# Formula Free electron gas in 2d (density of states) Density of states

## Density of states

`\( D \)`Unit

`\( \frac{1}{\mathrm J} \)`

Density of states of an electron gas - with free electrons which do not interact with each other and which are partially confined in a potential. In this case, the gas is confined in a two-dimensional surface and the density of states is independent of the energy \( W \) of the electrons.

The density of states gives the states per energy interval, (in this case) for both spin directions of the electron. To get the density of states for only one spin direction, you must multiply the density of states by \(\frac{1}{2}\). And, to get the density of states \( g \) per energy interval AND per area, multiply the density of states by \(\frac{1}{A}\):`\[ g ~=~ \frac{1}{\pi} \, \frac{2m}{\hbar^2} \]`

In the two-dimensional electron gas, there are the same number of states in each energy range: \( D \sim W^0\).

## Area

`\( A \)`Unit

`\( \mathrm{m}^2 \)`

Rectangular area of a solid (for example a metal) in which the 2d electron gas is located:

`$$ A ~=~ L_{\text x} \, L_{\text y} $$`With \( L_{\text x} \) and \( L_{\text y} \) as edge lengths of the solid.## Mass

`\( m \)`Unit

`\( \mathrm{kg} \)`

Mass of the Fermi particle. In the case of an electron gas, it is the (effective) mass of the electron.

## Reduced Planck's constant

`\( \hbar \)`Unit

`\( \mathrm{Js} \)`

Reduced Planck's constant is a physical constant and has the value:

`$$ \hbar ~=~ \frac{h}{2\pi} ~=~ 1.054 \, 571 \, 817 \cdot 10^{-34} \, \mathrm{Js} $$`