- #1
Einj
- 470
- 59
Hi everyone. Suppose we consider an electron in a two dimensional lattice, whose dispersion relation is given by:
$$
\epsilon(k_x,k_y)=-J(\cos(k_x a)+\cos(k_y a)),
$$ and where the wave vectors belong to the first Brillouin zone ([itex]k_i\in [-\pi/a,\pi/a][/itex]).
In this case it turns out that the Fermi energy is zero and the Fermi surface is a square defined by:
\begin{align}
&k_y=-k_x+\pi/a\;\;\;\;\text{ for the first quarter of the k-space} \\
&k_y=k_x-\pi/a\;\;\;\;\text{for the second quarter} \\
&k_y=-k_x+\pi/a\;\;\;\;\text{for the third quarter} \\
&k_y=k_x+\pi/a\;\;\;\;\text{for the fourth quarter}
\end{align}
I would like to compute the density of states near the Fermi surface. The density of states is given by:
\begin{align}
\nu (\epsilon)=\int\frac{d\vec l}{4\pi^2}\frac{1}{|\nabla \epsilon|}.
\end{align}
In the situation I am describing the we have, along the Fermi surface:
\begin{align}
|\nabla \epsilon|=|\sin(k_x a)|.
\end{align}
Since I should integrate from [itex]-\pi/a[/itex] to [itex]\pi/a[/itex] the integral is clearly divergent. This is the so called Van Hove divergency. How do I deal with it?
I am expecting a logarithmic divergency around [itex]\epsilon\simeq0[/itex], how do I find it?
Thank you all
$$
\epsilon(k_x,k_y)=-J(\cos(k_x a)+\cos(k_y a)),
$$ and where the wave vectors belong to the first Brillouin zone ([itex]k_i\in [-\pi/a,\pi/a][/itex]).
In this case it turns out that the Fermi energy is zero and the Fermi surface is a square defined by:
\begin{align}
&k_y=-k_x+\pi/a\;\;\;\;\text{ for the first quarter of the k-space} \\
&k_y=k_x-\pi/a\;\;\;\;\text{for the second quarter} \\
&k_y=-k_x+\pi/a\;\;\;\;\text{for the third quarter} \\
&k_y=k_x+\pi/a\;\;\;\;\text{for the fourth quarter}
\end{align}
I would like to compute the density of states near the Fermi surface. The density of states is given by:
\begin{align}
\nu (\epsilon)=\int\frac{d\vec l}{4\pi^2}\frac{1}{|\nabla \epsilon|}.
\end{align}
In the situation I am describing the we have, along the Fermi surface:
\begin{align}
|\nabla \epsilon|=|\sin(k_x a)|.
\end{align}
Since I should integrate from [itex]-\pi/a[/itex] to [itex]\pi/a[/itex] the integral is clearly divergent. This is the so called Van Hove divergency. How do I deal with it?
I am expecting a logarithmic divergency around [itex]\epsilon\simeq0[/itex], how do I find it?
Thank you all