Heat capacity from dispersion relation

[dark_matter_is_neat]

Homework Statement

At low temperatures, a 2-d d-wave superconductor can be described as a gas of non-interacting fermions that follow dispersion relation $E(k) = \sqrt{a^{2}k_{x}^{2}+b^{2}k_{y}^{2}}$ where a and b are positive constants. The fermion number is not conserved. Determine how the specific heat of the system depends on temperature.

Relevant Equations

$U = \int_{0}^{\infty} D(E)*E*\frac{1}{e^{\frac{E}{k_{B}T}} + 1} dE$
$C = \frac{dU}{dT}$

For me the part of the problem that is giving me issues is obtaining the density of states, since typically how you would calculate D(E) as D(E) = $\frac{A}{2 \pi} *k*\frac{dk}{dE}$ but this shouldn't work since this assumes angular symmetry in k space which this dispersion relationship doesn't have. This dispersion relation essentially makes an ellipse in k space so the density of states should be $D(E) = \frac{A}{4 \pi^{2}} * \frac{dk_{x}dk_{y}}{dE}$ which I'm not really sure how to calculate. Once I get the density of states it should be pretty trivial to get the temperature dependence of the heat capacity, since after appropriate usage of u substitution the expression for U will work out to just be a constant times some power of T.

 

[dark_matter_is_neat]

Since the problem is really just asking about the temperature dependence of the heat capacity I don't really need to consider the pre-factors, So rewriting $k_{x}$ and $k_{y}$ in terms of k as if in polar coordinates, $E = k\sqrt{a^{2}cos^{2}(\phi) + b^{2}sin^{2}(\phi)}$. So E is proportional to k and thus $\frac{dk}{dE}$ is just some constant and won't add any temperature dependence. From the k-space area element, you get k times some constant and since E is proportional to k, you get E times some constant.

So in the integrand for U you get a factor of $E^{2}$, so when substituting the variable of integration for $\frac{E}{k_{B}T}$, you get a factor of $T^{2}$. After this substitution the integral will just be a constant, so U will just be some constant times $T^{2}$.

$C = \frac{dU}{dT}$, so $C = \frac{d}{dT}(constant*T^{2})$, so C is proportional to T.