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\sect{Fermions in magnetic field - Landau}

Consider $N$ spinless electrons that have mass $m$ and charge $e$  
in a 2D box that has an area $A$ at zero temperature.
A perpendicular magnetic field $B$ is applied. The purpose
of this question is to find the magnetization of the system. 

\Dn

(1) What are the threshold value $B_{n}$ 
for which there are exactly $n$ filled Landau levels.

\Dn

(2) Find the energy $E(B)$ and the magnetization $M(B)$ 
for strong field $B>B_1$. Give an optional semicalssical 
derivation to the result assuming that each electron is doing 
a cyclotron motion with minimal one-particle energy.

\Dn

(3) Find the energy $E(B)$ and the magnetization $M(B)$ for $B_{n+1}<B<B_{n}$. 
In particular find $E(B_{n})$ and explain why they are all equal to $E(0)$. 
Write what is the value of the magnetization at the endpoints of each interval.

\Dn

(4) Give a semicalssical derivation to the 
drops of $M(B)$ at the threshold values $B_{n}$, 
using the Hall formula for the current along the Edge.  

\Dn

Keywords: Landau levels; Landau magnetism; The de Haas van Alphen (dHvA) oscillations; The quantum Hall effect.


