Handout 16 Electrical Conduction in Energy Bands · 2015. 3. 25. · Electrical Conduction in Energy Bands In this lecture you will learn: • The conductivity of electrons in energy

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ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Handout 16

Electrical Conduction in Energy Bands

In this lecture you will learn:

• The conductivity of electrons in energy bands • The electron-hole transformation• The conductivity tensor• Examples• Bloch oscillations


Inversion Symmetry of Energy Bands

Recall that because of time reversal symmetry:

rr knkn

,,

* kEkE nn

We know that:

kEkv nkn

1

Now let go to in the above equation:k

k

kvkv

kv

kE

kE

kEkv

nn

n

nk

nk

nkn

1

1

1

xk

a

a

Energy

xk

a

a

Energy

2


Current Density for Energy Bands

For a free electron gas the current density was given as:

kvkfkd

ekvkfV

eJk

3

3

all 22

2

In Drude model, the electron current density was given as:

venJ

Now we want to find the current density due to electrons in energy bands

The current density due to electrons in the n-th band can be written in a manner similar to the free-electron case:

kvkf

kde

kvkfV

eJ

nn

nk

nn

FBZ3

3

FBZ in

22

2

xk

a

a

Energy

xk

a

a

Energy


Current Density for a Completely Filled or Empty Bands

xk

a

a

Energy

xk

a

a

Energy

Completely filled bands do not contribute to electrical current or to electrical conductivity

where I have used the fact:

kvkv nn

Ef

Completely empty bands do not contribute to electrical current or to electrical conductivity

Only partially filled bands contribute to electrical current and to electrical conductivity

Of course, if for all in FBZ: 0kfn

k

0

22

FBZ3

3

kvkfkd

eJ nnn

Consider a completely filled band for which for all in FBZ

Application of an external field will not change anything!

1kfn

k

0

22

22

FBZ3

3

FBZ3

3

kvkd

ekvkfkd

eJ nnnn

3


Current Density and Electron-Hole Transformation

kvkfkd

e

kvkfkd

ekvkd

e

kvkfkd

e

kvkfkd

eJ

nn

nnn

nn

nnn

12

2

12

22

2

112

2

22

FBZ3

3

FBZ3

3

FBZ3

3

FBZ3

3

FBZ3

3

Consider the expression for the current density for a partially filled band:

The final result implies that since the current density of a filled band is zero, the current density for any band can always be expressed in two equivalent ways:

a) As an integral over all the occupied states assuming negatively charged particles (as in (1) above)

a) As an integral over all the unoccupied states assuming positively charged particles (as in (2) above)

(1)

(2)

0

xk

a

a

Energy

xk

a

a

Energy

Ef


Current Density and Electron-Hole Transformation

The Electron Choice:The current density is given by:

The Hole Choice:The current density is given by:

xk

a

a

Energy

xk

a

a

Energy

Ef

kvkfkd

eJ nnn

FBZ3

3

22

kvkfkd

eJ nnn

1

22

FBZ3

3

Current is understood to be due to negatively charged electrons This choice is better when the electron number is smaller than the hole number

Current is understood to be due to positively charged fictitious particles called “holes”

This choice is better when the hole number is smaller than the electron number

One has two choices when calculating current from a partially filled band:

4


Metals, Semiconductors, and InsulatorsMaterials can be classified into three main categories w.r.t. their electrical properties: Metals: In metals, the highest filled band is partially filled (usually half-filled)Semiconductors: In semiconductors, the highest filled band is completely filled (at least at zero temperature) Insulators: Insulators are like semiconductors but usually have a much larger bandgap

xk

Energy

FBZ

Metal

Ef

xk

Energy

FBZ

Semiconductor

Ef

xk

Energy

FBZ

Insulator

Ef


Inclusion of Scattering in the Dynamical Equation

In the presence of a uniform electric field the crystal momentum satisfies the dynamical equation:

Ee

dttkd

Now we need to add the effect of electron scattering.As in the free-electron case, we assume that scattering adds damping:

ktkEe

dttkd

Steady State Solution:

The boundary condition is that: ktk

0

Note: the damping term ensures that when the field is turned off, the crystal momentum of the electron goes back to its original value

Ee

ktk

In the presence of an electric field, the crystal momentum of every electron is shifted by an equal amount that is determined by the scattering time and the field strength

Energy

xk

Ex

Conduction band

hh valence band

ℓh valence band

Energy

xk

Ex

Conduction band

hh valence band

ℓh valence band

5


Electrical Conductivity: Conduction Band

oToocc kkMkkkEkE

..

21

2

oc kkMkv

.1

The velocity of electrons is:

Consider a solid in which the energy dispersion for conduction band near a band minimum is given by:

Energy

k

Conduction band

ok

The current density is:

kvkf

kdeJ cc

kc

o

near3

3

22

In equilibrium, for every state with crystal momentum that is occupied, the state is also occupied and these two states have opposite velocities.

Therefore in equilibrium:

okk

okk

0

22

near3

3

kvkfkd

eJ cck

co



Conduction band

Energy

k

ok

Now assume that an electric field is present that shifts the crystal momentum of all electrons:

xk

yk

Electron distribution in k-space when E-field is zero

xk

yk

Electron distribution is shifted in k-space when E-field is not zero

xEE x ˆ

Ee

ktk

Distribution function: kfc

Distribution function:

E

ekfc

Ee

Since the wavevector of each electron is shifted by the same amount in the presence of the E-field, the net effect in k-space is that the entire electron distribution is shifted as shown

kfc

ok

ok

E

6


Current Density:

Do a shift in the integration variable:

E

EMenJ

EMkfkd

eJ

Ee

kkMkfkd

eJ

Ee

kvkfkd

eJ

c

ck

c

ock

c

cck

c

o

o

o

.

.

.2

2

.2

2

22

12

1

near3

32

1

near3

3

near3

3

Where the conductivity is now a tensor given by: 12 Men

xk

ykE

e

ok


xEE x ˆ

kvE

ekf

kdeJ cc

kc

o

near

3

3

22

E

ekfc


Electrical Conductivity Example: Conduction Band of GaAs

Consider the conduction band of GaAs near the -point:

e

e

e

m

m

m

M

100

010

0011 Isotropic!

This implies:

e

z

y

x

e

z

y

x

e

e

e

cz

cy

cx

c

men

E

E

E

E

men

E

E

E

m

m

m

en

J

J

J

EMenJ

2

2

2

,

,

,

12

100

010

001

.

7


Electrical Conductivity Example: Conduction Band of Silicon

t

t

m

m

m

M

100

010

0011

In Silicon there are six conduction band minima (valleys) that occur along the six -X directions. 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 that for this valley:

z

y

x

t

t

cz

cy

cx

c

E

E

E

m

m

m

en

J

J

J

EMen

J

100

010

001

6

.6

2

,

,

,

12

The factor of 6 is there because only 1/6th of the total conduction electron density in Silicon is in one valley


Electrical Conductivity Example: Conduction Band of Silicon

E

E

E

E

men

E

E

E

mm

mm

mm

en

J

J

J

z

y

x

e

z

y

x

t

t

t

cz

cy

cx

2

2

,

,

,

4200

0420

0042

6

After adding the current density contributions from all six valleys, the resulting conductivity tensor in Silicon is isotropic and described by a conductivity effective mass

To find the conductivity tensor for Silicon one needs to sum over the current density contributions from all six valleys:

Isotropic!

te mmm21

311

Conductivity effective mass

8


Electrical Conductivity: Valence Band

oToovv kkMkkkEkE

..

21

2

ov kkMkv

.1

The velocity of electrons is:

Consider a solid in which the energy dispersion for valence band near a band maximum is given by: Energy

k

Valence band

ok

The current density is (using the electron-hole transformation):

kvkf

kdekvkf

kdeJ vv

kvv

kv

oo

1

22

22

near3

3

near3

3

In equilibrium, for every state with crystal momentum that is unoccupied, the state is also unoccupied and these two states have opposite velocities.

Therefore in equilibrium:

okk

okk

01

22

near3

3

kvkfkd

eJ vvk

vo



Valence band

Energy

k

ok

Now assume that an electric field is present that shifts the crystal momentum of all electrons in the valence band:

xk

yk

Hole distribution in k-space when E-field is zero

xk

yk

Hole distribution is shifted in k-space when E-field is not zero

xEE x ˆ

Ee

ktk

Distribution function: kfv

1 Distribution function:

E

ekfv

1

Ee

Since the wavevector of each electron is shifted by the same amount in the presence of the E-field, the net effect in k-space is that the entire electron distribution (and hole distribution) is shifted as shown

kfv

1

ok

ok

E

9


Current Density:

Do a shift in the integration variable:

E

EMepJ

EMkfkd

eJ

Ee

kkMkfkd

eJ

Ee

kvkfkd

eJ

v

vk

v

ovk

v

vvk

v

o

o

o

.

.

.12

2

.12

2

12

2

12

1

near3

32

1

near3

3

near3

3

Where the conductivity is now a tensor given by: 12 Mep

xk

ykE

e

ok


xEE x ˆ

kvE

ekf

kdeJ vv

kv

o

12

2 near

3

3

E

ekfv

1


Electrical Conductivity Example: Heavy-Hole Band of GaAs

Consider the heavy-hole band of GaAs near the -point:

hh

hh

hh

m

m

m

M

100

010

0011 Isotropic!

This implies:

hh

hh

z

y

x

hh

hh

z

y

x

hh

hh

hh

hh

hhz

hhy

hhx

hhhh

mep

E

E

E

E

mep

E

E

E

m

m

m

ep

J

J

J

EMepJ

2

2

2

,

,

,

12

100

010

001

.

10


Electrical Conductivity Example: Light-Hole Band of GaAs

Consider the light-hole band of GaAs near the -point:

h

h

h

m

m

m

M

100

010

0011 Isotropic!

This implies:

h

h

hh

mep

EEMepJ

2

12 .

The total valence band conductivity of GaAs can be written as the sum of the contributions from the heavy-hole and the light-hole bands:

h

h

hh

hh

mep

mep

22


The Phenomenology Of Transport

The presence of external fields, and scattering, the following relations work for electrons in any energy band near the band edge (assuming parabolic bands):

ktkEe

dttkd

on ktkMtkv

.1

tkvkfkd

etkvkfkd

etJ nnnnn

1

22

22

FBZ3

3

FBZ3

3

The first two can also be written as:

kvtkvMEe

dtkvtkvd

M nnnn

..

Problem: One needs simple models for current transport so that non-specialists, like circuit designers, can understand devices and circuits without having to understand energy bands

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Drift Velocity and Mobility for ElectronsWe define the drift velocity for the electrons in the conduction band (for parabolic bands) as:

kvtkvtv cce

The drift velocity is independent of wavevector for parabolic bands and satisfies:

tvM

Eedttvd

M ee

..

Once the drift velocity is calculated, the electron current density is:

tventvkf

kde

kvkvtkvkfkd

etkvkfkd

etJ

eec

cccccce

FBZ3

3

FBZ3

3

FBZ3

3

22

22

22

(1)

(2)

0

EEMekvtkvtv ecce

..1

In steady state:

e = mobility tensor

Electrons in the conduction band are to be thought of as negatively charged particles. In case of multiple electron pockets, current density contributions are calculated separately for each and added in the end.


We define the drift velocity for the “holes” in the valence band (assuming parabolic bands) as:

kvtkvtv vvh

The drift velocity is independent of wavevector and satisfies the equation:

tvMEe

dttvd

M hh

..

Once the drift velocity is calculated, the hole current density is:

tveptkvkfkd

etJ hvvh

12

2FBZ

3

3

Holes in the valence band are to be thought of as positively charged particles. In case of degenerate valence band maxima, the heavy and light hole current density contributions are calculated separately and added in the end.

(1)

(2)

Where realizing that the inverse effective mass tensor will have negative diagonal terms for valence band, I have multiplied throughout by a negative sign, with the result that the charge “-e” becomes “+e”

EEMetv hh

..1 In steady state: h = mobility tensor

Drift Velocity and Mobility for Holes

12


xk

Energy

FBZ

a

a

The Case of No Scattering: Bloch Oscillations

Consider an electron in a 1D crystal subjected to a uniform electric field. The energy band dispersion and velocity are:

xEE o ˆ

In the absence of scattering, the crystal momentum satisfies the dynamical equation:

0

tkteE

tk

eEdt

tkd

xo

x

ox

akVEkE xsssxn cos2

The time-dependent velocity of the electron is:

atktEae

Va

atkVatv

xo

ss

xssn

0sin2

sin2

akVadkkdE

kv xssx

xnxn sin2

1

Periodic!


The Case of No Scattering: Bloch Oscillations

atkt

EaeVatv

dttdx

xo

ssn 0sin2

A periodic velocity means that the electron motion in real space is also periodic:

00 txTtxdtdttdxT

o

where the period T is:oEae

T2

xk

a

a

xk

a

a

xk

a

a

xk

a

a

xk

a

a

0t4T

t 2T

t 4

3Tt Tt

x0

0t 4T

t 2T

t

43T

t Tt

Reciprocal space:

Real space:

13


Conductivity of Electrons in Graphene

yk

xk

22

K

oyyoxxpc

o

kkkkvEkE

k

Conduction band dispersion

k

kv

kk

kkv

kkkk

ykkxkkvkEkv

o

o

oyyoxx

oyyoxxckc

ˆˆ122

The dynamical equation for the crystal momentum still works:

ktkEe

dttkd

Ee

ktk


Conductivity of Electrons in Graphene

xk

yk

Electron distribution in k-space when E-field is zero

xk

yk

Electron distribution is shifted in k-space when E-field is not zero

xEE x ˆ

Ee

ktk

Distribution function: kfc

Distribution function:

E

ekfc

Ee

kfc

ok

ok

Energy

k

ok

kvEe

kfkd

eJok

near2

2

222

Current density can be obtained by the familiar expression:2 spins2 pockets or valleys

o

oc

kEek

kEekvtkv

Velocity magnitude remains the same but the velocity direction changes

Handout 16 Electrical Conduction in Energy Bands · 2015. 3. 25. · Electrical Conduction in Energy Bands In this lecture you will learn: • The conductivity of electrons in energy

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