Advanced Solid State Physics

classically seen due to electric interaction as we shall see in the following. Imagine a two electron
system described by the stationary Schrödinger equation

−

~^2

2 m

(∇^21 +∇^22 ) +V 1 (r 1 ) +V 2 (r 2 ) +V 1 , 2 (r 1 ,r 2 )Ψ =EΨ (76)

. Neglecting the electron-electron interaction termV 1 , 2 enables us to use the product wavefunction

Ψ(r 1 ,s 1 ,r 2 ,s 2 ) =ψ(r 1 )χ(s 1 )·ψ(r 2 )χ(s 2 ) (77)

. Because electrons are fermions the wavefunction has to be antisymmetric under an exchange of the
two particles so the possible solutions are either symmetric in spinχSand antisymmetric in position
ψAor vice versa (figure 65). With this picture in mind one sees that ferromagnetism will emerge
if the energy of the symmetric spin state (antisymmetric in position) is lower than the energy of
the antisymmetric spin state (symmetric in position). It is now somehow plausible to see how two
spins may align parallel. Because then their wavefunction is antisymmetric in position and therefore
the electrons are farther away from each other resulting in a more favourable configuration from an
electrostatic point of view. After this rough semiclassical picture of how one can understand the
emergence of ferromagnetism a short overview of a mathematical theory of ferromagnetism shall be
given. TheHeisenberg modelconsists of the hamiltonian

Hˆ=−

∑

i,j

Ji,jS~ˆi·S~ˆj−gμBB

∑

i

S~ˆ

i (78)

. The first term describes the exchange interaction between spins which raises or decreases the energy
depending on their alignment. One should not forget that this exchange interaction is of electric
nature as described above. The second term accounts for the interaction with an external magnetic
field. We will simplify this model by considering only nearest neighbor interaction (subscript nn).

Further we will remoe the nonlinearity inSby replacing theSˆ~nnoperator by its expectation value.
Thereby introducing the mean field hamiltonian

Hˆmf=

∑

i

S~ˆ

i(

∑

nn

Ji,nn<S >~ +gμBB) (79)

. It is possible to rewrite the first term as a magnetic field due to the neigboring spins.

B~mf=^1

gμB

∑

nn

Ji,nn<S >~ (80)

The magnetization is given as magnetic moment per volume

M~ =gμBN

V

~ (81)

B~mf= V

Ng^2 μ^2 B

nnnJM~ (82)