Band Theory of Soids5!1!13

7/29/2019 Band Theory of Soids5!1!13

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IntroductionBloch stated this theory in 1928. According to this theory, thefree electrons moves in a periodic field provided by the lattice.

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.

Band Theory of Solids


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Electron in a periodic potential

( Bloch Theorem )

A crystalline solid consists of a lattice which is composed of a

large number ofpositive 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 Blochtheorem.

The periodic potential V (x) may be defined by means of the

lattice constant a as V (x) = V ( x + a )


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One dimensional periodic potential in crystal.

a+++ +

0V

+ +++++

+ +++++

Periodic positive ion cores Inside metallic crystals.


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electrontheofmotiontheofstatetherepresentskwherea)(xU(x)U

lattice.crystalaofyperiodicitwithperiodicais(x)UWhere

)(x)exp(ikxU(x)

equationrSchrodingetheofsolution

ldimensionaonetheshown thathasBloch

0a)]V(x[Eh

m8

dx

d

0V][Eh

m8

dx

d

equationr waveSchrodingeFrom

kk

k

k

2

2

2

2

2

2

2

2


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KroningPenney Model

According to Kroning - Penney model the electrons move in a

periodic potential field provided by the lattice.

The potential of the solid varies periodically with the periodicity of

space lattice.

Consider Schrdinger equation for this case, we can find the

existence of the energy gap between the allowed values of energy

of electron.

For one dimensional periodic potential field.

0)]([8

2

2

2

2

xVEh

m

dx

d


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+++++

V0

0 a-b

12

0][8

)1(

2

2

2

2

Eh

m

dx

dregionfor

0][8

)2(

02

2

2

2

VEh

m

dx

dregionfor

)2.....(0

)1.....(0

2

2

2

2

2

2

dx

d

dx

d2

2

2

2

2

02

8

8[ ]

where

mE

h

mV E

h

0,)(

0,0)(

0

xbvxv

axxv


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.....(a)(x)eU(x)ikx

k

)(x).....(bUa)(xUkk

.....(c)(x)eNa)(xikNa

kk

According to Bloch, the solution of a Schrodinger equation

Where Uk(x) is the periodicity of the lattice i.e,.

According to Bloch theorem

By using above a, b, and c Bloch conditions, the solutions of

equations (1) & (2) becomes


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2mEh

2

strengthBarriercalledisbV

barrierpotentialtheofpowerscatteringtheisp

bV

h

ma4where..P

acosa

asinPcoska

0

02

2


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+1

-1

+1

-1

a

ac osa

asineP


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Conclusions

1. The motion of electrons in a periodic lattice is characterizedby the bands of allowed energy separated by forbiddenregions.

2. As the value ofa increases, the width of allowed energy

bands also increases and the width of the forbidden bandsdecreases. i.e., the first term of equation deceases on theaverage with increasing a .

3. Let us now consider the effect of varying barrier strength P. if

V0b is large ,i.e. if p is large ,the function described by the lefthand side of the equation crosses +1 and -1 region as shownin figure. thus the allowed bands are narrower and theforbidden bands are wider.


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a0

p

If P tends to infinite the allowed band reduces to one

single energy level


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0p

a

4. If P tends to zero

No energy levels exist:

all energies are allowed to the electrons.

22

2

22

2

2

2

2

2

22

2

22

22

2

1

2)

2(

1

)2(

)2

)(8(

)2(

2

coscos

mvm

p

h

p

m

hE

m

h

E

m

hE

kmE

mEk

k

k

kaa


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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.


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Allowedbands

Energy gap

First

Brillouin zone

E

k

Energy gap

a

a

2

a

3

a

a

2

a

3

E-k Diagram


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Origin of Energy band formation in Solids

When we consider isolated atom, the electrons are tightly

bound and have discrete, sharp energy levels.

When two identical atoms are brought closer the outer mostorbits of these atoms overlap and interact.

If more atoms are brought together more levels are formed andfor a solid of N atoms , each of the energy levels of an atomsplits into N levels of energy.

The levels are so close together that they form an almostcontinuous band.

The width of this band depends on the degree of overlap ofelectrons of adjacent atoms and is largest for outer most

atomic electrons.


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N energy levels

N atoms

E

E1

E2

E3

E2

E1

E1


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The energy bands in solids are important indetermining many of physical properties of solids.

The allowed energy bands

(1) Valance band

(2) Conduction band

The band corresponding to the outer most orbit iscalled conduction band and the next inner band iscalled valence band. The gap between these twoallowed bands is called forbidden energy gap.


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Classifications of materials into Conductors,Semiconductors & Insulators:-

On the basis of magnitude of forbidden band thesolids are classified into insulators, semiconductors

and conductors.

Insulators:

In case of insulators, the forbidden energy band is

very wide as shown in figure.

Due to this fact the electrons cannot jump from

valance band to conduction band.

In insulators at 00k and the energy gap between

valance band and conduction band is of the order.


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Forbidden gap

Valance band

Conduction band

INSULATORSForbidden gap

Valance band

Conduction band

SEMI CONDUCTORS

Valance band

Conduction band

CONDUCTORS


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SEMI CONDUCTORS

In semi conductors the forbidden energy ( band ) gap is very smallas shown in a figure.

Ge and Si are the best examples of semiconductors.

Forbidden ( band ) is of the order of 0.7ev & 1.1ev.

CONDUCTOS:

In conductors there is no forbidden gap. Valence and conduction

bands overlap each other as shown in figure above.The electrons from valance band freely enter into conduction band.

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Band Theory of Soids5!1!13