Advanced Solid State Physics

There are two limits where one can solve this problem:
High temperatures / Landau diamagnetism (weak magnetic fields)
For high temperatures the split gets smaller, so the states are close together. This calculation is called
Landau diamagnetism. Usually the calculation is done in a grand-canonical ensemble.
We can use the grand-canonical potentialΩto obtain the Gibbs-free energyGand then get the
magnetization by taking the derivative.

Ω =−kBTlnZG

G= Ω +μN

M=−

∂G

∂H

χ=−

∂^2 G

∂H^2

In this case, the logarithm of the partition function can be calculated as shown below. After that, the
discrete energies have to be plugged in and after that it is necessary to sum over all Landau levels and
put in the degeneracy of the levels. This is a difficult calculation^8 so only the result is presented here
(equ. 75).

Ω =−kBT

∑

k

2 ln

(

1 +eβ(μ−Ek)

)

(73)

En=

(

n+

1

2

)

~ωc (74)

Ω =−kBT

Am

π~^2

~ωc

∑

n

ln

(

1 +eβ(μ−En)

)

(75)

The factor containing A comes from the degeneracy of the Landau levels. The result, that is obtained
by evaluating the sum, shows that the susceptibility is about a third of the susceptibility for Pauli
paramagnetism.

Ω(H)≈Ω(0) +

1

3

μ^2 BD(EF)H^2 (kBT >>~ωc)

χ=−

∂^2 G

∂H^2

χ≈−

1

3

μ^2 BD(EF)

χLandau≈−

1

3

χPauli

Low temperatures / Landau diamagnetism (strong magnetic fields)
In this limit the energy associated with the steps between the Landau levels are much bigger than
kBT (kBT <<~ωc). Equation (75) stays valid, but when one evaluates it in this limit, the solution
turns out to be proportional to a cosine:

Ω∝

∑inf

k=1

cos

(

e~k

mEFH

−

π

4

)

(^8) the calculation can be looked up in the internet: http://physics.ucsc.edu/~peter/231/magnetic_field/magnetic_field.html