Scaling of C trick Sommerfield Model

Physics Q&A
1

In Lecture 4 about the Sommerfield Model there is a piece about a useful trick of scaling the proportionality between C and T.
Two separate examples are given however I don’t see why we consider g(\varepsilon_f) as “roughly constant” in the 3D free electron example but we don’t do the same for the density of states in the graphene model. The only difference I can come up with is that:

  • For free electrons in 3D g(\varepsilon) \propto \sqrt{\varepsilon} while for graphene we have g(\varepsilon) \propto \varepsilon.

But won’t both these densities of states be constant for some value \varepsilon_f? So how come one system has C_e \propto T and the other has C_g \propto T^2

2

The difference is that in graphene E_F = 0, while in a metal E_F ≫ k_BT.

3

But I still don’t really see why this gives C \propto T^2 instead of just T because wouldn’t g(\varepsilon) just also be a constant in that case?