6 Band Theory of Solids

Band Theory of Solids

AUTONOMOUS

INTRODUCTION

• Bloch stated this theory in 1928. According to this theory, the free electrons moves in a periodic field provided by the lattice. This theory is also called Band theory of solids.

• The energy band theory of solids is the basic principle of semiconductor physics and it is used to explain the differences in electrical properties between metals, insulators and semiconductors.

Electron in a periodic potential – Bloch theorem

• A crystalline solid consists of a lattice which is composed of a large number of positive ion cores at regular intervals and the conduction electrons move freely throughout the lattice.

• The variation of potential inside the metallic crystal with the periodicity of the lattice is explained by Bloch theorem.

+ + + + ++ +

+ + + + ++ +

+ + + + ++ +

+ + + + ++ +

+ + + + ++ +

• The potential of the solid varies periodically with the periodicity of space lattice and the potential energy of the particle is zero near the nucleus of the +ve ion in the lattice and maximum when it is half way between the adjacent nuclei which are separated by interatomic spacing distance ‘a’.

X

V

V

One dimensional periodic potential in crystal.

Periodic positive ion cores Inside metallic crystals.

+ + + + ++ +

+ + + + ++ +

+ + + + ++ +

+ + + + ++ +

+ + + + ++ +

Bloch’s Theorem• Bloch’s Theorem states that for a

particle moving in the periodic potential, the wavefunctions ψ(x) are of the form

• uk(x) is a periodic function with the periodicity of the potential– The exact form depends on the potential

associated with atoms (ions) that form the solid

)()(

function periodic a is )( ,)()(

axuxu

xuwhereexux

kk

kikx

k

• The one dimensional Schrödinger equation

• The periodic potential V(x) may be defined by means of the lattice constant a as V(x)=V(x+a)

0][8

2

2

2

2

VE

h

m

dx

d1

0)]([8

2

2

2

2

axVE

h

m

dx

d

Bloch has shown that the one dimensional solution of the Schrödinger equation is

)()exp()(

3

)()exp()(

rUikrr

DIn

xUikxx

kK

kk

lattice. crystal a ofy periodicit with periodic a is (x) UWhere k

2

I

• Let us consider a linear chain of atoms of length L in one dimensional case with N number of atoms in the chain

• This is refered to as Bloch condition.Similarly, the complex conjugate of eq(4)

)4.().........exp()()(

)exp()()exp()(

)(exp{)()(

)3....().........()(

ikNaxNax

ikxxUikNaNax

NaxikNaxUNax

NaxUxU

kk

kk

kk

kk

)()()()(

)23()4(

)5).......(exp().()(

**

*

xxNaxNax

andFromEq

ikNaxNax

kkkk

kk

• This means that the electron is localized around any particular atom and the probability finding the electron the electron is same throughout the crystal .

Bloch’s Theorem

The probability of finding an electron at any atom in the solid

is the same!!!

• Each electron in a crystalline solid “belongs” to each and every atom forming the solid

)()( axVxV

• Behaviour of an electron in a periodic potential:(The Kronig-Penny Model):

• This model treats the potential found in actual crystal to the point of getting an exact solution of the Schrödinger equation. It assumes that the potential energy of an electron in a linear array of positive nuclei has the form of a periodic array of square wells as shown in fig.

X=0 X=a

X=b

Potential barrier between the atoms.

We will eventually letV and b 0 in the problem.

The Kronig-Penney Model

U2(

x) U1(x)

x

V

• The potential energy is equal to zero in the regions 0<x<a, and in the potential V0 in the regions - b<x<0.Each of the potential energy wells may be considered..

• The wave functions associated with this model can be calculated by solving Schrödinger equations for the two regions:

2........00)(2

1..............002

022

2

22

2

xbforVEm

dx

d

axforEm

dx

d

• Let us define real quantities α and β by

• Now ,since the wave function must have Bloch form ,we may expect that

• Substituting eq (4) in eq(2) we get the following the equation for uk(x)

3......).........(;)(22

0202

22 VE

EVmand

mE

4.).........()( xUex kikx

axforukdx

duik

dx

ud 00)(2 1

22121

2

5

00)(2 22222

2

xbforkdx

duik

dx

ud

0

0)()(

2

)()(1

xbforDeCeu

axforBeAeuxikxik

xkixKi

6

The soln of these equations may be written as

7

Where A,B,C,D are the constants .These solutions must be subjected to the Following boundary condition

bxaxbxax

xxxx

dx

du

dx

duuu

dx

du

dx

duuu

212

0

2

0

10201

;)()(

;)()(

1

8

• The first two condition are imposed because of the requirement of continuity of the wave function Ψ and its derivative dΨ/dx at x=0,and hence of u and du/dx;the remaining two conditions are required because of the periodicity of uk(x).

• The application of these boundary condition to eq(7) leads to the following four linear homogenous equations involving the constants A,B,C,D:

• A+B=C+D

bikbikakiak DeCeBeAe

ikDikCkBikAi)()()()(

),()()()(

9

The coefficient A,B,C,D can be determined by solving these equation s,and Wave functions calculated.this leads to the following equations;

)(coscoscoshsinsinh2

22

baKabab

10

• This equation quite complicated ,.Kronig and Penny considered the possibility that Vo tends to infinity and b approaches zero in such a way that the product Vob remains finite .

• The quantity lim(Vob) representing the barrier strength.

• In this possibility , the equation (10) becomes

kaaaSinbmV

coscos2

0

If we define the quantity P by

20

bamV

p

11

• Eq (11) reduces to

Kaa

ap coscos

sin

12

This is the condition for the solutions of the wave equation to exist.

We see that this is satisfied only for those values of αa for which its Left hand side lies between +1and -1;this is because its right hand sideMust fall in this range .such values are represent the wave like solutions and are allowed.

•Consequence of this equationcan be understood with fig.

π

The Kronig-Penney Model

a

)cos()sin(

aa

aP

1

-1

Regions where the equation is satisfied, hence wherethe solution exists.

In general, as the energy increases (a increases), each successive band gets wider, and each successive gap gets narrower.

Boundaries are for αa = n.

No solutionexists, k2 < 0

0 2π 3π-π

-π

-2π

• The part of the vertical axis lying between the horizontal lines represents the range acceptable to the left-hand side

aa

ap

cossin

• Conclusions:

• **Allowed ranges of αa which permits a wave mechanical solution to exist are shown by the shadow portions. thus the motion of electrons in a periodic lattice is characterized by the bands of allowed energy separated by forbidden regions .

• ** As the value of α increase the width of the allowed energy bands also increase and the width of the forbidden band decreases.

• ** if the potential barrier strength P is large ,the function described by the right hand side of the equation crosses +1 and -1 region at steeper angle. Thus the allowed bands become narrower and forbidden bands become wider .

• If P tends to infinite the allowed band reduces to one single energy level :

a0

p

0p

a

If P tends to zero no energy levels exist, all energies are allowed to the electrons.

22

2

22

2

2

22

2

22

222

22

2

1

2)

2(

1)

2(

)2

)(8

(

)2

(

2

coscos

mvm

p

h

p

m

hE

m

hE

m

hE

km

E

mEk

k

k

kaa

Brillouin zones

• The Brillouin zone is a representation of permissive values of k of the electrons in one, two or three dimensions.

• Thus the energy spectrum of an electron moving in the presence of a periodic potential fields is divided into allowed zones and forbidden zones.

Allowed bands

Energy gap

First Brillouin zone

E

k

Energy gap

a

a

2a

3

a

a

2

a

3

E-k diagram :

Insulators, Semiconductors, Metals

• The last completely filled (at least at T = 0 K) band is called the Valence Band

• The next band with higher energy is the Conduction Band – The Conduction Band can be empty or

partially filed

• The energy difference between the bottom of the CB and the top of the VB is called the Band Gap (or Forbidden Gap)

Insulators, Semiconductors, Metals

• Consider a solid with the empty Conduction Band

• If apply electric field to this solid, the electrons in the valence band (VB) cannot participate in transport (no current)

Insulators, Semiconductors, Metals

• The electrons in the VB do not participate in the current, since– Classically, electrons in the

electric field accelerate, so they acquire [kinetic] energy

– In QM this means they must acquire slightly higher energy and jump to another quantum state

– Such states must be available, i.e. empty allowed states

– But no such state are available in the VB!

This solid would behave as an insulator

Insulators, Semiconductors, Metals

• Consider a solid with the half filled Conduction Band (T = 0K)

• If an electric field is applied to this solid, electrons in the CB do participate in transport, since there are plenty of empty allowed states with energies just above the Fermi energy

• This solid would behave as a conductor (metal)

Band Overlap• Many materials are

conductors (metals) due to the “band overlap” phenomenon

• Often the higher energy bands become so wide that they overlap with the lower bands– additional electron energy

levels are then available

Band Overlap• Example: Magnesium (Mg; Z =12):

1s22s22p63s2

– Might expect to be insulator; however, it is a metal

– 3s-band overlaps the 3p-band, so now the conduction band contains 8N energy levels, while only have 2N electrons

– Other examples: Zn, Be, Ca, Bi

Insulators, Semiconductors, Metals• There is a qualitative difference

between metals and insulators (semiconductors)– the highest energy band “containing”

electrons is only partially filled for Metals (sometimes due to the overlap)• Thus they are good conductors even at very low

temperatures• The resisitvity arises from the electron

scattering from lattice vibrations and lattice defects

• Vibrations increases with temperature higher resistivity

• The concentration of carriers does not change appreciably with temperature

Insulators, Semiconductors, Metals• The difference between Insulators and

Semiconductors is “quantitative”– The difference in the magnitude of the

band gap

• Semiconductors are “Insulators” with a relatively small band gap– At high enough temperatures a fraction

of electrons can be found in the conduction band and therefore participate in transport

Insulators vs Semiconductors

• There is no difference between Insulators and Semiconductors at very low temperatures

• In neither material are there any electrons in the conduction band – and so conductivity vanishes in the low temperature limit

Insulators vs Semiconductors

• Differences arises at high temperatures– A small fraction of the electrons is thermally

excited into the conduction band. These electrons carry current just as in metals

– The smaller the gap the more electrons in the conduction band at a given temperature

– Resistivity decreases with temperature due to higher concentration of electrons in the conduction band *

21mnq

Conduction

Electrical current for holes and electrons in the same direction