law, Stefan-Boltzmann law. Factors affecting the limiting ...

What we learnt last lecture:

• Black body radiation: Planck’s law, Wien’s displacement law, Stefan-Boltzmann law.

• Light source. AM1.5• Solar cell in dark and under illumination.• Factors affecting the limiting efficiency: Band gap and

spectrum. Shockley-Queisser limit• Requirements for an ideal solar cell.

Solar cell

Ambient radiation

Spontaneous emission

Solar radiation

Spontaneous emission

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Electrons and Holes in Semiconductor

Lecture 3

References:1. Physics of Solar Cells. Jenny Nelson. Imperial College Press, 2003.2. Photovoltaic Materials, Series on Properties of Semiconductor

Materials, Vol.1, Richard H. Bube, Imperial College Press, 1998.3. Third Generation Photovoltaics: Advanced Solar Energy Conversion.

Martin A. Green, Springer, 2006.4. Wikipedia (http://en.wikipedia.org/wiki/Main_Page).

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http://en.wikipedia.org/wiki/Main_Page

http://en.wikipedia.org/wiki/Main_Page

1. Why Semiconductors?2. Semiconductor in Equilibrium3. Semiconductor under Bias

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Why Semiconductors?

• Metal, Semimetal, Semiconductor, Insulator• Band Gap• Conduction Band and Valence Band• Charge Carriers: Electron and Hole• Exciton • Photoconductivity• Band Structure• Density of State (DOS)

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Why Semiconductors?

Requirements for a suitable photovoltaic material: (1) absorb visible light, (2) possess a band gap for photon absorption and charge separation,(3) be able to transport charges efficiently.

Semiconductors: Band Gap enables excited electrons to remain in higher energy levels for long enough to be exploited.

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fs

µs

Conduction band

Metal Semiconductor Insulator

Fermi Level

Valence band

Ban

dgapOverlap

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Bonds and Bands in Crystals

1 2 N Crystal 1 2 N Crystal

Metal Semiconductor or InsulatorEn

ergy

band gap

Conduction band

Valence band

Molecular orbitalHOMO & LUMO

BandConduction band & Valence band

Atomic orbital

Valence Band (VB): The highest occupied band, contains the valence electrons.Conduction Band (CB): The lowest unoccupied band.

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Semiconductor: VB is completely full and separated from the CB by an energy gap of 0.5~4 eV. All valence electrons in VB are involved in bonding and cannot be easily removed. Not a good heat and electric conductor.Insulator: VB is completely full and separated from the CB by a large energy gap. Negligible conductivity at room temperature.Semimetal: A small overlap in the energy of the CB and VB. However, the bottom of the conduction band is typically situated in a different part of momentum space. Unlike metal, it has two types of charge carriers.

Bonds and Bands in Crystals

Metal: VB is partly full or overlaps with CB. Valence electron is easy to be excited, scattered into a neighboring state in the presence of empty states at similar energies. Good heat and electric conductor.

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Electrons, Holes and Conductivity

The dynamics of the holes are equivalent to those of a particle with the opposite charge, wavevector and energetics to the missing electron.

CB

VBhole

electron

CB

VBT=0

CB

VBT>0

The vacancy can be filled by another electron involved in neighboring bonds. Then the vacancy moves to the neighboring bond.

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Semiconductor and Photoconductivity

Two classic examples of photoconductors: photocopying (xerography) and infrared detection applications.

Photoconductivity: an optical and electrical phenomenon in which a material becomes more conductive due to the absorption of electro-magnetic radiation.

hν > Eg

CB

VB

Eg

e-

h+

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Excitons

Exciton: a bound state of an electron and an imaginary particle hole in an insulator or semiconductor, and is a Coulomb-correlated electron-hole pair. It is an elementary excitation, or a quasiparticle of a solid. Excitons provide a means to transport energy without transporting net charge. The excitons must be split before doing work.

Exciton levels Eg

Valence band

Conduction band

-

+

Coulombic interaction

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Electron States in Semiconductors

Band Structure Diagram: The map of energies E against wavevector k for the two or three most important directions in the crystal.

Energy levels of atom/molecule can be obtained by solving Schrödinger’s equation. The electron states in crystalline material with infinite periodic lattice structure can be described by Bloch wavefunction.

Band gap

Ener

gy, E

Wavevector, k

Valence band

Conduction band

<110> <000> <100>

Brillouin zone boundary

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Parabolic band approximation: If k is near a central minimum of a band, it is possible to approximate the energy (band) E(k) with a parabola.

€

E k( ) = EC 0 +h2 k − kC 0

2

2mC*

mC* and mV* are effective mass of electron and hole, and

Electron States in Semiconductors

€

E k( ) = EV 0 −h2 k − kV 0

2

2mV*

€

1mC* =

1h2∂ 2EC k( )∂k 2

€

1mV* = −

1h2∂ 2EV k( )∂k 2

Band gap

Ener

gy, E

Wavevector, k<110> <000> <100>

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Direct and Indirect Band Gaps

Fundamental band gap (Eg): The minimum amount of energy promoting an electron from the valence band (VB) into the conduction band (CB).Direct band gap: The CB minimum and VB maximum occur at the same value of k.Indirect band gap: The CB minimum and VB maximum occur at different values of k. The phonon gives up its momentum to the electron at the moment of photon absorption so that both energy and momentum are conserved.

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Direct and Indirect Band Gap Semiconductors

€

α ∞hν − Eg

hν

InAs is a direct bandgap sc with Eg=0.35 eV. As hν<Eg, α=0; As hν>Eg,

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€

α ∞hν − Eg( )

2

hν

Germanium has a indirect bandgap at 0.66 eV and an direct bandgap at 0.80 eV. For an indirect bandgap sc, as hν>Eg,


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Direct absorption is much stronger than the indirect absorption.

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Density of States of Semiconductors

In the parabolic band approximation, the DOS for the CB,

€

gC E( ) =12π 2

2mC*

h2

3 / 2

E − EC 0( )3 / 2

€

gV E( ) =12π 2

2mV*

h2

3 / 2

EV 0 − E( )3 / 2

The DOS for the VB,

To calculate the densities of electrons and holes from the density of states, we need to know how the states are filled. That is the distribution function f(k,r). f is the probability that at a point r the electron state of waveveckor k occupied.

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Semiconductor in Equilibrium

• Thermal Equilibrium• Fermi energy and Fermi Level• Fermi Dirac Distribution• Boltzmann Approximation• Electron/Hole Density in Equilibrium• Doping

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Semiconductors in Equilibrium

Fermi Dirac Statistics: The average probability that an electron state at energy E will be occupied at temperature T.

Fermi energy (EF): the energy up to which states are filled.

Thermal equilibrium: no net exchange of particles or energy between different points in the semiconductor or between the semiconductor and its surroundings.

€

f0 E,EF ,T( ) =1

exp E − EF

kBT

+1

is the Fermi Dirac distribution function.

€

f0 E,EF ,T( )If EF is far enough from both band edges, f0 can be well approximated by the Maxwell-Boltzmann form.

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E

E

E

E

EF

EF

EF

EF

g(E)

f(E)

n(E)

n(E)

1

0

T=0

(T=0)

T>0

(T>0)

The total density of electrons in CB and holes in VB are

€

n = gC E( ) f0 E,EF ,T( )dEEC

∞

∫

€

p = gV E( ) 1− f0 E,EF ,T( )( )dE−∞

EV∫

Semiconductors in Equilibrium

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In the conduction band, as E>>EF,

€

f0 E,EF ,T( ) ≈ exp EF − EkBT

In the valence band, as E<<EF,

€

1− f0 E,EF ,T( ) ≈ exp E − EF

kBT

Then the electron density

€

n = NC expEF − EC

kBT

NC is the effective conduction band density of states and

€

NC = 2 mC* kBT2πh2

3 / 2

Electron Densities in Equilibrium

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€

p = NV expEV − EF

kBT

For the hole density

NV is the effective valence band density of states and

€

NV = 2 mV* kBT2πh2

3 / 2

For any given material at a given temperature the product np is a constant:

€

np = NCNV exp−Eg

kBT

Intrinsic carrier density (ni): It is equal to the density of electrons thermally excited into the CB of a pure semiconductor in thermal equilibrium.

€

ni2 = np = NCNV exp

−Eg

kBT

Hole Densities in Equilibrium

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€

n = ni expEF − Ei

kBT

p = ni expEi − EF

kBT

It is sometimes convenient to write n and p in terms of ni and an intrinsic potential energy Ei,

€

Ei =12EC + EV( ) − 1

2kBT ln

NC

NV

=12EC + EV( ) − 3

4kBT ln

mC*

mV*

where

Ei is an energy close to the center of the band gap and is equal to the Fermi level of the pure semiconductor in equilibrium.

Electron and Hole Densities in Equilibrium

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Electron affinity of the semiconductor (χ): the least amount of energy required to remove an electron from the solid. It is measured from the conduction band edge to the vacuum level EVAC.

Vacuum level (EVAC): the energy to which an electron must be raised to be free of all forces from the solid.

Electron and Hole Densities in Equilibrium

€

EC = EVAC − χ

EV = EVAC − χ − Eg

EF = EVAC − χ + kBT ln n /NC( )= EVAC − χ − Eg − kBT ln p /NV( )

EC

EV

EF

χ

Evac

-kBT ln(n/NC)

-kBT ln(p/NV)

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Impurity and Doping of Semiconductors

Intrinsic Semiconductor: Semiconductor of perfect crystal containing no impurities. Intrinsic semiconductor has ni electrons and ni holes available for conduction and conductivity σ is given by

€

σ = qµnn + qµp p

Because of small ni the conductivity of an intrinsic semiconductor is generally very small. It increases with increasing temperature and with decreasing band gap since ni varies as exp(-Eg/2kBT).

Intrinsic Silicon, ni=1.02×1010 cm-3, σ=3×10-6 Ohm-1 cm-1 at 300K.Germanium, Eg=0.74 eV, σ=2×10-2 Ohm-1 cm-1,Gallium arsenide, Eg=1.42 eV, σ=1×10-8 Ohm-1 cm-1.

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The impurity or defect introduces bonds of different strength to those which make up the perfect crystal, and therefore changes the local distribution of electronic energy levels.

The altered energy levels are localized, unlike the intrinsic conduction and valence band states. If the impurity energy level occur within the band gap they can affect the electronic properties of the semiconductor:Introducing occupied impurity energy levels above Ei increases the Fermi level, which increases the density of electrons relative to holes in equilibrium. Unoccupied level below Ei reduce the Fermi level and increase the density of holes relative to electrons.

Doping: The density and the nature of the carriers in the semiconductor can be controlled by adding definite amounts of impurities with energy levels close to the conduction or valence band edge.


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n-type Semiconductor: With occupied energy levels between Ei and EC, a semiconductor has been doped by replacing some of the atoms in the crystal lattice with impurity donor atoms, to increase the density of electrons relative to holes. The principle charge carriers are negative. p-type Semiconductor: With unoccupied energy levels between EV and Ei, a semiconductor has been doped by replacing some of the atoms in the crystal lattice with impurity acceptor atoms, to increase the density of holes relative to electrons. The principle charge carriers are positive.

++

n-type doping p-type doping

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In the band picture, the donor electrons have been promoted from a level in the band gap, Vn below EC, into the CB. The acceptor energy level appears in the band gap close to the VB edge.

The density of carriers can be controlled by varying the density of dopants. If Nd>>ni, and the donors are fully ionized at room temperature, then

€

n ≈ Nd

And since at equilibrium

€

p = ni2 /Nd

€

np = ni2


EC

EV

EFVn

donor level

EC

EV

EFVp

acceptor level

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CB

VB

EF

Intrinsic

CB

VB

EF

donor level

n-type

CB

VBEF

acceptor level

p-type

In an intrinsic semiconductor, EF lies roughly in the middle of the band gap. The effect of doping is to shift the Fermi level away from the center of the band gap, towards EC in n-type material, and towards EV in p-type material.


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Advantage: Unlike heating and illumination, doping is a way to increase the conductivity of the semiconductor at equilibrium without requiring a constant input of energy.Disadvantage: Heavy doping causes imperfections in the crystal structure which appear at states in the band gap close to the CB and VB edges, having the effect of adding a tail to the CB and VB density of states functions, and so effectively reducing the band gap. As a photovoltaic material, heavy doping is likely to increase the density of defect states which act as centers for carrier recombination or trapping.

Amorphous semiconductor: Have a similar structure to the crystal on the short range, but have no long range order. The small deviation of bond length and orientation give rise to a spread in energy level so that the DOS near the CB minimum and VB maximum tends to decay with energy into the band gap, rather than cutting off abruptly.


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Semiconductor under Bias

• Semiconductor under Bias• Quasi Thermal Equilibrium• Quasi Fermi Level• Effective Temperature and Hot Electron

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To understand the operation of a solar cell, we need to understand what happens to the semiconductor when it is displaced from equilibrium by exposure to light or electrical injection of electrons and holes through an applied electric bias. In either case, n and p have been disturbed so that they are no longer described by the Fermi Dirac distribution.

Quasi thermal equilibrium: If the disturbance is not too great, or not changing too quickly, the populations of electrons and holes relax to achieve a state called quasi thermal equilibrium. The population of electrons within the CB distribute themselves as though they were at equilibrium, with a common Fermi level and temperature. The holes within the valence band arrange themselves as though they shared another different Fermi level and possibly a different temperature.

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€

fc k,r( ) ≈ f0 E,EFn,,Tn( )fv k,r( ) ≈ f0 E,EFp,,Tp( )

Quasi Fermi levels (EFn and EFp): The new apparent Fermi levels for electrons and holes under quasi thermal equilibrium. EFn and EFp are position dependent. At any point in a semiconductor, there is a quasi thermal equilibrium and we can define local quasi Fermi level.

In quasi thermal equilibrium,


Assuming quasi thermal equilibrium and EC-EFn>>kBT, and EFp-EV>> KBT, the electron and hole densities in a semiconductor under bias:

€

n = ni expEFn − Ei

kBTn

p = ni expEi − EFp

kBTp

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€

n = NC exp− EC − EFn( )

kBTn

p = NV exp− EFp − EV( )

kBTp

Or

Tn and Tp are the electron and hole effective temperatures, and may be different from ambient temperature.

Hot electron: Electrons with Tn>T are called hot electrons and can arise in situations where the carriers gain excess kinetic energy from strong electric fields.

fs

fs

µs

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Electron and Hole Densities under Bias

CB

VB

EF

Equilibrium

n0 electrons

p0 holes

CB

VB

EFn >EF

Under bias

>>n0 electrons

>>p0 holesEFp<EF

Fermi level splits:

€

EFn ≠ EFp ≠ EF

€

Δµ = EFn − EFp

€

np = ni2 exp Δµ

kBT

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SummaryWhy Semiconductors: Band Gap, Conduction Band and Valence Band, Charge Carriers: Electron and Hole, Exciton, Photoconductivity, Band Structure, Density of State (DOS)Semiconductor in Equilibrium: Fermi Level, Fermi Dirac Statistics, Boltzmann Approximation, Electron and Hole Density, Doping, n-type Semiconductor, p-type Semiconductor

€

n = ni expEFn − Ei

kBTn

p = ni expEi − EFp

kBTp

€

n = NC expEF − EC

kBT

€

p = NV expEV − EF

kBT

Semiconductor under Bias: Quasi thermal equilibrium and Quasi Fermi Level

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