fermi dirac distribution & quantum theory

8/3/2019 Fermi Dirac Distribution & Quantum Theory

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AUTONOMOUS


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` In a metallic crystal the free electrons posses different

energies except the restriction put forward by Pauli's

exclusion principle.

` According to quantum theory, at absolute zero, the freeelectrons occupy different energy levels continuously without

any vacancy in-between filled states.

` This can be understood by dropping the free electrons of a

metal one by one into the potential well.

` The first electron dropped would occupy the lowest available

energy, Eo (say),and the next electron dropped also occupy

the same energy level.

` The third electron dropped would occupy the next energy

level. That is the third electron dropped would occupy the

energy level E1 (>E0) and so on because of Pauli's exclusion

principle

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` If the metal contains N(even) number of electrons, they will be

distributed in the first N/2 energy levels and the higher energy

levels will be completely empty as shown in fig. below,

3

E

0

E

1

EF 0


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` The highest filled level, which separates the filled and empty

levels at OK is known as the Fermi level and the energy

corresponding to this level is called Fermi energy (EF).

` Fermi energy can also be defined as the highest energy

possessed by an electron in the material at 0K .At 0K the

Fermi energy EF is represented as EF0.

` As the temperature of the metal is increased from 0K to TK,

then those electrons which are present up to a depth of KBT

from Fermi energy may take thermal energies equal to KBT

and occupy higher energy levels.

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Whereas the electrons present in the lower energy levels i.e.,

below KBT from Fermi level, will not take thermal energies

because they will not find vacant electron states.

The probability that a particular quantum state at energy E is

filled with an electron is given by Fermi-Dirac distribution function

f(E), given by

)/)exp((1

1)(

TKEEEf

BF!


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` At T=0K , the curve has step like character with f(E)=1 for

energies below EF0 and f(E)=0 for energies above EF0 . This

represents that all the energy states below EF0 are filled withelectrons and all those above it are empty.

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A graph is plotted between f(E) and E, at different temperatures

T1K, T2K, T3K is shown in fig.

F(E)

Energy (E)

p

E f

0


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As the temperature is raised from absolute zero to T1K, the

distribution curve begins to departs from step like function and

tails off smoothly to zero. Again with further increase in

temperature to T2Kand to T3K, the departure and tailing ofthe curves increases.

This indicates that more and more electrons may occupy

higher energy states with increase of temperature and as a

consequence number of vacancies below Fermi Level

increases in the same proportion.

At non zero temperatures, all these curves passes through

a point ,whose f(E) =1/2,at E=EF. So EF lies half way betweenthe filled and empty states.


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` According to classical theory, the free electrons in a metal haverandom motions with equal probability in all directions. Butaccording to quantum theory the free electrons occupy differentenergy levels , up to Fermi level at OK.

` So they posses different energies and hence they possesdifferent velocities. The different velocities of these free electronsof a metal can be seen in velocity space.

` At OK, the electrons present in Fermi level possess maximumvelocity, represented as VF , We assume a sphere of radius VF atthe origin of velocity space as shown in fig. below,

9

0

VF

VZ

VX

VYW

hen, E=0


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Each point inside the sphere represent velocity of a free electron.

This sphere is called Fermi sphere. The Fermi surface need not

always be spherical.

The vectors joining different points inside the sphere from origin

represent velocity vectors.

In the absence of external electric filed the velocity vectors

cancel each other in pair wise and the net velocity of electrons inall directions is zero.

Now if we apply an external electric field (E) along X- Direction

on these electrons , Then a force eE acts on each electron

along negative X-direction. Only those electrons present near

the Fermi surface can take electrical energy and occupies

higher vacant energy levels.

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For rest of the electrons the energy supplied by electrical force is

too small so they unable to occupy higher vacant energy levels .

Hence the electric field causes the entire equilibrium velocity

distribution to be shifted slightly by an amount in the opposite

direction to the field as shown fig .below,

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0

VZ

VX0VF

0

VY

E


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In Quantum theory, the velocity of a free electron can be

represented as

..(1)

Where and =Propagation or wave vector.

Differentiating equation (1) with respect to time gives acceleration

(a)

..(2)

m

Kv J!

T2

h!J

P

T2!K

dt

dK

mdt

dva

J!!

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The force on an electron due to applied electric field is eE, this

is equated to the product of mass and acceleration of the electron.

Hence

(or) .(3)

(or) . (4)

Integrating equation (4)gives

eEdt

dK!J

dteE

dKJ

!

J

eEtKtK ! )0()(

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(5)

Let the mean collision time, mean free path of a free electron

present at Fermi surface is represented as

F

F

F V

PX !


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For an electron at Fermi level, consider and

(t)-K (0)= K in equation (5)

Then (6)

Using Equation (6)

The applied electric field enhances the velocity of electrons

present near the Fermi level. The increase in velocity

causes current density (J) in the material, given by

.(7)

Ft X! (

!!(

F

FF

V

eEeEK

PX

JJ

)( v(

)( vneJ (!

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Where n is the number of electrons that participate in conduction per

unit volume of metal. Using equation (1)

The value of V is substituted in equation (8), we have

.(8)

Where m* is the effective mass of free electron.

Substituting equation (7) in equation (9) gives

..(9)

*m

KneJ

(!J

EVm

ne

V

eE

m

neJ

F

F

F

F

!!

PP

**

2

J

J

(


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From Ohms law

,W

here = Electrical conductivity.

So, (10)

Using equation (11) electrical conductivity of a metal can be calculated.

A Similar equation may be obtained from the band theory for electrical conductivity

as

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F

F

F

m

ne

Vm

neX

PW

**

22

!

!

F

eff

m

enXW

*

2

! (11)


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Where is the effective number of electrons per unit volume of

material.

Thus in case of quantum theory the electrical conductivity is due

to the electrons which are very close to Fermi surface only.

This expression is in agreement with experimental conclusions.

effn

fermi dirac distribution & quantum theory

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