Fermi energy as a function of magnetic field for free electrons in 2d

The general condition for the Fermi energy is,

$$n=\int\limits_{-\infty}^{E_F}D(E)dE.$$

Here $n$ is the electron density and $D(E)$ is the density of states. For free electrons in a magnetic field at very low temperatures, the number of completely filled Landau levels will be the integer part of $\frac{n}{D_0}$. The highest occupied Landau level is partially filled and is at the Fermi energy.

EF [eV] 

B [T]

Electron density: n =

[m-2]

Maximum B field: Bmax =

[T]

g - factor =

At very high magnetic fields, only the lowest Landau level is occupied. The Landau energy increases linearly with magnetic field but the Zeeman energy decreases linearly with the magnetic field so for $g\approx 2$ these two effects about cancel each other out and the Fermi energy remains about constant.

As B → 0, the Fermi energy converges to the free electron result,

$$E_F=\frac{\pi\hbar^2n}{m}.$$