## Fermi surface of a two-dimensional square lattice

For the free electron model in two dimensions, electron states inside the Fermi circle will be occupied and electron states outside the Fermi circle will be empty. The radius of the Fermi circle increases as the electron density increases. The electron density $n$ is given in units of electrons/unit cell.

This page shows four representations of the Fermi circle. First, the Fermi circle is plotted in an extended zone scheme and the line changes color as it passes through the Brillouin zones. In the reduced zone scheme, the segments of the Fermi circle are translated by a reciprocal lattice vector so that they come into the first Brillouin zone. Fermi energy is plotted in a dispersion relation using the empty lattice approximation. The parts of the Fermi circle in each Brillouin zone are also plotted in repeated zone schemes. The Fermi circle is drawn with a dark color on the side in contact with the occupied states and a light color on the side in contact with the empty states. The states at the Fermi surface are electrons if the dark line is on the inside of a closed contour. The states at the Fermi surface are holes if the light line is on the inside of a closed contour.

$n=$2.0 electrons/unit cell
$\Large \frac{k_Fa}{\pi}=$1.128  $\Large \frac{E_F}{\left[ \frac{\hbar^2\pi^2}{2ma^2}\right]}=$1.273

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Extended Zone Scheme

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Reduced Zone Scheme

 $\Large \frac{E}{\left[ \frac{\hbar^2\pi^2}{2ma^2}\right]}$ $\Gamma$       $X$       $M$          $\Gamma$
 Your browser does not support the canvas element. 1st Brillouin ZoneRepeated Zone Scheme Your browser does not support the canvas element. 2nd Brillouin ZoneRepeated Zone Scheme Your browser does not support the canvas element. 3rd Brillouin ZoneRepeated Zone Scheme Your browser does not support the canvas element. 4th Brillouin ZoneRepeated Zone Scheme