How to Derive the Density of States in Anisotropic Conduction Bands?

[cscott]

Homework Statement

have band dispersion

$$\epsilon = \epsilon_c + \frac{h^2 k_x^2}{2 m_x} + \frac{h^2 k_y^2}{2 m_y} + \frac{h^2 k_z^2}{2 m_z}$$

Show density of states is

$$g(\epsilon) = \frac{m^{3/2}}{\pi^2 h^2} \sqrt{2|\epsilon - \epsilon_c|}$$

Homework Equations

$$2 \frac{d\vec{k}}{(2\pi)^3} = g(\epsilon) d\epsilon$$

or

$$g(\epsilon) = \frac{1}{4 \pi^3} = \int \frac{|d\vec{S}|}{|(grad)_k \epsilon|}$$
over a constant energy surface where the gradient is always perpendicular to the surface.

The Attempt at a Solution

For the first method I don't know how to solve for total magnitude of k so I can't procede.

Second method I get what appears to be a "not nice" integral, where

$$|(grad)_k \epsilon| = \sqrt{ \frac{h^4 k_x^2}{m_x^2} + \frac{h^4 k_y^2}{m_y^2} + \frac{h^4 k_z^2}{m_z^2}}$$

I don't see how to do this nicely in spherical coords over the Fermi surface.

 

[fzero]

You can rewrite $$|\nabla_k \epsilon|$$ in terms of $$\epsilon-\epsilon_c$$. Then remember that the integral that you have to do is over a constant energy surface.