Density of states in anisotropic conduction band valley

[johng23]

I need to calculate the density of states for a dispersion relation which is like the free electron dispersion, but with one effective mass in the kx, ky directions, and a different effective mass in kz. So I need to integrate the inverse gradient of E(k) over a surface of constant energy, ie and ellipsoid. I am confused about how to perform this integral.

Looking up some mathematical results on ellipsoids, it seems that you need to define some elliptic integrals which don't have an analytic solution. But I also found a PDF from some other semiconductor course which outlines this exact problem, and then gives a simple result that the density of states is proportional to sqrt(m*1 m*2 m*3) for the case where you have a general ellipsoid. Am I missing some major simplifying trick in the calculation?

Thanks.

 

[johng23]

Well, I found in another thread that you need to express the gradient in terms of the energy, and then you can integrate over the constant energy surface. So I got as far as expressing the gradient as something of the form $$\sqrt{aE+b|k|^{2}}$$, then I need to integrate the inverse of that over the ellipsoid of constant energy. Couldn't make any progress there. The assignment is over now, so it doesn't matter, but if anyone knows how you do this I would like to know, because I don't expect to get a solution set for this class.