- #1
cscott
- 782
- 1
Homework Statement
have band dispersion
[tex]\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}[/tex]
Show density of states is
[tex]g(\epsilon) = \frac{m^{3/2}}{\pi^2 h^2} \sqrt{2|\epsilon - \epsilon_c|}[/tex]
Homework Equations
[tex]2 \frac{d\vec{k}}{(2\pi)^3} = g(\epsilon) d\epsilon[/tex]
or
[tex]g(\epsilon) = \frac{1}{4 \pi^3} = \int \frac{|d\vec{S}|}{|(grad)_k \epsilon|}[/tex]
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
[tex]|(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}}[/tex]
I don't see how to do this nicely in spherical coords over the Fermi surface.
Last edited: