Problem: Free Energy with Gap at the Fermi Energy

   

Consider a ``generic" density of states which describes an energy gap in the spectrum of electrons:

 

where the gap between the lower and upper band is , and the density of states of the unperturbed system can be approximated by a constant, , where V is the volume of the system. The gap opens at the Fermi energy .

(a) Calculate, at zero temperature, the leading term in the change of the total energy of the electrons as is increased from zero to a finite value.

  (b) Investigate the contribution of the TS term in the free energy, F=E-TS, at finite but small ( ) temperatures.