1. FERMI-DIRAC
DISTRIBUTION

2. 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 free electrons 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

3. 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,
EF 0 E1 E0

4. 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.

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

6. A graph is plotted between f(E) and E, at different temperatures T1K, T2K, T3K is shown in fig.
E f 0
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 with electrons and all those above it are empty.
Energy (E) 

7. Fermi Function at T=0 and at a finite temperature
E<EF
E>EF EF
E<EF
E>EF
0.5
fFD(E,T) E

8. Fermi-Dirac distribution: Consider T  0 K
For E > EF :
For E < EF : E EF
f(E)

9. Temperature dependence of Fermi-Dirac distribution

10. 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 T2K and to T3K, the departure and tailing of the 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 between the filled and empty states.

11. QUANTUM FREE ELECTRON THEORY OF ELECTRICAL CONDUCTION

12. According to classical theory, the free electrons in a metal have random motions with equal probability in all directions. But according to quantum theory the free electrons occupy different energy levels , up to Fermi level at OK.
So they posses different energies and hence they posses different velocities. The different velocities of these free electrons of a metal can be seen in velocity space.
At OK, the electrons present in Fermi level possess maximum velocity, represented as VF , We assume a sphere of radius VF at the origin of velocity space as shown in fig. below, VZ VF VX
When, E=0 VY

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

14. 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, E VZ VF VX VY

15. 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)

16. 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)
Integrating equation
(or) …………. (4)

17. ……………………(5)
Let the mean collision time, mean free path of a free electron present at Fermi surface is represented as

18. 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)

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

20. From Ohm’s law
, Where = 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
……………………(11)

21. 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.