Increase in thermal energy of system of N free electrons when their temperature is increased from 0 K to T K is
△ U = U(T) - U(0)
where f(ε) = fermi Dirac distribution function
D(ε) = density of state i.e of orbitals per unit energy range
Total number of free electrons is given by the expression
Multiplying both sides by Ef
With the help of equation 2, equation 7 can be written as
Here the first term gives the energy required to tale the electron from εf to the orbital of energy ε> εf and the second term gives the energy required to bring the electrons from the orbitals with energy ε <εf to the εf.
Now, the electronic heat capacity (C el ) = d ( ∆ U) /dT
The graph of df(ε)/ dT versus ε shows that df(ε)/ dT is significant only at ε = εf. So, it is good approximation to evaluate D(ε) at εf.
With this value, equation 4 becomes
This equation tells us that electrons heat capacity at low temperature is directly proportional to the absolute temperature. This equation shows that the heat capacity of the electrons is proportional to the temperature . At low temperatures, the heat capacity approaches a constant value, which is proportional to the number of free electrons. This is because at very low temperatures, the electrons occupy the lowest available energy levels up to the Fermi energy, and the number of occupied energy levels does not change with temperature.
It is important to note that this equation only applies to the electronic contribution to the heat capacity, and other contributions from lattice vibrations or impurities may need to be taken into account for a more complete calculation.
This note is a part of the Physics Repository.