Band structure effects on heat capacity I am looking for resources or explanations how the electronic band structure influences the heat capacity. I can only find explicit calculations for free fermions etc., but I am wondering what effects would flat bands, linear dispersions etc. have?
Thanks!
 A: I think you'll find your answer in all of the introductory solid state physics books (Kittel, Ashcroft & Mermin, etc).
Basically, irrespective of how you calculate the band structure, you can calculate the total energy of the electrons in your material as:
\begin{equation}
E=\int^\infty_0 \epsilon D(\epsilon) f(\epsilon,T) d\epsilon
\end{equation}
where $D(\epsilon)$ is the energy-dependent density of states and $f(\epsilon,T)$ is the temperature-dependent probability that a state at a given energy will be filled (i.e. the Fermi-Dirac distribution for electrons).  Then the differential electronic heat capacity can be calculated as
\begin{equation}
C_{el}=\frac{dE}{dT}=\int^\infty_0 \epsilon D(\epsilon) \frac{df(\epsilon,T)}{dT} d\epsilon.
\end{equation}
Similarly, the change in the energy when the material is heat from 0 to T is
\begin{equation}
\Delta U=E(T)-E(0).
\end{equation}
Without actually doing any integrals, or working with any real material band structures, we can make some observations.  First of all, flat bands have larger densities of states, so a change in temperature will correspond to a larger change in heat, meaning $C_{el}$ is larger than for steeper bands.  Relatedly, linear bands have a larger density of states at low energies than parabolic bands, but it switches at higher energies, so for this comparison it depends on the doping, etc.  Also, it depends on the dimensionality of the material (a parabolic 2D band structure has different density of states than in 1D or 3D).   This integral for the temperature-dependent heat capacity is tractable to solve for some basic band structures (i.e. linear or parabolic), and it's even easier to do numerically, so I recommend trying it out.
Then you can move on to phonon heat capacity, etc!
Edit based on comment suggestion:
Of course, above I assumed that the density of states is constant with temperature, but in general it is not.  For instance, band gaps change a little with temperature (generally getting smaller with increasing temperature since atoms are on average farther apart and bound together more weakly), and if the material is near a phase change, that would also have a large effect on $dE/dT$.  So to improve the calculation, simply apply the chain rule:
\begin{equation}
C_{el}=\frac{dE}{dT}=\int^\infty_0 \epsilon \left[ D(\epsilon,T) \frac{df(\epsilon,T)}{dT} + \frac{dD(\epsilon,T)}{dT} f(\epsilon,T) \right] d\epsilon.
\end{equation}
But usually, $dD/dT$ is small enough to ignore.
