solar cell physics

PHYSICS OF SOLAR CELLS:

PART I

Solar Cells Instructor: Dr. Alessia Polemi

OUTLINE: •  Structure of semiconductors •  Energy band structure and carrier concentration •  pn junction (very short review)

INTRO

Semiconductor solar cells devices 1) Semiconductors absorb light and deliver a portion of the energy of the absorbed photons to carriers: e- and h+. 2) A semiconductor diode separates and collects the carriers and conducts the generated electrical current in a specific direction.

à a solar cell is simply a semiconductor diode designed to absorb and convert light energy from the sun into electrical energy.

Conventional solar cell

•  Sunlight is incident from the top

•  Metallic grid •  Antireflective coating •  n-type and p-type are

brought together to form a junction.

•  The diode’s other electrical contact is formed by a metallic layer on the back of the solar cell.

STRUCTURE OF SEMICONDUTORS

Solar cells can be fabricated from various semiconductor materials •  most commonly silicon (Si) – crystalline, polycrystalline, and

amorphous •  GaAs •  GaInP •  Cu(InGa)Se2 •  CdTe

Properties of semiconductors

Solar cell materials are chosen on the basis of: •  how well their absorption characteristics match the solar

spectrum •  their cost of fabrication

Silicon is the most popular choice •  its absorption characteristics are a fairly good match to the

solar spectrum •  Silicon fabrication technology is well developed (electronics

industry) à IC

Properties of semiconductors

Structure of Solids The solids used in photovoltaics can be broadly classified as crystalline, polycrystalline, or amorphous. Crystalline à single-crystal materials; Polycrystalline à materials with crystallites (crystals or equivalently grains) separated by disordered regions (grain boundaries); Amorphous àmaterials that completely lack long-range order

Crystalline: -  long-range order represented by the lattice and a basic building block (the unit

cell) -  Fixed positions (there are only 14 crystal lattices possible in a three-dimensional

universe)

Structure of Solids Polycrystalline: -  composed of many single-crystal regions = grains, that exhibit long-range order -  the transition regions between the grains are termed grain boundaries -  grain boundaries have a significant influence on physical properties (they can

getter dopants or other impurities, store charge in localized states arising from bonding defects)

-  grain boundaries are broadly classified as open or closed. An open boundary is easily accessible to gas molecules; a closed boundary is not.

-  Diffusion coefficients are generally an order of magnitude larger along such boundaries than those observed in single-crystal material

Amorphous solids: -  disordered materials that contain large numbers of structural and bonding

defects -  no long-range structural order (there is no unit cell and lattice) -  composed of atoms or molecules that display only short-range order, at best -  there is no uniqueness in the amorphous phase, e.g. there are a myriad of

amorphous silicon-hydrogen (a-Si:H) materials that vary according to Si defect density, hydrogen content, and hydrogen-bonding details.

Structure of Solids

Phonon spectra

•  Because of the interactions among its atoms, a solid has vibrational modes. The quantum of vibrational energy is called phonon.

•  Phonons can be involved in heat transfer, carrier generation (thermal or in conjunction with light absorption), carrier scattering, and carrier recombination processes.

•  They behave like particles, e.g. when an electron in a solid interacts with a vibrational mode, the event is best viewed as an interaction between two types of particles, electrons and phonons

Phonon spectra

•  Phonons have a dispersion relationship Epn(k) [k wave vector of the vibrational mode]. Analogous to the dispersion relationship for light (Ept=hc|k|/2π)

•  Epn(k) is more complicated and gives what is called phonon

spectrum or phonon energy bands in a solid. In the case of both phonons and photons, k has the interpretation of particle momentum.

Phonon spectrum for Si and GaAs

Phonon bands in two crystalline solids: (a) silicon, and (b) gallium arsenide

•  O refers to optical branches (these modes can be strongly involved in optical properties)

•  A refers to acoustic branches (so called because frequencies audible to the human ear)

•  T and L refer to the transverse and longitudinal modes

SINGLE-CRYSTAL, MULTICRYSTALLINE, AND MICROCRYSTALLINE SOLIDS

Phonon spectra

Si

1) Of the order of 108 cm-1

2) Notice! Superimpose the plot for all photon energies Ept<3eV (solar spectrum) versus k onto phonon spectrum

•  the momentum of the photons constituting the majority of the solar spectrum is very small compared to the momentum of phonons

•  phonon energies are of the order of 10-2eV,10-1eV. Photon energies, at least those in the NIR, VIS, NUV range, are of the order of 1eV

Phonon spectra

NANOPARTICLES AND NANOCRYSTALLINE SOLIDS

•  As particle or grain size becomes smaller, the surface-to-volume ratio increases and surface-stress effects on bulk and surface phonon modes become more important

AMORPHOUS SOLIDS

•  There is no Brillouin zone in reciprocal space because there is no unit cell in real space (there is no crystal lattice)

•  It is difficult to distinguish between acoustic and optical phonons •  Phonons play the same critical roles in electron transport, heat

conduction, etc., as they do in crystalline solids

Phonon spectra

ENERGY BAND STRUCTURE AND

CARRIER CONCENTRATION

Energy Band Structure We have already looked at how by solving SE for periodic crystalline structures we obtain electronic properties

∇2ψ +2m!2

E −V( )ψ = 0

Motion of the e- in the crystal is like that of an e- in free space if its mass, m, is replaced by an effective mass m∗

E

Energy Band Structure

Frequently, the detailed E(k) plot is not needed. In many applications:

•  BandGap EG, •  CB edge EC, •  VB edge EV,

•  and local vacuum level EVL (energy needed to escape the material) or equivalently electron affinity χ = energy to promote

an e- from the bottom of the CB to the vacuum level

Software wxAMPS

Single crystal

Even polycrystalline and amorphous materials exhibit a similar band structure (over short distances, the atoms are arranged in a periodic manner and an electron wavefunction can be defined). However, there are a large number of localized energy states within the mobility gap that complicate the analysis.


Localized states: -  acceptor-like -  donor-like -  amphoteric in nature

Software wxAMPS

Energy Band Structure Acceptor states: neutral when empty and negative when occupied by an electron (ionized) Donor states: neutral when occupied by an electron and positive when empty (ionized)

Software wxAMPS

Amphoteric gap states: can be occupied by none, one , o r two va lence electrons. Their charge state depends on their occupancy

Localized (gap) states can serve as sources of carriers for the bands = doping It is possible to have so many gap states present in a material that they form a band within the energy gap. Such a band within the energy gap is termed an intermediate band (IB).

Software wxAMPS


∇2ψ +2m!2

E −V( )ψ = 0


Very resourceful equation!

Excitons are multi-electron solutions that may be viewed as an electron bound to a hole via Coulombic attraction

Possible solutionà Excitons: can be involved in the light absorption process in solar cell materials


1)   Binding energy is dictated by the Coulombic attraction à materials that polarize more have lower binding energies (i.e., the binding energy correlates inversely with the dielectric constant)

2)   They can be created by photon absorption

3)   They can be mobile in a solid. When they move, energy moves, but not net charge. Since they are not charged, they can only move by diffusion.

4)   If excitons are produced by light absorption in a solar cell

and are to be utilized, then some process must also be present to convert the exciton into at least one free negative–positive charge carrier pair

Energy Band Structure Where are e- and h+ located in terms of energy bands?

Density of states

Carrier Concentration

Density of states in the CB gC E( ) =mn* 2mn

* E −Ec( )π 2!3

cm−3eV −1"# $%

Density of states in the VB gV E( ) =mp* 2mp

* EV −E( )π 2!3

cm−3eV −1"# $%

@thermal equilibrium (i.e. at a uniform temperature with no external injection or generation of carriers), probability of finding an e- (or h+) is given by the Fermi function


T=0

f (E) = 1

1+ exp E −EF

kBT"

#$

%

&'

kB = Boltzmann constant

EF =!2

2m3π 2NV

!

"#

$

%&

2/3

EF

for e-


n0 = gC(E) f (E)Ec

Ec+χ∫ dE

⎟⎟⎠

⎞⎜⎜⎝

⎛ −−≈

TkEEEf

B

Fexp)(gC(E) = 4π (2mn* )3/2h−3(E −Ec )

1/2

= A(E −Ec )1/2

Density of states in the CB Fermi-Dirac function ≈ Boltzmann function


n0 = Electron concentration in the CB Nc = Effective density of states at the CB edge Ec = Conduction band edge, EF = Fermi energy kB = Boltzmann constant, T = Temperature (K)

n0 = NC exp −(EC −EF )kBT

"

#$

%

&'

Effective Density of States at CB Edge

Nc = Effective density of states at the CB edge, me* = Effective mass of the electron

in the CB, k = Boltzmann constant, T = Temperature, h = Planck’s constant

NC = 22πmn

*kBTh2

!

"#

$

%&

3/2



p0 = Hole concentration in the VB Nv = Effective density of states at the VB edge

Ev = Valence band edge, EF = Fermi energy kB = Boltzmann constant, , T = Temperature (K)

p0 = NV exp −(EF −Ev )kBT

"

#$

%

&'

Effective Density of States at VB Edge

Nv = Effective density of states at the VB edge, mh* = Effective mass of a hole in

the VB, k = Boltzmann constant, T = Temperature, h = Planck’s constant

NV = 22πmh

*kBTh2

!

"#

$

%&

3/2

n0p0 = ni2 = NCNV exp −

EG

kBT"

#$

%

&'

Mass Action Law

The np product is a “constant”, ni2, that depends on the material properties NC, NV,

Eg, and the temperature. If somehow n is increased (e.g. by doping), p must decrease to keep np constant.

Mass action law applies

in thermal equilibrium

and

in the dark (no illumination)

ni = intrinsic concentration

Intrinsic semiconductors

In un-doped (intrinsic) semiconductor, n. of electrons in CB and n. of holes in VB are equal

no = po = ni (ni =intrinsic carrier concentration)

ni = NCNV exp −EG

2kBT"

#$

%

&'

The Fermi energy in intrinsic semiconductor

Ei =EV +EC

2+kBT2ln NV

NC

!

"#

$

%&

typically very close to the middle of the bandgap


ni: very small compared with the densities of states and typical doping densities (ni ≈ 1010 cm−3 in Si) and intrinsic semiconductors behave very much like insulators à they are not good conductors of electricity

ni ∝ exp −EG

2kBT#

$%

&

'(

Intrinsic concentration.swf


n. of e- and h+ in their respective bands, and hence the conductivity, can be controlled through the introduction of specific impurities, or dopants, called donors and acceptors

Extrinsic semiconductors

All impurities introduce additional localized electronic states into the band structure, often within the bandgap. -  If the energy of the state ED introduced by a donor atom is

sufficiently close to the conduction band edge (within a few kBT), there will be sufficient thermal energy to allow the extra e- to occupy a state in the CB. The donor state will then be positively charged (ionized).

-  Similarly, an acceptor atom will introduce a negatively charged (ionized) state at energy EA.


Usually donors and acceptors are assumed to be completely ionized so that no≈ND in n-type material and po≈NA in p-type material. The Fermi energy: EF = Ei + kBT ln

ND

ni, EF = Ei − kBT ln

NA

ni



-  Very large concentration of dopants à the dopants can no longer be thought of as a minor perturbation to the system, i.e. effect on the band structure à bandgap narrowing (BGN) and thus increase of intrinsic carrier concentration

-  BGN is detrimental to solar cell performance; solar cells are typically designed to avoid this effect

-  BGN may be a factor in the heavily doped regions near the solar cell contacts

Bandgap narrowing

Extrinsic Semiconductors: n-Type

(a) The four valence electrons of As allow it to bond just like Si but the fifth electron is left orbiting the As site. The energy required to release to free fifth-electron into the CB is very small. (b) Energy band diagram for an n-type Si doped with 1 ppm As. There are

donor energy levels just below Ec around As+ sites.

As has 5 electrons


edhd

ied eN

NneeN µµµσ ≈⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

2

Nd >> ni, then at room temperature, the electron concentration in the CB will nearly be equal to Nd, i.e. n ≈ Nd

A small fraction of the large number of electrons in the CB recombine with holes in the VB so as to maintain np = ni

2

n = Nd and p = ni2/Nd

np = ni2

Extrinsic Semiconductors: p-Type

(a) Boron doped Si crystal. B has only three valence electrons. When it substitutes for a Si atom one of its bonds has an electron missing and therefore a hole. (b) Energy band

diagram for a p-type Si doped with 1 ppm B. There are acceptor energy levels just above Ev around B- sites. These acceptor levels accept electrons from the VB and

therefore create holes in the VB.

B has 3 electrons


haea

iha eN

NneeN µµµσ ≈⎟⎟

⎠

⎞⎜⎜⎝

⎛+=

2

Na >> ni, then at room temperature, the hole concentration in the VB will nearly be equal to Na, i.e. p ≈ Nd

A small fraction of the large number of holes in the VB recombine with electrons in the CB so as to maintain np = ni

2

p = Na and n = ni2/Na

np = ni2

Semiconductor energy band diagrams

Energy band diagrams for (a) intrinsic (b) n-type and (c) p-type semiconductors. In all cases, np = ni

2. Note that donor and acceptor energy levels are not shown. CB = Conduction band, VB = Valence band, Ec = CB

edge, Ev = VB edge, EFi = Fermi level in intrinsic semiconductor, EFn = Fermi level in n-type semiconductor, EFp = Fermi level in p-type semiconductor. χ is the electron affinity. Φ, Φn and Φp are the work functions for the intrinsic, n-

type and p-type semiconductors

Compensation Doping Compensation doping describes the doping of a semiconductor with both donors and acceptors to control the properties.

ad NNn −=ad

ii

NNn

nnp

−==

22

iad nNN >>−More donors than acceptors

More acceptors than donors ida nNN >>−

da NNp −=da

ii

NNn

pnn

−==

22

PN JUNCTION:

DARK CURRENT

(very short review)

Basic of pn junction Remember: this is the simplest solar cell possible

Properties of the pn junction

(a) The p- and n- sides of the pn junction before the contact.

(b) The pn junction after contact, in equilibrium and in open circuit (depletion region, SCL). Built-in field E0 tries to drift holes back into p and electrons into n.

(c) Carrier concentrations along the whole device: at all points, npoppo = nnopno = ni

2

pn Junc&on

Depletion Widths NaWp = NdWn

Donor concentration

(d) Net space charge density ρnet across the pn junction. Charge neutrality.

Acceptor concentration

(e) The electric field across the pn junction is found by integrating ρnet in (d).

pn Junc&on

ερ )(net x

dxd

=E

Field (E) and net space charge density Net space charge density

Eo = −eNdWno

ε=eNaWpo

ε

Emax =−eNaNdWo

ε Na + Nd( )

Field in depletion region

∫−=x

Wpdxxx )(1)( netρ

εE

(f) The potential V(x) across the device. Contact potentials are not shown at the semiconductor-metal contacts now shown.

(g) Hole and electron potential energy (PE) across the pn junction. Potential energy is charge Î potential = q V

pn Junc&on

E = − dVdx

V x( ) = E x( )dx =∫

Vo x ≥Wn

V0 −eNd

2εx +Wn

2( ) 0 ≤ x ≤Wn

eNa

2εx −Wp

2( ) Wp ≤ x ≤ 0

0 x ≤Wp

%

&

'''

(

''''

eV0 ≈ EG

eV0 = kBT lnNaNd

ni2

Some algebra

pn Junc&on

Depletion region width

( ) 2/12

⎥⎦

⎤⎢⎣

⎡ +=

da

odao NeN

VNNW ε

where Wo = Wn+ Wp is the total width of the depletion region under a zero applied voltage

By establishing continuity at the junction:

LEFT: Consider p- and n-type semiconductor (same material) before the formation of the pn junction, separated from each and not interacting.

ENERGY BANDS

Open circuit pn-junction 1)   Same EF 2)  Away from M, in the n-type, Ec-EFn

is the same 3)  Away from M, in the p-type, EFp-

Ev is the same 4)  The bandgap must be the same Ec-

Ev in the two materials 5)   Ec and Ev must bend in the SCL

a)  e- diffuse from n to p, they deplete the n-side near the junction à Ec must move away from Efn

b)  h+ diffuse from p to n, they deplete the p-side near the junction à Ev must move away from Efn

Energy band diagrams for a pn junction under (a) open circuit and (b) forward bias

P0 = e−qVo /kBT P = e−q Vo−V( )/kBT

Energy band diagrams for a pn junction under reverse bias (Shockley model)

Bias.swf

Forward biased pn junction and the injection of minority carriers. The negative polarity of the supply reduces the potential barrier V0 by V. Potential barrier against diffusion is reduced to V0-Vàmore injection of minority carriers

Forward Biased pn Junction

-  Vo + + V -

Forward Bias: Diffusion Current

⎥⎦

⎤⎢⎣

⎡−⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛= 1exp

2

hole, TkeV

NLneDJ

Bdh

ihD

Einstein relation:

Dh = µhkBT/e,De = µekBT/e

Assuming Boltzmann statistics + some algebra

Hole diffusion current in n-side in the neutral region

Similar expression for JD,elec in the p-region.

The total current anywhere in the device is constant. Just outside the depletion region it is due to the diffusion of minority carriers.

Forward Bias: Total Current


Ideal diode (Shockley) equation

J = Jo expeVkBT

!

"#

$

%&−1

(

)*

+

,-

Jo =eDh

LhNd

!

"#

$

%&+

eDe

LeNa

!

"#

$

%&

'

()

*

+,ni

2

Reverse saturation current

Jo depends strongly on the material (e.g. bandgap) and temperature


J = J0 expeVηkBT!

"#

$

%&−1

(

)*

+

,-

Diode (Shockley) equation

η =1÷ 2Ideality factor

Schematic sketch of the I-V characteristics of Ge, Si, and GaAs pn junctions.

V

I

0.2 0.6 1.0

Ge Si GaAs

Ge, Si and GaAs Diodes: Typical Characteristics

1 mA

Forward Bias: Recombination Current

Some minority carries also recombine in the SCLà supply provides also for the electrons and holes lost in SCLà Recombination Current

Typical I-V characteristics of Ge, Si and GaAs pn junctions as log(I) vs. V. The slope indicates e/(ηkBT)

Typical I-V characteristics of Ge, Si and GaAs pn junctions

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