Effect of Temperature on Fermi Function

EFFECT OF TEMPERATURE ON FERMI FUNCTION

The effect of temperature on Fermi function F (E) can be discussed with respect to equation (1)

(I) At 0 Kelvin

At 0 kelvin, the electrons can be filled only upto a maximum energy level called Fermi energy (E_F0), above EF0 all the energy levels will be empty. It can be proved from the following conditions.

(i) When E<E_F, equation (1) becomes

 [i.e 100% chance for the electron to be filled within the Fermi energy level]

(ii) When E> E_F, equation (1) becomes

 [(i.e) Zero % chance for the electron not to be filled within the Fermi energy level]

(iii) When E=E_F, equation (1) becomes

 [(i.e) 50% chance for an electron to be filled and not to be filled within the Fermi energy level]

This clearly shows that at 0 Kelvin all the energy states below E_F0 are filled and all those above it are empty.

The Fermi function at 0 kelvin (E_F0) can also be represented graphically as shown in Fig. 1.10. It is seen from the figure that the curve has step-like character at 0 kelvin.

(i.e) F (E) = 1 below E_F0 and

and F (E) = 0 above E E_F0

(II) At any temperature T kelvin

When the temperature is raised slowly from absolute zero, the Fermi distribution function smoothly decreases to zero as shown in Fig. 1.11.

Explanation: Due to the supply of thermal energy the electrons within the range of K_BT below the Fermi level (E_F0) alone takes the energy =K_BT and goes to higher energy state. Hence at any temperature (T), empty states will also be available below E_F0.

Note: Electrons lying very deep (far below) the Fermi energy level (E_F0) will not go to excited state

Also under classical limit

When K_B T<< E_F; F(E) is said to be degenerate function.

K_B T>>EF; F(E) is said to be Non-degenerate function.