So we have been given a dispersion relation of the form: $$ E=6-2(\cos k_xa+\cos k_ya) $$ and asked to calculate the density of states.

The equation for the density of states is (eq 2.48 from here http://www.tcm.phy.cam.ac.uk/~pbl21/princ-condmat/v2.pdf):

$$ g(E)=\int \frac{ds}{2\pi^2}\frac{1}{|\nabla_k E(\mathbf{k})|} $$

where we integrate over a surface (or curve in 2d) of constant energy.

Let's work out the bits individually:

  • $|\nabla_k E(\mathbf{k})|=|\left(2a\sin (k_xa),2a\sin (k_ya)\right)|=2a\sqrt{\sin^2k_xa+\sin^2k_ya}$

  • $ds=\sqrt{dk_x^2+dk_y^2}$ where $dE=0=2a\sin (k_xa) dk_x+2a\sin (k_ya) dk_y$ and hence we get $ds=\sqrt{\sin^2k_xa+\sin^2k_ya} \frac{dk_x}{\sin(k_ya)}$

  • The bound of the integral are $0$ and $\cos^{-1}(3-E/2)$ where E goes from 2 to 10.

  • the total integral is then


I then put this into mathematica and got either errors or complex answers


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.