1

3 Electrons and Holes in Semiconductors

3.1. Introduction

Semiconductors : optimum bandgap

Excited carriers → thermalisation

Chap. 3 : Density of states, electron distribution function, doping, quasi thermal equilibrium

electron and hole currents

Chap. 4 : Charge carrier generation, recombination, transport equation

3.2. Basic Concepts

3.2.1. Bonds and bands in crystals

Free electron approximation to band

Ref. CLASSIC “Bonds and Bands in Semiconductors” J. C. Phillips, Academic 1973

Atomic orbitals → molecular orbitals → bands

Valence band : HOMO

Conduction band : LUMO

2

Semiconductors : eVEg 35.0

Semi-metals : eVEg 5.00

Si : sp3 orbital, diamond-like structure

3.2.2. Electrons, holes and conductivity

Semiconductors

At KT 0 , all electrons are in valence band – no conductivity

At elevated temperature,

some electrons in conduction band, and some holes in valence band

3

3.3. Electron States in Semiconductors

3.3.1. Band structure

Electronic states in crystalline

Solving Schrödinger equation in periodic potential (infinite)

Bloch wavefunction

rk

k rrk i

i eu, (3.1)

for crystal band i and a wavevector k, which is a good quantum number.

The energy E against wavevector k is called by crystal band structure.

Typical band diagram plotted along the 3 major directions in the reciprocal space.

111,011,001,000 LMX

a)

Conduction Band

Valence Band

Electrons

a)’

Electron in CB

b) b)’

Hole in VB

Fig. 3.4. Electron in CB and Hole in VB

4

Point a

k

is called the Brillouin zone boundary.

For zinc blende structure,

aaaL

aaK

aX

,0

2

3

2

3,00

2,000

3.3.2. Conduction band

At the vicinity of the minimum kE in the conduction band 0ckk , approximation

*

2

0

2

02 c

c

cm

EEkk

k

(3.2)

where 00 cc EE k and *

cm is a parameter with dimensions of mass, and defined by

2

2

2*

11

k

kE

m

c

c

(3.3)

is called the parabolic band approximation.

eg. *

cm for GaAs 0

* 067.0 mmc , for Si 0

*

// 19.0 mmc 0

* 92.0 mmc ,

where 0m is the static electron mass.

*

cm is determined by the atomic potentials, should actually be a tensor,

but is usually treated as a scalar, and is called the effective mass of electron.

Momentum of an electron and the applied force are related as

dt

d

dt

d kpF (3.4)

Following eq. (3.2), the velocity and momentum can be obtained as

*

01

c

c

mE

kkkv k

(3.5)

0

*

ccm kkvp (3.6)

3.3.3. Valence band

At the vicinity of the maximum kE in the valence band 0vkk

*

2

0

2

02 v

v

vm

EEkk

k

(3.7)

5

where 00 vv EE k and *

vm is the effective mass of hole.

*

01

v

v

mE

kkkv k

(3.8)

0

*

vvm kkvp (3.9)

2

2

2*

11

k

kE

m

v

v

(3.10)

eg. *

vm for GaAs 0

* 5.0 mmv , for Si 0

* 54.0 mmv ,

*

cm is the curvature of the band.

3.3.4. Direct and indirect band gaps

Whether or not minimum of conduction band and maximum of valence band occur at the same

wavevector 00 vc kkk

Reciprocal lattices for (a) fcc and (b) bcc crystals

6

When the band gap is indirect, transition requires involvement of phonons for conservation of

momentum.

(Sze)

3.3.5. Density of states

According to Pauli’s exclusion principle, each quantum state only supports one electron (two,

considering spins). Also, from Heisenberg’s uncertainty principle,

hxp ~ ∴ hxk ~ ∴ 2~xk

Therefore, for a crystal of volume LLL

Lx

k 22

~

∴

3

2

L states in this unit volume.

Hence, the density of electron states per unit crystal volume considering spin is

kkk3

3

3

2

2ddg

(3.11)

If the band structure kE is spherically symmetric and isotropic around 0k

dkkkgdg 23 4kk (3.12)

Therefore,

dEdE

dkkkgdEEg 24 (3.13)

∴ dE

dkkEg 2

34

2

2

(3.14)

7

From eq. (3.2) *

2

0

2

02 c

c

cm

EEkk

k

∴ 02

*

2 2c

c EEm

k

(3.15)

Similarly, for the valence band (hole)

∴ EEm

k vv 02

*

2 2

(3.15’)

Differentiating (3.15)

21

0

21

2

*2c

c EEm

k

∴ 21

0

21

2

*2

2

1

c

c EEm

dE

dk

Therefore, from (3.14)

21

0

23

2

*

2

21

0

23

2

*

3

21

0

21

2

*

02

*

3

21

0

21

2

*

2

3

2

3

2

2

12

2

4

2

2

124

2

2

2

2

14

2

24

2

2

c

c

c

c

cc

cc

cc

c

EEm

EEm

EEm

EEm

EEm

kdE

dkkEg

(3.16)

Similarly,

21

0

23

2

*

2

2

2

1EE

mEg v

vv

(3.17)

Question : Derive the DOS for 2D and 1D materials. (Box 3.3)

Excitons

A bound state of a pair of electron and hole of same k .

The wavefunction can be described as

rkrk ,, heex

Satisfying a hydrogenic effective mass equation

exexex

s

ex Eq

rr

42

22

*

2

* : reduced effective mass of e-h pair

exE : binding energy of the exciton

8

22

2

0

0

*

l

Ryd

mE

s

ex

3.3.6 Electron distribution function

Fermi function in semiconductors

1exp

1

Tk

EEEf

B

F

This can be expressed in terms of rk,f although the energy is only dependent on the

wavevector.

krkkkr 33 , dfgdn

The total electron density is given by

kkrkkr

CBc dfgn 3, (3.20)

k

krkkrVB

v dfgp 3,1 (3.21)

3.3.7 Electron and hole currents

Using the above formalism

k

krkkkrCB

c

c

n dfgm

qJ 3

*,

(3.22)

k

krkkkrVB

v

v

p dfgm

qJ 3

*,1

(3.23)

3.4. Semiconductor in equilibrium

3.4.1. Fermi Dirac statistics

Equilibrium : no net charge exchange. In this case, it is independent of position.

TE,Ef,f F ,0 krk (3.24)

1

1,0

TkEEFBFe

TEE,f (3.25)

3.4.2. Electron and hole densities in equilibrium

The number density of electrons En with energy in the range between E and EE

dETEE,fEgdEEn F ,0 (3.26)

9

Then the total density of electrons in a conduction band of minimum energy cE is

cE

Fc dETEE,fEgn ,0 (3.27)

and holes in a valence band with maximum (minimum) energy vE is

vE

Fv dETEE,fEgp ,1 0 (3.28)

3.4.3. Boltzmann approximation

When the Fermi energy FE is located far enough from the band edge such that FEE ,

in the conduction band, (3.25) turns into

TkEE

TkEEFBF

BF

ee

TEE,f

1

1,0 (3.29)

Similarly, in the valence band FEE leading to

TkEE

TkEETkEE

TkEE

TkEEFBF

BFBF

BF

BF

eee

e

eTEE,f

1

1

11

1

11,1 0

(3.30)

Using these results, (3.27), (3.28) can be integrated

Tk

EEN

dEeEEm

dETEE,fEgn

B

cFc

E

TkEE

cc

EFc

c

BF

c

exp

2

2

1,

21

0

23

2

*

20

(3.31)

where

10

23

2

*

22

TkmN Bc

c (3.32)

For holes in valence band

vE

B

FvvFv

Tk

EENdETEE,fEgp exp,1 0 (3.33)

where

23

2

*

22

TkmN Bv

v (3.34)

And the product np is given as

TkE

vcBgeNNnp

(3.35)

This is actually a constant and the intrinsic carrier density in suffice the relation

TkE

vciBgeNNnpn

2

(3.36)

Introducing intrinsic potential energy iE , which is the Fermi level for intrinsic semiconductor,

Tk

EEnn

B

iFi exp (3.37)

Tk

EEnp

B

Fi

i exp (3.38)

where

*

*

ln4

3

2

1

ln2

1

2

1

v

cBvc

v

cBvci

m

mTkEE

N

NTkEEE

(3.39)

Electron affinity

Least amount of energy required to remove an electron from solid.

→ Better determined as energy difference between vacuum level and CBM.

11

vBgvaccBvacF

gvacv

vacc

NTkEENTkEE

EEE

EE

lnln

(3.40)

can all be derived by statistical mechanics.

Validity of Boltzmann approximation

The condition for the above relations to hold is

1

Tk

EE

B

vF and 1

Tk

EE

B

Fc

These relations hold regardless of cN and vN

→ basically, just statistical mechanical consideration.

3.4.4. Electron and hole currents in equilibrium

To find rnJ and rpJ in equilibrium, consider (3.22) ~ (3.24)

k

krkkkrCB

c

c

n dfgm

qJ 3

*,

(3.22)

k

krkkkrVB

v

v

p dfgm

qJ 3

*,1

(3.23)

TE,Ef,f F ,0 krk (3.24)

For parabolic band structure, i.e.,

*

2

0

2

02 c

c

cm

EEkk

k

(3.2)

Therefore, kE obviously is an even function of k . Hence,

TE,Ef,f F ,0 krk (3.24)

12

is also an even function of k and so is

dkkkgdg 23 4kk (3.12)

Consequently,

rkkk ,fgc

is an odd function of k . Therefore, integration of rkkk ,fgc in the entire k -space is 0.

Therefore,

0 rr pn JJ

which means there is no net current in the semiconductor in equilibrium.

3.5. Impurities and Doping

3.5.1. Intrinsic semiconductors

Semiconductors with carrier densities of n (electrons) and p (holes), if the mobilities are n

and p , respectively, the conductivity is given by

pqnq pn (3.41)

In intrinsic semiconductors,

inpn

where in is a small value at room temperature or lower.

Both intrinsic carrier density and conductivity are determined as mere function of temperature

Bandgap, location of Fermi level

At 300K for Si eVEg 12.1 , 3101002.1 cmni , 1161016.3 cm

In semiconductors, mobility increases as a function of temperature due to carriers thermally

excited across the band gap. (ref. What happens with metals?)

At 300K,

Ge : eVEg 74.0 ( eV66.0 ?), 112101.2 cm

13

GaAs : eVEg 42.1 , 1191038.2 cm

(From “Data in Science and Technology, Semiconductors” Springer 1991)

======================

Q. Estimate in for Ge and GaAs at 300K

A. Ge : 3131033.2 cmni GaAs : 36101.2 cmni

======================

3.5.2. n-type doping

Electrons as majority carrier : By replacing the atoms by those that would readily ionize to

provide free electron (leaving a positive ion). Such atoms are called donor atoms.

In case of Si, this can be accomplished by replacing some of the Si atoms by As, for example.

Typically, this substitution is done on the order of ppm or less.

(Sze)

14

Ionisation energy approximated by hydrogenic bond is

Rydm

mV

s

cn 2

0

2

0

*

s : dielectric function of the semiconductor (typically around 10)

Ryd : 13.6 [eV]

∴ eVRydVn 136.0100

1 or smaller

Therefore, room temperature is sufficient to ionise these atoms.

This is depicted in the right hand panel of Fig.3.10.

(Sze)

The density of carriers can be controlled by

1) density of dopants dN

2) donor level (choice of dopants)

3) temperature

Regarding 1), if id nN and the donors are fully ionised at room temperature, then

dNn (3.42)

and

15

d

i

N

np

2

(3.43)

since

TkE

vciBgeNNnpn

2

(3.36)

at equilibrium. Here, obviously,

pn

Hence, electrons are the majority carriers and holes the minority carriers.

Reference : Dielectric functions of matters

K. A. Mauritz, Univ. S. Mississippi, http://www.psrc.usm.edu/mauritz/dilect.html

Lorentz-Lorenz equation : oscillator

N

n

n

3

4

2

12

2

Kramers-Kronig relation

For complex function 21 i or i

where is a complex variable, analytic in the upper half plane and vanishes at

dP 2

1

1

dP 1

2

1

3.5.3. p-type doping

Holes as majority carrier : By replacing the atoms by those that would readily ionize to capture

free electron (turning into a negative ion). Such atoms are called acceptor atoms.

16

Basically, the same argument that was used for n-type can be applied to p-type doping.

In case of Si, this can be accomplished by replacing some of the Si atoms by B, for example.

When pV is small, the dopant atoms is ionised by removing a valence electron from another

bond, leaving a hole.

Regarding doping density, if ia nN and the acceptors are fully ionised at room temperature,

aNp (3.44)

a

i

N

nn

2

(3.45)

since

TkE

vciBgeNNnpn

2

(3.36)

at equilibrium. Here, obviously, np .

Hence, holes are the majority carriers and electrons the minority carriers.

17

The related energy levels are summarized in Fig.3.12.

3.5.4. Effects of heavy doping

Increase in carrier density

Adds tail to the CB and VB DOS : effectively reduces band gap-

Increase defect states : acts as centers for carrier recombination and trapping

3.5.5. Imperfect and amorphous crystals

Increase in carrier density

Modify electronic structures in general

Grain boundaries

18

3.6. Semiconductor under Bias

Actual devices under operation : non-equilibrium condition

3.6.1. Quasi thermal equilibrium

System disturbed from equilibrium

In quasi thermal equilibrium, for electrons and holes,

nFc TEE,ffn,, 0rk : for electrons

pFv TEE,ffp,, 0rk : for holes

nF TEE,fn,0 and pF TEE,f

p,0 are those of equilibrium, implying there is no net current.

rk,cf and rk,vf are general distribution functions, both r and k dependent.

The Fermi levels defined for this quasi thermal equilibrium is different from those at

equilibrium (and also)

nFE : electron quasi Fermi level

pFE : hole quasi Fermi level

Small correction factor introduced

rkrk ,,, 0 AnFc fTEE,ffn

(3.46)

nBnF

nBnFn

TkEE

TkEEnF ee

TEE,f

1

1,0 (3.47)

rk,Af is the part that is antisymmetric in k .

3.6.2. Electron and hole density under bias

In 3.4.3, relations

Tk

EEnn

B

iFi exp (3.37)

Tk

EEnp

B

Fi

i exp (3.38)

were introduced. These can be applied for the quasi Fermi levels

Tk

EEnn

B

iF

inexp (3.48)

Tk

EEnp

B

Fi

i

p

exp (3.49)

Also applying the results

19

Tk

EEN

dEeEEm

dETEE,fEgn

B

cFc

E

TkEE

cc

EFc

c

BF

c

exp

2

2

1,

21

0

23

2

*

20

(3.31)

23

2

*

22

TkmN Bc

c (3.32)

cE

B

Fv

vFvTk

EENdETEE,fEgp exp,1 0 (3.33)

23

2

*

22

TkmN Bv

v (3.34)

we can obtain

nB

cF

cTk

EENn nexp (3.50)

pB

Fv

vTk

EENp

p

exp (3.51)

where nT and pT are the electron and hole effective temperatures that could be different from

ambient temperature T . For instance,

TTn ‘hot’ electrons

In the following we do not consider ‘hot’ carriers, and assume

TTT pn

(‘Hot’ carriers will be considered in Chap 10.)

20

One thing to be noted for semiconductor under bias is that the Fermi levels for electrons and

holes are NOT identical, i.e.,

FFF EEEpn

and the Fermi levels are split. The difference between the quasi Fermi levels

pn FF EE (3.52)

The relationship between carrier densities was defined by

TkE

vciBgeNNnpn

2

(3.36)

Substituting (3.50) and (3.51) into (3.36)

Tk

i

TkEETkE

vc

TkEEE

vc

pB

Fv

v

nB

cF

c

BBpFnFBg

BpFnFgpn

eneeNN

eNNTk

EEN

Tk

EENnp

2

expexp (3.53)

In general, these quasi Fermi levels nFE and

pFE are not constant throughout the device and

are functions of position.

We consider local quasi equilibrium at any point in the semiconductor, and local quasi

Fermi levels.

3.6.3. Current densities under bias

If the distribution function is symmetric as discussed previously, there would be no current.

Hence, for currents rnJ and rpJ to be non-zero, the distribution function f needs to be

antisymmetric.

----------Box 3.4 Boltzmann Transport eq. and relaxation time approximation ------------------

Let the antisymmetric part of the distribution function f be Af .

Since cf is a function of tandrk, , by definition

t

ff

dt

df

dt

d

dt

df ccc

c

kr

kr (3.54)

where

dt

drv (3.55)

dt

dkF (3.56)

21

Define rk,E as the energy with respect to the conduction band minimum (CBM)

rk,EEE c (3.57)

where

rkrk ,,, 0 AnFc fTEE,ffn

(3.46)

We assume that

nFc TEE,ffn,, 0rk

From (3.47)

TkEETkE

TkEEETkEEFcF

BnFcB

BnFcBnFnn

ee

eeTE,EEfTEE,f

rk

rkrk

,

,00

1

1

1

1,,,

From physical consideration, we can assume that rk,E varies as a function of k , but not

much as function of r .

Therefore, for calculating cfr , let

TkEE

nFc

BnFc

neTEE,ff

,, 0rk

∴

n

nBnFcBnFc

Fc

BB

FcTkEETkEE

c EETk

f

Tk

EEeef

rrrr

0 (3.58)

For calculating cfk , note that nFc EE is a function of r but not of k , and that rk,E

varies as a function of k , but not much as function of r Hence, let rk,E be a function of k ,

but not r . Therefore, using,

TkEETkE

nFc

BnFcB

neeTEE,ff

rkrk

,

0 ,,

rk

rkkk

rk

rk

k

rk

kk

,, 0,

,,

ETk

f

Tk

Eee

eeeef

BB

TkETkEE

TkETkEETkEETkE

c

BBnFc

BBnFcBnFcB

From (3.5)

*

01

c

c

mE

kkkv k

Here, we can assume 00 ck (the result will be the same even if this is not the case).

Hence,

22

vrkkkTk

fE

Tk

ff

BB

c

00 ,

(3.59)

Substituting (3.58) and (3.59) into (3.54) together with dt

d

dt

d kpF (3.4)

t

fEE

Tk

f

t

f

dt

dEE

Tk

f

t

f

Tk

f

dt

dEE

Tk

f

dt

d

t

ff

dt

dkf

dt

d

dt

df

c

Fc

B

c

Fc

B

c

B

Fc

B

c

cc

c

nn

n

vFvvk

v

vkrr

rr

rkr

00

00

(3.60)’

Here, from physical consideration,

Fr cE (gradient of conduction band is the force on the electron)

∴ t

fE

Tk

f

dt

df cF

B

c

n

rv0 (3.60)

To solve this for cf , the following assumptions are made.

1) Intraband relaxation is much more frequent than interband

Intraband relaxation is dominantly phonon scattering with lattice and fast.

collisions

cc

t

f

t

f

2) The distribution relaxes exponentially towards the quasi equilibrium with time

constant

0ff

t

f c

collisions

c

(3.61)

This is called the relaxation time approximation.

Substituting (3.61) into (3.60),

00 ff

ETk

f

dt

df cF

B

c

n

rv

Under steady state, this equals 0,

00 ffE

Tk

f c

F

Bn

rv

nF

B

c ETk

ff rv

10 (3.62)

Therefore,

nF

B

A ETk

ff rv

0 (3.63)

23

(The sign might be wrong, but never mind : it is how the direction is considered)

Substituting for cf , the equation for current becomes

kr

k

r

k

r

kr

k

k

kk

k

kk

kkkvkk

kvkkkrkkk

krkkkr

CBc

c

B

F

CBc

c

cB

F

CBc

cB

F

CBF

B

c

cCB

Ac

c

CBc

c

n

dm

fgTk

qE

dm

fgm

q

Tk

Edfg

m

q

Tk

E

dETk

fgm

qdfg

m

q

dfgm

qJ

n

nn

n

3

2

*0

3

*0*

3

0*

3

0*

3

*

3

*

,

,

(3.64)

Since the term in brackets is NOT a function of bias, we rewrite this as nn where

n is the electron mobility

Then,

nFnn EnJ rr (3.65)

Similarly

pFpp EpJ rr (3.66)

-------------------------END of Box 3.4--------------------------------------------------------

The net current is given by

pn FpFnpn EpEnJJJ rrrrr (3.67)

3.7. Drift and Diffusion

3.7.1. Current equations in terms of drift and difusion

Tk

EENn

B

cFc exp (3.31)

By differentiating both sides

24

cF

B

c

B

cF

c

cB

cF

B

cF

cc

c

B

cF

cc

B

cF

B

cF

c

EETk

nNn

Tk

EEnN

N

n

Tk

EE

Tk

EENN

N

n

Tk

EENN

Tk

EE

Tk

EENn

n

nnn

nnn

rr

rrrr

rrrr

ln

exp

expexpexp

∴ cF

B

c EETk

nNnn

n rrr ln

∴ cBB

cB

cF NTknn

TkNnn

n

TkEE

nlnln rrrrr

∴ nn

TkNTkEE B

cBcFn rrrr ln (3.68)

Similarly, from (3.33)

pp

TkNTkEE B

vBvFp rrrr ln (3.69)

Note the relation (refer to Sze Fig. 4-32, below)

vBgvaccBvacF

gvacv

vacc

NTkEENTkEE

EEE

EE

lnln

(3.40)

The gradient in the conduction/valence band edge is

rr F qEc (3.70)

although, typically 0r .

Similarly

gv EqE rrr F (3.71)

although, typically 0r gE .

25

(Sze)

The electrostatic field is defined by

vacEq

rF 1

(3.72)

Substituting (3.68), (3.70) into (3.65), (3.66)

nFnn EnJ rr (3.65)

nn

TkNTkEE B

cBcFn rrrr ln (3.68)

rr F qEc (3.70)

26

∴

cBnn

BncBn

BcBn

BcBcnn

NTkqnnqD

nTkNTkqn

nn

TkNTkqn

nn

TkNTkEnJ

ln

ln

ln

ln

rrr

rrr

rrr

rrr

F

F

F

r

(3.73)

where

q

TkD B

nn (3.77)’

Similarly

vBgppp NTkEqppqDJ lnrrrr Fr (3.74)

where

q

TkD B

pp (3.77)”

As mentioned, typically 0r , 0r gE , and cN , vN (doping levels) are invariant.

Therefore,

nqnqDJ nnn Fr r (3.75)

pqpqDJ ppp Fr r (3.76)

The first terms are the diffusion current, and the second the drift.

Driving forces

Drift : potential gradient

Diffusion : concentration gradient

27

The total current for drift (without diffusion) is

Frrr pnqJJJ pnpn (3.78)

The total current for diffusion (without drift) is

pDnDqJJJ nnpn rrrrr (3.79)

3.7.2. Validity of the drift-diffusion equations

Assumptions

1) Electrons and hole populations each form quasi themal equilibrium with FE and T

2) pn TTT

3) Intraband phonon scattering dominant

4) Electron and hole states described by k

5) Boltzmann approximation TkEE BFc n , TkEE BvFp

6) Compositional invariance (uniformity of materials)

3.7.3. Current equations for non-crystalline solids

“Non-crystalline” can mean a number of things.

eg. defective crystals, amorphous, etc.

In any case, high DOS within band gap.

1) pn, : sensitive to T , F , illumination

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2) pn , : T , F , pn, dependent

3) conduction due to localized states between band gap

istateslocalized

ipn JJJJ (3.80)

i

iii

istateslocalized

i EgEfEJ (3.81)

i

iii

istateslocalized

i EgEfEJ

iE : mobility of carrier through localized states → mechanism dependent

iEf : (Fermi-Dirac) distribution function

iEg : DOS of localized states

Discussed in Chap 8.

3.8. Summary

Conduction band electrons : nearly free particles responsible for transport

Valence band holes

Equilibrium and Fermi level determining occupancy probability

Doping

Quasi Fermi levels