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I have the following dispersion relation:

$$\epsilon(\vec{k})=\frac{\hbar}{2}\left(\frac{k_x^2}{m_1}+\frac{k_y^2}{m_2}-\frac{k_z^2}{m_3}\right)$$

(note the minus sign in the third term). And I am asked to calculate the density of states $g(\epsilon)$ in the vicinity of the saddle point.

I was thinking in doing the following. I calculate the volume of the surface defined by $\epsilon=\,$constant, and the result will give me the number of $\vec{k}$ vectors inside this volume. Then I differentiate the volume, wich will be a function of the energy, with respect to the energy $\epsilon$. And this will give me the number of allowed $\vec{k}$ vectors inside the shell of thickness $d\epsilon$. Then I divide the result by the volume occupied by a single $\vec{k}$ vector and finally by the volume.

The problem is, the surface will be a hyperboloid (because of the minus sign). If it were a plus sign, it would be an elipsoid and the volume is easy calculated. Back to the hyperboloid: if I integrate over the entire $k$-space, the integral would diverge, right? So over which region (I know that it is in the vicinity of the saddle point) must I integrate? Is it over the first Brillouin zone?

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  • $\begingroup$ Regarding your question about the integration: as it is asked for the vicinity of the saddle point, you just have to integrate from $0$ to $E_F$, the Fermi level. First start by calculating $|\nabla E|$ as you will need it in $g(E)$, given by(generally): $g\propto \int \frac{dS}{8\pi^3} \left|\frac{\partial \mathbf{k}}{\partial E(\mathbf{k})}\right|$ $\endgroup$ – Phonon Dec 3 '14 at 1:39
  • $\begingroup$ @Phonon: why from 0 to the Fermi level? $\endgroup$ – Larara Dec 3 '14 at 1:53
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    $\begingroup$ Ah I may have mixed this with $c_v$ calculations possibly, where we integrate from $0 \to E_F.$ One possibility of saddle point is always the boundary of the 1st BZ, i.e. $k_{max}=\pi/a$, with this link you should be able to get started. $\endgroup$ – Phonon Dec 3 '14 at 2:04
  • $\begingroup$ If you're asked for the density of states in the vicinity of the saddle point that suggests that you are meant to impose some constraint on which $k$s you consider. My guess is that you are meant to consider the density of states along a path passing through the saddle point for each $\epsilon$. This would avoid the unboundedness of the dispersion relation. $\endgroup$ – By Symmetry Dec 3 '14 at 2:46
  • $\begingroup$ @BySymmetry: along a path or over an area? And will this really give me the density of states (because we're not integrating over the entire $k$-space)? $\endgroup$ – Larara Dec 3 '14 at 3:00

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