Here is a way to derive the increase in energy of a free-electron gas under a uniform magnetic field without calculating the explicit expression for free energy, as presented in Chapter 15 of Grosso & Parravicini.
One first considers a slice of the Fermi sphere with constant $k_z$. As the uniform magnetic field turns on, the allowed energy levels within each slice collapse into discrete Landau levels. Within $\pm\hbar\omega_c/2$ of the Fermi energy $E_F$, where $\omega_c$ is the cyclotron frequency, there is one Landau level at $E_n$, $\varepsilon$ away from $E_F$ and it is the only one responsible for the energy change in each slice. So far, so good.
Now begins the part I have trouble with. The authors write that if we consider the case $\varepsilon>0$, the electrons belonging to the energy interval $[E_n-\hbar\omega_c/2,E_F]$ are shifted to the Fermi energy and redistributed to other slices. The case of $\varepsilon <0$ gives a similar increase in net energy. Furthermore, the excess of states transferred to the Fermi energy in the slices with $\varepsilon >0$ are accommodated by an equal number of extra states available in the slices with $\varepsilon <0$, so for $\hbar\omega_c << E_F$, the Fermi energy is independent of the magnetic field.
Now my questions in specific forms
- In the above figure, why are the electrons in the colored interval "redistributed to other slices"? Because of translational symmetry along z, which is not broken by the magnetic field, every state should retain its value of $k_z$ right?
- Isn't the figure wrong in what it labels as "$E_F$"? Shouldn't it be really "$E_F - \hbar^2 k_z^2/2m$"? The Fermi energy is defined by the radius of the Fermi sphere so it should be the same whatever $k_z$ slice you consider.