L3 ECE-ENGR 4243/6243 09152015 Density of states 3D, 2D (quantum wells), 1D (quantum wires) p. 85, and 0d quantum dots

1

Fig. 1. Density of states as a function of energy in bulk, QW, Qwire, Qdot.Calculation of energy levels and wavefunctions in a multiple quantum well layer

(pp.30a to 43). Reference: S. Chuang, Chapter 3. 

2

3-D Density of states as a function of energy in bulk, 

3

3-D Density of states as a function of energy in bulk, 

4

3-D Density of states as a function of energy in bulk, 

5N(E)dE = dEE

mh 2/12/322

)2

(2

1

(15)

6

1.8.4 Derivation of carrier concentration expression using density of states in conduction and valence band, and occupancy of electrons and holes using Fermi- Dirac distribution [page 38/ECE 4211]

We started this method on page 14. The electron concentration in conduction band between E and E+dE energy states is given by

dn = f(E) N(E) dE. To find all the electrons occupying the conduction band, we need to integrate the dn expression from the bottom of the conduction band to the highest lying level or energy width of the conduction band.

That is, Eq. 76This equation assumes that the bottom of the conduction band Ec = 0. Substituting for

N(E) the density of states expression,

0

)()( dEEfENn

e h

kTm22=n kT

E

2

n

3/2f

An alternate expression results, if Ec is not assumed to be zero.

e h

kTm22=n kT

)E-E(

2

n

3/2fc

dEEm

kTEEn n

f

2/12/322

0

)2

(2

1

)/exp(1

1

dEEme 2/12/3

22)

2(

2

1

N(E)dE = (58)

(84A)

(84B)

Another form is n=ni exp[(Ef-Ei)/kT (84C).

7

1.8.4 Derivation of carrier concentration expression using density of states in conduction and valence band, and occupancy of electrons and holes using Fermi- Dirac distribution [page 38] We started this method on page 14. The electron concentration in conduction band between E and E+dE energy states is given by

dn = f(E) N(E) dE. To find all the electrons occupying the conduction band, we need to integrate the dn expression from the bottom of the conduction band to the highest lying level or energy width of the conduction band.

That is, Eq. 76This equation assumes that the bottom of the conduction band Ec = 0. Substituting for

N(E) the density of states expression,

0

)()( dEEfENn

e h

kTm22=n kT

E

2

n

3/2f

An alternate expression results, if Ec is not assumed to be zero.

e h

kTm22=n kT

)E-E(

2

n

3/2fc

dEEm

kTEEn n

f

2/12/322

0

)2

(2

1

)/exp(1

1

dEEme 2/12/3

22)

2(

2

1

N(E)dE = (58)

(84A)

(84B)

Another form is n=ni exp[(Ef-Ei)/kT (84C).

Graphical way to look at 3-D carrier concentrations, pp. 40-41/4211S-15

8

E

Ec

Ef

Ev

Density of states

N(E)

E

01/2 1 f(E)

dn electron conc. between E and E+dE

dp

E

(a)

dp hole conc. between E and E+dE

dp = N(E)[1-f(E)]Here, N(E) is for the

valence band

E c

E f Fermi Level

E v Density of states

N(E)

E

01/2 1 f(E)

dn electron conc.between E and E+dE

dp

E

E g

Ec - Ef of (a) > Ec - Ef of (b)

(b)

dp hole conc. betweenE and E+dE

Figure 28(a) and 28(b) on the following page illustrate the relationship between carrier concentrations and location of the Fermi level with respect to the band edges.In Fig. 28(a) the energy separation (Ec-Ef) between the conduction band edge and the Fermi level Ef is greater than in Fig. 28(b). As a result there are more electrons in case (b), obtained by integrating the dn plot. If n is larger, it means hole concentration is smaller. Similarly, one can plot the situation, when Ef is near the valence edge, now because of the shift of Ef , dp plot will have a larger magnitude, yielding a greater value of hole concentration p than electrons.

3-D Carrier Concentration in bulk, 

9

(15)

C. 2D density of state (quantum wells) page 84

10

)(2

EfdkdkV nz

yx

y

y

x

xyx L

dk

L

dk

Vdkdk

V /2/2

22

Carrier density in 2-D (quantum well), one dimensional quantization along z- axis. We sum states along k z using quantum number nz

(1)Without f(E) we get density of states in 2-D. Number of states per unit area

Quantization due to carrier confinement along the z-axis.

nz

nz

e EEHL

m)(

2

kt

dkt + kt

11

12

1D density of states (quantum wires) p. 85

13

(12)Without f(E) we get density of states

Quantization due to carrier confinement along the x and z-axes. Looking at the integration

)(*/2

2Ef

Ly

dk

Vnzkz

nxkx

y

Carrier density n or p

nzkz

nxkx

y

Ly

dk

V /2

2

2

22

2 y

zxy

y dk

LLL

dk

V13

dEE

m

LLL

dk

V zxy

y

2

1

2

14

/2

220

This simplifies

19

nzkz

nxkx

ydk

V 22

E

m

LL kxnxkznzzx2

21

Go back to Eq. (12), the density of states in a nano wire is

where E=E-Enx-Enz

L3 ECE-ENGR 4243/6243 09152015 D. 1D density of state (quantum wires) p85

14

(12)Without f(E) we get density of states

Quantization due to carrier confinement along the x and z-axes. Looking at the integration

)(*/2

2Ef

Ly

dk

Vnzkz

nxkx

y

Carrier density n or p

nzkz

nxkx

y

Ly

dk

V /2

2

2

22

2 y

zxy

y dk

LLL

dk

V13

dEE

m

LLL

dk

V zxy

y

2

1

2

14

/2

220

This simplifies

19

nzkz

nxkx

ydk

V 22

E

m

LL kxnxkznzzx2

21

Go back to Eq. (12), the density of states in a nano wire is

where E=E-Enx-Enz

Density of states in 0-D (Quantum dots)The k values are discrete in all three

directions

15

yk zk xk

nznynx EEEEV

)(2

Energy levels in nanowire (discrete due to nx and nz)Y-axis gives energy width

16

1. discrete value of nx=1 and then add nz= 1, 2, 3

2. discrete value of nx=2 and then add nz= 1, 2, 3

3. Add to each discrete value of nx, nz an energy width as shown in density of state plot.

L4 Derivation of current due to 1-D subband or channel i. page 91

17

V2π

egI

2

sy

Show conductance quantized as e2/h, g is 2 due to spin

18

Energy band diagrams in Heterojunctions

N-AlGaAsp-GaAs

Eg1

Eg2

EfN2

Efp1

Ec2

Ec1

Ev2

Ev1

q2 q1

Ec

Reference vacuum level

q2

q1

Ei Ei

Ec-Efp

Ei-Efn

19

p-AlGaAs n-GaAs

Ec

Ev

W

0-Xp0 Xn0

Ec

Ev

Ef

Eg ~ 1.9 eV

Eg = 1.424 eV

Energy band diagram in a heterojunction P-AlGaAs/n-GaAs.

Energy band/levels in Quantum dot lasers

20

21

Why quantum well, wire and dot lasers, modulators and solar cells?Quantum Dot Lasers:•Low threshold current density and improved modulation rate.•Temperature insensitive threshold current density in quantum dot lasers. Quantum Dot Modulators:•High field dependent Absorption coefficient (α ~160,000 cm-1) : Ultra-compact intensity modulator •Large electric field-dependent index of refraction change (Δn/n~ 0.1-0.2): Phase or Mach-Zhender ModulatorsRadiative lifetime τr ~ 14.5 fs (a significant reduction from 100-200fs). Quantum Dot Solar Cells: High absorption coefficent enables very thin films as absorbers. Excitonic effects require use of pseudomorphic cladded nanocrystals (quantum dots ZnCdSe-ZnMgSSe, InGaN-AlGaN)

Table I Computed threshold current density (Jth) as a function of dot size in for InGaN/AlGaN Quantum Dot Lasers

q and Jth

Quantum Dot Size 100100100Å 505050Å 353535Å

Defect Status

q Jth A/ cm2 =418nm

q Jth A/ cm2 =405nm

q Jth A/ cm2

=391nm

Negligible Dislocations

(ideal)

0.9 76

0.9 58

0.9 54

Traps N t=2.9x1017cm-

3 (Dislocations =1x1010 cm-2)

0.0068 10,118 0.0068 7,693 0.0068 7,147

Excitonic Enhancement (in presence of dislocations)

0.049 1,404 0.17 304 0.358 136

(Ref. F. Jan and W. Huang, J. Appl. Phys. 85, pp. 2706-2712, March 1999).

Quantum Dots (QDs)

• Nanocrystals confined in three dimensions.• Electronic and optical properties of the these

particles are dependent on the size and shape of the particles.

• The bandgap of the material is inversely proportional to the size of the dot due to the confinement.

• Example: Bandgaps of the QDs we prepare in our lab

Si QDs – ~1.24 eV where as bulk Si bandgap is 1.17 eV

Ge QDs – ~0.91 eV where as bulk Ge bandgap is 0.67/0.69 eV

• The mass and mobilities of the electrons in these are effective values and are different from the free electron mass and mobility.

Top Ravi[9], below is published results of this research[10]

Core

Cladding

22

Si QD absorption is greater than Bulk Si absorption

22B

• 99.999% pure Si/Ge powders. Typical size of the particles with 325 Mesh is ~44 µm.

• Perform high energy ball milling for 5 hours.

• Mix the ball milled powder with benzoyl Peroxide (oxidative agent ) in ethanol solvent. Sonicate for 48 hours.

• Then QDs are separated from the colloidal solution by performing the high-speed centrifuging at 3000, 6000, 9000 and 13000 rpm.

• The typical size of the particles obtained here ~20nm with 1-2nm cladding thickness. The size can be further reduced to ~4nm by reoxidizing and etching the cladding layer in 10ppm of 48% hydrofluoric acid and 100% ethanol.

QDs Colloidal Solution Preparation

Ball Mill Intrinsic Powder

Intrinsic Powder

Powder + Ethanol + Benzoy Peroxide

Keep Solution SonicatedHigh Sped Centrifuge

Reoxidize and Etch23

• Ball milling will reduce the size of micron sized powder particles to nanometers.

• Ball milling will change the surface morphology (crystallinity) of the particles.

• Ball milling causes the surface amorphization of the Si nanoparticles.• Amorphization will reduce the refractive index and increases the

absorption coefficient of the QDs, which causes the change in optical behavior.

• Wave number,

Where wavelength and n refractive index.

• Absorption coefficient ,

Where absorption coefficient and extinction coefficient of the material

Effect of Ball Milling [1]

24

• Raman spectrum of the Si powder before ball milling (Plot A: c-Si peak at 512 nm) and the spectrum after ball milling (plot B: A-Si peak at 475nm and c-Si peak at 518nm) and the self-assembled Si/SiOx on glass substrate (C). Raman Frequency undergoes a red shift as the diameter of the nanocrystal decreases [2]. Raman investigation suggests that the Si Core in these nanoparticles, 45% of the total Si-atom is composed of 34% crystalline and 11% of amorphous component.

Effect of Ball Milling (Cont’d)

25

• The Si/SiOx QDs are stable when the pH is between 7.8-6.7. But doesn’t self-assemble on the substrates. When the pH is in between 6.7-4.9 QDs in the solution are not stable will agglomerate/precipitate on the substrate. As the pH is ranging from 4.9 to 3.8 the dots are metastable and self-assemble uniformly on the substrates.

Effect of pH on stability of the QD [1]

26

•From the above the tables we can see the different component suboxides present in the cladding layer and the percent of Si/SiOx in the dot.

XPS Results [1]

27

•Below pH 5 the partial protonation of QD will provide a positive surface charge. •P-type substrate will have negative surface charge density.•N-type has positive surface density.•The positively charged QD will then assemble only on the negative surface charge of P-type substrate

Site-Specific Self-Assembly (SSA) [3]

28

Transport in Quantum dot channel (QDC) FET

29

p-Si

n nQuantum

dot channel

Gate

Drain

Source

Energy mini-bands in SiOx-Si QDSL

JEM, 2012.

EC

Dot Cladding

Dot Core

qVD

Energy band diagram showing the effect of drain voltage VD in populating upper mini-bands in a QD channel as Vg is increased.

0.21 eV 0.911 eV

0.14 eV

0.07 eV

0.408 eV

0.102 eV

0.028 eV

0.056 eV

0.084 eV

0.040 eV

0.163 eV

0.367 eV

4 nm

2nm

Si dot

SiOx

e

e

0.00E+00

5.00E-06

1.00E-05

1.50E-05

2.00E-05

2.50E-05

3.00E-05

3.50E-05

4.00E-05

-1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 3 3.5

ID(VD=0.500)

ID(VD=1.000)

ID(VD=1.500)

ID

Vgs

VVVV-VC

L

W=I DSjDSjTHiTHiG

m

i

n

jnoD }

2

1)({

0 0

''

Threshold shift VTH depends on transfer of charge to the mini-bands (i) in QDSL channel.

Quantum Dot Gate (QDG) FETs

Transfer characteristics of a fabricated QDGFET when VDS = 0.5 V

Output characteristics of a fabricated QDGFET

Cross sectional HRTEM image of QDGFET.

Karmakar et. al. Journal of Electronic Materials, 40, 2011.

2 Layers of GeOx - Ge quantum

GateSource Contact Drain Contact

ZnS/ZnMgSDrainSource

Si Substrate

SiO2 SiO2

3-State QDGFET with II-VI gate insulator

30

4-State/Mixed-Dot QDG FET

4-State QDGFET with GeOx-Ge and SiOx-Si quantum dots in gate region

31

Mixed-Dot QDGFET Quantum Simulations

Electrons tunneled to upper Si QDs when VG = 1.5V

Electrons confined in inversion channel when VG = 0V Electrons in lower Ge QDs when VG = 1.0V

Lingalugari et. al. Journal of Electronic Materials, 42, 2013.32

QDGFET Experimental Results

Lingalugari et. al. Journal of Electronic Materials, 42, 2013.

Transfer characteristics of a fabricated 4-state QDGFET when VD = 0.5 V

Output ID-VD characteristics of a fabricated QDGFET with Si and Ge QDs.

33

VDD

Vin1

Vout1

VSS

Vin2

Vout2

Data

DataWrite

Read

QDGFET based SRAM Cell

35

Results of an Inverter in the SRAM

Simulation Results of a 4-state QDGFET based Inverter

36

37

Si-SubstrateEc

Ev

EF

Vacuum

SiO

2

SiO

2

Si D

ot

Control Gate

qSi

qSiO2

qs

Ec

Eg

t1DNCt2

A

BC

DE

H

FG

Si Substrate

DrainSource

SiO2

Control Gate

Si Dots

Floating Gate

Fig .2. Schematic cross-section of a floating gate nonvolatile memory

Fig .4 energy band

Energy band diagram of a floating gate nonvolatile memory (FGNVM)

References[1] T. Phely-Bobin, D. Chattopadhyay, and F. Papadimitrakopoulos, Chem. Mater., 14, 1030

(2002).

[2]C. C. Yang and S. Li, J. Phys. Chem. B 2008, 112, 14193–14197, 2008.

[3] F. Jain and F. Papadimitrakopoulos, Patent # 7,368,370 (2008).

38