Solving Solid State Physics Problem with Strong Coupling Approximation

[paullondon]

Good evening ,


I will resolve a small problem but I don't know how I can do,

In l approximation of the strong coupling, the strucure of band of the electrons in a certain crystal is given by: E(k)=-[E1*cos(kx.a)+E2*cos(ky.b)+E3*cos(kz.c) ] Show that for a band very little filled, the electronic specific heat of electronic gas in this crystal is equivalent to the electronic specific heat such calculated for free electrons but with an effective mass m*=|detM|1/3 or M is the tensor of effective mass at least evaluated band is [ M-1 ]ij=(1/h²)*(d²E(k))/(dki dkj)

thank you for your assistance

 

[matteo83]

The concept of effective mass allows you to consider electrons in conduction band or holes in valence band as totally free particles. Since effective mass, as you can see by its definition, includes lattice periodic potential effects on carriers (E(K) depends on V(r)). So if carriers densities are smaller than effective densities of states of bands (very little filled band = rarefied gas) the system is equivalent to an ideal gas. Therefore you can computes specific heat like for an ideal gas.

Landau, Lif****z, "Statistical Physics" ,chapt.45. pp132

Matteo.