The internal energy is the integral of the internal energy spectral density. The delta functions in the density of states convert this integral in a sum.
$$u=D_0\hbar\omega_c\left(\sum \limits_{\nu = 0}^{\nu\uparrow_{\text{max}}}\nu+\frac{1}{2}-\frac{g}{4}+\sum \limits_{\nu = 0}^{\nu\downarrow_{\text{max}}}\nu+\frac{1}{2}+\frac{g}{4}\right)+\left(n-\text{Int}\left(\frac{n}{D_0}\right)D_0\right)E_F.$$Here the first sum up to the highest fully occupied Landau level with spin up electrons $\nu\uparrow_{\text{max}}$ and the second sum is over the fully occupied Landau levels with spin down electrons. The last term is the contribution to the energy density due to the partially filled landau level at the Fermi energy.
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$u(B=0)=$ J m-3 |
As $B\rightarrow0$, the internal energy density approaches the value for free electrons in two dimensions, $u = \frac{nE_F}{2},\quad E_F = \frac{\pi\hbar^2n}{m}$. At very high magnetic fields, only the lowest Landau level is occupied. For $g=2$, the Landau energy that increases linearly with magnetic field cancels the Zeeman that decreases linearly with magnetic field and the internal energy goes to zero.