Electron and phonon contribution dependence on temperature for heat capacity

[PhysForumID]

Hi there!
So I'm aware that the phonon contribution is proportional to T^3 and for electrons it is T (this is for metals where the first result comes from the Debye model). I was wondering where the electron contribution is derived from and why it is such a low dependence.

 

[kanato]

The T comes from the Sommerfeld model. Keep in mind that these are valid at *low* temperatures, so in that situation the electron contribution is larger than the phonon contribution (ie. T > T^3 for small T).

 

[kloptok]

The contribution from conduction electrons to the heat capacity can be calculated by considering the conduction electrons as a perfect quantum gas of electrons, which are fermions and thus obey Fermi-Dirac statistics. Calculating the energy of this gas and taking the derivative of the energy wrt temperature you will find that the heat capacity has a T-dependence. (This is the so called Free Electron Model, not entirely correct since electron interactions are ignored. An improvement is the Nearly Free Electron Model which takes lattice interactions into account by modeling the lattice as a periodic potential that the electrons move in, giving rise to for example the energy bands.)

A guess as to why the electronic part is much smaller is that the Fermi temperature is very high in most metals (typically $$T_F \approx 10^4$$ K), and in the expression for the heat capacity I think you have a term $$T/T_F$$ which is vanishingly low for most temperatures. Also, for temperatures not very small, $$T^3 >> T$$.