼 is the absorption coefficient, and A, B as well as C are constant

coefficients to fit. This 1D method is further developed into a 3D method through replacing

the boundary acceptor CQD layer with the incorporation of smaller bandgap CQDs into the

CQD thin film of interest. Through fitting Eq. (5) in the ref. [8], the carrier transport parameters

including diffusion length (LD), diffusivity (D), carrier lifetime (τ), mobility (µ), and trap

density can be obtained.

CHAPTER 3. 40

Figure 3.4: Schematic of a PL quenching method for diffusion length measurements. An

acceptor layer of CQDs with smaller bandgap should be pre-coated. Adapted from ref. [8].

Figure 3.5: Schematic of transient photoluminescence study of carrier diffusion length,

diffusivity, and lifetime. At x = L the boundary condition can be either with or without

quenching. Adapted from ref. [50].

The abovementioned methodology is based on the theoretical understanding of the charge

carrier transport mechanisms in CQD thin films. Therefore, it is not applicable for p-n junction

associated photovoltaic devices as the charge carrier transport behavior are different. Similarly,

using the non-destructive transient PL technique, Lee et al. [50] developed a methodology to

measure exciton diffusion length and diffusivity in CQD thin films. As shown in Fig.3.5, a

pulse laser is used as the photoexcitation source for CQD thin films on a substrate. The

substrate can act as either a quenching layer or a non-quenching layer. Assuming the

CHAPTER 3. 41

independence of the carrier diffusivity on the exciton density and location, i.e. D(n,x,t) = D(t),

the exciton concentration can be expressed by the one-dimensional diffusion equation,

𝜕𝑛(𝑥,𝑡)

𝜕𝑡= 𝐷(𝑡)

𝜕2𝑛(𝑥,𝑡)

𝜕𝑥2−𝑛(𝑥,𝑡)

𝜏+ 𝑔(𝑥, 𝑡) (3.15)

A time-dependent 𝐷(𝑡) does not yield an analytical solution, therefore, constant diffusivity, D

is usually applied by researchers.

It should be noted that Eq. (3.15) is also a fundamental part of the PCR theory for carrier

transport parameter measurements of CQD thin films and solar cells in this thesis. Furthermore,

a constant time-independent diffusivity is also assumed for PCR. With quenching or no

quenching boundary conditions, the exciton concentration 𝑛(𝑥, 𝑡) can be solved from Eq.

(3.15). The obtained 𝑛(𝑥, 𝑡) is a function of multiple parameters including diffusion length,

lifetime, and diffusivity. Therefore, 𝑛(𝑥, 𝑡) can be used to extract the exciton diffusivity and

diffusion length through fitting experimental transient PL data to the derived models [50]. Of

course, when converted to the frequency-domain, the 𝑛(𝑥, 𝑡) can be used in PCR methodology

that provides further advantages as discussed in Sect.3.5. Coupled with the transient PL

technique, Lee et al. [50] derived the expression for exciton diffusion length, 𝐿𝐷,

𝐿𝐷 ≈2𝐿

𝜋√2(

𝜏

𝜏𝑞− 1) (3.16)

in which 𝐿 is the QD thin film thickness (Fig. 3.5), and 𝜏𝑞 is the fitted average exciton lifetime.

Compared with PCR as discussed in Chapter 6, this methodology does not consider the

influence of trap states, moreover, this transient PL based methodology is limited by low SNR

and cannot measure very short lifetimes.

CHAPTER 3. 42

3.3 Photocarrier Carrier Radiometry (PCR)

PCR is a dynamic spectrally-gated photoluminescence to measure optoelectronic

properties in materials and devices. As will be detailed in the following sections, PCR uses

modulated lasers to excite semiconductor samples, and demodulate the near-infrared radiation

signals to quantitatively characterize carrier transport parameters. In comparison with

photothermal techniques which detect both thermal-wave and carrier-density wave, the

theoretical interpretation of PCR signal is much easier with less unknown parameters involved,

corresponding to relatively higher measurement accuracy as PCR only detect the purely

optoelectronic carrier-wave. The following sections will discuss PCR instrumentations used in

for this thesis, as well as a general theory for measuring semiconductors with continuous band

structures. The detailed analytical models developed specific CQD thin films is elaborated in

Chapter 6.

3.3.1 Photocarrier Radiometry Instrumentation

As a nondestructive, frequency-domain, and spectrally gated photoluminescence, PCR

starts with a super-bandgap frequency-modulated laser beam to create a periodic carrier-

density-wave (CDW) in semiconductor materials and devices. The periodic light emissions

from radiative recombination are detected by a single detector, and the amplitude and phase of

the periodic signal were computed using a lock-in amplifier. Consequently, the PCR spectra

are the frequency-dependent amplitudes and phases from low frequency (𝜔𝜏 ≪ 1, in which 𝜔

is the modulation angular frequency and 𝜏 is the effective carrier lifetime) to high frequency

(𝜔𝜏 ≫ 1) that can be used for following theoretical fitting process to extract useful carrier

CHAPTER 3. 43

transport parameters. As a spectrally gated frequency domain PL and the instrumentation

consists of, as shown in Fig.3.6, three key sub-systems:

1. Excitation laser generation system,

2. Dynamic photoluminescence detection system, and

3. Lock-in amplifier signal computation system.

Figure 3.6: Schematic of experimental instrumental setup for photocarrier radiometry.

The excitation laser generation system consists of a function generator, a laser, and other

optics including various mirrors as shown in Fig. 3.6. The function generator used for this

research project is from Stanford research systems, Model DS340, synthesized function

generator. Unless indicated, the PCR system for CQD materials and device characterization

for this projects uses the near-infrared diode laser at wavelength 830 nm (Melles Griot, Model

CHAPTER 3. 44

No. 56ICS115) with the highest DC power of ~ 30 mW. The laser excitation is modulated in

sine wave from 10 Hz up to 2 MHz with an accuracy of ±25ppm determined by the function

generator. From function generator the output amplitude is 10.96 Vpp, indicating the peak-to-

peak voltage of the sine wave, and the DC offset is 0.90 V. GPIB interface was used for

communication between the function generator and the computer.

The illuminating photons from samples are collected by a single detector (PDA400,

ThorLabs), prior to which a long-pass filter is used to filter out the excitation laser. The

PDA400 is an InGaAs detector with a switchable gain for detecting light signals from DC to

10 MHz. The effective diameter of the InGaAs detector is 1 mm with wavelength response

from 800 nm to 1750 nm. For modulation frequencies from 10 Hz to 100 kHz, the gain was

set to 30dB. The 1000-nm long-pass filter is mounted in front of the InGaAs detector.

Lock-in amplifier is the key component that contributes to the high signal-to-noise ratio

(SNR). As shown in Fig. 3.6, the reference signal is from the function generator, and the input

signal is from the InGaAs single detector. Time constant τ was set to 1s.

3.3.2 Theory of Lock-in Amplifier Signal Computation

Due to the critical roles that a lock-in amplifier plays in the signal processing with high

SNR for the PCR system, herein, the principle of the lock-in amplifier is briefly discussed here.

The technique used in a lock-in amplifier is called phase sensitive detection that singles out

the signal with a specific desired frequency and phase. However, noise signal at other

frequencies are rejected and have no influence on the measurement. Therefore, a lock-in

amplifier is capable of detecting AC signals as small as a few nanovolts, even when the noise

source is thousands of times larger than the signal. Without using a lock-in amplifier,

supposing a 1 µV sine wave signal at 10 MHz, in order to detect the signal, an amplifier is

CHAPTER 3. 45

required. Current good noise amplifiers have around 3nV √𝐻𝑧 of input noise. Considering an

amplfier with a bandwidth of 200 MHz and a gain of 1000, the output is expect to be 1 mV of

signal and 43 mV of broadband noise (3nV √𝐻𝑧 ×√200 MHz×1000), in which case the higher

noise than signal entails a measurement faisure. With the application of lock-in amplifier uisng

a phase sensitive detector (PSD), which enable the detection of the signal at 10 MHz while

with a bandwidth as narrow as 1 Hz (even narrower can also be achieed, such as 0.1 Hz, if

longer time contant is used). Therefore, the output noise is now only 3 µV (3nV √𝐻𝑧 ×√1

MHz×1000) which is considerablely much lower than 1 mV of signal. With a SNR as high as

300, the signal can be measured.

Suppose a reference signal is a sine wave that can be 𝑉𝑅 sin(𝜔𝑅𝑡 + 𝜑𝑅) and it can be sync

out from a function generator. The terms 𝑉𝑅 and 𝜑𝑅 are the refence signal amplitude and phase,

respectively, and 𝑡 is the time. When the sine wave output from the function generator is used

to modulate the excitaiton laser system, the response signal from samples is at the same

freuqency 𝜔𝑅 and might be 𝑉𝐼 sin(𝜔𝑅𝑡 + 𝜑𝐼), in which 𝑉𝐼 and 𝜑𝐼 are the input (i.e. output

from the samples) signal amplitude and phase, respectively. Lock-in amplifer uses a mixer to

multiply these two signals together, and it geenrates

𝑉𝑀1 = 𝑉𝑅 sin(𝜔𝑅𝑡 + 𝜑𝑅) 𝑉𝐼 sin(𝜔𝑅𝑡 + 𝜑𝐼) =1

2𝑉𝐼𝑉𝑅 cos(𝜑𝑅 − 𝜑𝐼) +

1

2𝑉𝐼𝑉𝑅 sin(2𝜔𝑅𝑡 +

𝜑𝑅 + 𝜑𝐼) (3.17)

in which 𝑉𝑀1 is the signal output from the mixer 1 (mixter 2 will be introduced later). The

resultant first term is a DC component, while the second term is an AC component osilated at

higher frequency 2𝜔𝑅. The AC component can be readily removed using a low pass filter,

which generates the filtered signal 𝑉𝑀1+𝐹𝐼𝐿𝑇

CHAPTER 3. 46

𝑉𝑀1+𝐹𝐼𝐿𝑇 =1

2𝑉𝐼𝑉𝑅 cos(𝜑𝑅 − 𝜑𝐼) (3.18)

Now it is evident to see the phase dependence, i.e. 𝑉𝑀1+𝐹𝐼𝐿𝑇 is proportional to the cosine of

the phase difference between the refence and the input signal (signal from samples). For the

sake of measuring 𝑉𝐼, a second mixer is used an shift the refernce signal 90o out of phase.

Therefore, reference sent to the second mixer is 𝑉𝑅 sin(𝜔𝑅𝑡 + 𝜑𝑅 − 𝜋/2). Following the

same procedure as above, the output of the second signal 𝑉𝑀2 is

𝑉𝑀2 =1

2𝑉𝐼𝑉𝑅 cos(𝜑𝑅 − 𝜑𝐼 − 𝜋/2) +

1

2𝑉𝐼𝑉𝑅 sin(2𝜔𝑅𝑡 + 𝜑𝑅 + 𝜑𝐼 − 𝜋/2) (3.19)

After filtering,

𝑉𝑀2+𝐹𝐼𝐿𝑇 =1

2𝑉𝐼𝑉𝑅 cos(𝜑𝑅 − 𝜑𝐼 − 𝜋/2) =

1

2𝑉𝐼𝑉𝑅 sin(𝜑𝑅 − 𝜑𝐼) (3.20)

Finally, the amplitude and phase of the input (signal from samples) can be determined, and

they are

𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 =2

𝑉𝑅√(𝑉𝑀1+𝐹𝐼𝐿𝑇)2 + (𝑉𝑀2+𝐹𝐼𝐿𝑇)2 (3.21a)

𝑃ℎ𝑎𝑠𝑒 = 𝜑𝑅 − 𝜑𝐼 = tan−1 (

𝑉𝑀2+𝐹𝐼𝐿𝑇

𝑉𝑀1+𝐹𝐼𝐿𝑇) (3.21b)

In-Phase: 𝑋 = 𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 ∙ cos(𝜑𝑅 − 𝜑𝐼) (3.21c)

Quadrature: 𝑌 = 𝐴𝑚𝑝𝑙𝑖𝑡𝑢𝑑𝑒 ∙ sin(𝜑𝑅 − 𝜑𝐼) (3.21c)

One important concept for lock-in measurements is the “time constant” τ, which can be

obtained by 𝜏 = 1/𝑓𝑐 and used to set the low-pass filer bandwidth [52]. The term 𝑓𝑐 is the

“cutoff” or “corner” frequency of the low-pass filter that has been used to remove 2𝜔𝑅

frequency components of 𝑉𝑀1 and 𝑉𝑀2, and to remove signals or noises at other frequencies

not equal to the reference frequency. The low-pass filter time constant and “rolloff” should be

selected carefully based on the nature of experiments. The filter rolloff is mostly taken one of

the four values: 6 dB/octave, 12 dB/octave, 18 dB/octave, and 24 dB/octave. Therefore, signal

CHAPTER 3. 47

with frequency 𝑓 ≪ 𝑓𝑐 will pass with unity gain, while single with frequency 𝑓 ≫ 𝑓𝑐 will be

attenuated as 𝑓−𝑛 for 6n dB/octave filter, for example, ∝ 𝑓−3 for 18 dB/octave filters. The

output is more steady and easier to be measured when the time constant increases.

3.3.3 General Theory of Photocarrier Radiometry

PCR is a full optical nondestructive dynamic spectrally gated frequency-domain PL

modality. It is also an evolution of photothermal radiometry (PTR). In comparison, PTR

detects both thermal-wave and electronic carrier-density wave (CDW) contributions, however,

lead to significant difficulties in PTR signal interpretation and computation due to a large

number of variables [53]. PCR was developed by Mandelis et al. [54] as a purely carrier-wave

laser-based detection methodology and eliminates the thermal-wave contributions. Because

PCR is only sensitive to the recombination of photoexcited carrier density waves, its signal

interpretation and computation are much simpler than that of PTR, therefore, increasing the

uniqueness and reliability of the obtained semiconductor material properties. As the detailed

theory has been elaborated in ref. [54], here, only some key conclusions that have been widely

used in this thesis will be discussed due to the limited scope of this chapter. Although the PCR

theory was developed in the scenario of Si wafers, most concepts and formulas can be directly

applied to other material systems such as GaAs [55], as well as CQD thin films and solar cells.

PCR detects photon emission from photovoltaic materials and devices. Therefore, the

discussion of its theory should start with excess carrier radiative and non-radiative

recombination mechanisms. As shown in Fig. 3.7, electrons are excited by photons with super-

bandgap energy, then the excited electrons and holes quickly thermalize and relax to the band

edge of the conduction and valence band, respectively, with the emission of phonons (or heat).

De-excitation of photoexcited carriers occurs through three recombination mechanisms:

CHAPTER 3. 48

radiative recombination, Auger recombination, and Shockley-Read-Hall (SRH) recombination.

An elaborate review of these recombination mechanisms can be found in refs. [56-58], while

will be briefly outlined here.

Radiative recombination is a direct band edge electron-hole recombination, emitting

photons with the bandgap energy of the semiconductor. As both electron and hole participate

in radiative recombination process, the recombination rate is proportional to the product of

electron and hole concentration.

Auger recombination denotes a direct nonradiative recombination of electrons and holes

via energy transfer to and emits another free carrier. It is a reverse process of MEG in QDs.

Depending on whether the energy is transferred to an electron or hole, the three-particle

interaction can be denoted by eeh or ehh. Auger recombination usually occurs under high

carrier condition with high injection levels. The recombination rate of an Auger recombination

is proportional to the product of the concentration of three particles involved.

Shockley-Read-Hall recombination is a two-step process with carriers trapped into defect

states and then followed by radiative and non-radiative recombination of these trapped states.

The photon emission from the SRH recombination is lower than the semiconductor bandgap

energy, depending on the energy level of defect states. SRH associated defect states can be

distinguished as, namely, recombination centers and traps. The capture coefficients of

electrons and holes for recombination centers are similar. However, for traps, the capture

coefficient for one carrier is higher when this carrier is trapped in, while the rate for capturing

another particle with the opposite sign for recombination is slow. It should be noted that SRH

recombination not only can happen between defects and conduction/valence band but also

CHAPTER 3. 49

occurs between defect states. The recombination rate of SRH is proportional to the product of

carrier concentrations and trap state density.

For QDs with high surface-to-volume ratios, the surface (interface) trap related surface

recombination should also be discussed. Surface recombination is defect involved. Therefore,

it is a type of SRH recombination. However, the SRH theory is derived based on a single well-

defined trap level. Interfaces or surfaces of semiconductor represent a termination of crystal

periodicity and induce a band of electronic states in the band gap. This inconsistency requires

additionally extended SRH recombination theory to deal with the continuum of surface states

across semiconductor bandgap [56].

For CQD thin films and solar cells, the radiative recombination and radiative component

of SRH recombination contribute to PCR, depending on the specific PL spectra of a sample as

shown in Fig. 6.2.

Figure 3.7: (a) Energy diagram of an n-type semiconductor with the illumination of

photoexcitation, and radiative and nonradiative recombination. Defects related states are also

depicted to carry radiative and non-radiative recombination. Adapted from ref. [54].

CHAPTER 3. 50

Figure 3.8: Schematic of one-dimensional Si wafer where an emission photon distribution is

yielded following laser excitation and carrier-wave generation. (a) A representative

semiconductor slab with thickness dz, centered at z. (b) Reflection photons from backing

support material. (c) Emissive IR photons from backing support materials at temperature Tb.

∆𝑁(𝑧, 𝜔) represents the depth- and frequency- dependent carrer-diffusion-wave, and L is the

thickness of the Si wafer. Other parameters can be found in the text. R1,2,b(λ) are reflectivity of

the front surface, back surface, and the backing support material. It should be noted that the

backing material is used to support the wafer but not necessary to be in contact with the sample.


The photoexcited carriers after ultra-fast decay to the respective band edge, accompanying

various combinations as discussed above, meanwhile diffuse within their statistical lifetime. If

the excitation laser is intensity modulated at a frequency f =ω/2π (ω is the angular frequency),

the photogenerated carrier density constitutes a spatially damped carrier-density-wave (CDW)

(or carrier-diffusion-wave). The CDW oscillates diffusely away from their generation source

due to its concentration gradient and recombines with a phase lag dependency on a delay time

that equal to the carrier statistical lifetime τ, a structure- and process-sensitive property [54,

59]. The schematic of photon excitation, absorption, and emission processes in a Si wafer is

illustrated in Fig.3.8. The one-dimensional geometry is suitable for thin semiconductor

CHAPTER 3. 51

materials or for the case of using spread laser beams of large spot size. The emission photon

power 𝑑𝑃𝑗(𝑧, 𝑡; 𝜆) at wavelength λ with a bandwidth dλ is given by [54]

𝑑𝑃𝑗(𝑧, 𝑡; 𝜆) = {𝑊𝑁𝑅[𝑇𝑇(𝑧, 𝑡); 𝜆] + 𝜂𝑅𝑊𝑒𝑅(𝜆)}𝑗𝑑𝜆; 𝑗 = 𝑟, 𝑡 (3.22)

where 𝑊𝑁𝑅[𝑇𝑇(𝑧, 𝑡); 𝜆] is the non-radiative related power per unit wavelength, 𝑊𝑒𝑅(𝜆) is the

radiative recombination generated photon power per wavelength, 𝜂𝑅 is the quantum yield for

radiative emission, 𝑇𝑇(𝑧, 𝑡) is the total temperature including the background temperature and

the temperature increase following photon absorption and heat generation and others, and the

subscript (r, t) denotes back-propagating (reflected) and forward-propagating (transmitted) as

shown in Fig. 3.8. The modulated super-band-gap laser photons impinges on the front surface

of the semiconductor and are absorbed within a short distance [α(λ)]-1 from the front surface,

where α(λ) is photon wavelength dependent absorption coefficient. The emission spectra are

within a broad wavelength range due to the various type of radiative recombination processes.

The final PCR expression was derived in one dimension by Mandelis et al. [54] as a depth

integration of excess carrier density:

𝑃(𝜔) ≈ 𝐹(𝜆1, 𝜆2) ∫ ∆𝑁(𝑧, 𝜔)𝐿

0𝑑𝑧 (3.23)

The term 𝐹(𝜆1, 𝜆2) is an instrumentation coefficient which depends on the spectral emission

bandwidth (𝜆1, 𝜆2), and the expression for 𝐹(𝜆1, 𝜆2) can be found in ref. [54].The term ∆𝑁 is

the excess free charge carrier density which depends on the material properties and carrier

transport nature. The next section discusses the derivation of ∆𝑁 with trap-mediated carrier

transport in CQD thin films.

CHAPTER 3. 52

3.4 Homodyne (HoLIC) and Heterodyne (HeLIC) Lock-in

Carrierography

HoLIC and HeLIC are the imaging evolution of PCR with the application of a CCD camera

rather than a single detector. Another difference, when compared with PCR, is that two laser

excitation systems are used for illumination. HoLIC can only construct low-frequency imaging

while HeLIC is able to image semiconductors at high frequencies. Therefore, HeLIC can

perform large-area, contactless, fast, all-optical, quantitative characterization of semiconductor

materials and devices. This section will discuss the instrumentation and signal processing

techniques of HoLIC and HeLIC and address the nonlinearity requirement for HeLIC. The as-

developed novel theoretical HeLIC signal generation models will be discussed in detail in

Chapter 8.

3.4.1 Instrumentation and Signal Processing Techniques used in

HoLIC and HeLIC Imaging

Instead of using InGaAs single detector, HeLIC uses high-speed NIR InGaAs camera for

signal collection. As shown in Fig.3.9, the InGaAs camera (Goodrich SU320 KTSW-

1.7RT/RS170) used through this study has the following features: 320×256 pixel active

elements, the spectral bandwidth of 0.9-1.7 μm, 120 fps frame rate, and exposure times tunes

between 0.13 and 16.6 ms. In comparison with PCR, two fiber-coupled diode lasers of 808 nm

wavelength were used for optical illumination. For acquiring homogeneous illumination, both

laser beams were spread and homogenized using diffusers to generate a 10 × 10 cm2 square

illumination area with small intensity variations (< 5 %). An optical long-pass filter

(Spectrogon LP-1000 nm) is mounted in front of the InGaAs camera, resulting in an effective

InGaAs camera bandwidth of 1-1.7 μm.

CHAPTER 3. 53

Figure 3.9: Experimental setup for homodyne (HoLIC) and heterodyne (HeLIC) lock-in

carrierography.

Due to the limitations of camera frame rate, a synchronous undersampling method is

employed through the application of a data acquisition module (NI USB-6259), which

produced a reference signal and an external trigger to the camera. Sixteen images per period

were scanned with a frame grabber (NI PCI-1427). The control is through homemade

LabVIEW program. To understand undersampling, the Nyquist-Shannon sampling theorem

should be reviewed:

“An analog signal with a bandwidth of fa must be sampled at a rate of fs>2fa in order to

avoid the loss of information.”

The term fs is the sampling rate. When fs = 2fa is satisfied, fs is called Nyquist rate. The

theorem is a bridge connecting continuous-time signals (analog signals) and discrete-time

signals (digital signals). In HoLIC, if the sampling method is used, for a harmonic signal with

frequency f, the sampling rate fs should be greater than 2f in order to precisely acquire the signal

CHAPTER 3. 54

information. Limited by the frame rate of 120 fps for our InGaAs camera, the highest frequency

in HoLIC can be calculated is 60 Hz. To ensure the imaging quality with sufficient exposure

time (i.e. the maximum exposure time of 16.7 ms), HoLIC imaging at 10 Hz is generally

performed.

Figure 3.10: Schematic of oversampling (a) and undersampling (b) signal processing methods.

For sampling, 16 samples are taken per one cycle (waveform), and one circle (waveform) is

skipped for undersampling.

CHAPTER 3. 55

Figure 3.11: Schematic of camera-based HeLIC imaging using an undersampling method (a)

and modulation laser frequency mixing mechanism for HeLIC imaging.

However, the low frequency (even 60 Hz) is not sufficiently high for high-frequency

imaging to generate measurable phase lags (τ ≈ 1/2πf) for materials with low carrier lifetimes.

CHAPTER 3. 56

Herein, undersampling method is used in HoLIC for higher frequency imaging. The

undersampling method is also known as harmonic sampling, bandpass sampling, IF sampling,

and IF to digital conversion. As shown in Fig.3.10, considering a 1 Hz sine wave, 16 imaging

is taken in one period, which is known as an oversampling process. The process of taking 16

images per period (or 1s, as shown in Fig. 3.10) corresponding to a frame rate of 16 Hz. If

undersampling is applied, with a skip of one waveform, the actual frame rate used is only 1

Hz. Following the same manna, through skip more waveforms (i.e. wait for more time to take

the next image), high-frequency imaging is achievable through the use of the highest frame

rate. Given the known reference frequency, the data collected using undersampling can be re-

calculated to oversampling data as shown in Fig. 3.10 (a). A complete schematic of HoLIC

using undersampling method is shown in Fig.3.11 (a) that one cycle is skipped between each

image. For example, four images are taken at 0, π/2, π, and 3π/2 phases. The main pulse train

plays the role of triggering the camera to start taking images, and the camera pulse train

initiates four images (16 in real experiments) to be taken during a fixed camera exposure period.

By shipping more cycles, higher modulation frequencies can be used while keeping the camera

frame rate unchanged. The collected reference and signal matrix are computed by lock-in

amplifier or data acquisition card shown in Fig.3.9.

With the application of the undersampling method, however, problems still arise when

high modulation of frequencies are used. They include the small timing errors and the

decreased resolution due to the limited camera exposure time. Therefore, a heterodyne

method was introduced for high-frequency HeLIC imaging through the superposition of two

modulated laser beams with a small frequency difference (the beat frequency) and the

camera measuring with a frame rate equal to the beat frequency. The experimental setup of

CHAPTER 3. 57

HeLIC is the same as that of HoLIC except that two laser excitations are modulated with a

small frequency difference as shown in Fig. 3.9. Figure 3.11(b) displays the image

generation mechanism of HeLIC imaging: two linearly combined laser irradiation

modulation with a beat frequency of 10 Hz is mixed. The interaction between the two

modulated excitation lasers and the sample is a nonlinear process that enables the generation

of an LIC image at the beat frequency as shown in Fig. 3.11(b). The camera modulation

frequency is always selected to be equal to the beat frequency, i.e., 10 Hz in this study. The

recorded image amplitudes carry information from the high modulation frequency (as

discussed in Sect. 8.2). Phase images cannot be obtained at the beat frequency due to the

close proximity of the two mixed two frequencies f1 and f2 (Fig. 3.9).

3.4.2 Requirements for HeLIC Response to Laser Excitation: Non-

linear Photoluminescence Processes

As shown in Fig. 3.11(b), HeLIC imaging requires the nonlinear combination of two carrier

density waves to create a carrier wave oscillated at the modulation frequency difference. This

section shows experimental evidence to the nonlinear response of the CQD solar cells. A

theoretical explanation to the nonlinearity is also discussed in this section with respect to

analytical models. Regarding the physics behind, the property of nonlinear response is due to

the various carrier recombination mechanisms as given in Sect. 7.3, which discusses the carrier

recombination mechanisms in CQDs.

Given the fact that the photoexcited exciton population has a linear dependence on the

incident photoexcitation intensity, for HeLIC imaging the photoexcitation laser beam was

modulated at two angular frequencies (𝜔1, 𝜔2) leading to the generation of two CDWs with a

small frequency difference Δω, mixed in a nonlinear signal processing device, a mixer, such

CHAPTER 3. 58

as a diode and a transistor [60]. The mixer creates a series of CDWs with new frequencies

including Δω. For our CQD solar cells, the sample itself acts as a mixer [61]. The nonlinear

coefficient γ of CQDs as a mixer can be obtained as shown in Fig. 3.12 (a), through fitting the

experimental DC image signal vs. laser photoexcitation intensity to I ∝ Nγ, in which I is the

average amplitude of all image pixels and N is the excitation laser power density as discussed

in Sect. 7.3. A nonlinear coefficient γ of 0.60 was extracted consistently with the requirement

for a non-linear process, γ ≠ 1, to generate a HeLIC signal [61, 62].

Figure 3.12: The nonlinear dependence of DC (a) and HeLIC (b) signals on photoexcitation

laser power density for our CQD solar cells with a typical structure: Au/PbS-EDT/PbS-

PbX2(AA)/ZnO/ITO as discussed in detail in Sect.7.3.

Theoretically, in HeLIC, as the laser excitation is modulated at two frequencies with an

angular frequency difference ∆𝜔, the excess photocarrier wave can be expressed as

∆𝑁(𝑥, 𝜔) = 2𝑛0(𝑥) + 𝐴(𝑥, 𝜔1)𝑐𝑜𝑠[𝜔1𝑡 + 𝜑(𝑥, 𝜔1)] + 𝐴(𝑥, 𝜔2)𝑐𝑜𝑠[𝜔2𝑡 + 𝜑(𝑥, 𝜔2)](3.24)

where 𝑛0(𝑥) is the DC component of the modulated excess CDW, 𝐴(𝑥, 𝜔𝑗) is the amplitude

of the cosinusoidally modulated CDW at 𝜔𝑗 ( 𝑗 = 1,2) and 𝜑 is the CDW phase. Equation

(3.24) is only approximate since PL emission response to photoexcitation intensity is a

fundamentally non-linear process with a non-linearity coefficient 𝛾 which was generally

CHAPTER 3. 59

measured to be between 0.5 and 2, and can be determined by plotting 𝐼 vs. 𝑁𝛾 , where 𝐼

represents the PL emission intensity and 𝑁 stands for the excitation laser power intensity.

Inserting ∆𝑁(𝑥, 𝜔) in Eq. (3.24) into Eq. (3.23) and considering the fully nonlinear response,

it can be shown that

𝑆(𝜔) = 𝐹 ∫ {2𝑛0(𝑥) + 𝐴(𝑥, 𝜔1)𝑐𝑜𝑠[𝜔1𝑡 + 𝜑1(𝑥, 𝜔1)] + 𝐴(𝑥, 𝜔2)𝑐𝑜𝑠[𝜔2𝑡 +𝑑

0

𝜑2(𝑥, 𝜔2)]}𝛾𝑑𝑥 (3.25)

Furthermore, the integrand can be expanded using the binomial theorem in the form

{2𝑛0(𝑥) + 𝐴(𝑥, 𝜔1)𝑐𝑜𝑠[𝜔1𝑡 + 𝜑1(𝑥, 𝜔1)] + 𝐴(𝑥, 𝜔2)𝑐𝑜𝑠[𝜔2𝑡 + 𝜑2(𝑥, 𝜔2)]}𝛾 =

∑ (𝛾𝑘)+∞

𝑘=0 ∑ (𝛾 − 𝑘𝑚

) [2𝑛0(𝑥)]𝛾−𝑘−𝑚{𝐴(𝑥, 𝜔1)𝑐𝑜𝑠[𝜔1𝑡 +

+∞𝑚=0

𝜑1(𝑥, 𝜔1)]}𝑚{𝐴(𝑥, 𝜔2)𝑐𝑜𝑠[𝜔2𝑡 + 𝜑2(𝑥, 𝜔2)]}

𝑘 (3.26)

Equation (3.25) can be further expanded using cos𝑘(𝜔𝑡) = 2−𝑘 ∑ (𝑘𝑚)𝑒𝑖(𝑘−2𝑚)𝜔𝑡∞

𝑚=0 . As only

signals modulated at the beat frequency ∆𝜔 = |𝜔2 − 𝜔1| contribute to HeLIC, the

demodulated HeLIC signal can be finally written as,

𝑆(∆𝜔) = 𝐹 ∑ ∑∏ (𝛾 − 𝑙)2𝑚+2𝑛+1𝑙=0

4𝑚+𝑛𝑚! (𝑚 + 1)! 𝑛! (𝑛 + 1)!

+∞

𝑛=0

+∞

𝑚=0

∫1

2𝑛0(𝑥)

𝛾𝑑

0

[𝐴(𝑥, 𝜔1)

𝑛0(𝑥)]

2𝑚+1

[𝐴(𝑥,𝜔2)

𝑛0(𝑥)]2𝑛+1

𝑒𝑖∆𝜑(𝑥)𝑑𝑥 (3.27)

For CQD-based thin films, ∆𝑁(𝑥, 𝜔) has been derived in Sect.6.2. In Fig.3.12(b), the

dependence of wideband (1-270 kHz) HeLIC images on the laser photoexcitation power was

investigated using a fixed 1-sun average intensity of modulated spread excitation beam while

changing the DC excitation intensity, i.e. 𝑛0(𝑥) in Eq. (3.27) changed from 0.2 to 1.1 sun. It

was found that the average amplitude of HeLIC images decreases with increasing DC

photoexcitation intensity which contrasts with its DC counterpart as shown in Fig. 3.12(b).

CHAPTER 3. 60

The decrease of the HeLIC signal with DC excitation is in agreement with Eq. (3.27) due to

the decreasing overall dependence on 𝑛0(𝑥): 𝑛0(𝑥)𝛾−2(𝑚+𝑛+2); 𝛾 = 0.6,𝑚, 𝑛 ≥ 0. As will be

discussed in detail in Sect.7.3, physically, for CQD systems featured with discrete energy

bands induced by spatial and energy disorder, the photogenerated excitons cannot dissociate

immediately into free charge carriers, in contrast to those generated in continuous energy band

semiconductors such as Si. Therefore, without the assistance of external forces including

interdot coupling and material interface effects, excitons are the dominant energy carriers in

CQD systems and act as the main radiative recombination sources [61]. Based on the foregoing

dominance of exciton dynamics in Sect.7.3, it was reported that the coefficient 𝛾 = 1 for

excitonic transitions while 𝛾 = 0.5 for carrier recombination induced by trap- or doping-

associated states. Hence, the physical meaning of Eq. (3.27) can be interpreted as an evolution

of the well-known exponential relation between photoexcitation and PL, 𝐼 ∝ 𝑁𝛾 , in the

framework of HeLIC. Therefore, the expectation of decreased HeLIC signal with increased

DC photoexcitation intensity from Eq. (3.27) results from three physical facts: unoccupied

trap-state density increases through enhanced photon absorption-mediated carrier ejection,

increased exciton density mediated carrier recombination, and the nature of radiative PL signal

collection in a heterodyne mode.

3.5 Comparison of Different Techniques and Advantages of PCR

and HeLIC

As discussed in Chapters 6 and 8, regarding charge carrier transport parameter

characterization, this thesis uses PCR and HeLIC and makes theoretical contributions to these

two novel techniques. Thereby, this section will compare general principles and features of

PCR and HeLIC with the commonly used techniques discussed above.

CHAPTER 3. 61

Based on the discussion and review of the carrier transport characterization methodologies

(SCCD/OCVD, PCD/μ-PCD, transient PL, PL imaging, PTR, and LIT) that are commonly

used by researchers, these techniques can be divided into either transient and quasi-steady-

state (frequency-modulated) techniques, or small-spot detection and large-area imaging

techniques.

SCCD/OCVD, PCD/μ-PCD, and transient PL are transient methodologies that start with

the application of a photoexcitation laser to pump the semiconductor surface and measure the

photoexcited excess carrier decay due to various carrier recombinations. The decay of excess

carrier concentration is monitored as a function of time through short-circuit current (SCCD),

open-circuit voltage (OCVD), conductance (PCD/μ-PCD), or PL emission (transient PL). This

excess carrier decay generally follows a minority carrier lifetime dependent exponential model

as shown by Eq. (3.8). Therefore, SCCD/OCVD, PCD/μ-PCD, and transient PL usually exhibit

an exponential decay spectrum. Depending on different recombination mechanisms in specific

samples and the characterization technique, effective carrier lifetimes can be extracted through

fitting an experimental decay spectrum to theoretical models. The common features and

disadvantages of these types of techniques can be summarized as follows.

1. The carrier lifetimes measured from these techniques are generally effective carrier

lifetimes. Limited by theoretical models, these techniques measure the overall excess carrier

density decay rate that is contributed by various recombination mechanisms, which is the

definition of effective carrier lifetimes. However, it should be noted that the transient PL

technique can measure diverse types of carrier lifetimes, which are determined by different

recombination mechanisms as shown in Fig. 3.2, while precise identification of the various

slopes in a PL decay spectrum is required.

CHAPTER 3. 62

2. These techniques cannot detect depth-resolved carrier transport properties, as they are

operated only at low frequencies. Therefore, the overall properties of a sample are measured,

making them unsuitable for particular property characterizations of p-n junctions in subsurface

regions, which, however, are essential for photovoltaic device efficiency optimization.

3. Transient methodologies are restrained by their relatively low SNR, especially when

carrier lifetimes are very low (for example in ns for some types of CQDs). Compared with

techniques such as PCR, HeLIC, lock-in thermography (LIT) and photothermal radiometry

(PTR) that use a lock-in amplifier, transient techniques always have low SNR values,

indicating that the experimental signal could be too small to measure when carrier lifetimes

are very short.

PTR and LIT use frequency-modulated photoexcitation. When compared with transient

techniques, PTR and LIT have two independent amplitude and phase channels, which ensure

high measurement reliability and accuracy. In addition, these techniques have high SNR values

due to the application of lock-in amplifiers. However, PTR and LIT are limited by the high

thermal diffusion length induced low resolution. Relying on the specific theoretical models

and the depth-resolution capability, PTR and LIT can measure multiple carrier transport

parameters. Generally, carrier lifetimes measured using these techniques are effective carrier

lifetimes.

The main advantage of large-area imaging over small-spot characterization is the capability

of an overall estimation of an entire photovoltaic device. However, the common transient PL,

frequency-modulated PL, and frequency-modulated PTR are either limited by low resolution,

the lack of depth-resolved carrier transport characterization capability, low SNR, low camera

frame rate, or by the high requirements of testing environments.

CHAPTER 3. 63

Compared with transient techniques, PCR overcomes the abovementioned disadvantages

and limitations. As invented by Mandelis et al. [54] in 2003, PCR is a spectrally gated dynamic

PL that applies frequency-modulated photoexcitation and collects radiative recombination

photons through an InGaAs single detector. Through fitting experimental frequency-dependent

PCR amplitudes and phases to theoretical models, multiple carrier transport parameters

including lifetime, diffusivity, and surface recombination velocities can be extracted [54, 63,

64]. With the use of a lock-in amplifier, PCR also has high SNR values. In addition, unlike

PTR that detects both radiative and nonradiative recombination, PCR only detects carrier

radiative recombination with the application of an InGaAs detector. Therefore, the theoretical

methodology is significantly simplified, increasing the measurement accuracy and uniqueness.

With the implementation of high frequency modulated photoexcitation, PCR can measure

carrier transport behavior at high rates and shows immense potential in the depth-resolved

characterization of photovoltaic properties. With the application of heterodyne techniques,

HeLIC overcomes the limitations of low camera frame rates and high exposure time

requirements for high-quality imaging. Therefore, ultrahigh-frequency (>100 kHz) imaging is

achieved for the first time. Compared with low-frequency-modulated (30 Hz) PTR and PL

imaging, high rate carrier transport behavior can be detected through HeLIC imaging.

Compared with transient techniques, PCR and HeLIC employ frequency-modulated

photoexcitation rather than pulse excitation. Comparing their theoretical models with other

commonly used transient techniques, they all begin with solving excess carrier continuous rate

equations to generate the analytical expression of excess carrier density. Therefore, the carrier

transport mechanisms described by transient techniques and PCR as well as HeLIC are the

same. However, theoretically, PCR and HeLIC need further mathematical manipulations to

CHAPTER 3. 64

transfer the excess carrier density from the time domain to frequency domain through various

techniques such as the Fourier transform. The capability of high-frequency detection is the

main advantage of PCR and HeLIC, which leads to the determination of high-rate carrier

transport dynamic behavior, while retaining high SNR values. The minority carrier lifetimes

extracted in this thesis using PCR and HeLIC are effective carrier lifetimes with the same

definition as those for the above-mentioned commonly used techniques. However, the

difference between measured carrier transport parameters, including lifetimes, from different

techniques can still be expected even when the definition of these parameters is the same for

each technique. This is because different approximations and assumptions are made during the

theoretical model development.

Furthermore, due to the application of frequency-modulated excitation and lock-in

amplifiers, PCR and HeLIC have higher SNR values (as high as 80 dB) [9] compared with μ-

PCD. This is the most prominent advantage of frequency-domain-associated techniques over

their time-domain counterparts. Additionally, the independent amplitude and phase channels

convey more material properties when compared with the PL techniques and the μ-PCD. The

amplitude channel not only contains information on carrier transport dynamics but also reflects

sample surface optical properties. In comparison, the phase channel is only determined by the

phase lag induced during various carrier recombination processes and is not influenced by the

photon excitation or emission intensity. More importantly, for PCR and HeLIC, the modulated

frequency determines the ac carrier diffusion length. In other words, when the modulation

frequency is much smaller than 1/τ, the ac diffusion length approximately equals the dc

diffusion length. However, when the modulation frequency is equal or higher than 1/τ, the ac

diffusion length is smaller than its dc counterpart [65]. Therefore, through increasing the

CHAPTER 3. 65

modulation frequency, the ac diffusion length can be further reduced; an effect that can be

applied to perform depth-resolved material property characterization. In this way, carrier

transport properties of the surface, sub-surface, pn junction, bulk, and back surface can be

distinctively characterized. This unique advantage of PCR and HeLIC can also increase image

resolution through increasing modulation frequencies.

3.6 Conclusions

For the characterization of CQD materials and solar cells, this chapter focuses on a review

of techniques that are extensively used for characterizing carrier transport properties. These

techniques include SCCD/OPVD, PCD (μ-PCD), TRPL, PCR, HeLIC, and methodologies

developed for carrier diffusion length measurements. PCR and HeLIC show apparent

advantages over other conventional techniques with respect to the all-optical, contactless,

nondestructive, high SNR, depth-resolution, and ultrahigh-frequency characterization of high-

rate carrier transport behavior features. However, the acquired parameters retain the same

definitions as the commonly used transient techniques. HeLIC overcomes the ubiquitous

limitations of camera-based techniques such as PL and LIT imaging. Therefore, ultrahigh-

frequency imaging is realized through HeLIC for the large-area characterization of high-rate

carrier transport behavior.

For future research interest, the development of large-area, all-optical, non-destructive,

contactless, ultrahigh-frequency imaging techniques for semiconductor material and device

characterization is the trend and promising for both fundamental carrier transport mechanism

study and for industrial product quality control and estimation.

66

Chapter 4

Quantitative Carrier Transport Study through Current-voltage

Characteristics

4.1 Introduction

Generally, to analyze current-voltage characteristics of CQD solar cells, the well-known

Shockley-Queisser (S-Q) equation is often used [1], which takes on the form 𝐽𝑖𝑙𝑙𝑢(𝑉) =

𝐽0 {𝑒𝑥𝑝 [𝑞𝑉𝑎

𝑛𝑖𝑑𝑘𝐵𝑇] − 1} − 𝐽𝑝ℎ, where 𝐽𝑖𝑙𝑙𝑢 is the current density under light illumination, 𝐽0 the

saturation current density, 𝐽𝑝ℎ a photocurrent density often treated as a constant short-circuit

current density, 𝑛𝑖𝑑 the ideality factor, 𝑘𝐵 the Boltzmann constant, 𝑞 the elementary charge, T

the absolute temperature, and 𝑉𝑎 the applied voltage. This equation was derived to model diode

behavior in an electrical circuit under the assumption of infinitely large material conductivity

(or carrier mobility) which is true for most solid-state p-n junctions consisting of highly

crystalline inorganic materials with a continuous carrier transport behavior in well-defined

lattice structures such as, Ge, Si, or GaAs. With high carrier mobility on the order of 102 - 103

cm2/Vs typical of Si solar cells. However, today’s CQD-based materials and devices feature

multiple energy disorder sources due to their high surface-to-volume ratio nanostructures,

variations in confinement energy and coupling, as well as thermal broadening. All of these

point to a non-continuous hopping/tunneling transport with low mobility ranging from 10-5 to

10-1 cm2/Vs [2, 3, 4, 5-10] for PbS CQDs passivated with various ligands. This huge difference

in mobility leads to questioning the validity of applying the S-Q model to such systems.

CHAPTER 4. 67

Although many authors have developed diffusion and drift principles ad hoc [11-14] that

are usually valid for semiconductor materials with well-defined continuous energy band

structure, carrier localization within a quantum dot and the effects of disorder which also

induces e.g. Anderson localized electrons [15]. With respect to electrical transport in colloidal

quantum dots, Guyot-Sionnest [16] summarized that “Many separate sources of disorder make

it extremely unlikely that QDS based on colloidal assemblies are anywhere close to exhibiting

band-like transport behavior”. Therefore, the evidence of low dot-to-dot transmission due to

variations in confinement energy and in coupling and thermal broadening, and

electron−electron repulsion points to hopping conductivity and diffusivity [16, 17]. In CQD

ensembles, exciton hopping transport in CQD solar cells involves exciton diffusion and

dissociation distinct from charge transport mechanisms which govern conventional inorganic

silicon cells [18]. However, most studies of CQD solar cells to date are based on the classical

S-Q equation that was derived under the assumption of continuous energy band semiconductor

systems where electron-hole pairs dissociate immediately upon their generation and travel at

high charge mobility. Crystalline Si is representative of these photovoltaic materials whereas

CQD and organic solar cells are excluded from that category [19, 20]. Würfel, et al. [21]

demonstrated that the Schottky equation cannot be applied to low-mobility materials in the

way it is used for inorganic solar cells, and device parameters extracted from the Schottky

equation such as ideality factor, series resistances, and shunt resistance lack real physical

meaning and provide very limited assistance toward the optimization of device fabrication. In

comparison, material parameters including carrier mobility, diffusion length, lifetime,

diffusivity, and trap states offer more important information for solar cell device structure

optimization, starting with the selection of materials. Unfortunately, in common with organic

CHAPTER 4. 68

photovoltaic materials, CQDs constitute low carrier mobility photovoltaic materials and have

high exciton binding energy, both impediments to improving solar efficiency in a

straightforward manner. In summary, a full understanding of charge carrier transport

mechanism within CQD devices is lacking and new theoretical I-V models based on the full

understanding is demanded.

Another impediment to solar cell device efficiency is the formation of a Schottky junction

at the CQD/anode interface which forms an electric field with the direction opposite to that of

the light incidence. Schottky junctions cause holes to accumulate at the CQD/anode interface

[20, 22], thereby reducing the surface recombination velocity (SRV) as observed in organic

photovoltaic devices [23]. All of these adverse processes handicap hole extraction at the anode

and lead to low solar conversion efficiency. An experimental consequence of hole

accumulation is the formation of anomalous (including S-shaped) current-voltage (I-V) curves,

which have been reported for heterojunction CQD solar cells with increasing frequency [24,

19, 25]. Nevertheless, the origins of the formation of these anomalous I-V curves have not

been well understood, nor have they been exploited toward designing higher performance solar

cells. Wang et al. [26] attributed them to the electron accumulation effect induced by an

exciton blocking layer. Wagenpfahl et al. [23] found the reduction of hole-associated surface

recombination velocities could also give rise to S-shaped I-V curves. One approach to studying

S-shaped I-V curves is through fitting experimental data to theoretical electric circuit models.

However, although Romero et al. [27] developed equivalent electric circuit models that can

quantitatively simulate three kinds of I-V curves in terms of forward and reverse diodes, they

did not consider actual physical mechanisms.

CHAPTER 4. 69

This chapter introduces the drift-diffusion I-V model in the framework of carrier hopping

transport in CQD solar cells. Based on this model, the anomalous I-V curves that significantly

retard CQD solar cell efficiency were quantitatively analyzed using a novel double-diode

electric circuit model to elucidate their origins: imbalanced carrier mobility and Schottky

barrier. In addition, interface effects on CQD solar cell open-circuit voltage dissipation were

quantitatively discussed. The temperature-dependent carrier hopping transport in CQD solar

cells were demonstrated through the I-V model and experimental temperature-dependent I-V

characteristics of the CQD solar cells under study. At last, performance factors of our device

architecture are discussed with improvement recommendations.

4.2 Derivation of Current-voltage Model from Hopping and

Discrete Carrier Transport

4.2.1 Carrier Hopping Diffusivity and Mobility in Quantum Dot

Systems

In a QD ensemble that QDs are separated by a mean distance from its nearest neighbors

and features own size and energy manifold. The rate equation for particle population Ni(x, t)

[particles/cm3] in QD (i) can be expressed as the net rate of carriers hopping into and out of a

QD [28]:

𝜕𝑁𝑖(𝑥,𝑡)

𝜕𝑡= −

𝑁𝑖(𝑥,𝑡)

𝜏− ∑ 𝑃𝑖𝑗𝑁𝑖(𝑥, 𝑡) +𝑗 ∑ 𝑃𝑗𝑖𝑁𝑗(𝑥, 𝑡)𝑗 (4.1)

in which 𝜏 is the effective lifetime of particles including excitons and free charge carriers, 𝑃𝑖𝑗

is the probability for a particle to migrate from a QD (i) to QD (j). Considering a flow of

particles hoping in and out of a virtual volume element dV = Adx in the colloidal medium, A is

CHAPTER 4. 70

a cross sectional area. As shown in Fig.4.1, the net particle flux 𝐽𝑒(𝑥, 𝑡) (particles/cm2s) is

given by

𝐴[ 𝐽𝑒(𝑥, 𝑡) − 𝐽𝑒(𝑥 + 𝑑𝑥, 𝑡)]𝑑𝑡 = −𝜕

𝜕𝑥 𝐽𝑒(𝑥, 𝑡)𝑑𝑉𝑑𝑡 (4.2)

Combining Eqs. (4.1) and (4.2), the net particle population rate entering dV in time dt is in the

form,


𝜕𝑡= −

𝜕𝐽𝑒(𝑥,𝑡)

𝜕𝑥−𝜕𝑁𝑖(𝑥,𝑡)

𝜏 (4.3)

The first term on the right-hand side denotes trans-volume particle hopping into and/or out of

dV, and the second term represents local particle de-excitation within dV through radiative and

nonradiative recombination. It is tough to calculate the summation terms over the entire

ensemble of particles in Eq. (4.1), therefore, the nearest neighbor hopping (NNH)

approximation will be adopted as shown in Fig.4.1 (b) the 1D NNH kinetics. Under external

optical excitation induced particle population gradient, the net flux 𝐽𝑒(𝑥, 𝑡) at QD (i) is a results

of flux out of QD (i+1) and QD (i-1) into the QD (i).Therefore, these fluxes are [28]:

𝐽𝑖+1,𝑖𝑛(𝑥 + ∆𝑥, 𝑡) =1

2𝑁𝑖+1(𝑥 + ∆𝑥, 𝑡)𝑓𝑖+1𝑒

−𝛾∆𝑥 (4.4 a)

𝐽𝑖−1,𝑖𝑛(𝑥 − ∆𝑥, 𝑡) =1

2𝑁𝑖−1(𝑥 − ∆𝑥, 𝑡)𝑓𝑖−1𝑒

−𝛾∆𝑥 (4.4 a)

It should be noted that the factor 1

2 accounts for the equal probability for a particle in QDi to

hop into QDi-1 and QDi+1 in a 1D geometry. ∆𝑥 is the hopping distance (equal to interdot

distance) as shown in Fig. 4.1 (b). The factor fj denotes particle hopping frequencies, and

𝑒−𝛾∆𝑥 represents the dot-to-dot crossing probability with a hopping transmission coefficient 𝛾.

According to the flux conservation, the net flux across the virtual area A at x is given by [28]

𝐽𝑖(𝑥, 𝑡) = 𝐽𝑖−1,𝑖𝑛(𝑥 − ∆𝑥, 𝑡) − 𝐽𝑖+1,𝑖𝑛(𝑥 + ∆𝑥, 𝑡) =1

2𝑒−𝛾∆𝑥 [

𝑁𝑖−1(𝑥−∆𝑥,𝑡)

𝜏ℎ,𝑖−1−𝑁𝑖+1(𝑥+∆𝑥,𝑡)

𝜏ℎ,𝑖+1] ∆𝑥

(4.5)

CHAPTER 4. 71

Here, 𝜏ℎ,𝑗 = 1/𝑓ℎ,𝑗 is defined as the hopping time, representing particles hopping between two

nearest neighbors.

Figure 4.1: (a) A virtual volume element dV = Adx, resembling a QD, illustrates the discrete

hopping transport of excitons and charge carriers. (b) Schematic of the discrete particle flux

into and out of three adjacent virtual planes. All planes have an area of A across the thickness

direction of a CQD solar cell. Adapted from ref. [28].

The hopping velocity 𝑣ℎ,𝑗 for a particle hoping from a QD to its nearest neighbor is defined

as ∆𝑥 = 𝐿 = 𝑣ℎ,𝑗𝜏ℎ,𝑗, in which 𝐿 is the effective QD-to-QD distance and 𝑣ℎ,𝑗 is the population

gradient associated hopping velocity that includes the drift velocity 𝑣𝑑𝑟,𝑗 when external or

internal electric field E exist. Therefore, for free charge carriers, the overall velocity 𝑣𝑇,𝑗

should include both types of hopping transport:

𝑣𝑇,𝑗 = 𝑣ℎ,𝑗 ± 𝑣𝑑𝑟,𝑗 (4.6a)

The overall velocity 𝑣𝑇,𝑗 is determined by the relative direction of the particle population

gradient and the internal or external electric field. The hopping velocity, 𝑣ℎ,𝑗(𝐸) , in the

presence of electric field can be further expressed as a function of the hopping velocity 𝑣ℎ,𝑗(0)

without electric field E and the electric field:

𝑣ℎ,𝑖±1(𝐸) = 𝑣ℎ,𝑖±1(0) ±𝑞𝐸𝐿

𝑚𝑒𝑥𝑣ℎ,𝑖±1(0) (4.6b)

CHAPTER 4. 72

in which q is the free carrier charge element, 𝑚𝑒𝑥 is the effective particle mass, the ± signs

account for the motion in the direction (+) or opposite to the direction (-) of the electric field.

It should be noted that the electric filed E changes the dipole moment �� = 𝑞�� of an exciton,

in which R is an effective length between the constituent electron and hole along the direction

of the field [29]. The exciton potential energy can be expressed by ∆𝑈𝑝𝑜𝑡 = ∆�� ∙ ��. Under

electric field, the length R is stretched and shrunk through two vectors: ∆�� = ±𝑞∆��. Therefore,

the net effect of the electric field on an exciton is a drift motion with a peak defer velocity in

the direction of the electric field [28]:

𝑣𝑑𝑟,𝑗 = (2∆𝑈𝑝𝑜𝑡

𝑚𝑒𝑥)1/2

= [2𝐸(

𝑑��

𝑑𝐸)∙��

𝑚𝑒𝑥]

1/2

(4.6c)

in which 𝑑��

𝑑𝐸 is a material-medium-dependent physical property and is defined as the gradient

of the exciton dipole moment in the electric field. Implementing Eq. (4.6c), the effective

exciton mobility is derived as

𝜇𝑒𝑥 = √2𝑑��

𝑑𝐸∙��𝐸

𝑚𝑒𝑥 (4.7a)

in which ��𝐸 is a unit vector which is in the direction of the electric field. Given very close

interdot distance (~ 1 nm) for most CQD materials, 𝑁𝑖±1(𝑥 ± ∆𝑥) in Eq. (4.5) can be expanded

while only keep the first (linear) order terms to yield particle flux for both excitons and free

charge carriers [28]:

𝐽𝑖(𝑥, 𝑡) ≈ [−𝑣ℎ𝑜𝐿 (𝜕𝑁𝑖(𝑥,𝑡)

𝜕𝑥) + 𝑣𝑑𝑟,𝑖𝑁𝑖(𝑥, 𝑡)] 𝑒

−𝛾𝐿 (4.7b)

CHAPTER 4. 73

where 𝑣ℎ𝑜 is the hopping velocity when the electric field E = 0. In CQD solar cells, CQD thin

films are always assumed as the nominal p-type energy absorption layer. When compared with

conventional p-type carrier transport in continuous semiconductors:

𝐽(𝑥, 𝑡) = −𝐷 [𝜕𝑁(𝑥,𝑡)

𝜕𝑥] + 𝜇𝐸𝑁(𝑥, 𝑡) (4.8)

Comparing Eq. (4.7b) and (4.8), the hopping diffusivity can be defined by

𝐷 ≡ 𝐷ℎ(𝑇) = [𝐿2

𝜏ℎ(𝑇)] exp (−𝛾𝐿) (4.9)

Corresponding, the hopping mobility of free charge carriers can be defined as

𝜇ℎ(𝑇) = 𝜇(𝑇)exp (−𝛾𝐿) (4.10)

with 𝜇(𝑇) = 𝜇𝑒𝑥 in Eq. (4.7a) for excitons, and 𝜇(𝑇) = [𝑞𝐿

𝑚𝑒𝑓𝑣ℎ0(𝑇)] for free charge carriers

using Eq. (4.6b). As indicated in Eqs. (4.9) and (4.10), both 𝐷ℎ and 𝜇ℎ are temperature-

dependent due to the thermally activated interdot hopping process. The hopping time constant

𝜏ℎ as defined in Eq. (4.9) is associated with the hopping probability Pij [30] and can be obtained

by

1

𝜏ℎ(𝑇)=

1

𝜏0exp (−

∆𝐸𝑗𝑖

𝑘𝑇) (4.11a)

in which, ∆𝐸𝑗𝑖 is the energy difference when a particle hopping from an initial QDi to the final

QDj, i.e. ∆𝐸𝑗𝑖 = 𝐸𝑗 − 𝐸𝑖. The term 𝜏0 is a characteristic time for particles tunneling/hopping

between neighbor QDs at a distance x, and has been given by ref. [30]

1

𝜏0= 𝑓𝑒𝑥𝑝(−𝛾𝐿) (4.11b)

CHAPTER 4. 74

Therefore, 𝜏0 is determined by the tunneling attempt frequency f and the probability 𝛾 which

is dependent on the energy barrier with an effective thickness 𝐿 that separates two QDs at

location i and j, i.e. the interdot distance.

Furthermore, Guyot-Sionnest [16] proposed that the carrier hopping mobility 𝜇ℎ(𝑇) is

dependent on the sum of the energy barriers Ea that encountered by hopping particles.

𝜇ℎ(𝑇) = (𝑞𝐿2𝐸𝑎

3ℎ𝑘𝑇) exp (−𝛾𝐿 −

𝐸𝑎

𝑘𝑇) (4.12)

This expression is structurally similar to Eq. (4.10) and predicts a thermal process with an

activation energy Ea estimated to be in the range of 10-50 meV [16], which is consistent with

experimental results obtained in this work. It is obvious to see that both transport processes are

spatially limited by the nearest neighbor QD energy state differences. Clearly, when compared

with conventional continuous energy band carrier transport, in the hopping picture, a very

different physical interpretation with respect to carrier diffusivity, mobility, critical transport

length is observed. These unusual discrete hopping transport of carriers in QD ensembles

results in the breakdown of the conventional Einstein relation. In view of Eqs. (4.9), (4.10),

(4.11), and (4.12), the ratio of mobility to diffusivity in the framework of “hopping” activation

transport in QD ensembles is given by [28]:

𝜇ℎ(𝑇)

𝐷ℎ(𝑇)= (

𝑞𝜏0𝐸𝑎

3ℎ𝑘𝑇) exp (−

𝐸𝑎−∆𝐸

𝑘𝑇) ≡ (

𝑞𝜏0𝐸𝑎

3ℎ𝑘𝑇) exp (−

∆𝐸𝑎

𝑘𝑇) (4.13)

in which 𝜇ℎ is the bulk mobility as measured in the external circuit. Equation (4.13) is a special

expression of the conventional Einstein relation that can be obtained when 𝐸𝑎 = ∆𝐸 = ℎ/𝜏0.

In other words, Equation (4.13) evolves to the conventional Einstein relation when the carrier

diffusion and drift rates are limited by the same thermal activation process, which is defined

CHAPTER 4. 75

as thermal velocity in continuous band semiconductors, and when the rate limiting step is the

dot-to-dot tunneling rate 𝜏0−1.

4.2.2 Current Density J(x) across CQD Solar Cells

Figure 4.2: Schematic of one CQD solar cell energy band structure.

Disorder sources in CQD ensembles, including variations in confinement energy, electron-

electron repulsion, coupling and thermal broadening, cause CQD-based materials and devices

to exhibit discrete hopping conductivity and diffusivity [16, 17]. Considering a general CQD

solar cell structure, in which CQD layers act as a nominal p-type light absorption layer, for

example, as shown in Fig.4.2, a CQD solar cell with the energy diagram of

TiO2/CQD/MoO3/Au/Ag. At equilibrium, 𝐽(𝑥, 𝑡) = 0 across the entire CQD layer thickness,

the carrier hopping drift current density equals hopping diffusion current density

𝐷ℎ𝑑𝑁(𝑥)

𝑑𝑥− 𝜇ℎ𝐸𝑁(𝑥) = 0; 0 ≤ 𝑥 ≤ 𝑑 (4.14)

CHAPTER 4. 76

in which the subscript i, denoting the discrete spatial location, has been adequately represented

by the coordinate variable x. The electric field can be derived through integrating 𝐸(𝑥) =

− 𝑑𝑉(𝑥)

𝑑𝑥 over the thickness of the solar cell and yields [28]

𝑁(0)

𝑁(𝑑)= exp (−

𝜇ℎ𝑉0

𝐷ℎ) (4.15)

The factor 𝑉0 is defined as the potential difference at x = 0 and d, i.e. 𝑉0 = 𝑉(0) − 𝑉(𝑑).

For a non-equilibrium situation where a non-zero photovoltage 𝑉𝑎 is generated upon

optical illumination of a QD medium and lead to the injection or extraction of excitons and

free charge carriers, Eq. (4.15) becomes [28]

𝑁𝑃𝑉(0)

𝑁𝑃𝑉(𝑑)= exp {−

𝜇ℎ

𝐷ℎ[𝑉0 − 𝑉𝑎]} (4.16)

The subscript PV indicates that the solar cell is under external bias. Defining ∆𝑁𝑃𝑉(0) =

𝑁𝑃𝑉(0) − 𝑁(0) as the excess carrier population under illumination, apply a quenching

boundary condition at x = d as proposed by Zhitomirsky et al. [14] that 𝑁𝑃𝑉(𝑑) − 𝑁(𝑑) = 0,

therefore, Eqs. (4.15) and (4.16) yield

∆𝑁𝑃𝑉(0) = 𝑁𝑃𝑉(0) [𝑒𝑥𝑝 (𝜇ℎ𝑉𝑎

𝐷ℎ) − 1] (4.17)

It can be expected that, for ultrathin CQD layers in several hundred nanometers, the population

difference of photons absorbed across the thickness of the layer is relatively small, therefore,

𝑁𝑃𝑉(0) ≈ 𝑁𝑃𝑉(𝑑) sometimes can be assumed. In a one-dimensional quantum dot ensemble,

each quantum dot characterized by its own size and energy manifold is separated by a mean

distance from its neighbors. The rate equation for the net carrier (electron, hole, or exciton)

flux entering one quantum dot within a time interval 𝑑𝑡 has been given as Eq. (4.3). With an

CHAPTER 4. 77

applied dc voltage, 𝑛𝑃𝑉 becomes time-independent, therefore, Eq. (4.3) can be further written

as

𝑑2𝑁𝑃𝑉(𝑥)

𝑑𝑥2− (

𝜇ℎ𝐸

𝐷ℎ)𝑑𝑁𝑃𝑉(𝑥)

𝑑𝑥−𝑁𝑃𝑉(𝑥)

𝐷ℎ𝜏= 0 (4.18)

where 𝜏 presents the total lifetime of the charge carrier, 𝐸 = 𝑉𝑏𝑖 −𝑉𝑒𝑥𝑡

𝑑 is the electric field

across the solar cell, 𝑉𝑏𝑖 is the built-in voltage, and 𝑉𝑒𝑥𝑡 is the photovoltage. The general

solution to Eq. (4.18) takes the form as

𝑁𝑃𝑉(𝑥) = 𝐴𝑒𝑄1𝑥 + 𝐵𝑒𝑄2𝑥 (4.19)

with

𝑄1,2 =1

2(𝐶0 ±√𝐶0

2 + 4𝐶1) (4.20)

and with definitions

𝐶0 =𝜇ℎ𝐸

𝐷ℎ= (

𝜇ℎ

𝐷ℎ)𝑉𝑏𝑖−𝑉𝑎

𝑑, 𝐶1 =

1

𝐿ℎ2 , 𝐿ℎ = √𝐷ℎ𝜏 (4.21)

where 𝐿ℎ is the carrier dc hopping diffusion length. Constants A and B can be solved with the

application of the abovementioned quenching boundary conditions, therefore, highlighting the

temperature dependence, the carrier diffusion current density is given by

𝐽ℎ,𝑑𝑖𝑓𝑓(0, 𝑇) = 𝑞𝑁0𝐷ℎ(𝑇)

𝐿ℎ(𝑇)(𝑐𝑜𝑠ℎ[𝑑/𝐿ℎ(𝑇)]

𝑠𝑖𝑛ℎ[𝑑/𝐿ℎ(𝑇)]) (𝑒𝑥𝑝 [

𝜇ℎ(𝑇)𝑉ℎ

𝐷ℎ(𝑇)] − 1) (4.22)

Correspondingly, the hopping drift current density is expressed as

𝐽ℎ,𝑑𝑟𝑖𝑓(0, 𝑇) = 𝑞𝑁0𝐷ℎ(𝑇)

𝐿ℎ(𝑇)𝜇ℎ(𝑇)𝐸 (𝑒𝑥𝑝 [


𝐷ℎ(𝑇)] − 1) (4.23)

CHAPTER 4. 78

The total carrier hopping density is the summation of diffusion and drift current density in the

form of

𝐽ℎ(0, 𝑇) = 𝐽ℎ,𝑑𝑖𝑓𝑓(0, 𝑇) + 𝐽ℎ,𝑑𝑟𝑖𝑓(0, 𝑇) (4.24)

4.3 Imbalanced Charge Carrier Mobility and Schottky Junction

Induced Anomalous J-V Characteristics of CQD Solar Cells

Although a sharp boost in CQD solar cell power conversion efficiency (PCE) has been

observed from 3% to 12 % in only 7 years [31], limiting factors are emerging at high frequency

to retards significant progress in CQD solar cell PCE improvement. One of the substantial

impediments to CQD solar cell device efficiency is the formation of a Schottky junction at the

CQD/anode interface which forms an electric field with the direction opposite to that of the

light incidence. This Schottky junction causes holes to accumulate at the CQD/anode interface

[20, 22], thereby reducing the surface recombination velocity (SRV) as observed in organic

photovoltaic devices [23]. These adverse processes handicap hole extraction at the anode and

lead to low solar conversion efficiency. An experimental consequence of hole accumulation is

the formation of anomalous (including S-shaped) current-voltage (I-V) curves, which have

been reported for heterojunction CQD solar cells with increasing frequency [19, 24, 25].

Nevertheless, the origins of the formation of these anomalous I-V curves have not been well

understood, nor have they been exploited toward designing higher performance solar cells.

Wang et al. [26] attributed them to the electron accumulation effect induced by an exciton

blocking layer. Wagenpfahl et al. [23] found the reduction of hole-associated surface

recombination velocities could also give rise to S-shaped I-V curves. One approach to studying

S-shaped I-V curves is through fitting experimental data to theoretical electric circuit models.

CHAPTER 4. 79

However, although Romero et al. [27] developed equivalent electric circuit models that can

quantitatively simulate three kinds of I-V curves in terms of forward and reverse diodes, they

did not consider actual physical mechanisms.

In this section, the temperature-dependent I-V characteristics are analyzed using a novel

double-diode electric circuit model and a carrier hopping transport model to interpret the

culprits of the notorious anomalous I-V characteristics of CQD solar cells. The open-circuit

deterioration induced by interface defects associated states are quantitatively analyzed. The

phonon-assisted carrier hopping transport behavior in CQD systems is addressed.

4.3.1 CQD Solar Cell Fabrication and Current-voltage

Characterization

PbS CQDs were prepared following the previous reports [32, 33]. Briefly, oleic acid (4.8

mmol), PbO (2.0 mmol) and 1-octadecene (ODE, 56.2 mmol) were mixed and heated to 95 oC

under vacuum. This was followed by the injection of bis(trimethylsilyl) sulfide and ODE at a

high temperature of 120 oC. After cooling, the PbS CQDs were successively precipitated and

re-dispersed using acetone and toluene, respectively. The products were stored in a nitrogen

glove box for further surface passivation treatments. CQD solar devices fabricated in this

manner have the structure of ITO/ZnO/PbS-TBAI CQD/PbS-EDT CQD/Au which has been

demonstrated by two groups to be stable for ca. 5 months [24], and at least 1 month [34],

respectively. S-shaped I-V curves at room temperature for a CQD solar cell with this structure

were also reported by Chuang et al. [24] but were not well-explained in terms of physical

optoelectronic processes. Devices were fabricated in air through a typical solution process [24,

32, 35]. TBAI and EDT denote tetrabutylammonium iodide and 1, 2-ethanedithiol,

respectively. They are exchange-ligands for PbS CQDs to passivate quantum dot surface trap

CHAPTER 4. 80

states and adjust the interdot distance which determines the coupling strength between two

neighboring dots. The exciton transition energy (effective band gap) of PbS CQD was

measured in a solid CQD thin film to be ca.1.4 eV. In the fabrication process, a ca.100-nm

ZnO nanoparticle layer was spin-coated onto a clean glass substrate with a pre-deposited ITO

electrode of 145 nm thickness. PbS CQD layers were deposited through a layer-by-layer spin-

coating process, after which a TBAI solution (10 mg/ml in methanol) was applied to the

substrate for 30 s, followed by successive methanol rinse-spin steps. An EDT solution (0.01

vol% in acetonitrile) and acetonitrile were used for the deposition of the PbS-EDT nanolayer.

As examined by scanning electron spectroscopy, the final thicknesses of PbS-TBAI and PbS-

EDT CQD layers were ca. 200 nm and 50 nm, respectively. In addition, a 120-nm-thick Au

anode was evaporated on top of PbS-EDT CQD layer. Fig. 4.3 shows a schematic of the

fabricated CQD solar cell. Its energy diagram is shown in Fig. 4.4 which illustrates that, under

illumination, excitons generated in PbS-TBAI dissociate into free electron and hole carriers

through interdot coupling strength and the electric field at the heterojunction interface. Free

electrons are swept onto the cathode within the depleted area. In addition, because of the

energy barrier formed at the PbS-TBAI/PbS-EDT interface, electrons are blocked from

flowing to the anode which can significantly increase Isc and Voc [24, 35]. Unfortunately, as

will be discussed later, such architecture leads to the formation of a Schottky barrier in PbS-

EDT, which prevents holes from being extracted to the external Au anode, resulting in hole

accumulation and formation of an electric field with reverse direction to the forward field in

the main heterojunction diode (PbS-TBAI/ZnO). In the following discussion, for the sake of

clarification, the subscript h refers to the heterojunction diode (ZnO/PbS-TBAI) and s refers

CHAPTER 4. 81

to the Schottky diode (PbS-EDT/Au). Fig. 4.5 shows the equivalent circuit: two diodes with

opposite electric field directions representing heterojunction and Schottky diode, respectively.

Figure 4.3: Schematic of the double-layer CQD solar cell with the structure: ITO/ZnO/PbS-

TBAI QD/PbS-EDT QD/Au.

Figure 4.4: Device energy band diagram under illumination. PbS-EDT acts as an electron

blocking layer and a Schottky barrier is formed for holes, thus prevent their extraction to the

Au anode.

Figure 4.5: Equivalent electric circuit of a double-diode model, consisting of a heterojunction

diode between ZnO and PbS-QD layers and a Schottky diode between PbS-EDT and Au.

CHAPTER 4. 82

The homojunction formed at the PbS-TBAI /PbS-EDT interface gives rise to an electric

field with forwarding direction, same as that of the heterojunction, but its strength is

diminished by the reverse Schottky barrier because of the low PbS-EDT layer thickness (50

nm). Therefore, to simplify the analysis and improve parameter fitting reliability by decreasing

the number of unknown parameters in the hopping transport theoretical model developed [28]

and used in this work, this homojunction is not considered as an independent diode in this

paper, but part of the heterojunction.

The I-V characteristic curve measurements were obtained under laser excitation with a

wavelength of 830 nm and excitation intensity of 100 mW/cm2. The samples were placed in

a Linkam LTS350 cryogenic chamber which can maintain a constant temperature in a range

from 77 K to 520 K. For this study, the I-V characterizations were performed at 300 K, 250 K,

200 K, 150 K, and 100 K.

4.3.2 Double-diode-equivalent Hopping Transport Model

CQD solar cell I-V curves can be quantitatively interpreted using a hopping transport

mechanism as discussed in Sect.4.2. Referring to the energy diagram in Fig. 4.4, any hole

accumulation at the PbS-EDT/Au interface due to the effective impedance presented by the

Schottky barrier can give rise to a local space charge layer (SCL) 𝑊 and an electric field with

opposite direction to the heterojunction electric field. Therefore, charge carrier hopping

transport in PbS-TBAI (heterojunction) dominated by electron transport, and in PbS-EDT

(Schottky barrier) dominated by hole transport, should be distinct. Influence of the Schottky

barrier on the net current density is expected to be strong and possibly dominant over the local

diffusion current density due to the thinness of the SCL, especially at low temperatures. To

CHAPTER 4. 83

extract carrier hopping transport parameters, the electron current across the heterojunction is

expressed as the sum of diffusion and drift currents according to Eq. (4.24):

𝐼ℎ = 𝑞𝐴𝑁0 {[𝐷ℎ(𝑇)

𝐿ℎ(𝑇)] (

𝑐𝑜𝑠ℎ[𝑑/𝐿ℎ(𝑇)]

𝑠𝑖𝑛ℎ[𝑑/𝐿ℎ(𝑇)]) + 𝜇ℎ(𝑇)𝐸ℎ} (𝑒𝑥𝑝 [


𝐷ℎ(𝑇)] − 1) (4.25)

Here, 𝐴 is the CQD solar cell area exposed to the light, 𝑁0 is the electron/hole population at

equilibrium without illumination, 𝐷ℎ(𝑇) is the electron hopping diffusivity, 𝐿ℎ(𝑇) is the

electron diffusion length, 𝑑 is the CQD layer thickness, 𝜇ℎ is the electron hopping mobility in

PbS-TBAI corresponding to the heterojunction, 𝑉ℎ is the electric potential across the entire

heterojunction, and 𝐸ℎ is the electric field. According to this mechanism, the heterojunction

gives rise to an electric field across the entire CQD nanolayer, while the Schottky diode

generates a reverse electric field in the local SCL with a nominal thickness of 𝑊. Similarly, as

a combination of diffusion and drift currents the hole current flowing across the Schottky diode

can be expressed as:

𝐼𝑠 = 𝑞𝐴𝑁0 {[𝐷𝑠(𝑇)

𝐿𝑠(𝑇)] (

𝑐𝑜𝑠ℎ[(𝑑−𝑊)/𝐿𝑠(𝑇)]

𝑠𝑖𝑛ℎ[(𝑑−𝑊)/𝐿𝑠(𝑇)]) + 𝜇𝑠(𝑇)𝐸𝑠} (𝑒𝑥𝑝 [

𝜇𝑠(𝑇)𝑉𝑠

𝐷𝑠(𝑇)] − 1) (4.26)

All parameters are analogous to those of the heterojunction in Eq. (4.25), while it is noted that

𝐸𝑠 is the electric field associated with the Schottky diode. The analysis of the double-diode

electric circuit model of Fig. 4.5 yields that at point 1

𝑉 = 𝑉ℎ − 𝑉𝑠 (4.27a)

at point 2,

𝐼𝑝ℎ = 𝐼 + 𝐼ℎ (4.27b)

and

𝐼 = 𝐼𝑠 (4.27c)

CHAPTER 4. 84

therefore,

𝐼𝑝ℎ = 𝐼𝑠 + 𝐼ℎ (4.27d)

Using Eqs. (4.25) to (4.27), the external voltage 𝑉 can be written as a function of the external

current 𝐼 and the free charge carrier hopping transport parameters:

𝑉 =𝐷ℎ(𝑇)

𝜇ℎ(𝑇)𝑙𝑛

{

𝐼𝑝ℎ−𝐼

𝑞𝐴𝑁0[𝐷ℎ(𝑇)

𝐿ℎ(𝑇)

𝑐𝑜𝑠ℎ[𝑑

𝐿ℎ(𝑇)]

𝑠𝑖𝑛ℎ[𝑑

𝐿ℎ(𝑇)]+𝜇ℎ(𝑇)𝐸ℎ]

+ 1

}

−𝐷𝑠(𝑇)

𝜇𝑠(𝑇)𝑙𝑛

{

𝐼

𝑞𝐴𝑁0[𝐷𝑠(𝑇)

𝐿𝑠(𝑇)

𝑐𝑜𝑠ℎ[𝑑

𝐿𝑠(𝑇)]

𝑠𝑖𝑛ℎ[𝑑

𝐿𝑠(𝑇)]+𝜇𝑠(𝑇)𝐸𝑠]

+ 1

}

(4.28)

4.3.3 Origins of Anomalous Current-voltage Curves

To investigate the effects of imbalanced charge carrier mobilities in the PbS-TBAI and

PbS-EDT, I-V curves were simulated using Eq. (4.28) and gradually reducing the hole mobility

µs from 1 cm2/Vs to 0.005 cm2/Vs, while other hopping parameters were kept unchanged as

shown in Fig. 4.7(a). Trial values for each parameter were selected referring to reported values

for different ligand-treated PbS CQDs. Kholmicheva et al. [37] studied MOA (8-

mercaptooctanoic acid), and MPA (3-mercaptopropionic acid) treated PbS CQDs using

photoluminescence (PL) spectroscopy and measured the corresponding exciton diffusivities to

be 0.003 cm2/s and 0.012 cm2/s, respectively. Carey et al. [38] estimated free electron and hole

diffusion length in the range from 30 to 230 nm for various PbS CQDs, including EDT-treated,

pure, CdCl2 treated, bromide treated, pure fused, and solution or solid-state-iodide-treated PbS

CQDs. For electron and hole mobility in CQD nanolayers used as CQD solar cell light

absorption and charge-carrier-transport layers, Carey et al. [38], and Tang and Sargent [18]

tabulated hole (electron) mobility for three types of ligand-treated PbS CQDs. PbS CQD

CHAPTER 4. 85

nanolayers without further ligand treatments yielded hole (electron) mobility of 7.2×10-4

cm2/Vs (1×10-3 cm2/Vs for electrons), while for CdCl2-treated and butylamine-treated CQD

nanolayers, it was reported to be 1.9 ×10-3 cm2/Vs (4.2 ×10-3 cm2/Vs for electrons) and 1.5

×10-3 cm2/Vs (2 ×10-4 cm2/Vs for electrons), respectively. To our best knowledge, the strength

of the hetero/homojunction-induced electric field in PbS CQDs has not been reported.

However, the electric field should be inversely proportional to device thickness. In InAs/GaAs

QD solar cells, Kasamatsu et al. [39] reported that the electric field was 46 kV/cm when the

device thickness was ca. 300 nm. This value increased to 193 kV/cm when the device thickness

was reduced to 50-nm. Due to the thinness of our Schottky diode SLC, a relatively higher

electric field value of 1×106 V/cm was chosen for simulations. To investigate the influence of

hole mobility µs on the solar cell I-V curves, parameters common to both heterojunction and

Schottky diode were considered to have the same values.

Figure 4.6: I-V characteristic curves of a CQD solar cell measured at 300K (a), 250K (b), 230K

(c), 200K (d), 150K (e), and 100K (f).

CHAPTER 4. 86

Figure 4.7: (a) Current-voltage curves at various µs, while other parameters are kept constant,

and (b) solar cell FF as a function of µs.

Fig. 4.7(a) shows shape changes which are consistent with our experimental solar cell I-V

curves from normal exponential to S-shaped, to negative exponential when µs decreases from

1 cm/Vs to 0.005 cm/Vs. Compared with the electron mobility µh in the heterojunction, a

higher µs value for the Schottky diode facilitates the extraction of charge carriers to the external

anode. As already mentioned, for this double-diode model there are four types of currents

CHAPTER 4. 87

contributing to the final device current, i.e. both heterojunction and Schottky diode contribute

diffusion and drift currents but in opposite directions. Most excitons dissociate within the

heterojunction (ZnO/PbS-TBAI) and the homojunction (PbS-TBAI/PbS-EDT) interfaces,

where the electric fields and/or interdot coupling strength are strong enough to split bound

electron-hole pairs. In addition, the diffusion current pertaining to the Schottky diode is

negligibly small. Consequently, it is expected that high µs can enhance hole extraction

efficiency, thereby improving solar cell performance.

In contrast, when µs is very small, such as in the case of low hole mobility within PbS-

EDT, holes accumulated at the PbS-EDT/Au interface induce a local SCL and electric field. A

similar phenomenon has been reported for a depleted-heterojunction PbS CQD solar cell [28].

In the present case, the decrease of µs may be due to multi-phonon assisted carrier hopping in

CQD thin films: at high temperatures, the high phonon density results in high hole mobility.

Temperature-dependence of charge carrier mobility was reported before [40]. It was also found

[41, 42] that carrier-mobility-controlled current density increased with temperature. As shown

in Fig. 4.7(b), the solar cell FF increases with µs and saturates above 0.2 cm2/Vs. Saturation

implies that all holes that flow to the anode metal are effectively extracted when hole mobility

is high enough, however, higher hole mobility cannot improve solar cell efficiency.

For our PbS CQD solar cell the diffusion current in the Schottky diode region is negligible.

To explore the influence of diffusivity, Fig. 4.8(a) shows that high diffusivity also gives rise

to anomalous solar cell I-V curves. This simulation used the same values of all other

parameters as in Fig. 4.7(a). High diffusion current at high diffusivity Ds compromises the one-

diode heterojunction I-V behavior. Consistently, Fig. 4.8(b) shows FF decrease with increasing

hopping diffusivity.

CHAPTER 4. 88

Figure 4.8: (a) Current-voltage curves at various Ds, while other parameters are kept

constant, and (b) solar cell FF as a function of Ds.

Besides imbalanced charge carrier mobility, the Schottky barrier is another pivotal role

playing a factor in the formation of anomalous CQD solar cell I-V curves. When holes arrive

at the Schottky diode side, Fig. 4.4, they are transported to the Au anode through a phonon-

assisted hopping transport mechanism. Therefore, with decreasing temperature, the reduced

phonon population depresses this phonon-assisted hole hopping process. Consequently, more

CHAPTER 4. 89

holes accumulate at the interface, leading to anomalous I-V curves as measured. Figs. 4.6(a)

to (f) show excellent match between experimental I-V curves and best fits to the theoretical

model using Eq. (4.28).

4.3.4 Open-circuit Voltage Origin of CQD Solar Cells

The Shockley-Queisser (SQ) limit [1] was derived for continuous-band semiconductors

where photon interactions with a solar cell induce the generation and recombination of free

electron-hole pairs. This mechanism is true for inorganic solar cells when carrier binding

energy is much smaller than the thermal energy kT, in which case lattice-bound excitons

dissociate into free electrons and holes at room temperature [43]. Consequently, Voc in the case

of ideal solar cells is the result of electron and hole quasi-Fermi energy level splitting, equal

to the voltage difference between device contacts [44-46]. Carried over to CQD solar cells, the

use of the SQ limit requires that exciton binding energy should be negligible. This is not the

case with most CQD solar cells which have high exciton binding energy, especially in CQD

thin films with low coupling strength and large interdot distances. Therefore, in most practical

CQD solar cells, an exciton must encounter a heterojunction where it can dissociate into free

electrons and holes. Unfortunately, heterojunctions incur additional energy losses by enhanced

exciton recombination, which, in turn, decreases Voc and device efficiency [47-49].

Specifically, the impact of heterojunctions on Voc loss is through the formation of charge-

transfer (CT) states (which are also named bandtail states [50]) located at acceptor-donor

interfaces (the ZnO/PbS-TBAI interface in this study), a mechanism also reported for organic

solar cells [47-49, 51]. As shown in Fig. 4.4, ECT is the energy gap between Ec of ZnO and Ev

of PbS-TBAI and excitons remain electrically neutral due to their high binding energy.

Hopping excitons diffuse and induce charge transfer across the heterojunction interface to form

CHAPTER 4. 90

a bound polaron pair (BP) with binding energy EB, which is significantly smaller than the

binding energy of a bulk exciton. These types of bound pairs are prone to dissociate into free

carriers. In the case of organic photovoltaic materials, EB is typically less than 0.5 eV and thus

results in non-thermodynamically limited exciton dissociation, always be lower than the bulk

exciton binding energy (1 eV) [47]. It is known [51-53] that CT states can also absorb photons,

but due to the much lower density of these interface states than bulk states, the CT absorption

coefficient is typically two to three orders of magnitude lower than that inducing bulk exciton

transitions [52, 53]. Furthermore, because excitons are bound with smaller energy gap (ECT) at

the heterojunction interface than with bulk energy gap (Eg), bulk materials dominate absorption

whereas heterojunction interfaces dominate recombination and dissociation. Consequently, the

recombination rate increases at interfaces because it can take place via lower energy bound-

pair states. Moreover, since small ECT does not limit dissociation, electron and hole quasi-

Fermi energy level splitting is no longer the major factor in determining Voc. The maximum

possible open-circuit voltage is determined by 𝑞𝑉𝑜𝑐𝑚𝑎𝑥 = E𝐶𝑇 , which occurs in the limit of

high-incident intensity and 0 K [47]. Voc is proven to be temperature-dependent and can be

expressed by the following expression [48, 51]

𝑉𝑜𝑐(𝑇) =𝐸𝐶𝑇

𝑞+𝑘𝑇

𝑞𝑙𝑛 [

𝐽𝑠𝑐ℎ3𝑐2

𝐹𝑞2𝜋(𝐸𝐶𝑇−𝜆)] +

𝑘𝑇

𝑞𝑙𝑛(𝐸𝑄𝐸𝐸𝐿) , (4.29)

where 𝐽𝑠𝑐 is the short-circuit current density, ℎ is Plank’s constant, 𝐹 is the florescence

emission intensity, 𝜆 is the reorganization energy associated with the CT absorption process,

and 𝐸𝑄𝐸𝐸𝐿is the electron luminescence external quantum efficiency. Voc losses occur through

radiative and non-radiative CT state recombination, labeled as ∆𝑉𝑂𝐶𝑟𝑎𝑑 and ∆𝑉𝑂𝐶

𝑛𝑜𝑛, respectively.

Voc loss to radiative CT state recombination ( ∆𝑉𝑂𝐶𝑟𝑎𝑑 ) is a thermodynamically imposed

mechanism for a given material system, whereas non-radiative recombination ( ∆𝑉𝑂𝐶𝑛𝑜𝑛) can be

CHAPTER 4. 91

avoided through many approaches, for example, by removing material defects. From Eq. (4.29),

∆𝑉𝑂𝐶𝑟𝑎𝑑 and ∆𝑉𝑂𝐶

𝑛𝑜𝑛can be obtained as [48]:

∆𝑉𝑂𝐶𝑟𝑎𝑑(𝑇) = −

𝑘𝑇

𝑞𝑙𝑛 [

𝐽𝑠𝑐ℎ3𝑐2

𝑓𝑞2𝜋(𝐸𝐶𝑇−𝜆)] (4.30a)

∆𝑉𝑂𝐶𝑛𝑜𝑛(𝑇) = −

𝑘𝑇

𝑞𝑙𝑛(𝐸𝑄𝐸𝐸𝐿) (4.30b)

Table 4.1: Summary of best-fitted parameters using Eq. (4.29).

Figure 4.9(a) shows best-fits to the temperature-dependent Voc using Eq. (4.29). The best-

fitting procedure was performed 300 times, followed by statistical analysis of the fitted

parameters to generate the final parameters as summarized in Table 4.1. The detailed procedure

to assure reliability of the measured parameters is discussed in the next section. The fitted ECT

= 1.1 eV exhibits high fitting uniqueness, leading to the maximum achievable Voc = 1.1 V. For

comparison, Eg of our PbS-TBAI is ca. 1.4 eV [24, 32] since the smaller ECT does not limit

exciton dissociation [47]. As shown in Fig. 4.9(b), the decrease in Voc through radiative and

non-radiative recombination was extracted using Eqs. (4.30a) and (4.30b). The maximum Voc

exhibits insignificant change within our experimental temperature range, consistent with

results reported by Gruber et al. [51]. Therefore, the maximum Voc determined by ECT is

henceforth considered to be constant. Both radiative and non-radiative recombination Voc

losses decrease when temperature decreases.

Parameters Sample size

(fitting times)

Mean Value SD 95 % confidence

intervals

ECT , eV

300

1.1 8.6×10-9 ±9.8×10-10

f , eV2 0.0069 0.0027 ±3.1×10-4

λ, eV 0.35 0.16 ±0.018

EQEEL 5.6×10-6 2.6×10-6 ±2.9×10-7

CHAPTER 4. 92

Figure 4.9: (a) Figure of the measured open-circuit voltage (Voc) and short-circuit current (Isc)

as a function of temperature. Equation (4.29) was used for the best-fitting of Voc. (b) Voc at

various temperatures. (c) The ratio 𝛥𝑉𝑜𝑐𝑟𝑎𝑑 / 𝛥𝑉𝑜𝑐

𝑛𝑜𝑛 as a function of temperature.

CHAPTER 4. 93

Furthermore, Fig. 4.9(c) shows that ∆𝑉𝑂𝐶𝑟𝑎𝑑 and ∆𝑉𝑂𝐶

𝑛𝑜𝑛 have similar magnitudes, although

radiative recombination seems to overtake non-radiative recombination as the dominant

recombination mechanism for Voc loss at low temperatures, as expected. The maximum Voc

will be theoretically achieved at 0 K [47]. It should be noted that if CT state emission is

negligible, the maximum Voc is determined by the bulk energy band gap 𝐸𝑔. Overall, according

to the above theoretical model, to increase the CQD solar cell Voc, ECT should be enhanced in

future QD photovoltaic device design. In addition, taking our solar cells as an example, it is

also suggested to passivate ZnO/PbS-TBAI interface traps states through proper chemical

ligands to decrease the non-radiative recombination induced Voc loss.

4.3.5 Temperature-dependent Carrier Hopping Transport and

CQD Solar Cell Performance

As shown in Fig. 4.9(a), despite the enhanced non-radiative recombination, short-circuit

current increases with increasing temperature. To extract carrier hopping transport parameters

through best-fits to the six I-V curves at different temperatures, Fig. 4.6, two independent best-

fitting computation programs were used to investigate the reliability and thus the uniqueness

of the best-fitted results in a statistical analysis. These programs have been used successfully

in earlier multi-parameter fits to experimental data from an amorphous/crystalline silicon solar

cell heterojunction [54]. The ‘mean-value best fit’ minimizes the mean square variance

between the experimental data and the theoretical values. The ‘statistical best fit’ uses the

fminsearchbnd solver [55] to minimize the sum of the squares of errors between the

experimental and calculated data. This program delivers different results due to different trial

starting points generated by the program itself, thereby creating standard deviations (SD) or

variances of the theoretical curve best-fitting procedure to the experimental points. To

CHAPTER 4. 94

investigate the reliability of the fitted parameters, this procedure was repeated several hundred

times and the 100 lowest variances were selected. Based on these 100 best-fitted results, the

variance and 95 % confidence interval were calculated. The statistical mean value was used as

reliability (uniqueness) measure and the variance as a precision measurement of the associated

parameter. It is seen that the results from the two independent best-fitting programs are in very

good-to-excellent agreement. In Table 4.2, the best-fitted values of both Dh and Ds decrease

monotonically with decreasing temperature, which is consistent with the multi-phonon assisted

hopping mechanism [16, 28, 56-58] and are close to the reported values [37, 58] between 0.003

and 0.012 cm2/s for PbS CQDs which were surface passivated with different ligands.

Figure 4.10: Arrhenius plots of (a) the ratio Th(T)/Dh(T) and (b) the ratio Ts(T)/Ds(T). The

mobility and diffusivities were calculated and fitted for the PbS-TBAI and the PbS-EDT

interface, respectively.

Similar to temperature-dependent electron mobility in a two-dimensional quantum dot

superlattice [40] (electron mobility increase with temperature), the electron and hole mobilities,

𝜇ℎ and μs, decrease monotonically with decreasing temperature, which is opposite to charge

carrier mobility trends in continuous energy band structures of e.g. inorganic photovoltaic

materials. It has been found [41, 42] that the hopping conductivity of CdSe or PbSe quantum

CHAPTER 4. 95

dot arrays increases with temperature in the range from 10 K to 523 K, due to the increase of

electron mobility by means of multi-phonon assisted hopping. Fig. 4.10 shows Arrhenius plots

of [𝑇𝜇𝑗(𝑇)

𝐷𝑗(𝑇)] , j = h,s, a combination of terms identified in Eq. (4.13) which replaces the

conventional Einstein relation in the CQD hopping transport theory. These figures

experimentally prove the validity of the hopping Einstein relation and extract heterojunction,

Fig. 4.10(a), and Schottky barrier inside the SCL, Fig. 4.10(b), activation energies Ea,h = 37.2

meV and Ea,s = 29.3 meV, respectively.

Figure 4.11: (a) The CQD solar cell FFs measured at various temperatures. (b) Maximum

power of as-studied CQD solar cell measured at various temperatures.

From the fitted parameters in Table 4.2 using Eq. (4.28), the following explanation emerges:

at low temperatures the charge carrier extraction efficiency is reduced due to the reduced hole

mobility, however, the enhanced influence of the Schottky barrier on hole extraction, Fig. 4.4,

reduces the hole current. The overall effect is a decrease in short-circuit current with decreasing

temperature. Hopping transport of charge carriers in CQD thin films is a multi-phonon-assisted

process [28, 56, 58]. As a consequence, the extraction efficiency of charge carriers, including

free electrons and holes, is suppressed at low temperatures owing to the reduced thermal

CHAPTER 4. 96

energy. In Fig. 4.11 (a), the FF calculated from 𝐹𝐹 = 𝑃𝑚𝑎𝑥

𝑉𝑜𝑐𝐼𝑠𝑐, where 𝑃𝑚𝑎𝑥 is the maximum

power calculated from the I-V curves, is reduced from 0.51 mW at 310 K to 0.13 mW at 100

K as a result of reduced charge carrier extraction efficiency.

The imbalance of carrier mobility in the heterojunction (PbS-TBAI) and in the associated

Schottky diode (PbS-EDT) gives rise to S-shaped and negative exponential current-voltage

curves, as shown in Fig. 4.6. In Table 4.2, the best-fitted values of carrier mobilities 𝜇ℎ and μs,

at 300K show that 𝜇ℎ is smaller than μs, implying sufficiently high charge carrier extraction

rate in the hole extraction layer (PbS-EDT). This results in holes being able to be extracted

efficiently and, as a result, the CQD solar cell behaves like a normal one-diode device. The

best-fitted results reveal better photovoltaic material made of PbS-EDT than of PbS-TBAI.

This is also consistent with the SCL lower mobility activation energy Ea,s across the Schottky

barrier, Fig. 4.10(b), than Ea,h across the heterojunction, Fig. 4.10(a). A similar conclusion in

terms of electron mobility was reported by ref. [58] using frequency-domain photocarrier

radiometry (PCR). However, due to the complicated device architecture, each of the two fitted

mobilities leading to the Einstein plots of Figs. 4.10(a) and (b) should not be unconditionally

interpreted to be exclusively associated with the PbS-TBAI or PbS-EDT interfaces. Although

at 250 K the fitted μs > 𝜇ℎ, the 250 K I-V characteristic shapes in Figs. 4.6(b) and 4.7(a) reveal

that low mobility of charge-carrier-induced hole accumulation on the anode side has already

set in, thereby reducing hole extraction efficiency. This trend is further demonstrated at even

lower temperatures by the fitted results which show that 𝜇ℎ and μs magnitudes reverse, with

𝜇ℎ becoming larger than μs, especially at 100 K. However, it should be noted that both

unbalanced mobility and Schottky diode contribute to the formation of anomalous I-V

characteristics, although our best-fitted results are not able to provide strong evidence of the

CHAPTER 4. 97

role of the SCL electric field, the fitted values of which show high standard deviation. The

existence of the reverse Schottky diode has been established in previous reports [24, 19]. It is

important to point out from the simulated curves shown in Fig. 4.7(a) that anomalous I-V

curves can appear even without the existence of a Schottky diode. Fig. 4.11(b) shows that the

calculated maximum power decreases with decreasing temperature. This is consistent with

both mechanisms of the imbalanced charge carrier mobilities and existence of Schottky diode,

which reduce solar efficiency through hole-accumulation-induced charge carrier extraction

reduction. Apart from optimizing the work functions of the anode metal and CQD nanolayers,

for instance, a smaller anode work function can alleviate the effect of Schottky diode effect in

our solar cells, applying a high charge carrier mobility layer next to the anode so as to reduce

the interfacial activation energy measured through the Einstein relation, Fig. 4.10(a), appears

to be a potentially effective method for the improvement of CQD solar efficiency. The

suggestion of applying high charge carrier mobility layer next to the anode is supported by the

results of Zhang et al. [59] that an increased CQD solar cell PCE was achieved by employing

a hole transport interlayer between the QD film and anode metal. Furthermore, based on our

simulations, it is also recommended to apply graded hole transport layers close to the anode

which 1) enhance the intrinsic electric field to increase the hole conductivity; 2) create

additional interfaces to improve exciton dissociation; 3) alleviate the Schottky diode effect to

remove hole accumulation influence; and 4) further block electron flow to the anode. However,

this strategy may complicate the device fabrication processes.

The photogenerated current, Iph, according to the electric circuit of Fig. 4.5, is the sum of

Isc and the current flowing across the heterojunction diode. Both of these currents are controlled

by temperature-sensitive exciton and free charge carrier hopping transport. The best-fitted

CHAPTER 4. 98

values of Iph at various temperatures are shown in Table 4.2. Iph and Isc exhibit the same trend

with temperature. Iph depends on the dissociation of excitons as follows: when excitons are

generated optically, they can dissociate into free electrons and holes through two paths. They

may diffuse to the heterojunction interface where the local electric field can separate electron-

hole pairs [60, 61]; or they can be decoupled during hopping diffusion between neighboring

quantum dots [62]. Inter-dot coupling strength is determined by ligand length: short ligand

length yields strong coupling strength. However, it is possible that exciton decoupling is also

thermal energy-related, as more ambient thermal energy induces higher exciton vibration

amplitude, increasing the decoupling probability. Excitons that do not dissociate undergo

recombination through radiative and/or nonradiative processes as discussed above.

Recombination contributes to the loss of excitons and consumes photogenerated current.

Therefore, Iph decreases at lower temperatures.

Electron (hole) lifetime 𝜏ℎ(τs) is calculated from statistically fitted values of Dh (Ds) and Lh

(Ls) through the equation: τh =Lh2

Dh. As summarized in Table 4.2, both 𝜏ℎ and τs decrease with

increasing temperature which is consistent with the temperature-dependent carrier lifetime

reported by Wang et al. [17, 63] and Mandelis et al. [28]. In these cases, excitons in both

coupled and uncoupled PbS CQDs were found to possess longer lifetimes at low temperatures

due to lower radiative and non-radiative recombination. Table 4.2 shows two calculated values

of 𝜏ℎ (as well as τs) for each temperature, obtained from ‘mean-value best fit’ and ‘statistical

best fit’. They exhibit small differences demonstrating high reliability and uniqueness of the

measurements resulting from the proposed model. The non-monotonic trend of the electron

hopping diffusion length 𝐿ℎ is the result of the trade-off between increased diffusivity and

decreased lifetime with increased temperature, through 𝐿ℎ = √𝐷ℎ𝜏ℎ . Strictly speaking, the

CHAPTER 4. 99

diffusion length 𝐿ℎ is a material property, and the intrinsic lifetime affecting factors should

include trap states and dot-to-dot coupling [64, 18, 57, 62]. The best-fitted 𝐿ℎ and 𝐿𝑠 values

agree with reported values for PbS CQDs in the range between 30 nm and 230 nm [38].

The fitted SCL width 𝑊 associated with hole accumulation at 300K is larger than that at

other temperatures. Note that the fitted depletion width 𝑊 through Eq. (4.28) is an effective

value across the PbS CQD layers. In other words, the actual SCL of PbS-TBAI and PbS-EDT

is determined through a competitive process between the depletion layers of the heterojunction

and the Schottky diode. To calculate the effective SCL, the depletion extent and width of both

diodes should be considered. The increase of W at high temperatures, Table 4.2, is consistent

with changes in the accumulated hole density at the PbS-EDT CQD/Au interface which acts

as a conventional junction depletion layer. Specifically, at low temperatures, due to reduced

hole extraction efficiency as discussed above, higher density of accumulated holes results in

higher density of occupied local QD energy states, and therefore a narrow depletion layer. In

contrast, at high temperatures, hole density at the interface decreases due to higher hole

extraction efficiency, thereby, alleviating the concentration gradient and resulting in reduced

density of occupied local QD energy states, and a wider depletion layer.

CHAPTER 4. 100

Tab

le 4

.2:

Sum

mar

y o

f b

est-

fitt

ed p

aram

eter

s.

T

emper

ature

, K

Fit

ted

par

amet

ers

300

250

230

200

150

100

Mea

n-

val

ue

bes

t fi

t

Sta

tist

ical

bes

t fi

t

Mea

n-

val

ue

bes

t fi

t

Sta

tist

ical

bes

t fi

t

Mea

n-

val

ue

bes

t fi

t

Sta

tist

ical

bes

t fi

t

Mea

n-

val

ue

bes

t fi

t

Sta

tist

ical

bes

t fi

t

Mea

n-

val

ue

bes

t fi

t

Sta

tist

ical

bes

t fi

t

Mea

n-

val

ue

bes

t fi

t

Sta

tist

ical

bes

t fi

t

Ho

pp

ing

dif

fusi

vit

y

Dh,

cm

2/s

3.6

6×

10

-

4

3.5

2×

10

-4

±1

.44

×1

0-5

2.5

2×

10

-

4

2.3

6×

10

-4

±2

.74

×1

0-5

1.1

7×

10

-

4

1.0

9×

10

-4

±2

.06

×1

0-5

7.7

4×

10

-

5

6.4

8×

10

-5

±1

.28

×1

0-5

4.7

4×

10

-

5

4.1

1×

10

-5

±6

.36

×1

0-6

3.6

1×

10

-

5

3.5

3×

10

-5

±1

.49

×1

0-6

DS,

cm

2/s

3.7

7×

10

-

4

3.9

4×

10

-4

±2

.46

×1

0-5

8.4

3×

10

-

5

8.3

2×

10

-5

±2

.88

×1

0-5

6.7

6×

10

-

6

2.0

7×

10

-5

±1

.60

×1

0-5

3.5

6×

10

-

7

5.8

7×

10

-7

±4

.52

×1

0-7

2.3

0×

10

-

7

2.5

1×

10

-7

±4

.83

×1

0-8

1.2

1×

10

-

7

1.2

8×

10

-7

±9

.57

×1

0-9

Dif

fusi

on

leng

th

Lh,

cm

3

.38×

10

-

6

3.9

3×

10

-6

±8

.63

×1

0-7

4.1

3×

10

-

6

3.7

3×

10

-6

±6

.56

×1

0-7

3.6

8×

10

-

6

3.9

6×

10

-6

±2

.75

×1

0-7

3.2

2×

10

-

6

3.8

5×

10

-6

±6

.41

×1

0-7

3.9

4×

10

-

6

4.1

8×

10

-6

±2

.53

×1

0-7

4.7

5×

10

-

6

5.1

1×

10

-6

±3

.66

×1

0-7

Ls,

cm

5

.12×

10

-

6

5.7

3×

10

-6

±6

.24

×1

0-7

3.0

8×

10

-

6

3.5

3×

10

-6

±5

.63

×1

0-7

5.1

2×

10

-

6

4.5

4×

10

-6

±6

.08

×1

0-7

3.3

7×

10

-

6

2.8

0×

10

-6

±5

.93

×1

0-7

2.6

5×

10

-

6

2.4

5×

10

-6

±3

.23

×1

0-7

5.7

1×

10

-

6

5.4

9×

10

-6

±2

.25

×1

0-7

Ho

pp

ing

mo

bil

ity

μh,

cm

2/V

s

3.4

1×

10

-

3

3.2

9×

10

-3

±1

.34

×1

0-4

1.3

3×

10

-

3

1.2

5×

10

-3

±1

.45

×1

0-4

3.8

3×

10

-

4

3.2

3×

10

-4

±6

.32

×1

0-5

2.0

9×

10

-

4

1.8

5×

10

-4

±3

.50

×1

0-5

8.1

5×

10

-

5

8.6

3×

10

-5

±1

.37

×1

0-5

3.8

9×

10

-

5

3.9

2×

10

-5

±9

.20

×1

0-7

μs,

cm

2/V

s

5.5

9×

10

-

3

5.8

4×

10

-3

±3

.64

×1

0-4

1.6

1×

10

-

3

1.6

6×

10

-3

±5

.48

×1

0-5

1.0

9×

10

-

4

1.5

7×

10

-4

±4

.80

×1

0-5

5.2

7×

10

-

6

7.9

8×

10

-6

±5

.84

×1

0-6

1.2

6×

10

-

6

1.3

2×

10

-6

±2

.52

×1

0-7

6.3

2×

10

-

7

6.6

4×

10

-7

±4

.11

×1

0-8

Sp

ace

char

ge

wid

th

W,

cm

2

.33×

10

-

5

2.2

4×

10

-5

±1

.08

×1

0-6

1.8

4×

10

-

5

1.8

2×

10

-5

±1

.66

×1

0-6

1.3

9×

10

-

5

1.5

7×

10

-5

±1

.83

×1

0-6

1.6

3×

10

-

5

1.7

8×

10

-5

±1

.65

×1

0-6

1.7

7×

10

-

5

1.6

×1

0-5

±1

.84

×1

0-6

1.6

4×

10

-

5

1.7

9×

10

-5

±1

.67

×1

0-6

Pho

to-

gen

erat

ed

curr

ent

I ph,

A

8.5

6×

10

-

4

8.5

7×

10

-4

±1

.08

×1

0-1

4

6.6

9×

10

-

4

6.7

0×

10

-4

±3

.11

×1

0-1

3

4.4

9×

10

-

4

4.5

0×

10

-4

±1

.22

×1

0-6

2.8

1×

10

-

4

2.7

5×

10

-4

±8

.74

×1

0-6

1.4

1×

10

-

4

1.3

7×

10

-4

±4

.3×

10

-6

5.8

2×

10

-

5

5.7

5×

10

-5

±1

.28

×1

0-6

Lif

etim

e

τ h,

s 3

.12×

10

-

8

4.3

9×

10

-8

6.7

7×

10

-

8

5.8

8×

10

-8

1.1

6×

10

-

7

1.4

4×

10

-7

1.3

4×

10

-

7

2.2

9×

10

-7

3.2

8×

10

-

7

4.2

5×

10

-7

6.2

5×

10

-

7

7.3

9×

10

-7

τ s,

s 6

.95×

10

-

8

8.3

3×

10

-8

1.1

3×

10

-

7

1.5

0×

10

-7

3.8

8×

10

-

6

9.9

6×

10

-7

3.1

9×

10

-

5

1.3

4×

10

-5

3.0

5×

10

-

5

2.3

9×

10

-5

2.6

9×

10

-

4

2.3

6×

10

-4

CHAPTER 4. 101

4.4 Conclusions

A theoretical model of carrier discrete hopping transport in CQD materials and

photovoltaic solar cells was introduced in the framework of non-continuous energy band

structure of CQD ensembles. For further investigation of the temperature-dependent thermal

energy associated carrier hopping transport mechanism in CQD solar cells. A double-diode

electric circuit model featuring a heterojunction (PbS-TBAI/ZnO) and a Schottky diode (PbS-

EDT/Au anode) with two electric fields of opposite directions was developed and used to

quantitatively interpret experimental I-V curves obtained from a fabricated CQD solar cell

with a structure: ITO/ZnO/PbS-TBAI QD/PbS-EDT QD/Au, which exhibits anomalous I-V

characteristics at temperatures below 300K. Detailed best-fits of I-V data to the theoretical

model and simulations revealed that imbalanced charge carrier mobility is one of two factors

giving rise to S-shaped and negative exponential I-V characteristics. The other factor is the

formation of a reverse Schottky barrier for holes adjacent to the hole-extracting anode. In

addition, the existence of charge-transfer (CT) states attribute to the loss of Voc through

enhanced radiative and non-radiative recombination processes in these states which provided

quantitative insight into the nature of the Voc temperature dependence.

The presented models and quantitative I-V analysis for the discrete carrier hopping

transport, imbalanced carrier mobilities and Schottky barrier induced anomalous I-V

characteristics, and the Voc deficit at interfaces can be used to measure device transport

parameters, especially hopping mobility, aimed at minimizing Schottky barrier in order to

maximize the short-circuit current and Pmax, toward the optimization of CQD solar cell

fabrication.

102

Chapter 5

Colloidal Quantum Dot Solar Cell Efficiency Optimization:

Impact of Hopping Mobility, Bandgap Energy, and Electrode-

semiconductor Interfaces

5.1 Introduction

As discussed in Chapter 2 CQD solar cells are presently attracting immense research

interest on a global scale. However, no comprehensive device efficiency optimization

strategies have been reported aiming at achieving higher PCE, specifically for CQD solar cells.

Researchers use common sense approaches instead, trying to improve CQD solar cell

efficiency through pursuing higher carrier mobility using disparate surface passivation

materials and increasing quantum dot size for lower bandgap energy in order to harvest the

solar spectrum in a wider wavelength range. This universal strategy, however, is typically valid

for conventional solar cells of high carrier mobility such as Si solar cells, rather than for low

carrier mobility systems of a discrete carrier transport nature, such as colloidal quantum dot

(CQD) and organic solar cells. Alarmingly, however, it has been reported that state-of-the-art

high PCE solar cells are actually achieved using materials generally not exhibiting the highest

CQD carrier mobility [1]. Furthermore, researchers reverted to using smaller dots with wider

bandgap energy when they found larger dots yielded even lower PCE. This raises the crucial

question of whether higher mobility and smaller bandgap CQDs can always produce higher

PCEs in CQD solar cells. How do CQD carrier mobility and bandgap energy determine solar

cell performance? This chapter addresses these critical issues with the fabrication and study of

CQD solar cells it was discovered that the photocurrent density is voltage-dependent and is

CHAPTER 5. 103

accompanied by low carrier mobility, which indicates that the conventional constant

photocurrent assumption may be invalid for CQD solar cells. Generally, to analyze current-

voltage characteristics of CQD solar cells, the well-known Shockley-Queisser (S-Q) equation

is often used [2], which is discussed in Sect.4.1. The huge difference in mobility, 10-5 to 10-1

cm2/Vs [3-6] for PbS CQDs while 102 - 103 cm2/Vs for Si solar cell, leads to questioning the

validity of applying the S-Q model to such systems. The discovery of external-voltage-

dependent photocurrent density in PbS CQD based solar cells enabled us to revisit and relax

the constant photocurrent density assumption in the well-known S-Q equation for CQD solar

cells which has been and continues to be the prevailing assumption among researchers to-date.

A similar voltage-dependent photocurrent was also reported by Würfel et al. [7] for organic

solar cells which also have much lower carrier mobility. Therefore, this chapter, for the first

time, develops a comprehensive analysis of the dependence of CQD solar cell current-voltage

characteristics on carrier mobility and CQD bandgap energy.

Figure 5.1: Schematic of the as-fabricated CQD solar cell sandwich structure. PbX2 and AA

represent lead halide and ammonium acetate, respectively, acting as exchange-ligands for PbS

CQDs.

CHAPTER 5. 104

Furthermore, as will be discussed in detail in Chapter 8, most researchers have reported

solar cell efficiencies based on small-spot (<0.1 cm2) testing, including professional

certification characterizations of solar cells towards an entry in the Solar Cell Efficiency Tables.

This, however, raises questions about the overall solar cell performance and stability

estimations. Furthermore, no imaging studies of CQD solar cells have been reported in efforts

to acquire an insightful physical picture of defect or contact effects on key solar cell

performance parameters. In order to study the contact/CQD interface influence on the

performance of CQD solar cells to further develop this device efficiency optimization strategy,

this chapter uses a large-area photovoltaic device non-destructive imaging (NDI) carrier-

diffusion-wave characterization technique (HoLIC), as discussed in details in Sect.3.4, to

obtain open-circuit voltage distribution and carrier collection efficiency images and were thus

able to elucidate the effects of the CQD/electrode interfaces on solar cell performance within

the framework of our drift-diffusion J-V model.

The presented theoretical model and large-area characterization technique can be of

significance for guiding CQD solar cell optimization with respect to CQD surface passivation

ligand selection and the determination of CQD energy bandgap (or quantum dot size), as well

as for solar cell fabrication quality control.

5.2 Derivation of Carrier Hopping Drift-diffusion J-V Model for

CQD Solar Cells

As it has already been discussed in Chapter 4, Disorder sources in CQD ensembles,

including variations in confinement energy, electron-electron repulsion, coupling and thermal

broadening, cause CQD-based materials and devices to exhibit discrete hopping conductivity

and diffusivity [8, 9]. Using intensity modulated illumination, the distribution and hopping

CHAPTER 5. 105

transport of excitons and charge carriers follow a diffusion-wave behavior. The theory of

particle-population-gradient-induced diffusive transport through spatial profiles of discrete

hopping into and out of a quantum dot was developed in detail in ref. [10]. In this chapter, the

prevailing assumption of constant photocurrent [10] was relaxed. Based on the photocarrier

hopping diffusion-wave theory to solve the carrier population rate equation, quenching and

surface recombination velocity (SRV) associated boundary conditions were assumed with

respect to both hopping diffusion and drift current densities for high-efficiency CQD solar cell

structures, thus overcoming the voltage limitation [10] which leads to a decreased J with V

when the applied voltage is higher than the built-in potential. Specifically, in a one-

dimensional quantum dot ensemble, each quantum dot characterized by its own size and

energy manifold is separated by a mean distance from its neighbors. Therefore, the rate

equation for the net carrier flux entering one quantum dot within a time interval 𝑑𝑡 can be

written as [10]:

𝜕𝑛𝑃𝑉(𝑥,𝑡)

𝜕𝑡= −


𝜕𝑥−𝑛𝑃𝑉(𝑥,𝑡)

𝜏 (5.1)

where 𝑛𝑃𝑉(𝑥) is the carrier concentration under illumination, 𝜏 is the lifetime, and 𝐽𝑒(𝑥, 𝑡) is

the carrier hopping flux in units of s-1cm-2. The subscript PV indicates that the solar cell is

under external bias. With an applied dc voltage, 𝑛𝑃𝑉 becomes time-independent, therefore, Eq.

(5.1) can be further written as

𝑑2𝑛𝑃𝑉(𝑥)

𝑑𝑥2− (

𝜇𝑒𝐸

𝐷𝑒)𝑑𝑛𝑃𝑉(𝑥)

𝑑𝑥−𝑛𝑃𝑉(𝑥)

𝐷𝑒𝜏= 0 (5.2)

where 𝐸 = 𝑉𝑏𝑖 −𝑉𝑎

𝑑 is the electric field across the solar cell, d is the CQD thin film thickness as

shown in Fig. 5.2, 𝐷𝑒 is the carrier hopping diffusivity, 𝜇𝑒is the hopping mobility, 𝑉𝑏𝑖 is the

built-in voltage, and 𝑉𝑎 is the photovoltage. Equation (5.2) is subject to a surface boundary

CHAPTER 5. 106

condition at x = 0 as shown in Fig. 5.2: 𝑛𝑃𝑉(0) − 𝑛(0) = ∆𝑁0, where ∆𝑁0 is the excess carrier

population generated by the photovoltaic effect. A second (quenching) boundary condition for

the CQD thin films at x = d is 𝑛𝑃𝑉(𝑑) − 𝑛(𝑑) = 0 , indicating an infinite SRV and the

immediate recombination of electrons and holes when they drift or diffuse to the interface.

Therefore, solving Eq. (5.2) with ∆𝑛𝑃𝑉(𝑥) = 𝑛𝑃𝑉(𝑥) − 𝑛(𝑥), it can be found that

∆𝑛𝑃𝑉(𝑥) =∆𝑁0(𝑒

𝑄2𝑑+𝑄1𝑥−𝑒𝑄1𝑑+𝑄2𝑥)

𝑒𝑄2𝑑−𝑒𝑄1𝑑 (5.3)

The excess carrier population ∆𝑁0 = 𝑛(0) (𝑒𝜇𝑒𝑉𝑎𝐷𝑒 − 1 ) can be derived through integrating

the electric field over the thickness of a solar cell under both dark and illumination conditions

[10]. Accordingly, the hopping drift and diffusion current densities [A/cm2] can be expressed

as

𝐽𝑒,𝑑𝑖𝑓𝑓 = −𝑞𝐷𝑒𝑑∆𝑛𝑃𝑉(𝑥)

𝑑𝑥|𝑥=0

= −𝑞𝐷𝑒𝑛(0) (𝑄1𝑒

𝑄2𝑑−𝑄2𝑒𝑄1𝑑

𝑒𝑄2𝑑−𝑒𝑄1𝑑) (𝑒

𝜇𝑒𝑉𝑎𝐷𝑒 − 1 ) (5.4)

𝐽𝑒,𝑑𝑟𝑖𝑓𝑡 = 𝑞𝜇𝑒𝐸𝑛(0) (𝑒𝜇𝑒𝑉𝑎𝐷𝑒 − 1 ) = 𝑞𝜇𝑒 (

𝑉𝑏𝑖−𝑉𝑎

𝑑) 𝑛(0) (𝑒

𝜇𝑒𝑉𝑎𝐷𝑒 − 1 ) (5.5)

with

𝑄1,2 =1

2(𝐶0 ±√𝐶0

2 + 4𝐶1) (5.6)

and with the definitions

𝐶0 =𝜇𝑒𝐸

𝐷𝑒= (

𝜇𝑒

𝐷𝑒)𝑉𝑏𝑖−𝑉𝑎

𝑑, 𝐶1 =

1

𝐿𝑒2 (5.7)

where 𝐿𝑒 is the dc hopping diffusion length. Eventually, the total dark current density 𝐽𝑒

(A/cm2) including drift and diffusion components can be expressed as

𝐽𝑒 = 𝐽𝑒,𝑑𝑖𝑓𝑓 + 𝐽𝑒,𝑑𝑟𝑖𝑓𝑡 = 𝑞𝑛(0) [−𝐷𝑒 (𝑄1𝑒

𝑄2𝑑−𝑄2𝑒𝑄1𝑑

𝑒𝑄2𝑑−𝑒𝑄1𝑑) + 𝜇𝑒 (

𝑉𝑏𝑖−𝑉𝑎

𝑑)] (𝑒

𝜇𝑒𝑉𝑎𝐷𝑒 − 1 ) (5.8)

CHAPTER 5. 107

Therefore, through adding electron and hole dark current densities and considering the

photocurrent density, the total carrier hopping current density under illumination can be

expressed as usual by

𝐽𝑖𝑙𝑙𝑢 = 𝐽𝑒 + 𝐽ℎ − 𝐽𝑝ℎ (5.9)

It should be noted that the hopping drift current density 𝐽𝑑𝑟𝑖𝑓𝑡 is 𝑉𝑎 dependent. Specifically,

the electric field (𝑉𝑏𝑖−𝑉𝑎

𝑑) leads to 𝐽𝑑𝑟𝑖𝑓𝑡 decreasing with 𝑉𝑎 , while the excess carrier

population 𝑛(0)𝑒𝜇𝑒𝑉𝑎𝐷𝑒 yields an exponential dependence of 𝐽𝑑𝑟𝑖𝑓𝑡 on 𝑉𝑎.

Figure 5.2: Schematic of the CQD solar cell energy band structure. PbX2 and AA represent

lead halide and ammonium acetate, respectively, acting as exchange-ligands for PbS CQDs.

In general, the boundary condition at x=d is not always an infinite SRV. There are four

types of surface recombination, namely, minority carrier hole recombination at the cathode,

majority carrier hole recombination at the anode, minority carrier electron recombination at

the anode, and majority carrier electron recombination at the cathode. Here, however, for the

CHAPTER 5. 108

sake of simplification, all types of surface recombination are treated in the same manner. The

surface recombination rate at d can be defined as

𝐽(𝑑) = 𝑆(𝑑)𝑛𝑃𝑉(𝑑) (5.10)

𝐽(𝑑) is a carrier flux in units of s-1cm-2. Solving Eq. (5.2) subject to boundary condition Eq.

(5.10), the total current density 𝐽𝑒 (A/cm2) is given by

𝐽𝑒 = 𝑞𝑛(0){−𝐷𝑒[𝑄1 − (𝑄1 − 𝑄2)𝑓] + 𝜇𝑒𝐸} (𝑒𝜇𝑒𝑉𝑎𝐷𝑒 − 1) (5.11)

with the definition

𝑓 ≡𝑒𝑄1𝑑[𝑆(𝑑)−𝜇𝑒𝐸+𝐷𝑒𝑄1]

[𝑆(𝑑)−𝜇𝑒𝐸](𝑒𝑄1𝑑−𝑒𝑄2𝑑)+𝐷𝑒(𝑄1𝑒𝑄1𝑑−𝑄2𝑒

𝑄2𝑑) (5.12)

Smaller surface recombination velocity S means more excess charge carriers across the

interface which should result in better solar cell performance and S = 0 corresponds to Ohmic

contact behavior as reported by Kirchartz et al. [11]. Also they found that surface

recombination played the role of a carrier recombination source in the way that a zero surface

recombination rate corresponds to preclude carrier recombination at the contacts (interface),

while an infinite recombination rate adds a new recombination pathway, leading to strongly

decreased Voc and PCE at high carrier mobility when carrier recombination is significantly

increased.

5.3 Experimental CQD Solar Cell Efficiency Optimization

The synthesis of oleic-acid-capped CQDs, ZnO nanoparticles, and CQD solar cells follows

the methods in Sect.4.3.1 and ref. [12] with the exception that PbS-TBAI CQD layer in Fig.4.3

was replaced by PbX2/AA-exchanged (PbX2: lead halide, AA: ammonium acetate) PbS inks

[13]. Figure 5.1 shows the sandwich structure of the CQD solar cells under study.

CHAPTER 5. 109

Following the device efficiency optimization strategy (See Sect. 5.5) based on the fact that

there are optimized bandgap energy and carrier mobility for a given type of CQD solar cells,

this section describes how the CQD solar cell efficiency was improved experimentally through

varying the CQD bandgap energy (dot size) and altering the CQD carrier mobility using

various surface passivation ligands and different ligand exchange methods[13]. Specifically,

instead of using PbX2/AA in solution for ligand exchange, two types of CQD solar cells using

solid-state layer-by-layer exchange with tetrabutylammonium iodide (TBAI) and solution

exchange with methylammonium lead iodide (MAPbI3) as ligands [13] were fabricated and

labeled PbS-TBAI and PbS- MAPbI3, respectively. The structures of these samples are shown

in Fig. 5.1 except that PbS- PbX2/AA is replaced by PbS-TBAI and PbS-MAPbI3 for our

control samples. As already reported [13], consistent with the efficiency optimization strategy,

the dependencies of CQD solar cell external quantum efficiency and current density on CQD

bandgap energy Eg were initially found to increase with the reduction of Eg, then decrease

when Eg became smaller than a threshold value between 1.28 eV and 1.38 eV. Limited by the

scope of the experiments, an exception was the current density dependence of PbS-PbX2/AA

which exhibited a monotonic increase with decreasing Eg without attaining a threshold value

yet. In comparison, Voc exhibited a positive linear dependence on the CQD bandgap energy,

and it was also exchange-ligand-dependent with PbS- PbX2/AA and PbS- MAPbI3 possessing

the highest and lowest Voc, respectively, at all bandgap energies. All of these experimental

results are in good agreement with our theoretical model predictions (Sect.5.5). Although

mobility measurements are not discussed here, the PbS- PbX2/AA CQD solar cells were

characterized with photothermal deflection spectroscopy and found to have fewer bandtail

states and higher CQD packing density compared with the controls, as well as higher

CHAPTER 5. 110

uniformity characterized by grazing-incidence small-angle X-ray scattering measurements.

Therefore, PbS- PbX2/AA CQD solar cells were found to have the highest PCE of all CQD

solar cells. Regardless of the fact that PbS- MAPbI3 CQD thin films had higher carrier mobility

than PbS-TBAI [14], PbS-TBAI-based CQD solar cells exhibited higher PCE values than that

of PbS- MAPbI3 which goes against the common sense that higher mobility corresponds to

better device performance. This perceived anomaly is, however, consistent with the theoretical

model of Sect. 5.5. Ultimately, the CQD solar cells were optimized with a certified efficiency

of 11.28 %, a bandgap energy of 1.32 eV and a PbS- PbX2/AA thickness of 350 nm.

J-V characteristics were obtained using a Keithley 2400 source measuring unit under

simulated AM1.5 illumination (Sciencetech class A) in a continuous nitrogen flow

environment. Furthermore, the calibration for spectral mismatch was carried out using a

reference solar cell (Newport). Finally, following experimental methods described in Sect.3.4,

LIC imaging of the CQD solar cells was performed in a room-temperature nitrogen

environment under 10 Hz modulation frequency.

5.4 Non-constant Photocurrent in CQD Solar Cells

The photocurrent density generated with illumination at short circuit can be expressed as

[15]

𝐽𝑝ℎ = 𝑞 ∫𝑏𝑠(𝐸)𝐸𝑄𝐸(𝐸)𝑑𝐸 (5.13)

where 𝑏𝑠(𝐸) is the incident spectral photon flux and EQE is the solar cell external quantum

efficiency which depends on the material absorption coefficient, charge separation efficiency,

and carrier collection ability in the device, but is independent of the incident optical spectral

distribution. It should be noted that 𝐽𝑝ℎ in Eq. (5.13) corresponds to the maximum photocurrent

density that can be collected. However, as discussed above, the assumption of a constant 𝐽𝑝ℎ

CHAPTER 5. 111

is not always true for CQD solar cells. Contrary to intuition, Fig.5.3(a) demonstrates that the

experimental current density Jillu under illumination is not equal to Jdark - Jsc; instead, the

current density difference (Jph, according to the S-Q equation discussed in the introduction)

between Jillu and Jdark is obviously voltage-dependent with a shape resembling Jillu. The

amplified dark current density Jdark is shown in Fig.5.3(b) that exhibits a typical exponential

J-V curve. This deviation can be attributed to the low carrier mobility in CQD and organic

solar cells [7, 16]. Solar cell efficiency is dominated by three main loss factors, namely, the

non-radiative recombination at the heterojunction interfaces or thin film/contact interfaces, the

inefficient collection of photogenerated excitons and charge carriers, and the parasitic

absorption of the contact layers [16]. Compared with high-mobility systems such as Si solar

cells, the much lower carrier hopping mobility significantly reduces the charge carrier

extraction rate. As a consequence, charge carriers or excitons recombine substantially at or

near the location where they are created [16]. In addition, low carrier hopping mobility causes

an almost open-circuit condition within the solar cell device which occurs even at short circuit,

leading to approx. 95% of the photogenerated carriers becoming lost to recombination as

reported by Würfel et al. [7] for organic solar cells with a carrier mobility equal to 10-6 cm2/Vs.

This phenomenon is well pronounced also in CQD solar cells with increased photoactive layer

thickness and/or under high illumination intensity conditions. Jph is constant only when the

carrier mobility is adequately high, comparable to that of commercial Si solar cells, so that the

driving forces for the transport of electrons and holes can be neglected. For CQD solar cells,

however, carrier hopping mobility and diffusivity are very small [3-6]. Furthermore, the higher

carrier concentration under illumination than in the dark additionally increases the conductivity

of the material [𝜎𝑒,ℎ = 𝑒𝜇𝑒,ℎ𝑛𝑒,ℎ, with 𝑛𝑒,ℎ being the carrier density of electron (e) or holes

http://www.nature.com/articles/ncomms7951#auth-1

CHAPTER 5. 112

(h)] for a given hopping mobility, so the influence of driving forces for electron and hole

extraction begins to emerge [7]. With this consideration in mind, we set out to develop an

analytical expression for the voltage-dependent Jph considering a carrier hopping drift and

diffusion transport mechanism.

Figure 5.3: (a) Experimental data and theoretical best-fits of current density vs. voltage under

illumination and in the dark; (b) The dark current density in (a) amplified. Comparison between

Jdark - Jsc, and Jdark, as well as Jillu, as a function of voltage, is also shown in (a). Equations (5.9),

(5.11) and (5.20) were used for the best-fits of the J-V characteristics. The best-fitted Jph at Va

= 0 (representing Jsc) is 24.9 mA and 7.9×10-7 mA under illumination and in the dark,

respectively.

For our solar cell sample ZnO/PbS-PbX2(AA)/PbS-EDT in Figs. 5.1 and 5.2,

photogenerated excitons dissociate into free electrons and holes when generated in CQD layers,

resulting in electric-field-dependent photocurrent with a fractional contribution, 𝜂′, a function

of hopping drift lengths. At voltage Va, the mean carrier hopping drift length ��𝑑𝑟𝑖𝑓𝑡 can be

expressed as:

��𝑑𝑟𝑖𝑓𝑡 = (𝜇ℎ𝜏ℎ + 𝜇𝑒𝜏𝑒)(𝑉𝑏𝑖−𝑉𝑎)

𝑑 (5.14)

Therefore, the fraction 𝜂′ can be extracted as the ratio of ��𝑑𝑟𝑖𝑓𝑡 to the total carrier hopping drift

transport length (the CQD solar cell thickness), i.e.

CHAPTER 5. 113

𝜂′ =��𝑑𝑟𝑖𝑓𝑡

𝐿𝐶𝑄𝐷+𝐿𝑍𝑛𝑂 (5.15)

According to the well-known Shockley-Queisser equation, Jph reverses the direction of Jdark

which is the net current density comprising drift and diffusion current densities and has a

direction same as that of a diffusion current density under forward bias. Therefore, the

direction of the built-in electric field (Ei) under equilibrium conditions is the positive direction

of the photocurrent. Based on the assumption that photocarriers will be fully extracted if their

hopping drift length is larger than the solar cell thickness, a case also addressed in refs. [13,

17], the hopping drift photocurrent density can be obtained through

𝐽𝑝ℎ,𝑑𝑟𝑖𝑓𝑡 = {

𝐽𝑝ℎ′ × 𝜂′, 𝑤ℎ𝑒𝑛 − 1 < 𝜂′ < 1

𝐽𝑝ℎ′ , 𝑤ℎ𝑒𝑛 𝜂′ ≥ 1

−𝐽𝑝ℎ′ , 𝑤ℎ𝑒𝑛 𝜂′ ≤ −1

(5.16)

where 𝐽𝑝ℎ′ is the maximum hopping drift photocurrent density that is constant at a given

illumination condition and is independent of the external voltage. Furthermore, CQD solar cell

efficiencies are deteriorated due to carrier transport toward wrong electrodes. Therefore,

researchers have tried to add additional energy barriers, for example, the extra PbS-EDT CQD

layer in Fig.5.2 was deposited to prevent electron diffusion to the Au electrode [13, 18].

Although the influence of carrier diffusion induced carrier loss is not significant in our CQD

solar cells because of the extra energy barrier introduced by PbS-EDT, for most other CQD

solar cell architectures these effects are substantial and better understanding is required. Hence,

in comparison with electric field induced 𝐽𝑝ℎ,𝑑𝑟𝑖𝑓𝑡 , the diffusion-associated photocurrent

𝐽𝑝ℎ,𝑑𝑖𝑓𝑓 is also studied and found to be constant across the entire external voltage range.

Specifically, hole diffusion to the ZnO is negligible due to the built-in energy barrier, while

CHAPTER 5. 114

there are no (or there are much smaller) energy barriers for electron diffusion in the CQD layer.

Thus, the mean diffusion distance is given by

��𝑑𝑖𝑓𝑓 = √𝐷𝑒𝜏𝑒 (5.17)

and

𝜂′′ =��𝑑𝑖𝑓𝑓

𝐿𝐶𝑄𝐷 (5.18)

Analogous to the derivation of hoping drift photocurrent density in Eq. (5.16), the hopping

diffusion photocurrent density can be given by:

𝐽𝑝ℎ,𝑑𝑖𝑓𝑓 = {−𝐽𝑝ℎ

′′ , 𝑤ℎ𝑒𝑛 �� ≥ 𝐿𝐶𝑄𝐷

−𝐽𝑝ℎ′′ × 𝜂′′, 𝑤ℎ𝑒𝑛 �� < 𝐿𝐶𝑄𝐷

(5.19)

Similarly, 𝐽𝑝ℎ′′ is the maximum diffusion photocurrent density. Equation (5.19) reveals the

negative hopping diffusion photocurrent, implying that 𝐽𝑝ℎ,𝑑𝑖𝑓𝑓 will diminish the active total

photocurrent. The negative 𝐽𝑝ℎ,𝑑𝑖𝑓𝑓is due to that carrier transport in a wrong direction toward

the incorrect electrodes, which offsets the drift photocurrent as shown in Fig.5.4(a) and (b).

Therefore, the total photocurrent density 𝐽𝑝ℎ can be obtained from adding 𝐽𝑝ℎ,𝑑𝑟𝑖𝑓𝑡 and 𝐽𝑝ℎ,𝑑𝑖𝑓𝑓.

Figure 5.4(a) shows the dependence of 𝐽𝑝ℎ on the external voltage using Eq. (5.16)

considering only 𝐽𝑝ℎ,𝑑𝑟𝑖𝑓𝑡 , while Fig. 5.4(b) is simulated using Eqs. (5.16) and (5.19) with the

addition of 𝐽𝑝ℎ,𝑑𝑖𝑓𝑓 which leads to lower 𝐽𝑝ℎ at the same mobility when compared with Fig.

5.4(a). It should be noted that 𝐽𝑝ℎ′ and 𝐽𝑝ℎ

′′ equal 35mA/cm2 for the simulation in Figs. 5.4(a)

and (b). The simulated 𝐽𝑝ℎ decreases with voltage except at high carrier mobilities (1 cm2/Vs

or higher) where constant 𝐽𝑝ℎ values across the entire voltage range are obtained as shown in

Figs. 5.4(a) and (b). Under reverse bias in Fig. 5.4(a), the external applied electric field has

the same direction as Ei, thereby helping to extract charge carriers and increase Jph until it

CHAPTER 5. 115

saturates to the maximum 𝐽𝑝ℎ′ − 𝐽𝑝ℎ

′′ . Therefore, high reverse voltage assists the extraction of

charge carriers, contributing to the overall 𝐽𝑝ℎ[7]. This is consistent with the saturated Jdark-

Jillu under reverse bias as shown in Fig. 5.3(a). The experimental Jdark-Jillu in Fig. 5.3(a) is

found to reduce under a forward bias, a behavior demonstrated by the simulated results in Figs.

5.4(a) and (b). The reduction in 𝐽𝑝ℎ in Fig. 5.4(a) is expected, as the net electric filed Enet

reduces with increasing Va. The resultant negative Jph results from insufficient drift

photocurrent to offset the negative hopping diffusion photocurrent density which is mirrored

by the negative Jdark-Jillu values when the forward voltage is larger than Voc. Therefore, some

special situations can be expected to arise for extremely small mobilities, such as negative Jph

for a mobility of 0.001 cm2/Vs (which should be much smaller for actual CQD solar cells) in

the entire Va range when the mobility is very small or the diffusion photocurrent is sufficiently

high. However, due to the restrictive assumptions as discussed below behind Eqs. (5.16) and

(5.19) and the carrier transport parameter values used for this simulation, the negative Jph for

the mobility of 0.001 cm2/Vs in Fig. 5.4(b) does not mean that CQD solar cells with such a

low mobility cannot materialize. In other words, Jph at 0.001 cm2/Vs can be simulated to be

positive simply by adjusting the simulation parameters, for example, considering a much

smaller 𝐽𝑝ℎ′′ in Eq. (5.19) or 𝐽𝑝ℎ

′′ = 0 as shown in Fig. 5.4(a). A very small 𝐽𝑝ℎ′′ is true for our

CQD solar cells as indicated by Fig.5.2 due to the energy barriers introduced by the PbS-EDT

CQD layer. Figures 5.4(a) and (b) also show that all Jph curves converge at a voltage

corresponding to the intrinsic voltage Vbi used in Eq. (5.14). Furthermore, lower carrier

hopping mobility 𝜇∗ lead to a lower voltage at which 𝐽𝑝ℎ starts to drop due to a lower carrier

hopping drift extraction efficiency (i.e. the reduced drift length) resulting from the reduced net

electrical field strength, in other words, lower mobility requires higher net electric field to

CHAPTER 5. 116

extract the photocarriers. These simulated results validate the fact that when carrier hopping

mobility is low, the driving forces (electric field and diffusion gradient) for carrier transport

start to impact 𝐽𝑝ℎ.

Figure 5.4: Simulated photocurrent density Jph (a) using Eq.(5.16) without Jph,diff, and (b) using

Eqs.(5.16) and (5.19) with Jph,diff at different effective carrier hopping mobilities; and (c) (Jillu

-Jdark)/𝐽𝑝ℎ𝑚𝑎𝑥 as a function of the external voltage at various effective mobilities using Eq. (5.20).

However, the experimental J-V characteristics in Fig. 5.3(a) illustrate that the difference

between Jdark and Jillu exhibits a nonlinear (exponential-like) dependence on the applied voltage,

contrary to the linear dependence shown in Fig. 5.4(a) and (b) using Eqs. (5.16) and (5.19).

According to the hopping drift-diffusion J-V model, 𝐽𝑝ℎ can be obtained through solving Eq.

(5.2) by implementing a voltage- and position-dependent carrier generation rate. For instance,

CHAPTER 5. 117

the photocurrent density 𝐽𝑝ℎ at 𝑥 = 0 can be obtained when 𝐽𝑝ℎ(0) = −𝑆(0)[𝑛𝑃𝑉(0) − 𝑛(0)]

that yields an exponential dependence of 𝐽𝑝ℎ on the applied voltage. Furthermore, free

electrons and holes dissociated from photogenerated excitons generated in CQD layers

contribute to electric-field-dependent and diffusion-related photocurrents with fractional

contribution 𝜂′ and 𝜂′′, respectively, as discussed above. However, the diffusion photocurrent

has been reported with negligible influence by Schilinsky et al. [19], especially for our CQD

solar cells with the additional PbS-EDT layer as shown in Fig.5.2. Therefore, considering the

drift photocurrent extraction efficiency and that the CQD layers are the main carrier transport

layer, an empirical ad hoc exponential dependence of photocurrent density on the applied

voltage to represent the aforementioned hopping drift-related photocurrent densities in Eq.

(5.16) is given according to refs.[7, 17]:

𝐽𝑝ℎ = 𝐽𝑝ℎ𝑚𝑎𝑥 [1 − 𝑒𝑥𝑝 (

(𝑉𝑎−𝑉𝑏𝑖)𝜇∗𝜏∗

𝑑2)] (5.20)

where 𝐽𝑝ℎ𝑚𝑎𝑥 is the maximum photocurrent density that can be extracted at a given illumination

level from Eq. (5.13). For the sake of simplification, electrons and holes are considered to have

the same transport parameters as defined by 𝜏∗ = 𝜏𝑒 = 𝜏ℎ, 𝐿∗ = 𝐿𝑒 = 𝐿ℎ, 𝐷∗ = 𝐷𝑒 = 𝐷ℎ, and

𝜇∗ = 𝜇𝑒 = 𝜇ℎ . Figure 5.4(c) shows the ratio (Jillu - Jdark) / 𝐽𝑝ℎ𝑚𝑎𝑥 as a function of 𝑉𝑎 using

Eq.(5.20), in agreement with the results reported for organic solar cells using a semiconductor

device simulation tool TCAD Sentaurus from Synopsys Inc. [7]. Furthermore, the excellent

fitting of the experimental J-V characteristics in Fig. 5.3 to the empirical expression of Eq.

(5.20) corroborates the validity of the ad hoc hopping drift-diffusion model proposed under

the assumption of nonlinear exponential dependence of Jph on Va. For a high carrier hopping

mobility such as 1 cm2/Vs, all photogenerated carriers can be extracted, resulting in a typical

inorganic solar cell behavior with a constant Jph across the entire voltage range of interest

CHAPTER 5. 118

except for the case when the external voltage becomes larger than the built-in voltage, Vbi,

Fig.5.2, and the photocurrent density drops to 0 mA/cm2. Similar to Figs. 5.4(a) and (b) under

linear-dependent photocurrent density, the photocurrent changes direction if the external

voltage increases beyond the value of the built-in potential as shown in Fig. 5.4(c). It should

be noted that due to low hopping mobilities, sometimes dark and illuminated J-V curves cross,

for example, the crossover point of Jdark and Jillu in Fig.5.3(a), because of the higher-carrier-

population-induced higher conductivity upon illumination [7]. This effect, however, is not

important in materials and devices with high carrier hopping mobility.

5.5 Impact of Hopping Mobility and Bandgap Energy

Using experimental parameter values, Figure 5.5(a) exhibits the simulated solar cell J-V

characteristics and their dependence on the carrier hopping mobility ranging from 0.001

cm2/Vs to 10 cm2/Vs. For the sake of better comparison with our experimental CQD solar cells,

the bandgap used for simulation is 1.32 eV, same as our experimentally optimized CQD energy

bandgap. The mobility of CQD thin films used in our CQD solar cells has been measured to

be ca. 2×10-2 cm2/Vs [20], which is also in agreement with the experimental results reported

by Yazdani et al. [21] for CQDs. As shown in Fig. 5.5(b), the simulated Voc and Jsc precisely

match with those measured from our CQD solar cell in Fig.5.3(a). Furthermore, with the

increase in mobility, Fig. 5.5(b), Voc decreases while Jsc increases then saturates at high

effective mobilities μ*. The reduced Voc with carrier mobility μ* has been intensively

investigated for organic solar cells, however, such studies are insufficient for CQD solar cells.

Wang et al. [22] and Shieh et al. [23] attributed the Voc loss to enhanced recombination with

dark charge carriers injected from contacts at high mobilities. While it also occurs in our model,

CHAPTER 5. 119

the fast extraction of charge carriers at high μ* is another reason for low Voc as predicted by

Mandoc et al. [24] and Deibel et al. [25].

Figure 5.5: (a) Simulated carrier-mobility-dependent J-V characteristics; (b) open-circuit

voltage Voc, and short-circuit current density Jsc; (c) fill factor FF; and (d) power conversion

efficiency PCE (d). The CQD thin film bandgap used in 1.32 eV same as our experimentally

optimized bandgap for the CQD solar cell in Fig. 5.3. The CQD solar cell carrier hopping

mobility was estimated from our previous study [20]. Equations (5.9), (5.11) and (5.20) were

used for the simulations.

Furthermore, Tress et al. [26] successfully simulated this trend of Voc decline with mobility

through various recombination mechanisms including Langevin recombination,

recombination via charge transfer states, and trap-assisted recombination. The essential

principle is the interplay between the high-mobility-boosted carrier extraction from high drift

current and the enhanced carrier loss resulting from increased carrier recombination rates at

CHAPTER 5. 120

high carrier mobilities. The developed drift-diffusion current-voltage model operates as a self-

consistent system considering the carrier concentration, surface recombination, and carrier

hopping mobility to interpret the dependence of Voc, Jsc, FF, and PCE on mobility. Specifically,

Voc is carrier concentration dependent and is determined by the energy difference between the

electron and hole quasi-Fermi levels [27]. At low carrier hopping mobilities, low

recombination rates yield high carrier concentrations at open circuit according to

𝑒𝑉𝑜𝑐 = ∆𝐸𝐹 = 𝑘𝑇𝑙𝑛 (𝑛𝑝

𝑛𝑖2) (5.21)

in which 𝑛 , 𝑝 , 𝑛𝑖 are the electron, hole, and intrinsic carrier concentrations, respectively.

Furthermore, within the framework of direct electron-hole recombination, the recombination

is given by

𝑅0 = 𝛽(𝑛𝑝 − 𝑛𝑖2) (5.22)

where 𝛽 is the recombination constant. Langevin theory gives a description connecting the

carrier mobility with recombination rate through 𝛽 =𝑒(𝜇𝑒+𝜇ℎ)

𝜖0𝜖𝑟 with 𝜖0𝜖𝑟 the permittivity of the

materials [28] and 𝜖𝑟 ≈ 20 for our CQD thin films [29]. Therefore, Tress et al. [26] derived

the open-circuit voltage 𝑉𝑜𝑐 as a function of mobility through 𝛽,

𝑉𝑜𝑐 =1

𝑒[𝐸𝑔 − 𝑘𝐵𝑇𝑙𝑛 (

𝛽𝑁𝑐𝑁𝑉

𝐺)] (5.23)

in which 𝑁𝑐 and 𝑁𝑉 are effective density of states on the order of 1019 cm-3 for CQDs [29] in

conduction and valence bands, respectively. The term 𝐺 is a carrier generation rate which

equals ca. 1×1022cm-3s-1 from ref. [26]. Therefore, 𝑉𝑜𝑐 decreases from 0.64 V to 0.49 V with

increasing carrier hopping mobility from 0.001 cm2/Vs to 10 cm2/Vs as simulated in Figs.

5.5(a) and (b). Taking 0.023 cm2/Vs as our solar cell’s mobility, the simulated Voc is found to

be close to the experimental value of 0.63 V. Although current CQD fabrication techniques

CHAPTER 5. 121

cannot enable mobilities in a wide range, in agreement with Fig. 5.5(b) for CQD solar cells

our previous study [30] found that Voc was reduced at higher temperatures which corresponds

to higher carrier mobility due to the nature of phonon-assisted carrier hopping transport in

these materials [31, 32]. Similar simulation results for Voc have also been reported for other

low-mobility solar cell systems [11, 25, 26] using an implicit solar cell simulator. As estimated

from fitting in Fig.5.3(a), the SRV used for Fig. 5.5 is 1×10-3 cm/s through Eq. (5.11). However,

when an infinite SRV is used, i.e. Eq. (5.8) with a quenching boundary condition indicating

immediately recombination of all carriers arriving at the contact, Voc decreases much more

dramatically at high mobilities, and this effect was demonstrated in refs.[24, 25] for organic

solar cells. However, Deibel et al. [25] found that a dramatic reduction of Voc can be avoided

if a finite surface recombination rate is considered. Infinite SRV is not reasonable, of course,

as discussed in ref. [22], also considering the significantly improved CQD surface quality of

our CQD solar cells through solution-ligand exchanges that leave few unsatisfied dangling

bonds on the CQD surface [13]. Furthermore, also from Fig. 5.5(b), Voc is directly proportional

to the bandgap energy Eg of the active photovoltaic CQD thin films, while the simulated eVoc

values are almost half the corresponding Eg, in agreement with the experimentally reported

results for CQD solar cells [33, 34], indicating that only half of the photon energy is harvested.

This is due to the energy loss of excitons and charge carriers to bandgap trap states and/or

bandtail states [13].The enhanced carrier collection efficiency of the photogenerated current at

higher carrier hopping mobility facilitates the increase of short-circuit current density. Short-

circuit current density was obtained at zero voltage using Eqs. (5.9), (5.11), and (5.20). From

Fig. 5.4, it is expected that Jsc rises with mobility due to the enhanced drift photocurrent density

resulting from increased mobility according to Eqs. (5.14) - (5.16). Theoretically, based on our

CHAPTER 5. 122

model, photocurrent saturation at high mobilities occurs because all photoexcited carriers are

extracted at short circuit as shown in Fig. 5.5(b), mirrored by the maximum Jph values at 0 V

for the high mobilities in Fig. 5.4. In practice, further enhancement of Jsc can be achieved

through enhancing photoexcitation intensity or absorption with small bandgap CQDs or

thicker CQD layers.

Furthermore, the trade-off between Jsc and Voc results in peaked FF and PCE with respect

to carrier hopping mobilities, Figs. 5.5(c) and (d). The simulated FF of 0.62 at ca. 0.023 cm2/Vs

in Fig. 5.5(c) is in agreement with the value of 0.63 estimated in our CQD solar cells

characterized in Fig. 5.3(a). Fill factor is a measure of the current-voltage characteristic shape

of solar cells. Compared with other parameters, FF can markedly elucidate carrier

recombination strength [35]. Before reaching the optimized mobility as shown in Fig. 5.5(c),

FF improves dramatically from the significantly enhanced carrier drift current and the

marginally decreased Voc as discussed above. The steep rise of FF in the low mobility regime

is attributed to the increased charge carrier extraction outside the device with mobility increase

while carrier recombination still remains at a relatively low level as per the drift-diffusion J-V

model and Fig. 5.5(b). After attaining the optimal carrier mobility, the decline in FF is an

indication that carrier recombination starts to overtake extraction. As for Figs. 5.5(c) and (d),

similar results of mobility-dependent FF as well as PCE are also found in low mobility organic

solar cells [26]. Consequently, the study of the competition between carrier extraction and

recombination, as well as the tradeoff between Voc and Jsc with mobility, is helpful for CQD

solar cell fabrication. Additionally, the simulated PCE of 9.3 % is comparable to the

experimental result of 10 % shown in Fig. 5.3(a). Through literature review, despite the fact

that relatively high field-effect [36-38] and terahertz radiation [39, 40] mobilities have been

CHAPTER 5. 123

reported on the order of 1-30 cm2/Vs for CQDs, the highest reported solar cell PCE to-date

was achieved by employing active materials with relative lower field-effect mobilities in a

range from 10-3 to 10-2 cm2/Vs [3, 41, 42]. Similar to our model, Zhitomirsky et al. [1]

attributed this to trap-state-limited carrier diffusion lengths, in other words, the low PCE values

in higher carrier mobility materials and devices are results of increased trap-state-assisted

recombination. Before the maximum PCE is attained, increasing the carrier mobility improves

the PCE, however, beyond this regime, increasing the hopping mobility simply enables a

higher rate of carrier recombination. Therefore, instead of intuitively pursuing higher carrier

mobility, a more effective suggestion for solar cell performance optimization should be to

reduce the trap states, which also reduce bimolecular recombination due to strengthened

interdot coupling and enhance the diffusion length, then to further increase carrier mobility. In

conclusion, the simulated results in Fig. 5.5 are in agreement with the finding that “low

mobility might help mitigate a particular loss mechanism in a certain material…” as reported

by Street et al. [28].

Table 5.1: Parameters used for the CQD solar cell simulations.

Parameter

Symbol Value Unit References

Carrier concentration at equilibrium n(0) 1×1016 cm-3 29, 43, 44

Solar cell thickness d 4.1×10-5 cm 13

Effective hopping diffusion length L* 350 nm 13, 45

Effective carrier hopping diffusivity D* Varied, 1×10-4-

0.01

cm2/

s

14, 43

Effective carrier hopping mobility μ* Varied, 1×10-3-10,

0.023 for our

CQDs

cm2/

Vs

29, 20, 46

Built-in voltage Vbi Varied, 0.1-1.5 V

Surface recombination velocity S 1×10-3 cm/s Estimated

from fitting

CQD bandgap energy Eg Varied, 0.5-3.6 eV

CHAPTER 5. 124

Figure 5.6: Theoretical simulations of CQD solar cell electrical parameters: (a) Voc and Jsc, (b)

PCE, and (c) FF as functions of CQD bandgap energy (Eg) for five different carrier hopping

mobilities. The maximum photocurrent 𝐽𝑝ℎ𝑚𝑎𝑥 is the same as Jsc at the mobility of 0.1 cm2/Vs.

The illumination intensity used for the simulation is AM1.5 spectrum at 1 sun intensity.

Equations (5.9), (5.11) and (5.20) were used for the simulations.

CQDs are promising in solar cell fabrication due to their dot-size-tunable bandgap energy,

thereby making the structural design for harvesting more solar energy much easier. Using the

parameter values in Table 5.1, a simulation of CQD bandgap-energy-dependent PCE, Fig.5.6,

was carried out using Eq.(5.13), for the sake of simplification, with an approximated average

EQE = 0.76 when the incident light energy is higher than the CQD thin film bandgap energy

according to our fitting shown in Fig.5.3; otherwise, EQE = 0. However, more precise

CHAPTER 5. 125

experimental EQE values as a function of wavelength can be found in ref. [13] for further

research investigations. Therefore, considering AM1.5 excitation, 𝐽𝑝ℎ𝑚𝑎𝑥

was obtained by

integrating the product of EQE, 𝑏𝑠(𝐸) , and the photon energy according to Eq. (5.13).

Subsequently, Jsc was calculated by combining Eqs. (5.9), (5.11), (5.13) and (5.20) at 0 V

external voltage. Smaller Eg facilitates absorption of photons with lower energy, leading to an

increased maximum photocurrent density 𝐽𝑝ℎ𝑚𝑎𝑥 , Fig. 5.6(a). However, due to the hopping

mobility-dependent Jsc, which equals Jph at short circuit, 𝐽𝑝ℎ𝑚𝑎𝑥 converges to Jsc only at 0.1

cm2/Vs across the whole simulated Eg range. The slight drop in Jsc at small Eg for all mobilities

is due to the scarcity of significant low-wavelength solar energy according to the nature of the

AM1.5 solar spectrum, i.e. 𝐽𝑝ℎ𝑚𝑎𝑥 starts to saturate as shown in Fig. 5.6(a). In addition, it is also

due to the decreased built-in voltage Vbi which is reduced when the CQD photovoltaic material

Eg decreases according to Eq. (5.20). Therefore, compared with higher mobilities, Jsc is

expected to start to decrease at higher Eg for low carrier mobilities as shown in Fig. 5.6(a). A

linear dependence of Voc on Eg was found and extracted by linear best fits of the experimental

data [13] from our CQD solar cells, Fig. 5.6(a), and it could be expressed as

𝑉𝑜𝑐 = 0.387𝐸𝑔 + 0.095 (5.24)

Here, the units of 𝑉𝑜𝑐 and 𝐸𝑔 are V and eV, respectively. A similar linear dependence of Voc on

Eg has also been reported by Bozyigit et al. [33] in the form of 0.27Eg+0.09 for ligand EDT

capped PbS CQD solar cells. Insofar as the trade-off between Jsc and Voc, an optimized CQD

bandgap energy for a maximized PCE value can be expected. In other words, although small-

bandgap CQDs facilitate solar energy absorption in a wider wavelength range, the reduced Eg

compromises Voc according to Eqs. (5.23) and (5.24). The simulated PCE in Fig. 5.6(b) are

through Eqs. (5.9), (5.11), (5.13), (5.20), and (5.24) and the calculated Jsc and Voc as discussed

CHAPTER 5. 126

above. Therefore, applying the given carrier hopping transport parameters as tabulated in Table

5.1, the simulation of PCE in Fig. 5.6(b) yields an optimized Eg of ~1.12 eV for a mobility of

0.02 cm2/Vs, a mobility estimated for our CQD systems [20]. Due to the nature of AM1.5

spectrum, multiple PCE sub-peaks are also observed, a feature consistent with the Jsc in Fig.

5.6(a) and the well-known Shockley-Queisser limit simulation. As discussed in Sect. 5.3, the

experimentally optimized Eg is 1.32 eV, which is close to the sub-peak labeled in Fig. 5.6(b).

It should be noted that, experimentally, only CQD bandgap values in a range between 1.28-

1.48 eV were tried [13]. The PCE simulation implies there is still room for further PCE

improvement using CQD materials at this mobility. The bandgap-dependent PCEs for other

mobilities in Fig. 5.6(b) reveal a blue-shift of the optimized Eg with mobility increase, which

is in agreement with ref. [7]. The shift to small bandgap is a result of relatively high carrier

drift current at high mobilities. Specifically, the reduced Eg diminishes the intrinsic electric

field and the drift current starts to decrease at smaller Eg for higher carrier mobilities when

compared with lower-mobility CQD solar cells. This overall non-monotonic behavior in Fig.

5.6(b) implies an increased PCE with the simulated carrier hopping mobility in a range where

the carrier extraction rate still surpasses the recombination rate. Furthermore, possible

simulation deviation of this model is expected due to the use of linear Eg –dependent Voc

through Eq. (5.24) which, however, is derived based on experimental data in a narrow Eg range,

probably not sufficiently accurate for a wide-range Eg simulation in this study. In addition, for

small Eg with high photoexcited carrier densities Voc should be further reduced as there will be

an exponential dependence of Voc on carrier concentration according to Eqs. (5.21) and (5.23).

The latter will lead to even lower Voc at smaller Eg than Eq. (5.24) predicts, resulting in a shift

of the optimized Eg to large Eg CQDs. Figure 5.6(c) shows a monotonic increase of FF with Ea

CHAPTER 5. 127

as well as with μ* in the range between 0.001 cm2/Vs and 0.1 cm2/V. As discussed above, the

increased FF values indicate that with increased bandgap, carrier extraction plays an

increasingly important role in carrier transport over recombination processes according to Fig.

5.6(c).

Figure 5.7: (a) Simulated CQD solar cell (a) Voc and Jsc, as well as FF and PEC (b), as functions

of the illumination intensity. Equations (5.9), (5.11) and (5.20) were used for the simulations.

The foregoing hopping drift-diffusion J-V model was further examined by studying the

effects of illumination intensity on the solar cell Voc, Jsc, FF, and PCE. Figure 5.7(a) validates

the enhanced Jsc with illumination intensity due to boosted photoexcited carrier densities. The

simulated Voc shows an exponential correlation with the excitation intensity as shown in Fig.

5.7(a) that is mirrored in the well-known relation 𝑉𝑜𝑐 = 𝐴 +𝑛𝑖𝑑𝑘𝑇

𝑞𝑙𝑛𝑋, in which A is a constant

and X represents the illumination intensity. According to recombination mechanisms including

the Langevin and trap-state-assisted Shockley-Read-Hall (SRH) recombination theories, the

carrier recombination rate changes proportional to the carrier concentration. For example,

through direct bimolecular recombination, Eq. (5.22), the carrier recombination rate grows

with carrier concentration proportional to the photoexcitation intensity.

With carrier recombination increase, Fig. 5.7(b) shows decreasing FF at high excitation

intensities, in good agreement with the experimental findings reported by [47, 48]. Because of

CHAPTER 5. 128

the increased carrier recombination rate at high photocarrier injection levels, PCE increases

only slightly from ~8.0% at 0.001 sun to ~10.1% at 1 sun illumination in agreement with our

CQD solar cells in Fig. 5.3(a), above which the PCE increase slows down and even saturates

at high illumination intensities. This implies that with the consideration of various carrier

recombination pathways, carrier radiative and nonradiative recombination through different

mechanisms such as direct biomolecule or SRH approaches can degrade CQD solar cell

performance significantly at high mobilities and/or high photocarrier injection levels. However,

one should be aware that at sufficiently high mobilities comparable to conventional Si solar

cells, this model should not be applied as the effects of electric and diffusion forces become

trivial and negligible. Approaches to reduce carrier recombination in CQD systems can be

through reducing exciton binding energy and/or through removing material trap states. High

exciton binding energy facilitates the probability for electrons and holes to recombine [49]. An

effective approach is to dissociate excitons through strengthening interdot coupling and/or

increase interface energy barriers through a heterojunction architecture. Strong interdot

coupling can be realized with the use of high-quality and monodispersed CQDs that remove

defects and trap states in CQDs. Therefore, improving CQD quality through various methods

as discussed in the introduction always contributes to improved CQD solar cell performance.

5.6 Impact of Electrode-semiconductor Interfaces Using

Homodyne Lock-in Carrierography

Equation (5.10) of the hopping drift-diffusion model above reveals the dependence of CQD

solar cell J-V characteristics on surface recombination velocity 𝑆(𝑑), a parameter determined

by the CQD thin film surface passivation or trap states at CQD semiconductor/Au electrode

interfaces, Figs.5.1 and 5.2. A better electrode coating with lower interface states leads to low

CHAPTER 5. 129

𝑆(𝑑) that results in higher CQD solar cell performance. To investigate the CQD/Au interface

influence on CQD solar cell performance, non-destructive imaging (NDI) of carrier population

distributions and key photovoltaic parameters was carried out using homodyne lock-in

carrierography (HoLIC) [50], a spectrally gated dynamic frequency-domain

photoluminescence imaging method as reported by Hu et al. [51] and also discussed in Sec.7.2.

The HoLIC images show the complexity and inhomogeneity of the electrode-coating-

associated surface recombination in our experimental CQD solar cells. As shown in Fig.5.8,

the J-V and P-V characteristics of one solar cell with lower PCE when compared with Fig.5.3

exhibited similar Voc values but much lower Jsc, and were best-fitted to the combination of Eqs.

(5.9), (5.11), and (5.20) with the consideration of Au-electrode-modified surface

recombination velocity.

Figure 5.8: (a) Experimental J-V characteristics; and (b) output power curves as a function of

photovoltage. Continuous lines are best fits to J-V characteristics and output power using Eqs.

(5.9), (5.11) and (5.20).

CHAPTER 5. 130

Figure 5.9: (a) A photograph of a CQD solar cell sample; and (b) its LIC image at open circuit

after the cell was flipped over. The excitation laser was frequency-modulated at 10 Hz at a

mean intensity of 1 sun. The eight Au-coated thin film electrodes on the top in (a) are electrical

contacts while dark brown regions are without Au contact layers. Both regions have an energy

structure as shown in Fig. 5.2. The Au electrode circumscribed with a dashed rectangle in (a)

and also shown in the flipped over orientation in (b) is further studied in Figs. 5.10 and 5.11.

For excitons and dissociated free carrier radiative recombination, the voltage-dependent

optical carrier flux corresponding to its electrical counterpart Eq. (5.9) was introduced by

Mandelis et al. [52] and subsequently used by Liu et al. [53] for mc solar cells as their Eq.(5.18);

it was further adapted here with a different optoelectronic coefficient 𝑚′ [= (𝜇∗

𝐷∗)′

] for hopping

transport in CQD solar cells:

𝐽[ℏ𝜔, 𝑉(ℏ𝜔)]𝑟 = 𝐽𝑅 − 𝐽𝑅0[𝑒𝑚′𝑉(ℏ𝜔) − 1] (5.25)

𝐽[ℏ𝜔, 𝑉(ℏ𝜔)]𝑅 can also be obtained experimentally through

𝐽[ℏ𝜔, 𝑉(ℏ𝜔)]𝑟 = 𝑞𝐶𝐿𝐼𝐶[𝐿𝐼𝐶(𝑉𝑜𝑐) − 𝐿𝐼𝐶(𝑉(ℏ𝜔))] (5.26)

where 𝑞 is the elementary electron charge, 𝐿𝐼𝐶 is the HoLIC signal at photovoltage 𝑉(ℏ𝜔),

𝐽[ℏ𝜔, 𝑉(ℏ𝜔)]𝑟 is the non-equilibrium radiative recombination current density, 𝐽𝑅 and 𝐽𝑅0 are

the relevant current-density-like quantities, and 𝐶𝐿𝐼𝐶 is a coefficient defined as [53]

𝐶𝐿𝐼𝐶 =|𝐼𝑖|(1−𝑅)𝜂

ℏ𝜔𝑖𝑛 [𝐿𝐼𝐶(𝑉𝑜𝑐)−𝐿𝐼𝐶(0))][1−𝜂𝑐𝑒(ℏ𝜔,𝑉=0,𝑇)

1−𝜆𝑖𝑛𝜆𝑒𝑚−1 ] (5.27)

CHAPTER 5. 131

where |𝐼𝑖| is the peak value of the incident modulated illumination intensity, 𝑅 is the surface

reflectance, 𝜂 and 𝜂𝑐𝑒(ℏ𝜔, 𝑉 = 0, 𝑇) are the quantum efficiency for exciton and charge carrier

photo-generation and photocarrier-to-current collection efficiency, ℏ𝜔𝑖𝑛 is the incident photon

energy, and 𝜆𝑖𝑛 and 𝜆𝑒𝑚 are, respectively, the incident and emitted photon wavelength. In a

manner similar to the electrical Eqs. (5.9) and (5.11), Eq. (5.25) links the exciton and free

carrier radiative recombination flux which is an optically measurable quantity to its electrical

parameter counterparts. The expressions for Jph, J0, Voc, and m’ have been derived by Liu et al.

[53] using the optical parameters in Eq. (5.25) and a photocarrier-to-current collection

efficiency which is the ratio of the photocarrier flux collected by the solar cell electrodes

(giving rise to the photocurrent) to the incident photocarrier flux. A photograph of our solar

cell is presented in Fig. 5.9(a), and the corresponding HoLIC image shown in Fig. 5.9(b)

reveals the inhomogeneity of the Au contact regions which are distinguishable from those

without Au layers. A dashed rectangle in Fig. 5.9(a) is circumscribed around the perimeter of

an Au layer, same as the one circumscribed in Fig. 5.9(b), is further studied in detail in Figs.

5.9-5.11. The LIC image-generating laser was introduced from the ITO/ZnO side, Figs.5.1 and

5.2. The image contrast originates in inhomogeneous exciton and free charge carrier

population distributions due to mechanical or electrical defect-induced photocarrier lifetime

variations. Defect-induced trap states act as thermal capture and emission centers of a

nonradiative recombination nature that diminish exciton and free charge carrier hopping

lifetimes, resulting in HoLIC image signal decreases. According to Fig. 5.9(b), regions (solar

cell pixel images) with electrodes on the right-hand side appear to have higher defect densities

or worse contacts than those on the left-hand side. These observations are consistent with our

experimental results that low solar cell efficiency is associated with J-V measurements of these

CHAPTER 5. 132

particular solar cell units. These results show that carrier diffusion-wave-based HoLIC

imaging has excellent potential for non-destructive inspection of CQD solar cells.

Figure 5.10. HoLIC images of the circumscribed CQD solar cell Au electrode in Fig. 5.9(a) at

open-circuit 0.64 V (a), 0.60 V (b), 0.56 V (c), 0.35 V (d), 0.20 V (e), and short-circuit (f). The

excitation laser was frequency-modulated at 10 Hz at a mean intensity of 1 sun.

For solar cell PCE optimization, dynamic carrier distribution visualization as a function of

applied external voltage, Fig. 5.10, is crucial for optimizing device fabrication with respect to

materials and nanoparticle deposition techniques. With decreasing external voltage, the HoLIC

image exhibits different trends in different regions within the entire solar cell unit image,

indicating highly inhomogeneous carrier hopping transport. These variations in performance

CHAPTER 5. 133

of each device lead to a poor overall solar cell behavior and should be considered seriously as

optimization issues for commercial CQD solar cell fabrication. Specifically, points A, B, C,

and the area inside the dashed rectangle were studied as shown in Fig. 5.10(a). Excitons and

charge carriers extracted into the external circuit through the Au and ITO contacts in Figs.5.1

and 5.2 can be probed through HoLIC signal differences between open-circuit and short-circuit

conditions, Figs.5.11(a) and (b). It can be observed that more photogenerated exciton and free

charge carriers are collected at points A and B than at point C and at regions close to the edge

of the contacts.

Table 5.2: Optical counterparts of CQD solar cell electrical parameters, obtained through

best-fitting of the experimental data in Figs.5.11(b) and (d) to Eq. (5.25).

Sample JR/ CLIC

(C·mV)

JR0/ CLIC

(C·mV)

m’

(V-1)

Point A 1.78×10-18 2.34×10-19 3.43

Point B 1.66×10-18 1.71×10-19 3.76

Point C 1.56×10-18 1.02×10-18 1.49

Selected Region 1.45×10-18 6.45×10-19 1.89

Excellent best-fits to our theoretical model Eq. (5.25) have been achieved as shown in

Figs.5.11(b) and (d), in addition to the best-fitted optical parameters shown in Table 5.2. The

results reveal that high-amplitude regions in Figs.5.11(a) and (c) yield higher optical

counterparts of Jph and along with lower optical saturation current densities. A conclusion can

be reached from Fig. 5.11(b) that defects compromise the optical Isc more significantly than

Voc which remains almost constant at all three selected points and inside the dashed rectangle

region. The HoLIC images thus suggest that material surface and interface treatments for

eliminating CQD surface defects may benefit Voc enhancement only in a limited manner

although they can raise Jsc significantly. Therefore, smaller Jsc values arising from higher

defect state density result in reduced maximum-output-powers Pmax as shown in Figs.5.11(c)

CHAPTER 5. 134

and (d), which present the maximum-output-power mapping and its voltage dependence,

respectively.

Figure 5.11: LIC of the circumscribed CQD solar cell Au electrode in Fig. 5.9(a): (a) LIC (Voc)

- LIC(Vsc); (b) [LIC (Voc) - LIC (V)] vs. V; (c) [LIC (Voc) - LIC(VPM)]VPM ; and (d) [LIC(Voc) -

LIC(V)]V vs. V characteristics. The excitation laser was frequency-modulated at 10 Hz at a

mean intensity of 1 sun. (b) and (d) are best-fitted to Eq. (5.25). Points A, B, C, and the dashed

rectangle region are shown in (a) and (c). It should be noted that values calculated for the

dashed rectangle region are based on averaging the LIC amplitudes over all pixels in this region.

CHAPTER 5. 135

Figure 5.12: Open-circuit voltage Voc LIC contour mapping of the circumscribed CQD solar

cell Au electrode in Fig. 5.9.

In comparison, the solar cell pixel Voc image, Fig. 5.12, is more homogeneous within the

Au electrode region as Voc is determined primarily by the work function difference between its

corresponding electrodes rather than device defects or defect-affected carrier hopping transport

behavior. Figures 5.11 and 5.12 clearly demonstrate the critical importance of electrode

influence on CQD solar cell performance. The respective maximum power and carrier

collection HoLIC images are critical to the evaluation of the CQD solar cell quality due to the

low carrier hopping mobility and diffusivity which result in low collection efficiency of

carriers generated far away (compared to a diffusion length) from the carrier extraction

electrodes.

5.7 Conclusions

A comprehensive study of CQD solar cell efficiency dependence on carrier mobility, CQD

energy bandgap energy, and electrode interface was performed. Optimized carrier hopping

CHAPTER 5. 136

mobilities and bandgap energies can be determined from studying their impact on CQD solar

cell efficiencies. Furthermore, the universally applied assumption of constant photocurrent was

relaxed and its variation with voltage and mobility was analyzed for CQD solar cells. This

voltage- and mobility dependent photocurrent density was demonstrated to originate from the

competition between carrier extraction rate and recombination rate (for example, through trap

states) in CQD solar cells. Large-area inspection of CQD solar cell carrier population

distribution, collection efficiency, and Voc using LIC NDT revealed a strong correlation

between Au electrode/CQD interface associated surface recombination effects and solar cell

performance, overcoming limits of conventional small-dot characterization methodologies.

The developed self-consistent hopping drift-diffusion model, together with large-area HoLIC

NDT pave the way for a comprehensive quantitative strategy for device fabrication toward

high-efficiency solar cells that can be of keen interest to the CQD solar cell community. The

presented efficiency optimization strategy is summarized below.

1. Attempts to enhance the carrier diffusion length by reducing trap state density either

through selecting proper interdot linking ligands that have lower lattice mismatch with the

CQD crystal lattice or through the use of solution-based ligand exchanges rather than the solid

state layer-by-layer method [13] should always be the first priority.

2. According to the achieved diffusion length, the proper CQD thin film thickness for

solar cells should be determined, and the present new drift-diffusion transport model that

introduces voltage-dependent Jph should be used to find optimized μ and Eg at a maximized

PCE for the given estimated parameters as shown in Table 5.1.

CHAPTER 5. 137

3. Only after the diffusion length improves, efforts to reduce CQD polydispersity or

strengthen interdot coupling to reach an optimized μ at a given CQD Eg should be implemented

guided by the parameter relationships shown in Fig.5.5.

4. Since μ cannot be characterized in as straightforward a manner as Eg, the preferred

procedure should be to use dot-size-tunable Eg, instead of varying μ, toward achieving

maximum PCE according to Fig.5.6.

5. Eliminating material surface and interface defects benefit Voc enhancement only in a

limited manner but can raise Jsc significantly. Therefore, selection of electrode metals and

contact procedures yielding optimal carrier density distributions, extraction efficiency, and Voc

as visualized by LIC imaging is an effective tool for further CQD solar cell efficiency

improvement.

138

Chapter 6

Photocarrier Radiometry Study of Quantitative Carrier

Transport in CQD Thin Films

6.1 Introduction

Understanding carrier transport dynamics and shedding light on energy dissipation

mechanisms in optoelectronics is essential to devise efficiency optimization. QD disorder in

the form of energy and/or geometry, originating in dot shape, size, composition, surface

chemistry, and capping ligands, as well as the degree of polydispersity and superlattice order

in thin films, disrupts the formation of continuous energy band structures in CQD ensembles.

Depending on the level of QD disorder, there are four possible carrier transport mechanisms

[1] as discussed in Sect.2.2.1. Phonon-assisted hopping is the most prevailing mechanism that

has been widely applied in studying various QD systems [2-8] in which carriers hop from one

dot to the next depending on the interdot distance, coupling strength, temperature, and the type

of carrier. Moreover, strengthening the capping-ligand-controlled interdot coupling has been

reported in PbSe QDs as originating in the Coulomb blockade dominated insulating regime

and into the hopping conduction dominated semiconductor regime [2], and has also been found

to assist exciton dissociation into free electrons and holes [9, 10, 11]. Furthermore, Lee et al.

[11] and Liu et al. [4] have observed a monotonic increase in hole mobility with increasing

QD size, while electron mobility exhibits a peak at a QD diameter of 6 nm, which can be

ascribed to the compromise between reduced activation energy (lower hopping energy barrier)

and weakened interdot coupling strength amongst larger QDs. The phonon-assisted hopping

CHAPTER 6. 139

transport mechanism predicts a temperature-dependent carrier mobility, diffusivity, lifetime,

conductivity, and conductance of QD devices [2, 3, 5, 7, 8, 12-14]. However, due to the large

specific surfaces of QDs, even with the application of capping ligands, QD trap states still

hinder the efficiency of CQD based electronic devices through acting as undesired radiative

and nonradiative recombination centers [4, 15-21]. Important as these effects are, a systematic

study of trap-state-modified carrier transport is still lacking.

Despite the importance of carrier dynamics to QD optoelectronic and electronic device

efficiency optimization, current characterization techniques are still not able to provide

sufficient feedback information about carrier transport kinetics in QD substrates and devices.

As discussed in Chapter 3, carrier mobility can be characterized by linearly increasing voltage

(CELIV) [20, 22, 23], time of flight (TOF) [22,24], transient photovoltage [20, 25], and by

using field effect transistors (FET) [20, 26, 27]. Nonetheless, these methods require thick QD

films and a completed device. Although Zhitomirsky et al. [28] introduced photoluminescence

(PL) quenching for carrier diffusion length measurement in CQD thin films, additional coating

and/or embedding different types of CQDs are compulsory. Nowadays, carrier lifetime is

measured mostly by Voc (open-circuit voltage) transient decay [22] and transient PL [20, 29,

30] for devices and substrates, respectively. However, due to the fragile nature of materials

comprising photovoltaic devices, especially organic and QD-based solar cells, and due to their

disadvantages as elaborated in Sect.3.5, most of these conventional techniques are suitable

neither for industrial in-line mass manufacturing of electronic devices at any and all fabrication

stages nor for optoelectronic process analysis involving light-carrier interactions.

Therefore, this chapter introduces the all-optical, fast, and non-destructive technique,

photocarrier radiometry (PCR) into CQD thin film characterization with its instrumentation,

CHAPTER 6. 140

general principles, and advantages as discussed in Sect. 3.5. A trap-state-mediated carrier

hopping transport model was developed and applied for the extraction of multiple carrier

transport parameters of different ligand-capped PbS CQD thin films. The temperature-

dependent carrier transport dynamics was investigated in perovskite-passivated PbS CQD thin

films. Combined with a carrier hopping transport model, PCR was shown to exhibit great

potential in QD materials characterization for fundamental physics research of carrier transport

dynamics, in addition to being an all-optical, nondestructive and promising technique for

industrial device quality control as discussed in Sect.3.5.

6.2 PCR Theory for CQDs: Trap-state-mediated Carrier

Transport Model

Figure 6.1: (a) Schematic of carrier hopping transport in PbS CQD thin films embedded in a

surface-passivation ligand matrix when excited by a frequency-modulated laser source. (b)

Illustration of carrier generation, dissociation, hopping transport, and trapping processes in a

CQD assembly. Se and Sh are the ground states for electrons and holes, respectively. Ea,1 and

Ea,2 are the activation energies associated with exciton binding energy (Eb) and trap-mediated

transition process, respectively. Eg and Eg, opt are, respectively, the electronic and optical band

gap energy.

Figures 6.1(a) and (b) exhibit the schematic of surface-passivated and laser-illuminated

PbS CQDs in a ligand matrix. Upon laser excitation, excitons will firstly be generated within

the CQDs and diffuse away through a carrier hopping mechanism [Fig.6.1(a)], during which

process, excitons may dissociate into free charge carriers. All these particles, including

CHAPTER 6. 141

excitons and their dissociated charge carriers, can recombine radiatively, or be bound to or

trapped in trap states and recombine radiatively or non-radiatively [Fig. 6.1(b)].

Therefore, the rate equation for the population 𝑁𝑖(𝑥, 𝑡) of charge carriers in quantum dot 𝑖

[31] must include the presence of trap states acting as thermal emission and capture centers.

Such trap states have been reported in thiol-capped PbS QDs [32], and in glass-encapsulated

PbS QDs [33], also several trap-related emission bands have been reported for PbS QDs in

polyvinyl alcohol [34]. Taking into consideration that those trap states acting as thermal

emission and capture centers, the carrier rate equation can be expressed as:


𝜕𝑡= −∑ 𝑃𝑖𝑗𝑗 𝑁𝑖(𝑥, 𝑡) + ∑ 𝑃𝑗𝑖𝑗 𝑁𝑗(𝑥, 𝑡) + ∑ {𝑒𝑖𝑘(𝑇)𝑛𝑇𝑘(𝑥, 𝑡) − 𝐶𝑖𝑘𝑁𝑖(𝑥, 𝑡)[𝑁𝑇𝑘 −

𝑚𝑘=1

𝑛𝑇𝑘(𝑥, 𝑡)]} −𝑁𝑖(𝑥,𝑡)

𝜏+ 𝐺0(𝑥, 𝑡) (6.1)

where 𝑘 denotes trap level, 𝑒𝑖𝑘 is the thermal emission rate of charge carriers from the trap

level 𝑘, 𝐶𝑖𝑘 is the charge carrier capture coefficient, 𝜏 is the carrier lifetime, 𝑁𝑇𝑘 is the trap

density of level 𝑘, 𝑛𝑇𝑘 is the trapped carrier density, 𝑃𝑖𝑗 (𝑃𝑗𝑖) is the hopping probability from

the 𝑖𝑡ℎ (𝑗𝑡ℎ) QD to the 𝑗𝑡ℎ (𝑖𝑡ℎ) QD. Here, 𝐶𝑖𝑘𝑁𝑇𝑘 is defined as the carrier-trapping rate 𝑅𝑇𝑘.

Go is the photocarrier generation rate. In the PbS CQD system under consideration, all the trap

states at distinct levels are considered to have the same effects on carrier transport behavior,

i.e. trapping and detrapping carriers. Although distinguishing them is possible using photo-

thermal deep level transient spectroscopy at various temperatures and can yield a more detailed

structure of trap levels, however, it is not necessary for the present optoelectronic transport

property characterization. Therefore, the rate equation for carrier population can be further

developed to

CHAPTER 6. 142


𝜕𝑡= −


𝜕𝑥+ 𝑒𝑖(𝑇)𝑛𝑇(𝑥, 𝑡) − 𝐶𝑖𝑁𝑖(𝑥, 𝑡)[𝑁𝑇 − 𝑛𝑇(𝑥, 𝑡)] −

𝑁𝑖(𝑥,𝑡)

𝜏+ 𝐺0(𝑥, 𝑡) (6.2)

where the charge carrier current density 𝐽𝑒(𝑥, 𝑡) is a function of the hopping diffusivity 𝐷ℎ and

can be written as

𝐽𝑒(𝑥, 𝑡) = −𝐷ℎ(𝑇)𝜕𝑁𝑖(𝑥,𝑡)

𝜕𝑥 (6.3)

Hopping diffusivity 𝐷ℎ is a fundamental photovoltaic electronic property, which depends on

the interdot distance 𝐿, charge carrier hopping probability 𝛾, and temperature 𝑇, through the

following relationship:

𝐷ℎ(𝑇) =𝐿2

𝜏0𝑒−𝛾𝐿−∆𝐸𝑗𝑖/𝑘𝐵𝑇 (6.4)

where 𝜏0 is the hopping time of a carrier from one QD to another, 𝛾 is the hopping

transmission coefficient, 𝐿 is the effective interdot distance, 𝑇 is the temperature, ∆𝐸𝑗𝑖 is the

energy difference of a hopping particle (exciton or dissociated carrier) between QD states (i)

and (j) [35], and 𝑘𝐵 is the Boltzmann constant. Since trapped charge carriers (𝑛𝑇) can be

emitted from trap states or re-captured, the kinetic equation for 𝑛𝑇 is given by

𝜕𝑛𝑇(𝑥,𝑡)

𝜕𝑡= −𝑒𝑖(𝑇)𝑛𝑇(𝑥, 𝑡) + 𝐶𝑖𝑁𝑖(𝑥, 𝑡)[𝑁𝑇 − 𝑛𝑇(𝑥 , 𝑡)] (6.5)

Combining Eqs. (6.2) - (6.5) yields an expression for the kinetics of the carrier population in

a QD ensemble involving the charge carrier generation, capture, and release from trap states,

as well as the carrier hopping diffusion:


𝜕𝑡+ {𝐶𝑖[𝑁𝑇 − 𝑛𝑇(𝑥, 𝑡)] +

1

𝜏(𝑇)}𝑁𝑖(𝑥, 𝑡) = 𝐺0(𝑥, 𝑡; 𝜔) + 𝑒𝑖(𝑇)𝑛𝑇(𝑥, 𝑡) + 𝐷ℎ(𝑇)

𝜕2𝑁𝑖(𝑥,𝑡)

𝜕𝑥2

(6.6)

CHAPTER 6. 143

There is much evidence for the existence of bright (or singlet) and dark (or triplet) states in

PbS QDs [33, 36-38]. Non-radiative recombination processes arise from charge carriers

trapped in both singlet and triplet states:


𝜕𝑡=

𝜕𝑁𝑠(𝑥,𝑡)

𝜕𝑡+𝜕𝑁𝑡(𝑥,𝑡)

𝜕𝑡 (6.7)

𝑁𝑠 and 𝑁𝑡 denote the carrier population in singlet and triplet states, respectively. An energy-

level relation between singlet and triplet states has been proposed [33]

𝑁𝑠(𝑥, 𝑡) = 𝑅𝑠𝑡𝑒−∆𝐸

𝑘𝐵𝑇𝑁𝑡(𝑥, 𝑡) (6.8)

where ∆𝐸 is the energy difference between the two split energy levels, and 𝑅𝑠𝑡 is an energy-

level degeneracy constant equal to 1/3. To simplify the notation, let

𝐴(𝑇) = 𝑅𝑠𝑡𝑒−∆𝐸

𝑘𝐵𝑇 (6.9)

Furthermore, for the harmonic laser excitation at frequency 𝑓𝑟𝑒 = 𝜔/2𝜋, 𝑁𝑡(𝑥, 𝑡), 𝑛𝑇(𝑥, 𝑡)

and 𝐺0(𝑥, 𝑡; 𝜔) can be written as,

𝑁𝑡(𝑥, 𝑡) =1

2𝑁𝑡(𝑥; 𝜔)(1 + 𝑒

𝑖𝜔𝑡) (6.10a)

𝑛𝑇(𝑥, 𝑡) =1

2𝑛𝑇(𝑥; 𝜔)(1 + 𝑒

𝑖𝜔𝑡) (6.10b)

𝐺0(𝑥, 𝑡) =1

2𝐺0(𝑥; 𝜔)𝛽𝑒

−𝛽𝑥(1 + 𝑒𝑖𝜔𝑡) (6.10c)

where 𝜔 is the modulation angular frequency and 𝛽 is the optical absorption coefficient.

The kinetics of the trapping rate Eq. (6.5) can be modified in the frequency domain to yield an

expression for the trapped carrier density 𝑛𝑇(𝑥;𝜔)

CHAPTER 6. 144

𝑛𝑇(𝑥; 𝜔) ≈ (𝐶𝑖𝑁𝑇𝜏𝑖

1+𝑖𝜔𝜏𝑖)𝑁𝑡(𝑥, 𝜔) (6.11)

where 𝜏𝑖 is defined by the carrier emission rate

1

𝜏𝑖(𝑇)= 𝑒𝑖(𝑇) (6.12)

Solving Eqs. (6.6)- (6.9) subject to frequency domain Eq. (6.10), and taking only the

modulated components gives

𝑑2𝑁𝑡(𝑥,𝜔)

𝑑𝑥2−

1

𝐷ℎ(𝑇)[𝑖𝜔 +

1

𝜏𝐸(𝑇)−

𝑅𝑇

[1+𝐴(𝑇)][1+𝑖𝜔𝜏𝑖(𝑇;𝑥,𝜔)]]𝑁𝑡(𝑥, 𝜔) = −

𝐺0𝛽𝑒−𝛽𝑥

𝐷ℎ(𝑇)[1+𝐴(𝑇)] (6.13)

Here, 𝜏𝐸(𝑇) is the effective carrier lifetime, defined as

1

𝜏𝐸(𝑇)≡

1

1+𝐴(𝑇)[

1

𝜏𝑡(𝑇)+

𝐴(𝑇)

𝜏𝑠(𝑇)] (6.14)

and 𝜏𝑡(𝜏𝑠) is the triplet (singlet) lifetime.

For CQD thin films with a thickness d (200 nm for CQD solar cell devices [13, 15]), charge

carriers at the boundaries should be quenched due to the high density of trap states. Equation

(6.13), therefore, can be solved with the boundary conditions: 𝑁𝑡(𝑥, 𝜔) = 0; 𝑥 = 0, 𝑑, viz.

𝑁𝑡(𝑥, 𝜔) = 𝐵1(𝜔, 𝑇)𝑒𝐾1𝑥 − 𝐵2(𝜔, 𝑇)𝑒

−𝐾1𝑥 + [𝐾2(𝑇,𝛽)

𝐾12(𝑇,𝜔)−𝛽

] 𝑒−𝛽𝑥 (6.15)

where the parameters are defined as

𝐾12(𝑇; 𝜔) =

1

𝐷ℎ(𝑇){𝑖𝜔 +

1

𝜏𝐸(𝑇)−

𝑅𝑇

[1+𝐴(𝑇)][1+𝑖𝜔𝜏𝑖(𝑇)]} (6.16a)

𝐾2(𝑇, 𝛽) =𝐺0𝛽

𝐷ℎ(𝑇)[1+𝐴(𝑇)] (6.16b)

𝐵1(𝜔, 𝑇) = [𝐾2(𝑇,𝛽)

𝐾12(𝑇,𝜔)−𝛽2

] (𝑒−𝐾1𝑑−𝑒−𝛽𝑑

𝑒𝐾1𝑑−𝑒−𝐾1𝑑) (6.16c)

CHAPTER 6. 145

𝐵2(𝜔, 𝑇) = [𝐾2(𝑇,𝛽)

𝐾12(𝑇,𝜔)−𝛽2

] (𝑒𝐾1𝑑−𝑒−𝛽𝑑

𝑒𝐾1𝑑−𝑒−𝐾1′ 𝑑) (6.16d)

The radiative emission (i.e., PCR) signal can be expressed as an integral of the charge carrier

population over the thickness of the active layer [39]:

𝑆(𝜔) = 𝐹(𝜆1, 𝜆2) ∫ 𝑁𝑖(𝑥,𝑑

0𝜔)𝑑𝑥 (6.17)

Here, 𝐹(𝜆1, 𝜆2) is an instrumentation coefficient which depends on the spectral emission

bandwidth [𝜆1, 𝜆2 ] of the near-infrared detector. From Eqs. (6.16) and (6.17), the final

expression for the PCR signal can be obtained

𝑆(𝜔)

𝐹(𝜆1,𝜆2)= [

𝐾2(𝑇,𝛽)

𝛽2−𝐾12(𝑇,𝜔)

] {(1+𝑒−𝛽𝑑)(1−𝑒−𝐾1𝑑)

2

𝐾1(1−𝑒−2𝐾1𝑑)−

1

𝛽(1 − 𝑒−𝛽𝑑)} (6.18)

It should be noted that when the trap state density 𝑁𝑇 = 0,

𝐾1(𝑇;𝜔) = √1+𝑖𝜔𝜏𝐸(𝑇)

𝐷ℎ(𝑇)𝜏𝐸(𝑇) =

1

𝐿ℎ(𝑇;𝜔) (6.19)

which is the conventional carrier diffusion wavenumber [7], and 𝐿ℎ(𝑇; 𝜔) is the effective

charge carrier hopping diffusion length.

6.3 CQD Thin Film Homogeneity and Optical Properties

The PbS CQDs were synthesized and purified using the same method as described in

Sect.4.3.1 and ref. [40]. These CQD thin films have a thickness of 200 nm as characterized by

scanning electron spectroscopy. As shown in the room temperature PL spectra, Fig.6.2, three

CQD thin film samples capped with the abovementioned three ligands have the same band-to-

band energy gap of 1.21 eV, while, for further investigation, perovskite MAPbI3 was also

applied to passivate CQD thin films with larger QD size, implying a smaller energy band gap

CHAPTER 6. 146

of 1.09 eV. To clarify, the perovskite MAPbI3 passivated PbS CQD thin film with larger dot

size is labeled PbS-MAPbI3-B throughout this paper, while the one with smaller dot size is

labeled PbS-MAPbI3. It is also shown in Fig. 6.2 that the PL peaks for each type of CQD thin

films depend on the QD size, as well as on surface capping ligands. In addition, secondary PL

emission peaks are also characterized, such as those at 0.81 eV (PbS-MAPbI3) and 0.83 eV

(PbS-TBAI), as well as the PL shoulder at 0.99 eV (PbS-EDT). These secondary PL emission

peaks originate from recombinations that occur through defect-induced donors/acceptors

arising from unpassivated surface states, structural defects, or other changes induced during

ligand exchange processes. Similar types of defect-induced donor/acceptor radiative emission

have also been reported in other materials, such as ZnO nanowires [41], MoS2[42], and InP

[43].

Figure 6.2: Photoluminescence (PL) spectra of four PbS CQD thin films surface passivated

with MAPbI3, EDT, and TBAI.

CHAPTER 6. 147

Figure 6.3: Homodyne lock-in carrierography images of PbS-MAPbI3 (a), PbS-MAPbI3-B (b),

PbS-EDT (c), and PbS-TBAI (d) measured at 10 Hz. Note all the samples were placed on an

aluminum platform for imaging.

As shown in Fig.3.6 the experimental PCR setup for CQD thin films. Constant

characterization temperatures in a range from 100 K to 300 K were maintained using a Linkam

LTS350 cryogenic chamber same as that described in Sect.4.3.1. Photothermal spectroscopy

was performed with the same PCR system at a fixed laser modulation frequency and scanning

temperature. CQD thin films are promising candidates for QD photovoltaic devices; however,

their efficiency is considerably limited by mechanical and electrical defects in CQD thin film

materials. Therefore, we performed homogeneity examination through homodyne lock-in

carrierography imaging (details are shown in Sects.3.4 and 8.2) [44], as shown in Fig.6.3. The

image contrast arises from the modulated carrier wave (including free charge carriers and

excitons) density distribution. Regions with high amplitude values originate from high carrier

CHAPTER 6. 148

density, consistent with high carrier transport parameters including carrier lifetime, diffusivity,

and low trap state density.

6.4 Carrier Transport Kinetics in Various CQD Thin Films

Excitons and free charge carrier transport in CQD thin films is a multi-fact-determined

process as discussed in Chapter 2. The temperature dependence of carrier transport kinetics is

discovered in Chapter 4 using hopping drift-diffusion current-voltage model for high-

efficiency CQD solar cells, herein, using PCR and the trap states involved carrier transport

PCR model, the temperature-dependent transport kinetics will be further investigated in CQD

thin films that surface-passivated with various ligands or consisting of various size of CQDs.

Furthermore, as the ligand-controlled carrier transport kinetics are also explored to reveal more

fundamental physics behind this energy transport behavior. In addition, carrier transport

parameters are obtained through fitting experimental data to theoretical models, therefore, the

fitting uniqueness and measurement reliability are analyzed at the end of this section.

6.4.1 Temperature-dependent Carrier Transport Kinetics

PCR can generate independent carrier diffusion-wave amplitude and phase channels

simultaneously from a single frequency scan, both of which can be used for data analysis

through a best fitting to increase the accuracy and reliability of the best-fitted parameters.

Detailed derivation of PCR amplitude and phase and the Matlab-based computational fitting

for parameter extraction are discussed in Sect. 6.5. Furthermore, a parametric theory as

discussed in Sect. 6.5 is used to examine the uniqueness and reliability of the best-fitted

parameters and demonstrates that all six parameters can be resolved in the framework of our

equations and experimental data sets. For example, Sect.6.5 shows the determinants and

sensitivity coefficients for parameters Dh and τE. Therefore, this validated methodology was

CHAPTER 6. 149

employed for parameter extraction through this study. The experimental and best-fitted PCR

amplitude and phase frequency scans of PbS-MAPbI3 at various temperatures (300 K-100 K),

using Eqs. (6.18), (6.21) and (6.22), are presented in Fig.6.4. Due to the reduced carrier-phonon

interactions at low temperatures and a concomitant increase in the radiative emission rate

accomplished by a decrease in the nonradiative decay rate, the PCR amplitude increases at low

temperatures. In addition, at low temperatures, the increased carrier lifetime yields an

increased PCR phase lag when compared with that at higher temperatures.

Figure 6.4: PCR amplitude (a) and phase (b) of MAPbI3-passivated CQD thin films (PbS-

MAPbI3) measured at various frequencies ranging from 10 Hz to 100 kHz and temperatures

between 100 K and 300 K.

CHAPTER 6. 150

Table 6.1: Best-fitted parameters for PbS-MAPbI3 CQD thin films at different temperatures.

Parameters Temperature (K)

300 250 200 150 100

Dh (cm2/s) 2.41×10-3

±2.75×10-4

5.41×10-4

±1.76×10-6

2.47×10-4

±1.72×10-5

3.62×10-5

±1.44×10-6

1.04×10-6

±3.93×10-13

τE (µs) 0.45 ±0.15 0.66 ±0.21 1.35 ±0.39 4.75 ±0.65 5.37±3.68×10-8

RT (s-1) 2.40×1013

±4.55×1012

2.02×109

±2.37×108

5.54×106

±6.93×105

1.08×105

±3.70×102

2.36×104

±8.99×10-4

𝑒𝑖 (s-1)

3.41×108

±1.22×108

2.97×107

±9.38×106

2.18×107

±3.45×105

4.55×105

±5.33×102

6.70×104

±0.00059

Lh (μm) 0.33 0.19 0.18 0.13 0.023

Figures 6.5(a), (c)-(f) and Table 6.1 show the measurements of five temperature-dependent

carrier hopping transport parameters: Dh, τE, RT, 𝑒𝑖, and Lh of PbS-MAPbI3.With the increase

in temperature from 100 K to 300 K, the best-fitted hopping diffusivity Dh increases

dramatically from 1.04×10-6 cm2/s to 2.41×10-3 cm2/s. The latter value is comparable to the

previously reported values of 0.012 cm2/s and 0.003 cm2/s [9] measured at room temperature

by transient PL spectroscopy for 3-mercaptopropionic acid (MPA) and 8-mercaptooctanoic

acid (MOA) passivated CQD thin films, respectively. The temperature-dependent behavior of

the carrier-wave diffusivity Dh is consistent with the phonon-assisted carrier hopping transport

mechanism. It should be noted that a tunneling transport mechanism is not taken into

consideration due to its non-phonon-assisted transport nature [1, 4, 45-48]. Hopping transport

of carriers within the CQD assembly is carried out through the temperature-dependent nearest

neighbor hopping (NNH) or Efros-Shklovskii variable-range-hopping (ES-VRH) [1, 3, 5, 14].

NNH does not occur at extremely low temperatures because, on average, hopping between

nearest neighbor states requires higher activation energy [5]. Hopping distance is always

optimized spontaneously to yield the highest carrier mobility, and the optimized distance

decreases with increasing temperature [5, 49]. With the temperature rising above a threshold

CHAPTER 6. 151

value, carrier hopping behavior switches from ES-VRH to NNH which has the same hopping

distance as the interdot spacing determined by thermal energy. Kang et al.[5] found that the

hopping distance was longer than the interdot spacing at lower temperatures, and the optimized

distance was equal to the nearest neighbor distance in a temperature range from 40 K to 75 K,

indicating a threshold temperature lower than the minimum temperature of this study. Eq. (6.4),

which does not assume a conventional Einstein relation, predicts an exponential increase in 𝐷ℎ

with increasing temperature, and an exponential decrease with increasing average barrier width

(ligand length).

Figure 6.5: Best-fitted hopping diffusivity Dh (a), and Arrhenius plot of Dh for the extraction

of the carrier hopping transport activation energy (b) of the MAPbI3-passivated (PbS-MAPbI3)

CQD thin film. For the same sample, (c)-(e) are the best-fitted effective exciton lifetime 𝜏𝐸,

carrier trapping rate RT , and Arrhenius plot of thermal emission rate 𝑒𝑖, respectively. (f) Carrier

hopping diffusion length Lh calculated from the best-fitted 𝜏𝐸 and Dh values.

Furthermore, Fig. 6.5(b) shows the activation energy (96.2 meV) obtained from the

Arrhenius plot of the trap-state-mediated hopping diffusivity. It should be reiterated that the

calculated activation energy is trap-state-mediated, i.e., an average energy barrier must be

CHAPTER 6. 152

overcome when carriers hop over the interdot energy barrier and hop out of trap states, as

shown in Fig. 6.1(b). The activation energy extracted from the Dh Arrhenius plot is consistent

with those obtained from the thermal emission rate 𝑒𝑖 as well as those from the photo-thermal

spectra which will be discussed later, corresponding to shallow trap states with an energy depth

much smaller than that of defect-related states measured by PL (Fig.6.2). These activation

energy values are mirrored by the shallow trap states of ca. 0.1 eV (from the conduction band)

obtained from photocurrent quenching [50, 51].

Figure 6.5(c) shows the same trends of carrier lifetime dependence on temperature as our

earlier reported results [7, 8]: carrier lifetime increases with decreasing temperature which is

due to the reduced non-radiative decay rate at low temperatures, a result of decreased phonon-

carrier interactions. Similar values of carrier lifetimes at room temperature can also be found

elsewhere [7, 8, 52-55] in a range from 0.01 µs to 5 µs. Carrier lifetime can be influenced by

many intrinsic QD properties including size, surface ligands, and QD composition. Figures

6.5(d) and (e) show the temperature-dependent carrier trapping rate RT and the thermal

emission rate 𝑒𝑖, respectively. Table 6.1 shows that ei increases from ~104 s-1 at 100 K to ~108

s-1 at 300 K. The activation energy of 106.3 meV, originating from shallow trap states, were

extracted from the Arrhenius plot of 𝑒𝑖 as shown in Fig.6.5(e) and agrees with the activation

energy measured from the hopping diffusivity Dh. At lower temperatures, more carriers are

localized at the excitation sites and the smaller population of phonons freezes these

photogenerated carriers in trap states, which is mirrored by the much lower hopping diffusivity

Dh when compared with the values obtained at room temperature. Therefore, it is reasonable

to conclude that, with the help of phonons at high temperatures, the more widely distributed

carriers are subject to a relatively higher carrier trapping rate, i.e., at high temperature more

CHAPTER 6. 153

trap states are empty which results in an increased RT. Furthermore, due to the higher ambient

thermal energy, the thermal emission rate from the trap states is higher as shown in Fig.6.5(e).

Uing the best-fitted Dh and 𝜏𝐸 values, the hopping diffusion lengths were calculated

through 𝐿ℎ = (𝜏𝐸𝐷ℎ)1/2 , Fig.6.5(f). Although 𝜏𝐸 decreases with temperature, the hopping

diffusion length still increases dramatically from 23 nm to 0.33 µm when the temperature rises

from 100 K to 300 K, because the diffusivity increase is stronger than the lifetime decrease.

Diffusion length is capping-ligand-dependent, for example, at room temperature, diffusion

lengths of PbS CQD thin films treated with different ligands vary widely: with partially fused

PbS CQDs (230 nm) [56], with CdCl2 (80 nm) [24], with ethanethiol (140 nm) [57], and with

3-mercaptopropionic acid (MPA, 100 nm-1000 nm) [58]. Notwithstanding the fact that the

hopping diffusion length can vary as a function of probe method, the Dh values at room

temperature obtained in this study additionally indicate the high photocarrier diffusion ability

of the perovskite photovoltaic material MAPbI3.

6.4.2 Carrier Hopping Activation Energy and Exciton Binding

Energy

40 50 60 70 80 90 100 110 120 130 140

0.1

1

10

100

Best fits

Am

pli

tud

e (m

V)

1/kT (eV-1)

PbS-TBAI

PbS-EDT

PbS-MAPbI3

PbS-MAPbI3-B

Figure 6.6: Temperature scans of the PCR amplitude for different ligands passivated PbS CQD

thin films. The continuous lines are the best fits to each set of data using Eq. (6.20).

CHAPTER 6. 154

Following the temperature-dependent carrier transport dynamics, this section discusses the

trap-state-mediated carrier hoping activation energies and the exciton binding energies. These

properties were studied using PCR photothermal spectra with Arrhenius plot analysis. Figure

6.6 shows the Arrhenius plots of the temperature-dependent PCR amplitude at temperatures

ranging from 90 K to 300 K. Taking into the consideration of both radiative and non-radiative

recombination pathways in PbS CQD thin films, the temperature-dependent dynamic PL (PCR

amplitude) intensity I(T) can be described by the following expression [59-61]

𝐼(𝑇) = 𝐼0

1+∑ 𝐴𝑖𝑒𝑥𝑝(−𝐸𝑖𝑘𝐵𝑇

)𝑖

(6.20)

where 𝐼0 is a normalizing factor, 𝐸𝑖 is the activation energy of the process (i), and 𝐴𝑖 is the

carrier transition rate for process (i). The activation energy is the energy difference between

the original and the final energy states within a carrier transition process. Here, we assume that

our exciton complexes are in the ground state with energy E0 and at least two carrier transition

channels with higher energy states E1 and E2, which should be overcome for the transition

process of excitons to occur. It should be noted that the distribution of excitons in these three

levels is govern by Boltzmann statistics featuring an equilibrium temperature behavior, which

leads to the derivation of Eq. (6.20) with i = 2. Therefore, the activation energy can be

expressed as: Ea,1 = E1- E0 and Ea,2 = E2- E0. The PL emission of PbS CQD thin films in the

entire experimental temperature range cannot be fitted using only one activation energy level

as different carrier dynamic transport processes dominate in different temperature ranges. The

best-fitted curves to the photo-thermal spectra of the four samples using Eq. (6.20), Fig. 6.6,

are the results of two strategies applied for the fitting: first, the entire thermal spectrum was

fitted across the entire temperature range, while the number of activation energy levels was

increased until a satisfactory fit was achieved. For PbS-EDT, PbS-MAPbI3, and PbS-MAPbI3-

CHAPTER 6. 155

B, two activation levels were found to be adequate. When three levels were attempted for these

samples, the third activation energy was identical to one of the first two activation energies.

Compared with other samples, PbS-TBAI exhibits two distinguishable trends in the entire

temperature range, which cannot be accounted for by Eq. (6.20). Therefore, the PbS-TBAI

data were split into two regimes using the dashed line boundary in Fig. 6.6. The two sub-ranges

were fitted separately, using two energy levels (high temperature end) and one energy level

(low temperature end). Second, to investigate the temperature-dependent trap effects on carrier

transport, each spectrum was divided into 5 parts with 100 K, 150 K, 200 K, 250 K, and 300

K being the average temperature (central temperature) in each range, and then each range was

fitted using only one energy level through Eq. (6.20). All the best-fitted activation energies

through these two strategies are summarized in Table 6.2.

Table 6.2: Activation energies at different temperatures for PbS CQD thin films passivated

with various ligands.

Samples

Activation energy fitted in separate temperature

range (one level fitting) (meV)

Activation energy fitted

across the whole

temperature range (meV)

Ea Ea,1 Ea,2 Ea,3

300 K 250 K 200 K 150 K 100 K

PbS-

MAPbI3 275.90 147.51 110.87 98.77 59.55 53.20 147.41

PbS-

MAPbI3-B 190.94 170.23 92.25 95.98 69.96 35.21 107.24

PbS-TBAI 233.81 236.27 77.92 46.45 52.25 51.55 273.94 25.91

PbS-EDT 172.94 148.36 113.63 101.29 59.35 45.21 129.94

CHAPTER 6. 156

Generally, a higher activation energy, accounting for the trap-state-related thermally

activated carrier transition process, dominates in the high-temperature range. In comparison,

at relatively low temperatures, lower activation energies are usually observed, which can be

ascribed to phonon energy [62], exciton binding energy [62-65], and exciton dislocation

binding energy [64]. As shown in Table 6.2, thermal quenching was observed across the entire

temperature range and the activation energy increases with temperature for all samples. The

temperature-dependent activation energy of PbS CQD thin films using the same method was

also reported by Wang et al. [66]. Therefore, as shown in Fig. 6.1(b), Ea,2 is associated with

shallow trap states, and the best-fitted values are close to the activation energies measured

from the carrier hopping diffusivity [Fig. 6.5(b)], and from the thermal emission rate [Fig.

6.5(e)]. As shown in Fig. 6.1 (b), contrary to deep level trap states, which operate as

recombination centers, carriers that are trapped in these shallow trap states do not recombine

but escape from these traps quickly. Nevertheless, it is difficult to identify the source of Ea,1

based on this study. In Table 6.2, with the exception of PbS-TBAI, activation energies Ea at

300 K are higher than either Ea,1 or Ea,2. Moreover, Ea at 100 K is almost equal to Ea,1 (except

for PbS-MAPbI3-B with bigger dot size), which is consistent with the elimination of shallow-

trap-state-related carrier transport processes as contributors to the activation energy Ea,2.

Nonetheless, the Ea,1 process is active throughout the entire temperature range regardless of

the type of surface capping ligands and QD size. Also, the value of Ea,1 is consistent with the

exciton binding energy in a range from 50 to 200 meV for QDs with a diameter of 1-2 nm [67].

In addition, the exciton dissociation occurs in the course of all carrier transport kinetics in our

experimental temperature range. Therefore, it is reasonable to assign Ea,1 to be the exciton

binding energy (Eb) as depicted in Fig. 6.1 (b). It should also be noted from Fig. 6.1 (b) that

CHAPTER 6. 157

the exciton binding energy in quantum confined systems is the energy difference between

exciton transition (optical gap, Eg, opt) and electronic bandgap (Eg), i.e., Eb = Eg - Eg,opt, which

can be approximated through the electron-hole Coulomb interaction [67] and affected by the

material dielectric constant. Electron-hole Coulomb interaction predicts that exciton binding

energy is proportional to 1/R [67], where R is the radius of the QDs. This is consistent with

the much smaller Ea,1 (i.e. Eb, 35.21 meV) of PbS-MAPbI3-B than that of other CQD thin films,

as exciton binding energies for smaller CQDs (PbS-EDT, PbS-TBAI, and PbS- MAPbI3) have

higher values ranging from 45.21 meV to 53.20 meV. This measured exciton binding energy

is similar to the activation energy of ca. 40 meV for exciton dissociation in the PbS CQD solar

cell as reported by Gao et al. [10] and in PbSe QD films measured by Mentzel et al. [68]. As

for PbS-TBAI, the activation energy does not exhibit a monotonic increase with temperature.

The additional activation energy Ea,3 of 25.91 meV might result from many possible

mechanisms, such as exciton delocalization energy [64] from donors or acceptors resulting

from the capping ligand TBAI. The identification of Ea,3 needs further investigation.

The foregoing discussion summarizes that activation energies Ea for CQD thin films arise

from two carrier transition channels except that for PbS-TBAI which has three channels, i.e.

exciton dissociation (Ea,1) and shallow-trap-related thermal activation (Ea,2). Consequently, the

extraction of Ea using Eq.(6.20) at only one activation energy level is subject to the assumption

that, in each temperature range (with a central temperature of 300 K, 250 K, 200 K, 150 K, or

100 K), only one carrier transition process (Ea,1 or Ea,2) is dominant. It should be noted that

deeper lying trap states require higher Ea for carrier transitions, as expected. Therefore,

comparing Ea values at different central temperatures (Table 6.2) with the corresponding Ea,1

and Ea,2 as discussed above, it must be kept in mind that the exciton dissociation process occurs

CHAPTER 6. 158

across the entire temperature range, while the activation energies (Ea,2) for trap-mediated

carrier transitions decrease with decreasing temperature. Furthermore, at the low temperature

of 100 K, Ea for all samples is approximately equal to Ea,1, indicating a negligible contribution

of Ea,2 to the overall activation energy at this temperature. All of these facts point to the

following conclusion: deep-lying trap states dominate carrier transport at higher temperatures,

while shallow trap states control carrier transport at low temperatures, in agreement with [44].

These effects may arise because carrier distributions are localized near their generation sites

at low temperatures due to the low values of Dh.

6.4.3 Ligand- and Size-dependent Carrier Transport Kinetics

Equation (6.18) describes the PCR signals generated by the carrier transport in CQD thin

films. Besides temperature, surface passivation ligands and the QDs geometry are also

substantial factors to carrier transport properties in CQD thin films [1, 3-6, 11, 14]. In addition

to passivating QD unsaturated surface bonds to minimize or eliminate surface trap states,

solution exchange ligands reduce the interdot spacing and enhance the coupling strength

between neighboring QDs. When trap states are not the dominant factors for carrier hopping

transport, smaller interdot spacing, according to Eq. (6.4), results in increased diffusivity.

Figure 6.7 shows the PCR amplitude and phase frequency scans at 100 K for CQD thin films

passivated with four different ligands and the best fits of Eq. (6.18) to each curve. The best-

fitted parameters of carrier transport properties in these CQD thin films are tabulated in Table

6.3. The interdot spacing values of PbS-TBAI and PbS-MAPbI3 CQD thin films measured by

grazing-incidence small-angle scattering (GISAXS) are 3.50 nm and 3.30 nm, respectively

[69]. In addition, Liu et al. [4] calculated the nominal EDT length to be ca. 0.43 nm which

should result in a smaller interdot spacing than the other ligands due to its smaller molecule

CHAPTER 6. 159

size. The small interdot spacing of PbS-EDT allows the increase of hopping diffusivity above

that of MAPbI3 passivated CQD thin films, consistent with Eq. (6.4). The slightly higher

carrier diffusivity of PbS-MAPbI3 than PbS-MAPbI3-B originates in the lower hopping

activation energy of PbS-MAPbI3 (59.55 meV) at 100 K than PbS-MAPbI3-B (69.96 meV)

with shallow, yet deeper lying, trap states, as shown in Table 6.2. On the contrary, at 300 K,

when trap states start to play key roles in carrier hopping transport, PbS-MAPbI3 carriers face

higher transport activation energy (275.90 meV) than PbS-MAPbI3-B carriers (190.94 meV),

Table 6.2, resulting in a smaller Dh (2.41×10-3cm2/s) than PbS-MAPbI3-B (1.80×10-2 cm2/s),

Table 6.4. In addition, the values of activation energies obtained from the photothermal spectra

also explain the slightly higher Dh of PbS-TBAI (Ea = 233.81 meV) than that of PbS-MAPbI3

CQD thin films, as both samples feature significant defect-related states, Fig.6.2, which seem

to become limiting factors of the carrier hopping diffusion at room temperature.

0.01 0.1 1 10 100

0.4

0.6

0.8

1

PbS-MAPbI3-B

PbS-MAPbI3

PbS-TBAI

PbS-EDT

Continuous lines are best fits

No

rmal

ized

Am

pli

tud

e (m

V)

Frequency (kHz)

(a)

0.01 0.1 1 10 100-50

-45

-40

-35

-30

-25

-20

-15

-10

-5

0

(b)

PbS-MAPbI3-B

PbS-MAPbI3

PbS-TBAI

PbS-EDT

Continuous lines are best fitsPh

ase

(deg

ree)

Frequency (kHz)

Figure 6.7:100 K PCR amplitudes (a) and phases (b) of CQD thin films passivated with four

different ligands, and the best fits to each curve using Eq. (6.18).

However, the PCR signal resolution (i.e., phase lag and amplitude decrease) for the PbS-

EDT samples is too low to be resolved and fitted for the extraction of these parameters even

when the modulation frequency is increased to 1 MHz. This is not a limitation inherent in the

CHAPTER 6. 160

PCR technique, but rather a consequence of the inability of our equipment to measure samples

with too poor transport properties.


various ligands. These parameters were evaluated for 100 K measurement.

Samples PbS-EDT PbS-TBAI PbS-MAPbI3-B PbS-

MAPbI3

Hopping

diffusivity

Dh (cm2/s)

1.62×10-6

±2.83×10-8

8.58×10-7

±3.74×10-13

8.82×10-7

±6.07×10-14

1.04×10-6

±3.93×10-13

Effective carrier

lifetime τE (µs)

2.78

±7.13×10-6

3.79

±1.42×10-8

7.66

±3.53×10-8

5.37

±3.68×10-8

Thermal emission

rate

ei (s-1)

6.53×104

±1.16

6.80×104

±0.00086

4.96×104

±0.0010

6.70×104

±0.00059

Trapping rate RT

(s-1)

4.72×104

±0.80

5.24 ×104

±3.71×10-4

2.04×104

±3.43×10-4

2.36×104

±8.99×10-4

Absorption

coefficient

𝛽 (cm-1)

8.57×107

±2.83×106

6.22×106

±1.43

7.80×106

±0.34

2.79×106

±0.56

Generation rate

G0 (cm-3s-1)

1.84×109

±7.22×107

1.81×108

±3.12

1.44×108

±2.50

3.04×107

±0.31

Diffusion length

Lh (μm) 0.017 0.018 0.026 0.024

Interdot spacing

(nm)

0.43 (nominal

ligand length) [4] 3.50 [69] 3.30 [69] 3.30 [69]


various ligands. These parameters were evaluated for 300 K measurement.

Samples PbS-TBAI PbS-MAPbI3-B PbS-MAPbI3

Hopping diffusivity

Dh (cm2/s)

5.09×10-3

±6.96×10-4

1.80×10-2

±2.30×10-3

2.41×10-3

±2.75×10-4

Effective carrier lifetime τE (µs) 0.16

±0.06

0.51

±0.15

0.45

±0.15

Thermal emission rate

𝑒𝑖 (s-1)

7.35×109

±4.93×109

8.33×1010

±5.78×1010

6.993×109

±1.35×109

Diffusion length Lh (μm) 0.29 0.96 0.33

CHAPTER 6. 161

The effective carrier lifetime , 𝜏𝐸 , and the thermal emission rate, ei, at 100 K, are

summarized for all ligands in Table 6.3. Consistently with the foregoing mechanism of the

temperature-dependent carrier hopping lifetime as shown in Table 6.1, values of 𝜏𝐸 for PbS-

TBAI (0.16 µs) and PbS- MAPbI3-B (0.51 µs) at 300 K greatly increase when the temperature

decreases to 100 K (3.79 s and 7.66 s, respectively) as shown in Table 6.3. On the contrary,

the thermal emission rate 𝑒𝑖 was found to significantly increase from ~104 s-1 at 100 K (Table

6.3) for all samples to ~1010 s-1 at 300 K (Table 6.4). Comparing all three samples, PbS-

MAPbI3-B at 300 K exhibits the highest thermal emission rate 𝑒𝑖. Returning to the effective

carrier lifetime 𝜏𝐸, MAPbI3-passivated PbS CQD thin films exhibit longer lifetime at 100 K

than PbS-EDT and PbS-TBAI, while PbS-MAPbI3-B lifetime remains the highest amongst all

the tested samples at both 100 K (Table 6.3) and 300 K (Table 6.4). This is not unexpected

because PbS-MAPbI3-B does not exhibit any defect states induced secondary PL emission

peak as shown in Fig.6.2.

Regarding the calculated diffusion length Lh, both MAPbI3-passivated samples have

similar Lh of ca. 24-26 nm at 100 K, a temperature at which the influence of trap states is not

significant. These values are higher than those of TBAI and EDT treated samples, Table 6.3.

Considering the QD size (~2 nm) and interdot spacing (~3 nm), the short diffusion length

indicates that carriers hop across only a few QDs before statistically recombining. At 300 K,

trap states limit carrier transport which is, nevertheless, assisted by phonon interactions in

overcoming the hopping activation energy. This trade-off between trap-state limitations and

phonon assistance results in longer diffusion lengths Lh at high temperatures. Tables 6.3 and

6.4 show that PbS-MAPbI3-B possesses the longest Lh at both high and low temperatures,

indicating that this material is optimal for solar cell performance improvement.

CHAPTER 6. 162

The carrier trapping rate RT as defined in Sect. 6.2 is proportional to the trap state density

and was measured to be on the order of 104 s-1 at 100 K (Table 6.3). Consistently with the PL

spectra in Fig. 6.2, PbS-MAPbI3, as shown in Table 6.3, was fitted with higher RT than PbS-

MAPbI3-B. In addition, the strong trap peak of the PbS-TBAI spectrum is also consistent with

its higher RT than that of PbS-EDT. MAPbI3-capped PbS CQD thin films exhibit relatively

lower carrier trapping rates than PbS-EDT and PbS-TBAI (Table 6.3) due to the smaller lattice

mismatch between PbS and the MAPbI3 perovskite material [30] which results in better surface

passivation. The absorption coefficient 𝛽 at 100 K (Table 6.3), is on the order of 106 cm-1 for

all samples except PbS-EDT with a slightly higher value of 8.57×107 (± 2.83×106). From the

same table, the exciton generation rate G0 at 100 K is between 3×107 cm-3s-1 and 2×109 cm-3s-

1 for all our PbS CQD thin films.

6.5 Fitting Uniqueness and Reliability – Parameter Extraction

from PCR

PCR can generate independent carrier diffusion-wave amplitude and phase channels

simultaneously from a single frequency scan, both of which can be used for data best fitting to

increase the accuracy and reliability of best-fitted parameters. For the PCR measurements

carried out in this Chapter, at each temperature, 120 points (60 points each for amplitude and

phase) were used in one computational fitting process to extract six to-be-measured parameters:

𝛽, 𝐷ℎ, 𝜏𝐸 , 𝑅𝑇 , 𝑒𝑖 and 𝐺0, from Eq.( 6.18). The expressions for the PCR amplitude and phase can

be derived from Eq. (6.18):

𝐴(𝜔) = √𝑆𝑅2(𝜔) + 𝑆𝐼

2(𝜔) (6.21)

𝜙(𝜔) = 𝑡𝑎𝑛−1 [𝑆𝐼(𝜔)

𝑆𝑅(𝜔)] (6.22)

CHAPTER 6. 163

where 𝑆𝑅(𝜔) and 𝑆𝐼(𝜔) are the real and imaginary parts of the PCR signal 𝑆(𝜔), respectively,

in Eq. (6.18).

Figure 6.8: The determinant of diffusivity Dh (a) and effective carrier lifetime 𝜏𝐸 (b) in the

PCR phase channel. Diamonds indicate frequencies at which linear dependence occurs; no

such linear dependencies were found for the amplitude channel of all parameters. (c) and (d)

are the sensitivity coefficients of 𝜏𝐸 in the amplitude and phase channel, respectively. Besides

the measured parameters in this figure, other parameters were also treated similarly to yield

the best-fitted values for all samples as shown in Tables 6.1, 6.3, and 6.4.

To extract these carrier transport parameters, the energy difference between singlet and

triplet states was first found to be 37.24 meV for PbS CQDs as reported earlier [27]. The best-

fitted parameters for charge carrier dynamics in MAPbI3-passivated CQD thin films with small

dots are tabulated in Table 6.1 as shown in Sect.6.4.1. The Matlab-based computational

CHAPTER 6. 164

program employs the fminsearchbnd solver [70] which minimizes the sum of the squares of

errors between the experimental and calculated data [7, 51].

Furthermore, a parametric theory was also applied to test the uniqueness of our best-fitted

parameters. First, the sensitivity coefficient 𝑓 was defined as the first derivative of function 𝑆

with respect to a specific parameter [71]. According to the measurement theory, the sensitivity

coefficients of the parameters to-be-measured (𝛽, 𝐷ℎ, 𝜏𝐸 , 𝑅𝑇 , 𝑒𝑖 and 𝐺0) can be obtained for

amplitude and phase channel as follows:

𝑓(𝐴) = [𝜕𝐴(𝜔)

𝜕𝛽 𝜕𝐴(𝜔)

𝜕𝐷ℎ 𝜕𝐴(𝜔)

𝜕𝜏𝐸 𝜕𝐴(𝜔)

𝜕𝑅𝑇 𝜕𝐴(𝜔)

𝜕𝑒𝑖 𝜕𝐴(𝜔)

𝜕𝐺0] (6.23)

𝑓(𝜙) = [𝜕𝜙(𝜔)

𝜕𝛽 𝜕𝜙(𝜔)

𝜕𝐷ℎ 𝜕𝜙(𝜔)

𝜕𝜏𝐸 𝜕𝜙(𝜔)

𝜕𝑅𝑇 𝜕𝜙(𝜔)

𝜕𝑒𝑖 𝜕𝜙(𝜔)

𝜕𝐺0] (6.24)

Mathematically, all these parameters should be linearly independent over the range of

observations (frequencies), the number of which should be larger than the volume of unknown

parameters. Taking the amplitude channel as an example, the following relation should be

valid:

𝐶1𝜕𝐴(𝜔)

𝜕𝛽+ 𝐶2

𝜕𝐴(𝜔)

𝜕𝐷ℎ+ 𝐶3

𝜕𝐴(𝜔)

𝜕𝜏𝐸+ 𝐶4

𝜕𝐴(𝜔)

𝜕𝑅𝑇+ 𝐶5

𝜕𝐴(𝜔)

𝜕𝑒𝑖+ 𝐶6

𝜕𝐴(𝜔)

𝜕𝐺0≠ 0 (6.25)

where 𝐶1 − 𝐶6 are coefficients, not all equal to zero. Eq. (6.25) is satisfied if, and only if, the

determinant of the 6×6 matrix of the sensitivity coefficient is not equal to zero, i.e.,

𝑑𝑒𝑡𝐴 =

[ 𝜕𝐴(𝜔1)

𝜕𝛽⋯

𝜕𝐴(𝜔1)

𝜕𝐺0

⋮ ⋱ ⋮𝜕𝐴(𝜔6)

𝜕𝛽⋯

𝜕𝐴(𝜔6)

𝜕𝐺0 ]

≠ 0 (6.26)

Although mathematically, if these six unknown parameters can be determined, measurements

at only six frequency points are sufficient, however, for better computational fitting reliability,

CHAPTER 6. 165

60 frequency points in both amplitude and phase channels were measured in this study. Eqs.

(6.23), (6.24) and (6.26), as well as the expression corresponding to Eq.( 6.26) for the PCR

phase channel, were used and the results showed that when the parameters at each temperature

were assigned the values in Tables 6.1, 6.3, and 6.4 as presented in the previous sections, the

determinants at all experimental frequencies were not zero. For example, as shown in Figs.

6.8(a) and (b) for Dh and 𝜏𝐸, when the range of an estimated parameter value was extended,

the PCR phase channel exhibited some values that do not satisfy linear independence. In

comparison, no such values were found for the PCR amplitude channel. Theoretically, these

results indicate that all six parameters can be resolved in the framework of our equations and

experimental data sets.

In addition to linear independence which indicates whether a parameter can be determined

reliably, the signal response sensitivity to each parameter is another important factor which

shows to what extent the PCR signal will change when the specific parameter value is varied.

To analyze the sensitivities of the measured carrier transport parameters, best-fits for PbS-

MAPbI3 at 100 K were used as the database. For, example, Figs. 6.8(c) and (d) show the

sensitivity coefficients of 𝜏𝐸 in the two channels. In the amplitude channel, larger sensitivity

is found when 𝜏𝐸 is smaller than 4 ns. In the phase channel, the PCR signal is more sensitive

to 𝜏𝐸 at high frequencies. The frequency-dependent parameter sensitivity provides important

clues for optimal experimental measurements, i.e., the proper frequency window can be chosen

through the analysis of parameter sensitivity coefficients for the calculation of specific

parameters with optimal accuracy. However, it should be noted that the sensitivity values of

different parameters are not comparable due to the difference in their units. Furthermore, the

values of other parameters influence the sensitivity coefficient of a specific parameter. From

CHAPTER 6. 166

the sensitivity coefficients in Fig. 6.8 and that of other parameters, it is interesting to see that

some of the parameters have a positive correlation with the PCR signal while others, a negative

correlation.

6.6 Conclusions

This chapter, first, introduces the Instrumentation of PCR system and discusses concepts

of modulated laser excitation, detector signal collection, and lock-in amplifier based signal

computation. The general theory of PCR was also discussed. Second, coupled with the fully

optical non-destructive PCR technique, a novel quantitative methodology was developed to

characterize carrier transport dynamics for QD systems by deriving a trap-state-mediated

carrier hopping transport model. Multiple materials and carrier transport parameters for PbS-

EDT, PbS-TBAI, PbS-MAPbI3, and PbS-MAPbI3-B CQD thin films were measured at

different temperatures. The observed monotonic dependence of effective carrier lifetime 𝜏𝐸,

hopping diffusivity Dh, carrier trapping rate RT, and hopping diffusion length Lh on the

temperature in the range from 100 K to 300 K is consistent with a phonon-assisted carrier

hopping transport mechanism in PbS CQD thin films. For all samples, trap-state-mediated

activation energies were found to be in a range between 100 meV and 280 meV. Photothermal

spectroscopy modified from the PCR system was also used to measure exciton binding

energies as a function of dot size. From PL spectroscopy, it was shown that perovskite

(MAPbI3) passivated thin films with larger dot size (bandgap energy: 1.09 eV) are free of

obvious defect states induced secondary PL emission. These thin films exhibited the highest

carrier lifetime and hoping diffusivity at 300 K, thus proving to be better photovoltaic materials

than PbS-MAPbI3, as well as TBAI, or EDT treated CQD thin films.

CHAPTER 6. 167

The PCR technique sheds lights on the temperature- and ligand-dependent carrier transport

dynamics in photovoltaic CQD thin films, hence, benefiting CQD solar cell efficiency

optimization through a better understanding of device energy dissipation physics and through

quantitative recombination process analysis in CQD surface trap states with a goal to

minimizing their effects through ligand passivation and bandgap energy engineering. The

results of this study can be further applied in directing high-efficiency CQD solar cell

fabrication in conjunction with the development of an improved PCR theory for photovoltaic

devices.

168

Chapter 7

Carrier Recombination Mechanism, Energy Band Structure, and

Inhomogeneity-affected Carrier Transport in Perovskite Shelled

PbS CQD Thin Films Using PCR and HoLIC

7.1 Introduction

Chapter 6 introduces the PCR technique through discussing trap-state-mediated carrier

transport PCR theory, and the dependence of carrier transport on CQD dimensions and

temperatures. However, the recombination mechanisms and band energy structure of CQDs

are not provided. Therefore, this chapter applies the theoretical models and findings discussed

in Chapter 6 to investigate the trap-state-mediated carrier transport mechanisms further.

In specific, charge carrier recombination processes and sub-bandgap energy states in

perovskite passivated CQD thin films for photovoltaic applications are discussed using PL

spectroscopy, excitation power dependent PL intensity, and photocarrier radiometry (PCR)-

based photothermal spectroscopy. Quantitative analysis of carrier transport properties was

carried out through PCR frequency scans. An energy band structure is proposed based on the

above energy states study. It should be noted that the as discussed recombination mechanism

leads to the theoretical explanation of the nonlinear response of radiative recombination to

laser excitation intensity, which is fundamental for HeLIC technique as discussed in Chapter

8. Furthermore, the sample inhomogeneity-associated variation of carrier transport in CQD

thin films is also discussed through the combination of PCR, HoLIC, and HeLIC.

Interpretation of HoLIC and HeLIC imaging contrast is addressed that image amplitudes can

CHAPTER 7. 169

reflect carrier density distribution, which is proportional to defects or trap states associated

effective carrier lifetime.

7.2 Experimental Details and CQD Thin Film Synthesis

The schematic of the LIC imaging setup is shown by Fig.3.9 and discussed in detail in

Sect.3.4. To characterize charge carrier recombination processes and sub-bandgap states

within CQD thin films, excitation laser intensity scans, PCR temperature (photothermal

spectroscopy) scans and laser modulation frequency scans were carried out with a conventional

PCR system as discussed in Sect.3.3. Specifically, incorporated within the PCR system,

photothermal spectra were obtained through linear temperature scans by measuring PCR

signals from the sample at a fixed laser modulation frequency while the temperature was

reduced from 300 K to 100 K at a sufficiently slow (quasi-equilibrium) rate of 5 oC/min using

the Linkam LTS350 cryogenic stage. An average 1 sun laser excitation intensity was used for

the measurements. The PL emission from the sample was collected by a single detector

connected to a lock-in amplifier. Different modulation frequencies were used for the

temperature scans.

The CQD thin films studied in this work are of the lead sulfide (PbS) kind and were surface

passivated with methylammonium lead triiodide perovskite (MAPbI3) to remove CQD surface

defect states and adjust interdot distances. The detailed fabrication process is available in

Sect.4.3.1, and the scanning electron microscopy (SEM) and transmission electron microscopy

(TEM) images of these perovskite (MAPbI3)-passivated CQD thin films can be found in ref.[1].

With excellent surface passivation of perovskite thin shells onto CQD surfaces, the CQD solar

cells exhibited improved open-circuit voltage and power conversion efficiency compared with

those without MAPbI3 treatment or those treated with other ligands [1]. Two CQD thin films

CHAPTER 7. 170

(samples A and B) were fabricated with different free-exciton PL emission peaks for LIC

imaging analysis towards qualitative thin film homogeneity and mechanical defect

characterization for the purpose of CQD solar cell efficiency optimization in lab fabrication

processes.

7.3 Charge Carrier Recombination Mechanism for PbS CQDs:

Nonlinear Response

Figure 7.1: Schematic of PbS QDs in MAPbI3 matrixes, i.e. MAPbI3-passivated PbS QDs. Due

to the slight lattice mismatch, chemical bonds can be satisfied at the interfaces between QD

and MAPbI3, and the unsatisfied bonds act as surface trap states, denoted by red circles.

To a great extent, semiconductor quality is determined by structural defects arising from

dangling bonds on QD surfaces, such as point defects including vacancies and interstitials

performing as very efficient trap states for electrons, holes, and excitons. They exhibit a strong

influence on optical and electrical properties of the host semiconductor materials. In PbS QDs,

potential point defects include Pb-vacancies (VPb), S-vacancies (Vs), Pb-interstitials (Pbi), S-

interstitials (Si) and antisites (PbS and SPb). These possible defects can be introduced during

the material fabrication process, acting as recombination centers. MAPbI3 has been reported

CHAPTER 7. 171

to be an excellent candidate for passivating PbS CQD surface defects because of the minimal

lattice mismatch between the MAPbI3 and PbS, a schematic shown in Fig. 7.1. As a result, no

defect-related peaks were found in PL spectra of MAPbI3-PbS CQDs in solution [1].

Figure 7.2: Near-band-edge photoluminescence via variable radiative and nonradiative

transitions. (a) Free-exciton recombination, (b) and (c) recombination of donor (D)- and

acceptor (A)-bound excitons (DX, AX), (d) donor-acceptor pair recombination (DA), (e)

recombination of a free electron with a neutral acceptor (eA), (f) recombination of a free hole

with a neutral donor (hD).

However, a better understanding of charge carrier recombination processes in CQD thin

films is necessary. As shown in Fig. 7.2, photogenerated excitons, and free electrons and holes

may undergo the following transitions in PbS CQDs: radiative free-exciton recombination

(FE), radiative acceptor- and donor-bound exciton (AX, DX) recombination, nonradiative

donor-acceptor pair (DA) recombination, radiative recombination of a free electron and a

neutral acceptor (eA), and radiative recombination of a free hole and a neutral donor (hD).

These recombination processes have also been encountered in other low-dimensional nano-

systems including ZnO nanocrystals [2], MoS2, MoSe2 and WSe2 monolayers [3], ZnSe

nanowires [4], InGaN/GaN multiple quantum wells [5], InAs/GaAsSb quantum dots [6], and

CHAPTER 7. 172

PbS quantum dots [7]. Unlike semiconductor materials with continuous energy bands, CQDs

exhibit excitonic behavior with photogenerated excitons having much higher binding energy

than that of excitons in semiconductor materials with continuous energy bands, and are

incapable of forming continuous band structures due to the dot size polydispersity and the

energy band disorder. Therefore, excitons are bound together and are not able to separate into

free electrons and holes without the help of external forces including interdot coupling effects

and electric fields [8]. Physical descriptions of processes (I) to (VI) in Fig. 7.2 have been

detailed by Schmidt et al.[9], while for MAPbI3-PbS CQDs, four unique conditions are

considered: first, their n-type conducting property (only for calculation simplification), which,

although not measured in this study, hinges on previous evidence of the n-type conducting

property of CQD-perovskite LEDs [10]; second, exciton recombination dominates the

transitions; third, excitons bound to ionized donors and acceptors can be neglected due to their

weak transition probabilities; and fourth, the excitation laser energy is higher than the CQD

bandgap Eg. Therefore, the rate formulas, adapted from Schmidt et al. [9], can be expressed by

𝑑𝑛

𝑑𝑡= 𝑔𝑃 − 𝑎𝑛2 (7.1)

𝑑𝑛𝐹𝐸

𝑑𝑡= 𝑎𝑛2 + 𝑃 − (

1

𝜏𝐹𝐸+

1

𝜏𝐹𝐸𝑛𝑟) 𝑛𝐹𝐸 − 𝑏𝑛𝐹𝐸𝑁𝐷0 (7.2)

𝑑𝑛𝐷𝑋

𝑑𝑡= 𝑏𝑛𝐹𝐸𝑁𝐷0 − (

1

𝜏𝐷𝑋+

1

𝜏𝐷𝑋𝑛𝑟 ) 𝑛𝐷𝑋 (7.3)

𝑑𝑁𝐷0

𝑑𝑡= 𝑔(𝑁𝐷 − 𝑁𝐷0)𝑛 − 𝑚𝑁𝐷0𝑃 − 𝑏𝑛𝐹𝐸𝑁𝐷0 + (

1

𝜏𝐷𝑋+

1

𝜏𝐷𝑋𝑛𝑟 ) 𝑛𝐷𝑋 − 𝑓𝑁𝐷0𝑛 (7.4)

where 𝑃 is the excitation laser intensity, 𝜏𝐹𝐸 and 𝜏𝐹𝐸𝑛𝑟 are the radiative and nonradiative

lifetimes of free excitons, respectively, and 𝜏𝐷𝑋 and 𝜏𝐷𝑋𝑛𝑟 are the respective radiative and

nonradiative lifetimes of donor bound excitons undergoing transition DX. n, 𝑛𝐹𝐸 , and 𝑛𝐷𝑋 are

the concentrations of free electrons, free excitons, and donor bound excitons, respectively.

Furthermore, 𝑁𝐷 and 𝑁𝐷𝑜 are the concentrations of donors and neutral donors, respectively.

CHAPTER 7. 173

𝑎, 𝑏, … , 𝑓 are the coefficients associated with the processes shown in Fig. 7.2, while 𝑔 and 𝑚

are coefficients for exciton generation and electron excitation, respectively. Solving Eqs. (7.2)

and (7.3) in the steady state, yields the luminescence intensities of free and bound excitons,

IFE, and IDX defined as

𝐼𝐹𝐸 ∝𝑛𝐹𝐸

𝜏𝐹𝐸=

𝐵

𝜏𝐹𝐸𝑛2 (7.5)

𝐼𝐷𝑋 ∝𝑏𝑁𝐷𝐵

1+𝜏𝐷𝑋𝜏𝐷𝑋𝑛𝑟𝑛2 (7.6)

where

𝐵 =𝑎

(1

𝜏𝐹𝐸+

1

𝜏𝐹𝐸𝑛𝑟)+𝑏𝑁𝐷

(7.7)

Provided the probabilities of free-to-bound transitions be proportional to the respective

transition rates, luminescence intensities of free-to-bound (donors) transitions, 𝐼ℎ𝐷 can be

expressed by

𝐼ℎ𝐷 ∝ 𝑛 𝑁𝐷0 (7.8)

Solving Eq. (7.1) in the steady state, yields 𝑛 ∝ 𝑃0.5. Using this relationship in Eqs. (7.5) to

(7.8), it is found that 𝐼 ∝ 𝑃 for excitonic transitions and 𝐼 ∝ 𝑃0.5 for free-to-bound transitions.

Therefore, the γ value can be used to characterize the type of a radiative charge carrier

recombination process. Experimentally, γ is generally measured to be between 1 and 2 for

excitonic emissions including transitions (a)-(c) in Fig. 7.2, and less than 1 for free-to-bound

(acceptors/donors) emissions as shown by transitions (d)-(f). For example, γ = 0.69 for sub-

bandgap recombinations and γ = 1.48 for band-edge associated recombinations for PbS-TBAI

QD/PbS-EDT QD devices [7], in which TBAI and EDT denote tetrabutylammonium iodide

CHAPTER 7. 174

and 1,2-ethanedithiol, respectively. Therefore, the value of γ can be used to physically interpret

the linear or nonlinear behavior of a charge carrier recombination process with respect to the

laser excitation and reveals the underlying radiative recombination process. It is an important

parameter for the HeLIC process, because nonlinear PL responses to the laser excitation, acting

as a nonlinear frequency mixer, are essential for PL based heterodyne LIC imaging [11-14].

As opposed to CQD thin films, excitons in polycrystalline and amorphous Si wafers, upon

their formation, immediately dissociate into free electrons and holes, with the dominant

recombination process being via defect states. In this situation, the nonlinear recombination

term 𝑎𝑛2 in the rate equation 𝑑𝑛

𝑑𝑡= 𝑔𝑃 − 𝑎𝑛2 − ℎ𝑛(𝑁𝐷 −𝑁𝐷0) − 𝑒𝑛𝑁𝐴0 (ℎ is the coefficient

for nonradiative transitions of free electrons to ionized donors, and 𝑁𝐴0 is the concentration of

natural acceptors) [9], can be neglected, resulting in 𝑛 ∝ 𝑃. Solving Eqs. (7.5) - (7.8) using the

relationship 𝑛 ∝ 𝑃, it follows that 𝐼 ∝ 𝑃2 for excitonic transitions and 𝐼 ∝ 𝑃 for free-to-bound

transitions. This conclusion is consistent with experimental LIC results for Si wafers with

exponent γ >1 [13].

Figure 7.3: PCR amplitude vs. excitation power at three different temperatures for sample A

(a) and sample B (b), at 10 kHz laser modulation frequency.

CHAPTER 7. 175

To identify charge carrier transition types of various radiative recombination processes for

the study of non-linear PCR responses, Fig. 7.3 shows the excitation power dependent PCR

signals of sample A and B at three temperatures (300 K, 200 K, and 100 K). Based on the

foregoing discussion of radiative recombination processes, experimental data were fitted

to 𝐼 ∝ 𝑃𝛾. At room temperature (300 K), the γ value of sample A was 0.86 which is close to,

but less than, unity and is indicative of donor/acceptor-related free-to-bound recombinations.

However, free-exciton-like transitions also exist. By comparison, the only dominant emission

peak (i.e. the excitonic recombination) of sample B results in a γ value of 0.94, and the small

deviation from unity is probably due to the very few sub-bandgap trap states involved in non-

radiative exciton trapping which compromises only slightly the strength of the PCR amplitude,

much less so than that of trap rich sample A. Both samples exhibited reduced γ values when

temperature decreased. The values of γ at 100 K for both samples are close to 0.7 which is

likely indicative of the onset of free-to-bound transitions (γ=0.5) starting to dominate the

radiative emission processes as opposed to free-exciton-like emissions (γ=1). This can be

explained by the following typical behavior of exchange coupled excitons. The presence of

bright and dark states can be attributed to the nonstoichiometry of QD surfaces [15], lattice

mismatch during ligand exchange processes, and incomplete surface bond termination or

chemical changes [16]. Although the presence of these exciton bright and dark states has not

been proven directly, evidence of the presence of these states is accumulating [17-20]. De

Lamaestre et al. [17] studied the temperature dependent PL intensity and decay rates of PbS

nanocrystals in a silicate glass and found a large energy splitting (ca. 30 meV) of the exciton

ground state fine structure which showed evidence for the existence of a triplet state. Nordin

et al. [18] has observed the PL emission from two active states with an energy separation of

CHAPTER 7. 176

ca. 6 meV that is close to the theoretically reported energy difference between triplet and

singlet states, ca. 10 meV. Gaponenko et al. [19] provided another piece of evidence using

steady-state and time-resolved PL, and developed a theoretical energy-level model considering

the lowest 1S-1S exciton state splitting that can present a consistent quantitative description of

experimental results. Gao et al. [20] discussed charge trapping in bright and dark states of

coupled PbS quantum dot films with analysis of temperature-dependent PL from dots of

different sizes or different surface passivation. Considering the state splitting, although the

physics behind the γ exponent decrease with decreasing temperature is not well understood, it

is well known that triplet exciton states with spin 1 (parallel spins) are energetically more

stable than singlet exciton states with spin 0 (antiparallel spins). Furthermore, triplet states

have a higher statistical weight of 3 (allowed values of spin components:-1, 0 and 1) than that

of singlet states (statistical weight = 1, spin component 0). Consequently, at low temperatures,

most excitons condense into triplet states, from which they cannot decay radiatively to the S =

0 ground state. However, triplet excitons have longer lifetimes and higher probability for

nonradiative recombination than deeper dark states. As a result, the contribution of excitonic

recombinations to dynamic PL (measured by PCR) decreases with reduced temperature in

favor of nonradiative triplet recombinations. It is concluded that the presence of large densities

of trap states can reverse the non-radiative recombination suppression rate at low temperatures

previously observed due to decreased phonon populations in PbS CQD thin films [21]. Instead,

enhanced nonradiative recombinations reduce the radiative emission, an effect which may

severely compromise the solar efficiency of photovoltaic solar cells fabricated using this type

of CQDs.

7.4 Energy Band Structure

CHAPTER 7. 177

7.4.1 Photoluminescence of CQD Thin Films

Figure 7.4 (a) shows PL spectra of perovskite MAPbI3-passivated PbS CQD thin films,

samples A and B with different dot sizes. The measurements were performed at room

temperature using an excitation laser wavelength of 375 nm, and two dominant radiative PL

emission peaks located at 1169 nm and 1553 nm were observed for Sample A. The 1169 nm

PL emission peak corresponds to the band-edge excitonic recombination (PbS is a direct

bandgap semiconductor) including transitions a and b/c, Fig. 7.2, but is dominated by the free

exciton recombination (transition a) at room temperature [22]. The 1553 nm emission peak

originates from recombinations that occur through donors/acceptors (transitions e/f, Fig. 7.2)

arising from unpassivated surface states, structural defects or other chemical changes induced

during ligand exchange processes. In contrast, only the band-edge excitonic recombination

peak (1232 nm) is observed for sample B. When compared with PL spectra of other CQD thin

films with different QD sizes, Fig. 7.4 (b), although the CQD and thin film synthesis processes

are the same, it is found that these PL peaks are quantum dot size dependent, indicating a

complexity of surface passivation mechanism which requires further investigation.

Figure 7.4: (a) Photoluminescence (PL) spectra of MAPbI3-passivated PbS (MAPbI3−PbS)

thin films (samples A and B) spin-coated on glass substrates. (b) PL spectra of MAPbI3−PbS

thin films fabricated through the same process as that of samples A and B but with different

QD sizes.

CHAPTER 7. 178

The surfaces of CQDs typically contain a lot of recombination centers because of the abrupt

termination of the semiconductor crystal periodicity even within a QD. The absence of sub-

bandgap-state-related emissions implies better surface passivation of sample B than A,

revealing the inevitable diversity of the lab fabrication processes. It is also reasonable that

sample A, with a smaller dot size (due to its wider bandgap from PL characterization) that

leads to higher surface-to-volume ratio, incorporates more trap states, making itself a more

difficult candidate for surface passivation. From Fig. 7.4 (a), energy bandgaps of samples A

and B were calculated to be ca. 1.1 eV and 1.0 eV, respectively. The emission from sub-

bandgap states in sample A is ca. 0.3 eV lower than the band-edge emissions. In Fig. 7.4 (a),

sample A has a smaller FWHM (full width at half maximum) than sample B, reflecting a

narrower quantum dot size distribution, as the broadening of PL peaks arises from the quantum

dot size polydispersity with specific spectral components originating from dots of specific

sizes. It should be noted that with the use of a long-pass filter, PL emission detected by the

InGaAs camera is in a wavelength range from 1000 nm to 1700 nm.

7.4.2 PCR Photothermal Spectra of CQD Thin Films

Further investigation of sub-bandgap states using PCR photothermal spectroscopy is

presented in Fig.7.5. From Figs. 7.5 (a) and (b), corresponding to samples A and B,

respectively, it is seen that sample A exhibits more and deeper troughs at all frequencies.

Furthermore, these troughs shift toward higher temperature with increasing modulation

frequency of the excitation laser which can be attributed to three dominant sub-bandgap (trap

state) levels: (I), (II) and (III). Thermal emission rates of carriers from trap states are

temperature dependent processes. When the emission rate matches (or is resonant with) the

modulation frequency (a process called “rate window”), a dynamic photoluminescence (PCR

CHAPTER 7. 179

signal) enhancement occurs due to the increased number of free de-trapped carriers

recombining radiatively, exhibiting a peak in the amplitude of the photo-thermal spectrum at

the resonance temperature. Amplitude peaks and phase troughs follow opposite trends, i.e.

when there is an amplitude peak in the amplitude spectrum, a trough forms in a corresponding

phase spectrum. The phase signal is more sensitive than the amplitude channel which can be

influenced easily by, for example, sample surface reflections and shallow surface states. The

phase locks on the rate of the photo-thermal emissions only and is little or not sensitive to

factors complicating the amplitude spectrum. For an optoelectronic sample with one or more

traps like the one in Fig. 7.5(a), the phase trough associated with a trap state characterized by

a fixed carrier de-trapping energy Ea, shifts to higher temperatures with increasing modulation

frequency. This is caused by the change in the rate-window resonance condition between the

faster thermal ejection rate of trapped carriers at the higher temperature [a Boltzmann factor,

Eq. (7.9)] and the modulation frequency, which is now satisfied at a higher frequency [23, 24].

The activation energy of sub-bandgap trap levels in sample A can be calculated through

Arrhenius-plot fitting of the photo-thermal emission rate, 𝑒𝑛 [23]:

𝑒𝑛(𝑇) = 𝑅𝑛𝜎𝑛𝑒𝑥𝑝 (−𝐸𝑎

𝑘𝑇) (7.9)

Here 𝑅𝑛 is a material constant and 𝜎𝑛 is the exciton capture cross section. At each trough,

𝑒𝑛(𝑇𝑡𝑟𝑜𝑢𝑔ℎ) = 2.869𝑓, in which 𝑓 is the pulse repetition frequency [24]. Figure 7.5(c) shows

the best-fitted activation energies for three dominant sub-bandgap levels in sample A. These

sub-bandgap trap levels exhibit similar activation energies ranging from 33.8 meV to 40.7

meV. Associated with the colloidal environment and surface states in PbS QDs, these shallow

level multi-energetic traps can capture excitons which undergo nonradiative recombinations

CHAPTER 7. 180

or phonon-mediated radiative emissions [21]. Activation energies for trap states in EDT

passivated PbS QDs have also been measured by Bozyigit et al. [25] using thermal admittance

spectroscopy to be around 100 meV for QDs with an energy bandgap of 1.1 eV. Activation

energy differences mostly originate from different exchange-ligands. Chuang et al. [7] also

reported a ~230 meV energy difference (activation energy) between band-edge and sub-

bandgap state emission using PL, which is close to ~ 260 meV following the model proposed

by reference [25] for an EDT treated PbS CQDs with an energy bandgap of 1.1 eV.

Figure 7.5: Photocarrier radiometry (PCR) photothermal spectra of MAPbI3-passivated PbS

(MAPbI3-PbS) thin films spin-coated on glass substrates, samples A (a) and B (b). (c)

Arrhenius plots of the PCR phase troughs, I, II, and III, as shown in (a), and best-fitted to Eq.

(7.9) for the extraction of activation energies for each sub-bandgap trap level.

As shown in Fig. 7.6, which incorporates information from PL spectroscopy and photo-

thermal temperature scans, there are two trap levels located in the PbS QD bandgap: a deep

level (𝐸𝑎 = 0.3 eV) and shallow levels (𝐸𝑎 in a range from 33.8 meV to 40.7 meV). As shown

in Fig. 7.6, when excitons are free roaming they may experience radiative recombination,

become captured in trap states or diffuse to the next QD through nearest-neighbor-hopping

(NNH). The coupling strength between two QDs induces exciton dissociation into free charge

carriers. Trapped excitons require overcoming an activation energy barrier to become de-

trapped and undergo radiative recombinations. For shallow states, the activation energy can be

provided by thermal energy, therefore, PCR photothermal spectroscopy with a lock-in rate

CHAPTER 7. 181

window can reveal shallow trap states. With regard to sample B, Fig. 7.5(b) shows that the

PCR thermal spectra are much smoother at all frequencies, indicative of very few sub-bandgap

states in this sample, and consistent with the PL spectroscopy results, Fig. 7.4 (a).

Figure 7.6: (a) Schematic of PbS QDs in MAPbI3 matrixes, i.e. MAPbI3-passivated PbS QDs.

Due to the slight lattice mismatch, chemical bonds can be satisfied at the interfaces between

QD and MAPbI3, and the unsatisfied bonds act as surface trap states, denoted by red circles

(a); (b) energy band structure (assumed n-type) of a PbS- MAPbI3 nanolayer, sample A,

featuring shallow and deep level trap states. Excitons are excited in the right QD and diffuse

through nearest-neighbor-hopping (NNH) to the next QD, during which process the coupling

strength between two QDs dissociate excitons into free charge carriers. Carriers may

experience radiative recombination or captured by different types of trap states, where non-

radiative recombination or de-trapping may occur.

7.5 Large-area Imaging and Carrier Transport of CQD Thin Films

7.5.1 Qualitative Large-area Imaging

Figure 7.7 (a) and (b) presents photos of MAPbI3-passivated PbS thin films. The thin films

were spin-coated on glass substrates with an area of 25×25 mm2 and stored together in a

nitrogen environment for further study. It is observed that these thin films are visually

homogeneous with few visible imperfections. From a series of CQD thin films with different

QD size, we have experimentally observed that the CQD thin film color changes slightly with

QD size. This is consistent with the expected change in quantum confinement which affects

the optical absorption coefficient of these quantum dots, thereby accounting for the slight color

CHAPTER 7. 182

difference between Figs. 7.7 (a) and (b). In contrast, homodyne and heterodyne LIC images

shown in Fig.7.7 (c) - (f) illustrates significant degrees of inhomogeneity.

Figure 7.7: Photos of MAPbI3 -PbS thin films, (a) sample A and (b) sample B. 1 kHz homodyne

LIC amplitude images of MAPbI3- PbS thin films, (c) sample A and (d) sample B. 20 kHz

heterodyne LIC amplitude images of MAPbI3-PbS thin films, (e) sample A and (f) sample B.

Note the very different signal strength scales associated with the two samples.

The physical origins of the LIC spatial contrast are due to the free photocarrier density

diffusion-wave distributions, which depend on charge carrier transport parameters, mainly the

effective exciton lifetime as well as the hopping diffusivity, de-trapping time, and trap state

density as shown in Sect.6.4. Specifically, large amplitudes and phase lags correspond to high

photocarrier density, a result of long local carrier lifetimes; however, for regions associated

with mechanical damages or intrinsic material defects, the lower amplitude is generally

expected because defects lead to a significant increase of nonradiative recombination rates

resulting in a reduction of carrier lifetimes. Comparison between regions 2 and 3 in Fig.7.7(d)

CHAPTER 7. 183

provides a direct example of the LIC image contrast arising from different photocarrier

diffusion-wave distributions, with region 3 being indicative of longer carrier lifetimes. This

may be due to different spin-coating and ligand passivation processes or unexpected surface

chemical reactions upon exposure to ambient air.

Regarding mechanical-damage-induced defects, as shown in Fig.7.7 (f), the area of a

scratched letter B on the front surface of sample B exhibits lower PCR amplitude values

compared with its neighboring regions. This is attributed to the damage-induced lower

photocarrier density diffusion-wave, leading to the enhanced probability of nonradiative

recombinations into defect states, thus shorter lifetimes. It should be noted that the scratch was

produced after the homodyne image of sample B was obtained and shown in Fig. 7.7(d). For

both A and B, homodyne and heterodyne LIC amplitude images show prominent

inhomogeneities in the charge carrier density distributions.

7.5.2 PCR Characterization of Carrier Transport Parameters

The spatial resolution of LIC images is determined by the ac hopping diffusion length

through

𝐿(𝜔) = √ 𝐷ℎ𝜏𝐸

1+𝑖𝜔𝜏𝐸 (7.10)

Shorter 𝐿(𝜔) yields higher spatial resolution that can be achieved through increasing the

angular modulation frequency 𝜔. Limited by the camera frame rate and exposure time, the

highest frequency achieved in this study with homodyne lock-in carrierography is 1 kHz for

high-quality images. However, better image resolution relies on higher laser modulation

frequencies [12-14, 26, 27], so heterodyne imaging was performed at 20 kHz. Higher

frequency imaging is limited by the low radiative emissions of our samples but can be attained

by using laser sources with higher power. Compared with Fig.7.7 (c), Fig.7.7 (e) exhibits

CHAPTER 7. 184

higher spatial resolution with more detailed features of charge carrier density wave

distributions. As for sample B, Fig.7.7 (f) with a laser frequency of 20 kHz exhibits limited

image resolution improvement when compared with Fig.7.7 (d). This is probably due to the

longer effective exciton lifetimes of sample B than A, as shown in Table 7.1 through PCR

frequency characterization and computational best-fits, leading to a much smaller change of

the longer hopping diffusion length 𝐿(𝜔), Eq. (7.10).

Figure 7.8: Phase diagram of PCR frequency scans in three different regions 1-3 as shown in

Fig.7.7 (c)-(d) and the best-fits of experimental data to Eq. (6.18) in Sect. 6.2.

Table 7.1: Summary of best-fitted parameters using Eq. (6.18) in Sect. 6.2.

Parameters Region 1

(Sample A)

Region 2

(Sample B)

Region 3

(Sample B)

Dh (cm2/s) 0.0181± 0.0061 0.0106 ± 0.0049 0.00870± 0.00437

τE (μs) 0.43 ± 0.10 1.56 ± 0.48 2.05 ± 0.20

ΔE (meV) 25.3 ± 3.1 25.4 ± 3.6 25.5± 3.2

τi (ns) 1.33 ± 0.62 0.75 ± 0.45 2.92 ± 0.78

NT (×1013 s-1) 20.05±2.30 2.99 ± 0.35 3.18 ± 0.38

(×107 cm-1) 4.55± 0.87 1.58 ± 0.49 1.30 ± 0.53

Go (×1025 cm-2/s) 4.85± 0.69 4.94 ± 0.72 5.03 ± 0.70

CHAPTER 7. 185

To investigate exciton transport parameters and material properties of MAPbI3- passivated

PbS CQD thin films, PCR frequency scans, and theoretical best-fits were performed. Sample

surface spots were selected as shown in Fig.7.7 (c)-(d) from regions 1 to 3. Equation (6.18) in

Sect.6.2 was used to fit the experimental PCR data for the extraction of parameters involved.

Figure 7.8 presents the PCR frequency scans of the three regions as well as the best-fits of each

curve. Region 1 exhibits a smaller phase lag than regions 2 and 3, indicating shorter lifetime

[21, 28], while region 3 exhibits slightly larger phase lag than region 2, consistent with the

slightly longer lifetime of region 3.

The best fitting program was implemented through a fminsearch solver minimizing the

square sum of errors between the experimental and calculated data. To establish the uniqueness

and reliability of our measurements, different starting points were generated automatically by

the solver for each fitting process, so that the best-fitted results fluctuated about their mean

values. The fitting procedure was repeated several hundred times and 100 sets of results with

the smallest variance were selected for statistical calculation as tabulated in Table 7.1. The

table summarizes the best-fitted carrier transport and material property parameters including

Dh, 𝜏𝐸, ΔE, 𝜏𝑖, 𝑁𝑇, , and Go for the three regions, providing room temperature measurements.

The CQD thickness of 200 nm was measured by scanning electron microscopy. Region 3

exhibits the highest effective exciton lifetime of 2.05 ± 0.20 μs, consistent with the LIC image

results. The relatively long lifetime could be the effect of dielectric screening similar to that

observed in other IV-VI semiconductor nanocrystals [29]. Similar theoretical results of exciton

lifetimes have also been reported to be in a range between 1 and 3 μs for PbS CQDs and PbSe

CQDs [30]. Additional same range experimental data have been reported in the literature from

time-resolved PL spectroscopy: between 4.97 and 2.74 μs for small PbSe CQDs with diameter

CHAPTER 7. 186

from 2.7 to 4.7 nm) [31]; 1 μs for oleic-acid-capped PbS CQDs [29]; and 2.5 – 0.9 μs for PbSe

CQDs, 3.2 -4.3 nm in diameter [32]. Furthermore, both our A and B samples exhibit Dh values

on the order of 10-2 cm2/s. Comparable results using transient PL spectroscopy for 3-

mercaptopropionic acid (MPA) ligands or 8-mercaptooctanoic acid (MOA) linked PbS CQDs

were also reported elsewhere [33]. Those PbS QDs were doped with metal nanoparticles to

introduce fixed exciton dissociation distances away from the location of their formation. The

best-fitted separation energy ΔE between singlet and triplet states is ca. 25 meV, which

indirectly provides another piece of evidence for the existence of dark and bright states in PbS

CQDs. Manifestations of the existence of these states in PbS CQD thin films with similar

separation energies were also reported elsewhere [21]. Consistent with the PL spectra, Fig. 7.4

(a), Table 7.1 exhibits that region 1 has the highest carrier trapping rate 𝑁𝑇, indicating the

highest trap state density amongst all three regions. Table 7.1 further presents the de-trapping

lifetimes, i, reflecting the time an exciton resides in a trap state before being released. The de-

trapping activation energies obtained from Fig. 7.5 (c) are tentatively attributed to phonon-

mediated photo-thermal interactions and the associated interface trap states appear in the

energy diagram of Fig. 7.6 (b). Room temperature NNH and the carrier diffusivity Dh are

determined by the interdot coupling strength. The absorption coefficient 𝛽 was evaluated

through best-fitting to be on the order of 107 cm-1 for the 800 nm excitation. For comparison,

an absorption coefficient on the order of 105 cm-1 for excited oleic acid-capped PbS CQDs

suspended in tetrachloroethylene (C2Cl4) [30] was measured using a UV-vis-NIR

spectrophotometer. Furthermore, using the same method, the PbSe CQDs bandgap absorption

coefficient was found to be ~ 106 cm-1 and decreased with the dot size [34]. It is hypothesized

that these large differences may arise from different surface passivation ligands, dot

CHAPTER 7. 187

dimensions, and sample state (solvent/solid). Finally, the exciton generation rate Go is an

excitation source determined parameter which essentially remains constant across the three

regions 1-3, as expected.

Apart from fitting uniqueness, sample stability is also essential for measurement accuracy,

hence, sample stability was examined under laser excitation for the duration of one complete

PCR frequency scan. Figure 7.9 illustrates the sample A PCR phase time dependence measured

at 100 kHz over 25 minutes. It exhibits a standard deviation of only 0.12o, which allows

concluding that the PCR phase is not time dependent and the excitation laser has negligible

influence on CQD thin films.

Figure 7.9: PCR phase dependence on time over 25 minutes, the duration of a PCR frequency

scan. Sample A at 100 kHz laser modulation frequency.

Coupled with the theoretical best-fits, a conclusion can be reached that in Fig.7.6, excitons,

undergo recombination, or dissociate into free charge carriers which can recombine radiatively

CHAPTER 7. 188

through donors/acceptors, or through trap state-induced nonradiative transitions, giving rise to

non-linear heterodyne LIC image responses to laser excitation. The combined PCR, LIC

imaging and photothermal temperature scans of perovskite-shelled PbS CQD thin films were

shown to yield quantitative information about key exciton transport processes like effective

lifetimes and other hopping transport parameters extracted from the theoretical exciton

diffusion-wave density trap model as discussed in Sect.6.2.

7.6 Conclusions

High-frequency InGaAs-camera-based HoLIC and HeLIC images of CQD thin films, as

well as temperature scanned photothermal emission rates, activation energies, and trap

densities were obtained to qualitatively characterize CQD nanolayer properties. It was

demonstrated that a MAPbI3-shelled PbS CQD thin film exhibits non-linear PCR signal

response that acts as an effective frequency mixer giving rise to heterodyne LIC images,

originating from free-to-bound and trap-state associated recombination. Furthermore,

quantitative analysis of exciton transport processes using PCR frequency scans yielded carrier

transport parameters including effective exciton lifetimes and diffusivities of MAPbI3-

passivated CQD. Combined with LIC imaging, PCR frequency scans, and photothermal

temperature scans can provide fast, quantitative, contactless, nondestructive evaluation of

charge carrier transport as well as material properties of CQD materials and electronic devices.

This combined analytical methodology can be used for improved control of PbS CQD solar

cell fabrication and performance/efficiency optimization.

189

Chapter 8

Heterodyne and Homodyne Lock-in Carrierography Imaging of

Carrier Transport in CQD Solar Cells

8.1 Introduction

With intensive researches on device architecture engineering [1-3], surface materials

chemistry [4-6], synthesis methodologies [7, 8], charge carrier dynamics [9-11], and

theoretical modeling [12-14], CQD solar cell power to electricity conversion efficiency has

boosted from only 3 % to today’s 13.4 % in a short 7-year period [15]. However, as discussed

in Sects. 5.1 and 5.6, conventional small-spot (< 0.1 cm2) testing, arises seriously questionable

overall CQD solar PCE and stability on a large-scale. Therefore, nowadays, large-area

photovoltaic solar cells prevail, the characterization of which fulfills various purposes

including shading effects, fundamental carrier transport dynamics, and mechanical and

electrical defect evaluation. Therefore, large-area characterization methodologies are needed

for CQD solar cell efficiency optimization. Spatially resolved photoluminescence (PL) and

electroluminescence (EL) constitute powerful methodologies for the characterization of silicon

wafers [16-18] and solar cells [18-21]. They yield measurements of minority carrier hopping

lifetime [16, 17, 20], open-circuit voltage [18, 20, 21], current density [21], series resistance

[21], fill factor [18, 21], and quality monitoring in different device fabrication steps [19].

Furthermore, due to the high signal-to-noise ratio (SNR), synchronous frequency-domain

imaging methodologies are emerging, such as the lock-in thermography (LIT) [22] that has

been used for determining series resistance and recombination current at a frequency of 20 Hz.

However, static (dc) PL and EL, as well as low-frequency LIT that limited by the low camera

CHAPTER 8. 190

frame rate, cannot monitor electronic transport kinetics and recombination dynamics. The

latter, however, are key parameters for the determination of photovoltaic energy conversion

and dissipation.

To address these critical issues, this chapter discusses large-area HoLIC qualitative

carrier distribution imaging of CQD solar cells, which was mostly used for material or device

homogeneity and quality estimation. Furthermore, what is addressed is the development of

large-area HeLIC quantitative imaging of carrier transport parameters in these CQD solar cells.

HoLIC imaging and PCR (discussed in Chapters 3 and 6) are the same type of dynamic

spectrally gated frequency-domain photoluminescence modality and can yield quantitative

information about carrier transport dynamics with accuracy and precision superior to the time-

resolved PL due to their intrinsically high signal-to-noise ratio (SNR) by virtue of lock-in

demodulation [23, 24]. HoLIC has advantages as an all-optical non-destructive imaging

technique for large-area photovoltaic device imaging, yet it is limited to the low modulation

frequency range (<1 kHz) due to the low frame rates of even the state-of-the-art conventional

cameras [9]. Using a single InGaAs detector, PCR can attain high-frequency characterization

(> 100 kHz), however, the fast large-area imaging capability is compromised. Through creating

a slow enough beat frequency component, HeLIC overcomes the high-frequency limitation of

conventional camera-based optical characterization techniques and the poor SNR at short

exposure times associated with high frame rates [25]. Therefore, with higher SNR than dc

photoluminescence (PL) imaging, HeLIC can attain a wide range of frequency-dependent ac

carrier diffusion lengths to generate depth-selective/resolved high-frequency imaging of

carrier transport parameters in large-scale devices.

CHAPTER 8. 191

Thereby, to study carrier transport dynamics in CQD solar cells and the effects of CQD

layer inhomogeneity (for example, induced at various fabrication stages) and contact/film

interface effects on solar cell performance, This chapter, for the first time, develops a large-

area quantitative characterization methodology through combining a J-V model with PCR and

HeLIC to quantitatively produce carrier lifetime, diffusivity, and drift and diffusion length

images for a high-efficiency CQD solar cell under frequency modulated excitation. The

proposed methodology overcomes the limitations of small-dot testing, providing a fast,

contactless, and kinetic property characterization technique that is also suitable for in-line solar

cell quality monitoring in the industrial photovoltaic manufacturing process.

8.2 Theories of Homodyne and Heterodyne Lock-in

Carrierography

Section 3.4.1 discusses the instrumental setups and various signal processing techniques

that used in HoLIC and HeLIC. With the demonstration of nonlinear response from CQD solar

cells (Sect.3.4.2), this section will quantitatively describe the working principles of HoLIC and

HeLIC.

The single detector based PCR, and the InGaAs camera based HoLIC and HeLIC methods

collect photons from radiative recombination of charge CDWs or excitons in CQD

photovoltaic materials and devices. The recombination rate (RR) is proportional to the product

of the concentration of electrons n and holes p in the form of RR = knp [26], in which k is a

material-dependent constant which can be obtained from the semiconductor’s absorption

coefficient. Using the CDW ∆𝑁(𝜔, 𝑥) in the frequency domain, the homodyne and heterodyne

CHAPTER 8. 192

signals can be modeled through a depth integral of the radiative recombining free photocarrier

densities:

𝑆(𝑡) = ∫ 𝑑𝑥 ∫ ∆𝑁(𝜔, 𝑥)[∆𝑃(𝜔, 𝑥) + 𝑁𝐴] 𝐹(𝜆)𝑑𝜆𝜆2

𝜆1

𝑑

0 (8.1)

where ∆𝑃(𝜔, 𝑥) is the excess hole CDW and is equal to ∆𝑁(𝜔, 𝑥) according to the quasi-

neutrality approximation, i.e. the photogenerated excess electron and hole concentrations are

identical across the thin film thickness. 𝐹(𝜆) is an instrumental coefficient that depends on the

spectral detection bandwidth (𝜆1, 𝜆2) of the near-infrared detector. 𝑁𝐴 is the equilibrium

majority carrier concentration determined by material doping resulting from in-air oxidation

of our CQD thin films and solar cells. For single frequency modulation, the excess carrier

density waves can be expressed as

∆𝑁(𝑥, 𝜔) = ∆𝑛0(𝑥) + 𝐴(𝑥, 𝜔)𝑐𝑜𝑠[𝜔𝑡 + 𝜑(𝑥, 𝜔)] (8.2)

Here ∆𝑛0(𝑥) , 𝐴(𝑥, 𝜔) and 𝜑(𝑥, 𝜔) are the dc component, ac amplitude and phase of the

photogenerated excess electron CDW ∆𝑁(𝑥, 𝜔), respectively. Therefore, considering lock-in

detection at only the fundamental frequency term, the homodyne lock-in carrierography signal

from Eq. (8.1) can be given by

𝑆ℎ𝑜(𝜔) = ∫ {𝐴2(𝑥, 𝜔)𝑐𝑜𝑠2[𝜔𝑡 + 𝜑(𝑥, 𝜔)] + [𝑁𝐴 + 2∆𝑛0(𝑥)]𝐴(𝑥, 𝜔)𝑐𝑜𝑠[𝜔𝑡 +𝑑

0

𝜑(𝑥, 𝜔)]}𝑑𝑥 (8.3)

In comparison, in HeLIC the incident laser excitation is modulated at two different angular

frequencies 𝜔1 and 𝜔2. The excess electron CDW is

∆𝑁(𝑥, 𝜔) = 2∆𝑛0(𝑥) + 𝐴(𝜔1, 𝑥)𝑐𝑜𝑠[𝜔1𝑡 + 𝜑(𝜔1, 𝑥)] + 𝐴(𝜔2, 𝑥)𝑐𝑜𝑠[𝜔2𝑡 + 𝜑(𝜔2, 𝑥)](8.4)

The reference beat frequency is∆𝜔 = |𝜔2 − 𝜔1| , indicating that radiative recombination

modulated at other frequencies will be filtered, and hence the corresponding signal can be

expressed as

CHAPTER 8. 193

𝑆ℎ𝑒(∆𝜔) = ∫ 𝐴(𝑥, 𝜔1)𝐴(𝑥, 𝜔2)cos [∆𝜔𝑡 + ∆𝑑

0𝜑(𝑥, 𝜔1, 𝜔2 )]𝑑𝑥 (8.5)

where ∆𝜑 = 𝜑(𝑥, 𝜔2) − 𝜑(𝑥, 𝜔1). Using the Fourier transformation, ∆𝑁 in frequency domain

is given by

∆𝑁(𝑥, 𝜔) = 4𝜋∆𝑛0(𝑥)𝛿(𝜔) + 2𝜋𝐴(𝑥, 𝜔1)𝑒𝑖𝜑(𝑥,𝜔1)𝛿(𝜔 − 𝜔1) +

2𝜋𝐴(𝑥, 𝜔2)𝑒𝑖𝜑(𝑥,𝜔2)𝛿(𝜔 − 𝜔2) (8.6)

The demodulated signal for PCR and HoLIC becomes,

𝑆ℎ𝑜(𝜔) = ∫ (2∆𝑛0(𝑥) + 𝑁𝐴)∆𝑁(𝑥, 𝜔)𝑑𝑥𝑑

0 (8.7)

whereas that for HeLIC becomes

𝑆ℎ𝑒(∆𝜔) = ∫ ∆𝑁∗(𝑥, 𝜔1)∆𝑁(𝑥, 𝜔2)𝑑𝑥𝑑

0 (8.8)

where * denotes complex conjugation. It should be noted that the HeLIC phase is very small

on the order of (10-3)o due to the small frequency difference ∆𝜔.

8.3 Carrier Transport Theory of CQD Solar Cells under

Modulated Photoexcitation

The nature of photocarrier generation, discrete hopping transport, and recombination in

CQD-based thin films was found to follow a hopping diffusion transport behavior [14, 27]

under frequency-modulated laser excitation that reveals details of carrier hopping transport

dynamics in these photovoltaic materials. Here, in order to extract charge carrier hopping

transport dynamics in CQD-based solar cells, light-matter interaction under modulated-

frequency excitation is investigated as an extension of conventional current-voltage

characterization of CQD solar cells under DC laser excitation [12, 14, 28]. Reviewing the CQD

solar cell structure in Fig.5.1 in Chapter 5 and Fig.8.1, due to the larger bandgap energy of

ZnO than the incident excitation photon energy as well as the thicker CQD layers than ZnO,

CHAPTER 8. 194

charge carriers and excitons are considered to be generated only in CQD layers, thus

contributing to the primary current within this type of solar cell.

Figure 8.1: Schematic of CQD solar cell sandwich structure (a), and the corresponding band

energy structure (b) also shows the illumination depth profile, the photocarrier density wave

distribution and the intrinsic and external electric fields.

Therefore, in a manner similar to the carrier hopping transport model under static excitation

[14], the rate equation for electrons in the nominal p-type CQD layers under dynamic

illumination can be written as

𝜕∆𝑛(𝑥,𝑡)

𝜕𝑡=


𝜕𝑥−∆𝑛(𝑥,𝑡)

𝜏+ 𝑔(𝑥, 𝑡) (8.9)

∆𝑛(𝑥, 𝑡) is the excess electron density and 𝐽𝑒(𝑥, 𝑡) is the electron current flux; 𝜏 is the nominal

minority carrier electron lifetime, and 𝑔(𝑥, 𝑡) is the carrier generation rate. Considering that

the ambipolar diffusion coefficient and mobility, 𝐽𝑒(𝑥, 𝑡) can be further defined by [29, 30]

𝐽𝑒(𝑥, 𝑡) = 𝐷𝑒𝜕∆𝑛(𝑥,𝑡)

𝜕𝑥+ 𝜇𝑒𝐸∆𝑛(𝑥, 𝑡) (8.10)

where 𝐷𝑒 is the diffusivity, 𝜇𝑒 is the mobility, and 𝐸 is the electric field (a constant value given

as the difference between the external and intrinsic electric fields), Eq. (8.9) is reduced to a

diffusion equation which can be solved using the Green function method and transferred to the

CHAPTER 8. 195

frequency domain through a Fourier transformation [14]. Upon harmonic optical excitation,

the photoexcited excess carrier distribution follows the Beer-Lambert Law:

𝑔(𝑥, 𝜔) =𝛽𝜂𝐼0

2ℎ𝜈𝑒−𝛽𝑥(1 + 𝑒𝑖𝜔𝑡) (8.11)

where 𝛽 is the optical absorption coefficient, 𝜂 is the quantum yield of the photogenerated

carriers, and ℎ and 𝜈 are the Plank constant and the frequency of incident photons, respectively.

𝐼0 denotes the incident photon intensity. In the one-dimensional geometry, the boundary

conditions at 𝑥 = 0 and 𝑑 , Fig. 8.1, can be written as functions of surface recombination

velocities (𝑆1and 𝑆2 at 𝑥 = 0 and 𝑥 = 𝑑, respectively,) and the excess carrier density at the

corresponding boundaries, Fig. 8.1.

𝐷𝑒𝜕∆𝑁(𝑥,𝜔)

𝜕𝑥|𝑥=0

= 𝑆1∆𝑁(0,𝜔) (8.12a)

−𝐷𝑒𝜕∆𝑁(𝑥,𝜔)

𝜕𝑥|𝑥=𝑑

= 𝑆2∆𝑁(𝑑, 𝜔) (8.12b)

where ∆𝑁(𝑥, 𝜔) is the Fourier transformed counterpart of ∆𝑛(𝑥, 𝑡), a carrier-density-wave

(CDW). Therefore, the final expression of excess carriers ∆𝑁(𝑥, 𝜔) can be obtained as follows

∆𝑁(𝑥, 𝜔) =𝜂𝐼0𝛽

4ℎ𝜈𝐷𝑒(1−𝑅𝑒1𝑅𝑒2𝑒−2𝐾𝑒𝑑)[𝛽2−(𝑄02+𝜎𝑒

2)]{([(1 + 𝜌𝑒) − 𝑅𝑒1(1 − 𝜌𝑒)] −

𝑅𝑒1[(𝜌𝑒 − 1) + 𝑅𝑒2(1 + 𝜌𝑒)]𝑒−(𝐾𝑒+𝛽)𝑑)𝑒−𝐾𝑒𝑥 + (𝑅𝑒2[(1 + 𝜌𝑒) − 𝑅𝑒1(1 − 𝜌𝑒)] +

[(1 − 𝜌𝑒) − 𝑅𝑒2(1 + 𝜌𝑒)]𝑒−(𝛽−𝐾𝑒)𝑑)𝑒−𝐾𝑒(2𝑑−𝑥) − 2(1 − 𝑅𝑒1𝑅𝑒2𝑒

−2𝐾𝑒𝑑)𝑒−𝛽𝑥} (8.13)

with the following definitions,

𝑄0 =𝜇𝑒

2𝐷𝑒�� [cm-1]; 𝜎𝑒 = √

1+𝑖𝜔𝜏

𝐷𝑒𝜏 [cm-1] (8.14a)

𝑅𝑒𝑗 =𝐷𝑒√𝑄0

2+𝜎𝑒2−𝑆𝑗

𝐷𝑒√𝑄02+𝜎𝑒

2+𝑆𝑗

, 𝑗 = 1, 2 (8.14b)

𝐾𝑒 = √𝑄02 + 𝜎𝑒2 − 𝑄0 [cm-1]; 𝜌𝑒 =

𝛽

𝐾𝑒 (8.14c)

CHAPTER 8. 196

As shown in Fig.8.1, the electric field �� = ��𝑖 + ��𝑒𝑥𝑡 (Ei > Eext) prevents the spreading of the

excess CDW density towards the 𝑥 = 𝑑 terminal of the solar cell. On the contrary, when the

net electric field switches its direction it facilitates the spreading of the CDW concentration

gradient, resulting in reduced energy barriers. Here, all discussion will be based on the

condition that �� prevents the spreading of the excess electron CDW density, i.e. �� and ��𝑖 have

the same direction, which is a general working condition for traditional solar cells.

Ultimately, when substitute Eq. (8.13) into Eqs. (8.7) and (8.8) under low injection levels,

i.e. 2∆𝑛0(𝑥) ≪ 𝑁𝐴 , the final expression of Eq. (8.7) after the necessary mathematical

manipulations becomes

𝑆ℎ𝑜(𝜔) ≈ 𝐴𝑁𝐴 [𝐵(1−𝑒−𝑑𝐾𝑒)+𝐶(𝑒−𝑑𝐾𝑒−𝑒−2𝑑𝐾𝑒)

𝐾𝑒+𝐷(𝑒−𝑑𝛽−1)

𝛽] (8.15)

Coefficients A, B, C, and D are defined according to Eq. (8.13), as follows,

𝐴 =𝜂𝐼0𝛽

4ℎ𝜈𝐷𝑒(1−𝑅𝑒1𝑅𝑒2𝑒−2𝐾𝑒𝑑)[𝛽2−(𝑄02+𝜎𝑒

2)] (8.16a)

𝐵 = [(1 + 𝜌𝑒) − 𝑅𝑒1(1 − 𝜌𝑒)] − 𝑅𝑒1[(𝜌𝑒 − 1) + 𝑅𝑒2(1 + 𝜌𝑒)]𝑒−(𝐾𝑒+𝛽)𝑑 (8.16b)

𝐶 = 𝑅𝑒2[(1 + 𝜌𝑒) − 𝑅𝑒1(1 − 𝜌𝑒)] + [(1 − 𝜌𝑒) − 𝑅𝑒2(1 + 𝜌𝑒)]𝑒−(𝛽−𝐾𝑒)𝑑 (8.16c)

𝐷 = 2(1 − 𝑅𝑒1𝑅𝑒2𝑒−2𝐾𝑒𝑑) (8.16d)

Correspondingly, the final expression of Eq. (8.8) is derived as

𝑆ℎ𝑒(∆𝜔) =1

2𝜋𝐴∗(𝜔1)𝐴(𝜔2) {−

𝐵∗(𝜔1)𝐵(𝜔2)[𝑒−𝑑[𝐾𝑒

∗(𝜔1)+𝐾𝑒(𝜔2)]−1]

𝐾𝑒∗(𝜔1)+𝐾𝑒(𝜔2)

+

𝐵∗(𝜔1)𝐶(𝜔2)[𝑒

−𝑑[𝐾𝑒∗(𝜔1)+𝐾𝑒(𝜔2)]−𝑒−2𝑑𝐾𝑒(𝜔2)]

−𝐾𝑒∗(𝜔1)+𝐾𝑒(𝜔2)

+𝐵∗(𝜔1)𝐷(𝜔2)[𝑒

−𝑑[𝐾𝑒∗(𝜔1)+𝛽]−1]

𝐾𝑒∗(𝜔1)+𝛽

+

𝐶∗(𝜔1)𝐵(𝜔2)[𝑒−𝑑[𝐾𝑒

∗(𝜔1)+𝐾𝑒(𝜔2)]−𝑒−2𝑑𝐾𝑒∗(𝜔1)]

𝐾𝑒∗(𝜔1)−𝐾𝑒(𝜔2)

+

𝐶∗(𝜔1)𝐶(𝜔2)[𝑒−𝑑[𝐾𝑒

∗(𝜔1)+𝐾𝑒(𝜔2)]−𝑒−2𝑑[𝐾𝑒∗(𝜔1)+𝐾𝑒(𝜔2)]]

𝐾𝑒∗(𝜔1)+𝐾𝑒(𝜔2)

−

CHAPTER 8. 197

𝐶∗(𝜔1)𝐷(𝜔2)[𝑒−𝑑[𝐾𝑒

∗(𝜔1)+𝛽]−𝑒−2𝑑𝐾𝑒∗(𝜔1)]

𝐾𝑒∗(𝜔1)−𝛽

+𝐷∗(𝜔1)𝐵(𝜔2)[𝑒

−𝑑[𝛽+𝐾𝑒(𝜔2)]−1]

𝛽+𝐾𝑒(𝜔2)−

𝐷∗(𝜔1)𝐶(𝜔2)[𝑒−𝑑[𝛽+𝐾𝑒(𝜔2)]−𝑒−2𝑑𝐾𝑒(𝜔2)]

−𝛽+𝐾𝑒(𝜔2)−𝐷∗(𝜔1)𝐷(𝜔2)[𝑒

−2𝛽𝑑−1]

2𝛽} (8.17)

where 𝐴∗, 𝐵∗, 𝐶∗, and 𝐷∗are the complex conjugates of A, B, C, and D, respectively, in Eq.

(8.16).

8.4 Quantitative Colloidal Quantum Dot Solar Cell Imaging

8.4.1 Device Fabrication and Characterization Details

Oleic-acid-capped CQDs and ZnO nanoparticles were synthesized following the previously

published method as discussed in Sect.4.3.1 and [31]. As shown in Fig.5.1 and 8.1, the CQD

solar cells have a sandwich structure of PbS CQDs surface-capped with two different ligands,

i.e. PbX2/XX (PbX2: lead halide, AA: ammonium acetate) and EDT (1, 2-ethanedithiol). The

oxygen plasma etching of CQD solar cell surface to add additional surface trap states for

further carrier lifetime study was performed through our plasma etch system (PE-50) at room

temperature with an oxygen flow rate of 10 cc/min in a vacuum environment at a pressure of

100 mTorr. J-V characteristics were obtained using a Keithley 2400 source measuring

instrument under simulated AM1.5 illumination in a nitrogen environment. The experimental

setups for camera-based HoLIC and HeLIC, as well as for single detector based PCR, are

identical to that discussed in Chapter 3.

8.4.2 Quantitative HeLIC Imaging of Carrier Transport in CQD

Solar Cells

Figure 8.2 (a) is a photograph of the as-synthesized multilayer CQD solar cell sample

(structure shown in Figs. 5.1 and 8.1) with metallic Au contacts on the top. The bottom Au

CHAPTER 8. 198

contact so labeled in Fig. 8.2 (a) is connected to an indium tin oxide (ITO) conducting layer.

Each golden circle-rectangle-shaped region, for example, the one circumscribed in a dashed

rectangle, represents a complete solar cell structure with top Au contact, while other regions

with brown color are without top contact Au deposition. The highest PCE of this type of solar

cell has been certified to be as high as 11.28% [31], while device performances vary from

sample to sample due to factors such as electrical and mechanical defects introduced during

various device fabrication processes, as well as film-inhomogeneity-associated short-circuit

effects. As shown in Fig. 8.2 (b), a HoLIC image of the corresponding solar cell sample in Fig.

8.2 (a) reveals the charge carrier distribution within an entire solar device, thereby elucidating

the influence of the Au electrode on charge carrier transport, and depicting the CQD solar cell

inhomogeneity that originates from various defects. The solar cell contact electrode A

circumscribed within the dashed rectangle in Figs. 8.2 (a) and (b) has a PCE of ca. 9%, Fig.8.3,

and has been further investigated using HeLIC and PCR.

Figure 8.2: (a) A photograph of the CQD solar cell sample under study, and (b) the

corresponding HoLIC image of this solar cell. The dashed-rectangle-circumscribed solar cell

A is selected for further studies as shown in Figs. 8.3 and 8.4. The HoLIC characterization was

carried out at 10 Hz. It should be noted that for carrierographic imaging, the sample was flipped

over with the top Au contact on the bottom, resulting in mirror image positions being assumed

in (a) and (b) by the dashed rectangles and inscribed solar cells.

CHAPTER 8. 199

Figure 8.3: Current-density-voltage characteristic of the CQD solar cell shown circumscribed

by a dashed rectangle in Fig.8.2(a).

For the investigation of contact effects on carrier lifetime, single-detector based PCR as

discussed in Chapter 7 was used to study the carrier lifetimes within regions with and without

Au contact deposition and to compare with the HeLIC imaging methodology. The PCR single-

element InGaAs detector can detect a spot area equal to the size of the circular tip of a solar

cell electrode unit (e.g. A and B in Figs. 8.2) and can measure the average carrier lifetime in a

spot region. The large phase lag in PCR phase with increased frequency indicates longer carrier

lifetimes. Therefore, as directly shown in Fig.8.4, according to the larger phase lag at high

frequencies, area C without contact Au deposition exhibits a longer carrier lifetime than its

counterparts in regions A and B, Figs. 8.2. Similarly, region A presents a slightly longer

lifetime than B. The lifetime difference between regions A, B, and the area C is manifested by

the quantitative fitting of experimental frequency-dependent PCR phases to Eq.8.15 and are

consistent with the results from the HeLIC high-frequency imaging method.

CHAPTER 8. 200

Figure 8.4: Frequency-dependent PCR phase spectra of the solar cell electrode units A, B, and

area C without Au contact (Figs. 8.2) at 200 K. Equation (8.15) is used for the best fitting of

each curve. The characterization spot area of the single-detector based PCR is the same as the

area of the circular Au contact tip.

Figure 8.5: High-frequency HeLIC images at 1 kHz (a) and 100 kHz (b) for the CQD solar cell

shown in Fig. 8.2.

The frequency-dependent AC diffusion length enables the characterization of photovoltaic

device properties at different depths using HeLIC. A comparison of HeLIC images at 1 kHz

and 100 kHz is presented in Fig. 8.5, which qualitatively reveals depth-resolved HeLIC images

through the evolution of CQD solar cell image patterns with increased frequency. In other

words, at the low frequency of 1 kHz, the longer AC diffusion length is expected to yield

CHAPTER 8. 201

different image contrast from HeLIC images at the higher frequency, 100 kHz, which

corresponds to a shorter AC diffusion length. Importantly, the HeLIC images reveal the

inhomogeneity of the CQD solar cell and the quality of each CQD solar cell unit.

Figure 8.6: The frequency-dependent average HeLIC image amplitudes of the CQD solar cell

shown in Fig. 8.2. The HeLIC images in Fig.8.5 are also included.

In comparison, as a large-area imaging technique, HeLIC can image an entire solar cell

sample. The dependence of HeLIC amplitudes on modulation frequency is shown in Fig. 8.6

in which each point is the average amplitude of an entire HeLIC image at the corresponding

frequency. The best fitting of the data in Fig. 8.6 into Eq.8.17 yields the overall carrier transport

parameters for the CQD solar cell measured at 200 K. For the best-fitting, as tabulated in Table

8.1, other parameters involved in Eq. (8.17) were taken from the literature or measured

experimentally. Specifically, carrier lifetime τ = 2.98 ± 0.06 µs, diffusivity De = 3.60 × 10-5 ±

4.00 × 10-6 cm2/s, diffusion length Ldiff = 99.10 ± 5.42 nm, and drift length Ldrif = 47.01 ± 6.04

nm. Compared with their room temperature (293 K) counterparts, except for τ, other carrier

CHAPTER 8. 202

transport parameters are smaller at 200 K: at 293 K, De is on the order of 10-3 cm2/s, and Ldiff

and Ldiff are around 400 nm. The carrier lifetime τ decreases to around 500 ns when the

temperature increases from 200 K to 293 K, a phenomenon attributed to increased nonradiative

recombination and consistent with phonon-assisted carrier hopping transport within spatial and

energy disordered CQD systems [11, 12, 14, 32].

Table 8.1: Summary of the parameters used for heterodyne lock-in carrierography best-fits to

Eq. (8.17).

Parameters E (Vcm-1) Sj (cm/s) 𝜂 𝐼0(Js-1cm-2) 𝛽(cm-1) d (cm)

Parameter

values used for

fitting

1.2×104

0 1 0.1 (1 sun) 107 360×10-7

References and

experimentally

obtained

parameters

𝑉𝑜𝑐𝑑𝑠𝑜𝑙𝑎𝑟 𝑐𝑒𝑙𝑙

ideal

situation

ideal

situation

experimentall

y measured

[27] [31]

The carrier transport parameter images for CQD solar cell region E, Fig. 8.7, were

reconstructed using the same method mentioned above. These HeLIC images were taken at

various frequencies between 400 Hz and 270 kHz. Therefore, the carrier lifetime image of

electrode E at 200 K was constructed as shown in Fig. 8.7(b). Regions with Au contacts exhibit

shorter τ of ca. 2.3 µs than the surrounding regions. This can be ascribed to the enhanced

interface-induced defects and traps that decrease carrier lifetime through increased non-

radiative recombination. This finding is consistent with the PCR phase study of carrier

lifetimes in regions with and without Au coating as shown in Fig. 8.4. For comparison, at 293

K the carrier lifetime τ image of the same electrode also yielded a shorter τ of ca. 0.5 µs in Au

regions than the lifetime outside the Au/CQD interfaces, Fig. 8.7(b). Comparison between the

lifetime images at 293 K and 200 K revealed that the increased carrier lifetime at the low

CHAPTER 8. 203

temperature is due to the reduced carrier-phonon interactions which act as necessary phonon-

mediation pathways for trap state related non-radiative recombination [9, 11, 32].

Figure 8.7: (a) 400 Hz HeLIC image of the CQD solar cell region E, Fig. 8.2, and its carrier

lifetime τ image (b) at 200 K. (c) For comparison, the carrier lifetime image of the same

electrode E at 293 K. (d)-(f) are images of carrier diffusivity, diffusion, and drift lengths,

respectively, at temperature 200 K.

The measured carrier lifetimes from HeLIC images are comparable to those measured from

transient photovoltage for PbS CQD solar cells (3-6 μs) [33], impedance spectroscopy for PbS

CQD thin films (3 μs) [34], time-resolved PL spectroscopy for PbS-capped CQDs (1 μs) and

for CdSe nanocrystals (0.88 μs) [35], and from absorbance spectroscopy for PbS CQDs (1-1.8

CHAPTER 8. 204

μs) [36]. Additionally, for a similar type of solar cells to ours except for PbS-PbX2/AA (Fig.5.1)

replacement by PbS-TBAI (tetrabutylammonium iodide surface passivated PbS CQDs), Wang

et al. [37] reported a carrier lifetime of ca. 0.5 μs using impedance spectroscopy. Compared

with these literature carrier lifetime data, the slightly lower carrier lifetime measured in this

paper may be attributed to the different types of samples, also it should be noted that the CQD

solar cells characterized in this study are not of our highest efficiency which is expected to

have higher carrier lifetimes. In addition, the carrier lifetimes characterized by various

transient methodologies were obtained through fitting the time-dependent PL (or other

electrical parameters such as photovoltage) decay spectrum to a simplified exponential decay

model which is also commonly used for lifetime extraction for thin films. It is apparent that

the complexity of carrier transport behavior via various pathways in CQD solar cell devices

was ignored, inevitably leading to deviations from the actual carrier lifetime in solar cell

devices.

Furthermore, for electrode E (Fig. 8.2) at 200 K, carrier diffusivity, and diffusion and drift

lengths were also obtained as shown in Figs. 8.7(d)-(f). Specifically, the carrier diffusivity, Fig.

8.7(d), was imaged to be on the order of 10-5 cm2/s which is much smaller than its room

temperature counterpart (ca. 10-3 cm2/s). Fig. 8.7(d) also shows the effects of the Au/CQD

interfaces on the carrier diffusivity with a lower average De in the Au region. Interface-induced

trap states can trap, de-trap, or recombine carriers, a process that inhibits carrier hopping

diffusion transport. Therefore, with the extraction of τ and De, Ldiff = De was also reconstructed

to be approx. 120 nm, which is much shorter than ca. 400 nm at room temperature. The reduced

Ldiff at low temperature is attributed to the decreased availability of thermal energy for the

phonon-assisted carrier hopping transport within the CQD thin film [12, 14, 32]. With the

CHAPTER 8. 205

presence of interface trap states or defects, carriers in pure CQD layers can be transported

about 30 nm longer than those in Au regions, hopping across ~ 10 more QDs. As shown in Fig.

8.7(f), the effects of interface traps are also substantiated through carrier drift length Ldrif

images using HeLIC, i.e., lower carrier drift lengths of ca. 50 nm in Au regions are obtained

than in other regions.

8.5 Further HeLIC Carrier Lifetime Imaging of CQD Solar Cells

8.5.1 HeLIC Imaging at Various Frequencies

Depth-selective/resolved high-frequency (1 kHz-100 kHz) HeLIC images of CQD solar

cells with a structure shown in Fig.5.1 without Au electrode deposition were obtained as shown

in Fig.8.8. For comparison, HeLIC images taken at different frequencies are presented with all

ranges (the differences between two neighboring values on the color scales of Fig. 8.8) equal

to one-fifth of the difference between the maximum and minimum pixel amplitudes of each

image. Therefore, similar to Fig.8.5, it was found that with increasing modulation frequency

in the HeLIC images the low amplitude pattern A fades while the high amplitude pattern B

spreads out. Different HeLIC images at various modulation frequencies can exhibit different

image contrast emerging from electronic property variations with depth. Another phenomenon

that should be noted is that HeLIC image amplitudes decrease with the modulation frequency,

therefore, degrading the image quality as shown in the Fig.8.9 the HeLIC image at 270 kHz

for the same solar cell in Fig. 8.8. To obtain an optimized contrast, the image range of 270 kHz

image is not set in the same way for images in Fig. 8.8 because of the low-amplitude of 270

kHz image. Hence, the contrast of Fig. 8.9 should not be compared with those in Fig. 8.8.

Despite the low amplitude, benefiting from the high signal-to-noise ratio of HeLIC as

CHAPTER 8. 206

discussed in Sect.3.4, 270 kHz image can still be used for carrier transport parameter imaging

construction as shown in Fig.8.10.

Figure 8.8: HeLIC images of a CQD solar cell at different modulation frequencies 1 kHz (a),

10 kHz (b), 50 kHz (c), and 100 kHz (d) as labeled.

Figure 8.9: 270 kHz HeLIC images of the same CQD solar cell shown in Fig. 8.8.

CHAPTER 8. 207

8.5.2 HeLIC Lifetime Imaging of CQD Solar Cells with/without

Plasma Etching

103

104

105

0.03

0.06

0.09

0.12

0.15

0.18

0.21

0.24 (a)

Region 3

Region 4

HeL

IC (

mV

)

Frequency (Hz)

: 0.56 s

: 0.63 s

0.3 0.4 0.5 0.6 0.7

0.0

0.2

0.4

0.6

0.8

1.0(d)

Lifetime (s)

With 15s plasma etching Without plasma etchingN

orm

aliz

ed C

ou

nts

Figure 8.10: (a) Frequency-dependent HeLIC image average amplitude for regions 3 and 4 of

the CQD solar cell shown in Fig. 8.8(a) without plasma etching; and (b) lifetime imaging of

the same CQD solar cell. (c) Furthermore, bar-plotted lifetime statistical distribution for the

above CQD solar cell without plasma etching and another CQD solar cell of the same type

except with 15 s plasma etching.

The mechanisms of carrier generation, hopping transport, and recombination under

frequency-modulated excitation for CQD solar cells have been developed as Eq. (8.17).

Following the same procedure, the lifetimes of CQD solar cells with and without 15 s plasma

etching were obtained, Fig. 8.10. Furthermore, the average lifetimes in regions 3 and 4,

rectangle-circumscribed in Fig. 8.8(a), were best-fitted to Eq. (8.17) through taking the average

CHAPTER 8. 208

HeLIC amplitudes of pixels in each region at each frequency (ranging from 1 kHz to 270 kHz).

As shown in Fig. 8.10(a), frequency-dependent HeLIC amplitudes and their best-fits yielded

average lifetime measurements for regions 3 and 4 equal to 0.56 μs and 0.63 μs, respectively.

The slightly higher value in region 4 is in agreement with the HeLIC lifetime image as shown

in Fig. 8.10(b) for the same CQD solar cell through best-fitting of 27 HeLIC images to Eq.

(8.17) including those shown in Figs. 8.8 and 8.9. It is interesting to see that the lifetime image

in Fig. 8.10(b) resembles the HeLIC images in Fig. 8.8. The dependence of HeLIC amplitude

on modulation frequency can directly reflect carrier lifetime through a simplified model fk~

1/(2πτ), in which fk is the knee frequency at which the amplitude starts to drop as shown in Fig.

8.10(a).

Furthermore, oxygen plasma dry etching created more trap states in our CQD solar cells,

paving an additional way to investigate the influence of defects and trap states on carrier

lifetimes and to demonstrate the validity of this HeLIC methodology. Specifically, oxygen gas

plasma is efficient in breaking chemical bonds of PbS CQDs and the ligands, creating extra

dangling chemical bonds on the surface of quantum dots and the ligand agents. Moreover, the

oxygen species including ionized oxygen can be added into CQDs acting as oxygen interstitial-

associated defects. These effects on the CQD solar cell from oxygen plasma dry etching are

expected to reduce carrier lifetime through enhanced non-radiative recombination induced by

the increased material trap states and defects. The lifetime distribution image of an oxygen

plasma etched CQD solar cell is shown Fig. 8.10(c), and is plotted as a histogram in Fig.

8.10(d) along with the carrier lifetime distribution of the intact CQD solar cell. The two

dominant lifetime peak distributions as shown in Fig. 8.10(d) for both CQD solar cells with

and without plasma etching may originate from the nature of the sample itself as regions with

CHAPTER 8. 209

distinct image patterns were deliberately chosen for characterization. This pattern of CQD

solar cells is in agreement with the HeLIC amplitude image contrast (Fig. 8.8), where two

different prominent regions can be observed. The difference in FWHM of each lifetime

distribution can be attributed to the homogenization nature of the plasma-etched CQD solar

cell or the plasma etching effects as manifested by the significant narrowing of the FWHM

after plasma etching.

8.5.3 HeLIC Lifetime Imaging for Interface Effects on CQD Solar

Cells

Furthermore, to study the influence of surface recombination velocity on CQD solar cell

performance, CQD/electrode interface effects were also investigated. Fig. 8.2 (a) is a

photograph of the CQD solar cell (structure shown in Fig.5.1 and 8.1) under study with the Au

contacts on top. In addition to J-V characteristics of the solar cell unit A as shown in Fig.8.3,

CQD solar cell unit B was also characterized with a PCE as high as 8.38 %. Using the same

lifetime extraction methodology, the lifetime images of two adjacent CQD solar cells were

obtained as shown Fig. 8.11 (a) and the histogram of the imaged lifetimes is presented in Fig.

8.11 (b). Apparently, regions with Au electrodes exhibit slightly shorter lifetime that can be

ascribed to increased trap state densities formed at the Au/PbS-EDT interfaces, which act as

nonradiative recombination centers and compromise carrier hopping lifetimes. As shown in

Fig. 8.11 (a), the CQD solar cell region A corresponds to region A in Fig. 8.2 which has a high

PCE value of ca. 9.0 %, Fig.8.3. In comparison, the J-V characteristic demonstrated a poor

performance of region D in Fig. 8.2 with a very low efficiency less than 1%. These results are

in agreement with the lifetime images in Fig. 8.11 (a) that region A is more homogeneous than

region D with fewer fabrication-associated defects.

CHAPTER 8. 210

Figure 8.11: (a) Lifetime image of two adjacent CQD solar cell units A and D which reveals

the device homogeneity and influence of electrode contacts on carrier hopping transport, and

(b) the barplot of carrier hopping lifetime image in (a). It should be noted that the top Au

contacts as shown in Fig. 8.2 (a) were on the bottom through flipping the sample over for all

the HeLIC imaging.

8.6 Conclusions

This chapter introduces large-area imaging techniques HoLIC and HeLIC for CQD solar

cell imaging. The Instrumentation and signal processing principles including sampling,

undersampling, and heterodyne are reviewed with details. The nonlinear response of PL

emission to laser excitation intensity which is a prerequisite for HeLIC was further investigated

experimentally. The theories for HoLIC and HeLIC were discussed with great efforts.

Combined with the as-developed carrier transport model under dynamic excitation, the

mathematical expressions for HoLIC, HeLIC, and PCR were derived. HeLIC as a large-area

imaging technique that can perform ultra-high frequency (270 kHz for the as-studied CQD

solar cells) is the state-of-the-art advanced imaging technique for photovoltaics to generate

qualitative imaging of carrier transport parameters that are essential parameters for CQD solar

cell efficiency optimization. The combination of HoLIC, HeLIC, and PCR as emerging

dynamic quantitative interface non-destructive imaging methodologies shows great potential

CHAPTER 8. 211

for fundamental photovoltaic optoelectronic transport studies and industrial in-line or off-line

manufactured solar cell device characterization.

Specifically, it was demonstrated that, compared with regions without Au contact,

enhanced trap state density at the Au/CQD interface results in lower minority carrier lifetime

(ca. 0.5 μs and 2.3 μs at 293 K and 200 K, respectively, in agreement with literature transient

photovoltaic results). The dependence of HeLIC images on modulation frequencies manifests

the potent applications of HeLIC for probing solar cell surface and sub-surface (including p-n

junctions) properties. In addition to the elucidation of large-area defect-related device

homogeneity that originates from various fabrication stages, the carrier hopping lifetime

imaging using HeLIC shows the dominant effective carrier lifetime of ca. 0.60 μs, which

reduces to ca. 0.36 μs after 15 s plasma etching that created more surface trap states.

212

Chapter 9

Conclusions and Outlook

9.1 Conclusions

The present research work adds to the versatility of carrier transport dynamics and current-

voltage mechanisms in CQD systems, to the ultrahigh-frequency testing of carrier transport,

and to the quantitative large-area ultrahigh-frequency imaging via miscellaneous theoretical,

conceptual, and experimental advances. All these scientific and engineering efforts contribute

to reaching the final goal of CQD solar cell efficiency optimization.

9.1.1 Advances in Carrier Transport and J-V Mechanisms of

CQD Systems

Due to the significant electronic, electrical, and optical difference between CQD systems

(low carrier mobility and discrete energy band system) and conventional inorganic Si systems

(high carrier mobility and continuous energy band system), this thesis revisited the

conventional Si solar cell working principles and introduced carrier discrete hopping transport

in CQD systems. Based on the nature of hopping transport behavior, novel carrier drift-

diffusion current-voltage models were developed for different CQD solar cell working

conditions. Therefore, the phonon-assisted carrier hopping transport property was

demonstrated from temperature-dependent CQD solar cell current-voltage characteristics, the

study of which led to the quantitative transport mechanism derivations of the double-diode

mechanism, imbalanced carrier mobilities, and Schottky barrier effects. These novel

mechanisms interpret the S-shaped current-voltage characteristics which have been found to

CHAPTER 9. 213

deteriorate CQD solar cell efficiency considerably. Meanwhile, open-circuit voltage deficit

was further quantitatively studied with the finding of the existence of charge transfer states at

p-n junction interfaces that were quantitatively modeled and ascribed to the high radiative and

nonradiative recombination probability at interfaces. Furthermore, carrier recombination

processes in CQD thin films was also quantitatively described, which led to a complete CQD

energy band structure. Moreover, rather than being constant, voltage- and mobility-dependent

photocurrent was found experimentally in high-efficiency CQD solar cells and was modeled

quantitatively, which demonstrated the invalidity of constant photocurrent assumption. The

study of CQD energy bandgap and carrier mobility demonstrated optimized energy bandgaps

and mobilities for higher solar cell efficiency.

9.1.2 Advances in Ultrahigh-frequency Diagnostics of Carrier

Transport

Carrier transport dynamics in photovoltaic solar cells are essential for fundamental

physical understating of energy transport and loss in solar cells, and for solar cell carrier

efficiency optimization. Based on the understanding of carrier transport mechanisms and

working principles of CQD solar cells. This thesis introduces a trap-state-mediated carrier

transport model for single detector based PCR high-frequency characterization of carrier

lifetime, hopping diffusivity, mobility, and diffusion length in CQD systems. The highest

frequency applied is 1 MHz. However, higher frequency can be achieved depending on

experimental Instrumentation.

High-frequency PCR characterization of carrier transport parameters further demonstrated

the phonon-assisted carrier hopping transport in CQD systems. The carrier transport

dependencies on dot size, ligand, temperature, and hopping activation energy were

CHAPTER 9. 214

investigated. It was found that large CQDs tend to have fewer trap states, while perovskite-

passivated CQDs exhibit long lifetime and high diffusivity due to a low trap state density.

These transport behavior are reflected by their trap-state-mediated hopping transport activation

energies.

9.1.3 Advances in Quantitative Large-area Ultrahigh-frequency

Imaging

Traditional small-spot testing techniques arise doubtful overall photovoltaic materials and

device quality and stability estimation. This thesis developed analytical methodologies for

HoLIC and HeLIC, which are all-optical, large-area, fast, and non-destructive imaging

techniques. Overcoming the limitations of low frame rate and long exposure time even for the

state-of-the-art IR cameras, ultrahigh-frequency imaging of CQD solar cells was achieved for

the first time through heterodyne technologies and an excess carrier-diffusion-wave model.

Therefore, large-area imaging of CQD solar cells at high modulation frequencies can be

obtained now, and the quantitative imaging of carrier transport parameters including carrier

lifetime, diffusivity, and diffusion and drift lengths was obtained through the development of

a novel HeLIC signal generation methodology. The acquired high-frequency images and the

carrier transport parameter imaging are essential for CQD solar cell homogeneity and quality

estimation, for fundamental physical carrier transport study, and for studying CQD/contact

interface influence. Due to the enhanced trap states at CQD/contact interfaces, carrier lifetime

was found to reduce.

Qualitative HoLIC large-area imaging was also achieved for CQD thin films and solar cells.

HoLIC can estimate large-area device homogeneity and quality, which reveals a preliminary

CHAPTER 9. 215

assessment of the carrier lifetime, photocarrier collection efficiency, and output power of an

entire photovoltaic device.

9.2 Outlook

This thesis has shown the advances in high-efficiency CQD solar cells, novel CQD solar

cell working principles, and high-frequency, large-area, and nondestructive characterization

techniques. For future investigation in this area, the following fields are of great interest and

necessity.

Further study of the energy band evolution from localized energy states to extended

energy bands with the increase of interdot coupling strength. This needs to surface

passivate CQDs with short exchange ligands to achieve closely packed CQD

ensembles. This study can be conducted through theoretical carrier transport study and

PCR and HeLIC carrier transport parameter characterization.

Study of hot carriers in CQD materials and devices. Develop or adopt PCR system

through combining it with photothermal techniques for the testing of radiative and

nonradiative (thermal) emission of hot carriers. The applications of these hot carriers

can be very effective in increasing both current density and open-circuit voltage.

In this thesis, the lifetime discussed and measured is identified as effective lifetime,

however, for future investigation, if necessary, different lifetimes including bulk

lifetime and surface lifetime should be distinguished from further development of

theoretical carrier transport models.

To realize complete depth-selective/resolved imaging of HeLIC, a wider modulation

frequency range can be tried for the precise investigation of p-n junction or interface

properties that are essential for high-efficiency CQD solar cell optimization.

CHAPTER 9. 216

All the as-developed theoretical models can be helpful as references for solar cells of

other types. Therefore, using these mechanisms and models with the proper adoption

of PCR and HeLIC signal generation models for other types of solar cells is also a

promising research field.

217

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