Handout 14 Statistics of Electrons in Energy Bands...1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 14 Statistics of Electrons in Energy Bands In this lecture

1

ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Handout 14

Statistics of Electrons in Energy Bands

In this lecture you will learn:


Example: Electron Statistics in GaAs - Conduction Band

Consider the conduction band of GaAs near the band bottom at the -point:

e

e

e

m

m

m

M

100

010

0011

This implies the energy dispersion relation near the band bottom is:

e

ce

zyxcc m

kE

m

kkkEkE

22

222222

Suppose we want to find the total number of electrons in the conduction band:

We can write the following summation:

FBZin

2k

c kfN

2


fc

KTEkEc EkEfkffc

exp1

1

Where the Fermi-Dirac distribution function is:

We convert the summation into an integral:

KTEkE

kdVkfN

fckc

exp1

1

222

FBZ3

3

FBZ in

Then we convert the k-space integral into an integral over energy:

?

?FBZ3

3

exp1

1

22 fc

fc

EEfEgdE

KTEkE

kdVN

We need to find the density of states function gc(E) for the conduction band and need to find the limits of integration


FBZin

2k

c kfN

Another way of writing itEf


Density of States in Energy BandsEnergy

xk

a

a

sE ssV4 akVEkE xsssx cos2

sss VE 2

sss VE 2

Consider the 1D energy band that results from tight binding:

We need to find the density of states function g1D(E):

dEEgL

dEdEdk

Ldk

Ldk

L

sss

sss

sss

sssx

VE

VED

VE

VE

xa

xa

a

x

k

2

21

2

20FBZ in

22

42

22

akaVdkdE

xssx

sin2

2212

12

sssD

EEVaEg

E

Eg D1

sEsss VE 2 sss VE 2

3


?

?FBZ3

3

exp1

1

22 fc

fc

EEfEgdE

KT

EkE

kdVN


e

ce

zyxcc m

kE

m

kkkEkE

22

222222

Energy dispersion near the band bottom is:

Electrons will only be present near the band bottom

(parabolic and isotropic)

Since the electrons are likely present near the band bottom, we can limit the integral over the entire FBZ to an integral in a spherical region right close to the -point:

point3

2

FBZ3

3

8

42

22 fcc EkEfdk

kVkf

kdVN

Ef



point

3

2

8

42 fc EkEfdk

kVN

Since the Fermi-Dirac distribution will be non-zero only for small values of k, one can safely extend the upper limit of the integration to infinity:

03

2

8

42 fc EkEfdk

kVN

emdkk

dkdE 2

ce EEm

k 22

and

We have finally:

e

ce

zyxcc m

kE

m

kkkEkE

22

222222

cEfcfc EEfEgdEVEkEfdk

kVN

03

2

8

42

We know that:

Ef

4


We have finally:

Where the conduction band density of states function is:

ce

c EEm

Eg

23

222

2

1

cEfcfc EEfEgdEVEkEfdk

kVN

03

2

8

42

E

Egc

cE


The density of states function looks like that of a 3D free electron gas except that the mass is the effective mass and the density of states go to zero at the band edge energy

emcE

Ef


ce

c EEm

Eg

23

222

2

1

cEfc EEfEgdEn

fEEf

EfE

Egc Ef

KT

fEE

eEEf f

If then one may approximate the Fermi-Dirac function as an exponential:

KTEE fc

KTEE

KTEE

EEf f

ff exp

exp1

1

KTEE

NEEfEgdEn fcc

Efc

c

exp

cE

Where:

23

222

KTm

N ec

Maxwell-Boltzman approximation


Effective density of states (units: #/cm3)

5


Example: Electron Statistics in GaAs - Valence Band and Holes

Ef

• At zero temperature, the valence band is completely filled and the conduction band is completely empty

• At any finite temperature, some electrons near the top of the valence band will get thermally excited from the valence band and occupy the conduction band and their density will be given by:

• The question we ask here is how many empty states are left in the valence band as a result of the electrons being thermally excited. The answer is (assuming the heavy-hole valence band):

• We call this the number of “holes” left behind in the valence band and the number of these holes is P:

KTEE

Nn fcc exp

FBZ in

12k

fhh EkEf

fhhk

fhh EkEfkd

VEkEfP

12

212FBZ

3

3

FBZ in



parabolic approx.

fhh EkEf

kdVP

12

2FBZ

3

3

hh

vhh

zyxhh m

kE

m

kkkEvkE

22

222222

Energy dispersion near the top of the valence band is:

Holes will only be present near the top of the valence band

Since the holes are likely present near the band maximum, we can limit the integral over the entire FBZ to an integral in a spherical region right close to the -point:

point

3

2

18

42 fhh EkEfdk

kVP

Ef

6



03

2

18

42 fhh EkEfdk

kVP

Since the Fermi-Dirac distribution will be non-zero only for small values of k, one can safely extend the upper limit of the integration to infinity:

point

3

2

18

42 fhh EkEfdk

kVP

hh

vhh

zyxhh m

kE

m

kkkEvkE

22

222222

We know that:

hhmdkk

dkdE 2

EEm

k vhh 2

2

and

We have finally:

vE

fhh

fhh

EEfEgdEV

EkEfdkk

VP

1

18

42

03

2

Ef



Where the heavy hole band density of states function is:

EEm

Eg vhh

hh

23

222

2

1

E

Eghh

vE

We have finally:

vE

fhh

fhh

EEfEgdEV

EkEfdkk

VP

1

18

42

03

2

Note that the mass that comes in the density of states is the heavy hole effective mass and the density of states go to zero at the band edge energy , and the density of states increase for smaller energies

hhm vE

Ef

7


EEm

Eg vhh

hh

23

222

2

1

vE

fhh EEfEgdEp 1

fEEf

EfE

Egc

Ef

If then one may approximate the Fermi-Dirac function as an exponential:

KTEE vf

KTEE

KTEE

EEf f

ff exp

exp1

11

KTEE

NEEfEgdEp vfhh

E

fhh

vexp1

cE

Where:

23

222

KTm

N hhhh

Maxwell-Boltzman approximation for holes


vE

Eghh

Effective density of states (units: #/cm3)


Example: Electron Statistics in GaAs - Valence Band and HolesIn most semiconductors, the light-hole band is degenerate with the heavy hole band at the -point. So one always needs to include the holes in the light-hole valence band as well:

Ef

fEEf

EfE

Egc

cEvE

Eghh Eg h

v

v

vv

E

fv

E

fhhh

E

fh

E

fhh

EEfEgdE

EEfEgEgdE

EEfEgdEEEfEgdEp

1

1

11

KTEE

Np vfv exp

Where:

23

222

KTm

N hv

and 322323hhhh mmm

EEm

EgEgEg

vh

hhhv

23

222

2

1

Density of states effective mass

8


Example: Electron Statistics in GaAs – Electrons and Holes

Ef

At any temperature, the total number of electrons and holes (including both heavy and light holes) must be equal:

c

vvcf

vcf

c

v

fcc

vfv

NNKTEE

E

KTEEE

NN

KTEE

NKTEE

N

np

log22

2exp

expexp

Because the effective density of states for electrons and holes are not the same (i.e. Nv ≠ Nc), the Fermi level at any finite temperature is not right in the middle of the bandgap.

But at zero temperature, the Fermi-level is exactly in the middle of the bandgap

fEEf E

fE

Egc

cEvE

Egv


Example: Electron Statistics in GaAs – Electrons and Holes

Ef

At any temperature, the total number of electrons and holes (including both heavy and light holes) must be equal:

where ni is called the intrinsic electron (or hole) densityinnp

Note that the smaller the bandgap the larger than intrinsic electron (or hole) density

KT

ENNn

nKTEE

NN

nKTEE

NKTEE

N

nnp

nnp

gcvi

ivc

cv

ifc

cvf

v

i

i

2exp

exp

expexp

2

2

2

9


Electron and Hole Pockets in GaAs

Ef

• At any non-zero temperature, electrons occupy states in k-space that are located in a spherically symmetric distribution around the -point

• This distrbution is referred to as the “electron pocket” at the -point

• At any non-zero temperature, the holes (heavy and light) also occupy states in k-space that are located in a spherically symmetric distribution around the -point

• This distribution is referred to as the “hole pocket” at the -point

Hole pocket

Electron pocket


Shape of Fermi Surface/Contour and Mass Tensor: 2D Example

xkyk

Energy

xkyk

Energy

EF EF

kF

yy

y

xx

xcc m

k

mk

EkE22

2222

m

k

mk

EkE yxcc 22

2222

xxyy mm

When the energy dispersion relation is anisotropic, the distribution of carriers in k-space, and the Fermi surface/contour, are not spherical/circular but become ellipsoidal/elliptical

10


Constant Energy Surfaces

e

cc mk

EkE2

22

Constant energy surfaces are in the reciprocal space and are such that the energy of every point on the surface is the same.

For example, the conduction band energy dispersion:

All points in k-space that are equidistant from the origin (-point) have the same energy. Constant energy surfaces in 3D are spherical shells, and in 2D are circles, with the origin as their center.

xk

yk

zk

Equation of a Constant Energy Surface with Energy Eo:

cozyxoe

c EEm

kkkEmk

E 2

22222 2

2

Equation of a sphere in k-space of radius = co EE

m

22


Constant Energy Surfaces

Now consider the energy band dispersion: zz

z

yy

y

xx

xcc m

km

k

mk

EkE222

222222

Now the equation of a constant energy surface with energy Eo is:

Equation of an ellipsoid in k-space with semi-major axes given by:

coxx EE

m

22

cozz

z

yy

y

xx

xo

zz

z

yy

y

xx

xc EE

mk

m

k

mk

Emk

m

k

mk

E 2

222222222 2222

coyy EE

m

2

2

co

zz EEm

2

2

xk

yk

zk

Fermi-Surfaces are Examples of Constant Energy Surfaces:

xk

yk

zk

Fczz

z

yy

y

xx

xc EE

mk

m

k

mk

E 222

222222 Ec+EF

11


Silicon: Electrons in the Conduction Band

coc EkE

t

t

m

m

m

M

100

010

0011

In Silicon there are six conduction band minima that occur along the six -X directions. These are also referred to as the six valleys. For the one that occurs along the -X(2/a,0,0) direction:

0,0,

285.0

ako

Not isotropic!

mℓ = 0.92 mmt = 0.19 m

This implies:

t

ozz

t

oyyoxxcc m

kkm

kk

mkk

EkE222

222222

fc

kkEkEf

kd

o

near

3

3

22

Expression for the electron density in the valley located at along the -X(2/a,0,0) direction can be written as:

Ef



Define:

ozzt

z

oyyt

yoxxx

kkmm

q

kkmm

qkkmm

q

mq

EqE

mkk

m

kk

mkk

EkE

cc

t

ozz

t

oyyoxxcc

2

22222

222222

This implies:

fc

q

tt

fckk

EqEfqd

m

mmm

EkEfkd

o

0 near 3

3

3

near 3

3

22

22

Therefore, expression for the electron density in the valley located at along the -X(2/a,0,0) direction can be written as:

Ef

Dispersion is isotropic in q-space

12



fc

q

tt EqEfqd

m

mmm

0 near 3

3

3 22

03

2

3 8

42 fc

tt EqEfdqq

m

mmm

Where: ce

c EEm

Eg

23

222

2

1

Total electron density in the conduction band consists of contributions from electron density sitting in all the six valleys:

03

2

3 8

426 fc

tt EqEfdqq

m

mmmn

cEfc EEfEgdEn

31326 tte mmmm and:Density of states effective mass

Ef



KTEE

NEEfEgdEn fcc

Efc

c

exp

Where:

23

222

KTm

N ec

31326 tte mmmm And:

Six electron pockets in FBZ:

There are six electron pockets in Silicon - one at each of the valleys (conduction band minima)

The electron distribution in k-space in each pocket is not spherical but ellipsoidal since the electron masses in different directions are not the same

Ef

13


Germanium: Electrons in the Conduction Band

In germanium there are eight conduction band minima that occur at the L-points

The L-point is at the edge of the FBZ, so one-half of each electron pocket is not in the FBZ and therefore one-half of the electron distribution in each L-valley should not be counted in the sum for calculating the number of electrons:

FBZin

2k

fc EkEfN

The other way to look at the problem is to realize that the other-half of each pocket is also located in the FBZ on the opposite side – so in reality there are four complete pockets of electrons in the FBZ

FBZ


Handout 14 Statistics of Electrons in Energy Bands...1 ECE 407 – Spring 2009 – Farhan Rana – Cornell University Handout 14 Statistics of Electrons in Energy Bands In this lecture

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