handout 7 statistics in energy bands · 2017. 12. 22. · statistics in energy bands in this...

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ECE 407 – Spring 2009 – Farhan Rana – Cornell University

Handout 7

Properties of Bloch States and Electron Statistics in Energy Bands

In this lecture you will learn:

• Properties of Bloch functions

• Periodic boundary conditions for Bloch functions

• Density of states in k-space

• Electron occupation statistics in energy bands


Bloch Functions - Summary

• Electron energies and solutions are written as ( is restricted to the first BZ):

• The solutions satisfy the Bloch’s theorem:

and can be written as a superposition of plane waves, as shown below for 3D:

• Any lattice vector and reciprocal lattice vector can be written as:

• Volume of the direct lattice primitive cell and the reciprocal lattice first BZ are:

rkn

, and kEn

reRr knRki

kn

,

.,

j

rGkijnkn

jeV

Gkcr

.

,1

k

332211 bmbmbmG

3213 . bbb

332211 anananR

3213 . aaa

2


Bloch Function – Product Form ExpressionA Bloch function corresponding to the wavevector and energy band “n” can always be written as superposition over plane waves in the form:

k

j

rGkijnkn

jeV

Gkcr

.

,1

The above expression can be re-written as follows:

rueV

eGceV

eV

Gkcer

knrki

j

rGijkn

rki

j

rGijn

rkikn

j

j

,.

.,

.

..,

1

1

1

Where the function is lattice periodic: ru kn

,

ru

eGceGcRru

kn

j

rGijkn

j

RrGijknkn

jj

,

.,

.,,

satisfiesBloch’s theorem

re

Rr

knRki

kn

,.

,

rue

Vr kn

rkikn

,

.,

1Note that:


Allowed Wavevectors for Free-Electrons (Sommerfeld Model)

xLyL

zL zyx LLLV

We used periodic boundary conditions:

zyxLzyx

zyxzLyx

zyxzyLx

z

y

x

,,,,

,,,,

,,,,

The boundary conditions dictate that the allowed values of kx , ky , and kz, are such that:

z

zLki

yy

Lki

xx

Lki

Lpke

Lmke

Lnke

zz

yy

xx

21

21

21

n = 0, ±1, ±2,….

m = 0, ±1, ±2,….

p = 0, ±1, ±2,….

32V There are grid points per unit volume

of k-space

xL2

yL2

zL2

xk

yk

zk

3


Bloch Functions – Periodic Boundary Conditions

• Any vector in the first BZ can be written as:k

332211 bbbk

where 1 , 2 , and 3 range from -1/2 to +1/2:

Reciprocal lattice for a 2D latticeDirect lattice

1b

1a

2b

2a

k

21

21

1 21

21

2 21

21

3



• Consider a 3D crystal made up of N1 primitive cells in the direction, N2 primitive cells in the direction and N3 primitive cells in the direction

1a

2a

3a

Assuming periodic boundary conditions in all three directions we must have:

rreaNr aNki

11.11

rreaNr aNki

22.22

rreaNr aNki

33.33

Volume of the entire crystal is: 3321332211 . NNNaNaNaNV

1a

2a

22 aN

11 aN

221

2211

:crystal2DtheFor

NN

aNaNA

4



The periodic boundary condition in the direction implies:

1

11

111

1111

.

2. :that recall 22

integer an is 2.

111

Nm

bamN

mmaNk

e

kjkj

aNki

Since:21

21

1 221

11 N

mN

m1 can have N1 different integral values between –N1/2 and +N1/2

Reciprocal lattice for a 2D lattice

Direct lattice

1b

1a

2b

2a

1a

k 332211 bbbk


Bloch Functions – Periodic Boundary Conditions Similarly, the periodic boundary conditions in the directions of and imply:

3

33

2

22

333222

..

&

2.&2.

1&1 3322

N

m

Nm

maNkmaNk

ee aNkiaNki

m1 can have N1 different integral values m2 can have N2 different integral values m3 can have N3 different integral values

Reciprocal lattice for a 2D lattice

Direct lattice

1b

1a

2b

2a

2a

k

332211 bbbk

3a

Since any k-vector in the FBZ is given as:

there are N1 N2 N3 different allowed k-values in the FBZ

There are as many different allowed k-values in the FBZ as the number of primitive cells in the crystal

22&

223

332

22 N

mNN

mN

5


Density of States in k-SpaceReciprocal lattice for a 2D lattice

332211 bbbk

222

22

2

22

Nm

NNm

223

33

3

33

Nm

NNm

221

11

1

11

Nm

NNm

1

2a

2

2a

1b

2b

11

2aN

22

2aN

Question: Since is allowed to have only discrete values, how many allowed k-values are there per unit volume of the k-space?

k

Volume of the first BZ is:

3213 . bbb

• In this volume, there are N1 N2 N3allowed k-values

• The number of allowed k-values per unit volume in k-space are:

3

33

321

3

321

2

2

V

NNN

NNN

where V is the volume of the crystal

3D Case:


Density of States in k-Space

1b

2b

1b1D Case:

Length of the first BZ is:1

112

b

• In the first BZ, there are N1 allowed k-values • The number of allowed k-values per unit length in k-space are:

22 11

11

1 LN

N

Length of the crystal: 1111 NaNL

2D Case:

Area of the first BZ is:

2

2

2122

bb

• In the first BZ, there are N1N2 allowed k-values • The number of allowed k-values per unit area in k-space are:

222

212

21

22 A

NNNN

Area of the crystal: 2212211 NNaNaNA

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States in k-Space and Number of Primitive Cells

k

Energy

a

a

k

Energy

a

a

1D Case:

1b

a

a

Length of the first BZ is:a

b2

11

• In the first BZ, there are N1 allowed k-values • The number of allowed k-values per unit length in k-space are:

22 11

11

1 LN

N

Length of the crystal: aNNaNL 11111

LaN 22

1

Reciprocal lattice is:

There are N1 allowed k-values in k-spaceThere are N1 allowed k-values per energy bandThere are as many allowed k-values per energy band as the number of primitive cells in the entire crystal

xaa ˆ1


xk

yk

)0,(1a

),0(2a

Energy

FBZ

States in k-Space and Number of Primitive Cells2D Case:

1

2a

2

2a

1b

2b

11

2aN

22

2aN

Reciprocal lattice is:

• In the first BZ, there are N1N2 allowed k-values

There are N1N2 allowed k-values per energy band

There are as many allowed k-values per energy band as the number of primitive cells in the entire crystal

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Statistics of Electrons in Energy Bands

k

Energy

a

a

k

Energy

a

a

kE1

kE2

kE3

1D Case:

The number of allowed k-values per unit length in k-space is L / 2therefore:

Suppose I want to find the total number of electrons in the n-th band – how should I find it?

The probability that the quantum state of wavevector is in the n-th energy band is occupied by an electron is given by the Fermi-Dirac distribution:

k

TKEkEnfne

kf

1

1

Then the total number N of electrons in the n-th band must equal the following sum over all the allowed values in k-space in the first BZ:

FBZin all

2k

n kfN

spin

kfdk

LkfN n

a

akn

222

FBZ in all


FBZ in all

2k

n kfN

2D Case:

The number of allowed k-values per unit area in k-space is:

kfkd

AkfN nFBZk

n

2

2

FBZ in all 222

22A

Therefore:

Need to find the total number of electrons in the n-th band

3D Case:

The number of allowed k-values per unit volume in k-space is:

kfkd

VkfN nFBZk

n

3

3

FBZ in all 222

32V

Therefore:

Statistics of Electrons in Energy Bands

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Band Filling at T0K for a 1D lattice

k

Energy

a

a

k

Energy

a

a

Question: suppose we have 2 electrons per primitive cell. How will the bands fill up at T0K? Where will be the Fermi level?

2 electrons per primitive cell 2N1 total number of electrons

Number of k-values per band = N1Number of quantum states per band = 2xN1

spin First band will be completely filled. All higher bands will be empty

Question: Suppose we have 3 electrons per primitive cell. How will the bands fill up at T0K?

3 electrons per primitive cell 3N1 total number of electrons

First band will be completely filled. Second band will be half filled. All higher bands will be empty

Ef for 3 electrons per primitive cell



Suppose the number of primitive cells = N1


Band Filling at T0K for a 2D lattice

Question: suppose we have 2 electrons per primitive cell. How will the bands fill up at T0K? Where will be the Fermi level?

2 electrons per primitive cell 2N1N2 total number of electrons

Number of k-values per band = N1N2Number of quantum states per band = 2xN1N2

spin First band will be completely filled. All higher bands will be empty

Suppose the number of primitive cells = N1N2

Important lesson:In an energy band (whether in 1D, 2D or 3D) the total number of quantum states available is twice the number of primitive cells in the direct lattice. How the bands get filled depends on the number of electrons per primitive cell.

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Fermi Surfaces (3D) and Contours (2D) in Solids

xk

yk

Energy

EF

kF

Fermi circle for a free electron gas in 2D

What happens in solids when the energy bands are more complex?

xk

yk

Energy

FBZ

First energy band of a 2D lattice


xk

yk

Energy

FBZ

Fermi contours for different electron densities corresponding to the energy band shown on the left

FBZ

First energy band of a 2D lattice

xk

yk

xc

b ˆ2

11

xca ˆ11

yc

b ˆ2

22

yca ˆ22


FBZ

10



Fermi surface of a simple cubic direct lattice shown inside the first BZ

Fermi surface of a FCC lattice shown inside the first BZ (the figure shows the Fermi surface of Copper)


Band Filling at T0K for Silicon

Electron Configuration: 1s2 2s2 2p6 3s2 3p2

Atomic number: 14

• The electrons in the outermost shell can move from atom to atom in the lattice – they are not confined to any individual atom. Their energies are described by the energy bands

• The electrons in the inner shells remain confined to individual atoms

Number of electrons in the outermost shell: 4

Silicon:

• Silicon lattice is FCC

• There are 2 Silicon atoms per primitive cell (2 basis atoms)

There are 4 electrons contributed by each Silicon atom and so there are 8 electrons per primitive cell that are available to fill the energy bands

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2

2

1

1

1

1

Band Filling at T0K for Silicon• There are 8 electrons per unit cell available to fill the energy bands

• Recall that in each energy band the number of states available is twice the number of primitive cells in the crystal

• In Silicon, the lowest 4 energy bands will get completely filled at T0K and all the higher energy bands will be empty

Ef

1

1

1

1

FBZ (for FCC lattice)Silicon Energy Bands


2

2

1

1

1

1

Ef

1

1

1

1

FBZ (for FCC lattice)Silicon Energy Bands

Valence band

Conduction band

Energy Bands in Silicon• The highest filled energy band is called the valence band. In silicon the valence band is double degenerate at most points in the first BZ• The lowest empty energy band is called the conduction band• In energy, the valence band maximum and the conduction band minimum need not happen at the same point in k-space (as is the case in Silicon)• The lowest energy of the conduction band is called Ec and the highest energy of the valence band is called Ev

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k

Energy

a

a

xk

yk

Energy

FBZ

handout 7 statistics in energy bands · 2017. 12. 22. · statistics in energy bands in this...

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