Magnetism, Free Electrons and Interactions

Magnetism

Magnets

Zero external field
Finite external field

• Types of magnetic systems • Pauli paramagnetism in metals • Landau diamagnetism in metals • Larmor diamagnetism in insulators • Ferromagnetism of electron gas • Spin Hamiltonian

Paramagnets Diamagnets Ferromagnets

• Mean field approach • Curie transition

Antiferromagnets Ferrimagnets

…

Pauli paramagnetism

ε_↑= p²/ 2m−e=B/ 2mc

ε = p²/ 2m+e=B/ 2mc

Let us first look at magnetic properties of a free electron gas.

^↓G

d³p

Electron are spin-1/2 particles
#of majority spins:

N_↑,↓=V

₃f (ε_↑,↓

)

∫

In external magnetic field B – Zeeman splitting
#of minority spins:

(2π=)

ε_↑= p²/ 2m−e=B/ 2mc

ε_↓= p²/ 2m+e=B/ 2mc

Magnetization (magnetic moment per unit volume):
- minority spins
Fermi level

e=

M =

(N_↑− N_↓)

: aligned along the field and proportional

2Vmc

to B in low fields
- magnetic succeptibility

χ

M = χB

- majority spins
- paramagnetism

χ > 0

Pauli succeptibility

Landau quantization

G

ε_↑= p²/ 2m−e=B/ 2mc

ε_↓= p²/ 2m+e=B/ 2mc

A free electron in magnetic field:

ˆ

B & z

G

2
2

µ+e=B/2mc

G

⎛⎜⎝
⎞⎟⎠

=

ieA

A_y= Bx; A = A = 0

V

g e=B

Schrödinger equation:

x

z

−

∇+

ψ = εψ

N_↑− N_↓=

g(ε)dε ≈

V

∫

2m

=c

2 _µ−e=B/2mc

2 mc

n,k_z

B=1T corresponds to e=B / mc =1K ×k_Bprovided m is free electrons’s mass

Solutions: labeled by two indices

G

ψ

_nk(r) = exp(ik_yy +ik_zz)ϕ_n(x − =ck_y/ eB)

For any fields,

e=B/ mc ꢀ µ

ϕ_n

- wave functions of a harmonic oscillator
Magnetic succeptibility:

^Energies:ε_nk= =²k_z²/ 2m+(e=B/ mc)(n+1/ 2)

2

- strongly degenerate!!

e=

⎛

⎞

χ_P

=

g

⎜⎝
⎟⎠

2mc

We “quantized” momenta transverse to the field
(Landau levels)

1

Landau diamagnetism

Electrons in metals

A free electron in magnetic field: moves along spiral trajectories and create magnetic field themselves.
We know that there are diamagnetic metals. How can we explain their existence?
This magnetic field is directed antiparallel to the external one
Different spectra of electrons in metals and free electrons.

m^*

Example: renormalized electron mass
Diamagnetism
This only affects orbital motion, not Zeeman splitting

dk

Energy:

E = 2

ε_nkf ε

(

)

∑

s

nk

∫

2π

= −V ⁻¹

n

χ_P^*= χ_P; χ_L^*= χ_L(m/ m^*)²

M

∂E /

∂

B

Magnetization:

2

13

e=

1

⎛⎝
⎞

Result for the succeptibility:

We can explain paramagnetic and diamagnetic metals!

χ_L= −

g = − χ_P

⎜

⎟

⎠

2mc

3

But there is no way we can explain ferromagnetic and antiferromagnetic metals in non-interacting electron model.
Total succeptibility:

χ

=

χ_P

+

χ_L

=

2χ_P/3

>

0

: paramagnetic!
Try insulators?

Larmor diamagnetism

e²

e²ZN

Consider an ionic insulator with filled shells. Electrons are localized at the ions.

∆E_g=

B²

r²

=

B²r²

G

2

∑

i

Electron Hamiltonian:

12mc²

2

g

G

G G G

G

⎛⎜⎝
⎞⎟⎠

=

ieA

e

=

G

i g

ˆ

H

= −

∇+

+

BS; A

=

B

×

r
2m

=

c

mc

1 ∂²E e²Z

Succeptibility:

c_Ar²

- ionic charge

Z

r small: consider terms with the field as a perturbation.

χ = −

= −

V ∂B²
6mc²

g

c_A_{- #of atoms per unit volume}

Correction to the energy in the ground state:

ˆ

∆E_g= g.s H g.s

Diamagnetism!
Total spin and total momentum of electrons in a filled shell are zero;

r²

If the ionic shells are not filled – can get paramagnetic contribution due to other terms.

only the term with

contributes.

e²
8mc²e²
12mc²

∆E_g=

B²x²+ y²

=

B²

r²

(

)

Can explain paramagnetic and diamagnetic insulators, but not ferromagnetic and antiferromagnetic ones!

∑

i

g

Ferromagnetism for localized electrons

Exchange interaction

Try to visualize for localized electrons (atoms; artificial atoms – quantum dots; defects etc)
Two electrons: antisymmetric wave function! Take an electron level

ε

The simplest model:
Wave function of two electrons:

δ

Discrete electron states; spacing (each level is doubly degenerate)

δ

1

S = 0,1

for

ψ (r ,r ) =

ϕ (r )ϕ (r ) ±ϕ (r )ϕ (r )

[

]

1

2

1

2

1

2

To put an electron into the system costs electrostatic energy U and exchange

energy -J

Energy splitting due to interaction: spin-dependent!

^{Same spin:}U +δ − J

ε →ε +C ± J / 2

Energy to pay:

^{Opposite spin:}U

C = d r d r U(r − r ) ϕ₁(r ) ²ϕ₂(r ) ²

1

2

1

2

1

2

∫

J <δ ^{Non-magnetic state}

J >δ Ferromagnetic state

J = 2 d r d r U(r − r )ϕ₁(r )ϕ₁^*(r )ϕ₂^*(r )ϕ₂(r )

1

2

1

2

1

2

1

2

∫

G G

ˆ

Spin Hamiltonian

H = const − JS₁S₂

2

Itinerant ferromagnetism

If we can not get ferromagnetism with free electrons, try interacting electrons.
Working the interaction terms out for free electrons (see Advanced

Quantum Mechanics, lecture 4)

35

3N

=

²k_F²2m

Hartree-Fock approximation:

Kinetic energy:

E_kin

=

NE_F

=

5

E_int= g V g

int

3

k_F

G G e²

G

Potential energy:

^NB:N =Vk_F³/(3π ²)

E_int= − Ne²

2

12

≈

drdr '

ψ_i(r) ²ψ _j(r ') −ψ_i(r)ψ _j(r ')ψ_i(r ')ψ _j(r)

*

⎡

⎤

G

4

π

∑
∫

⎢

⎥

⎦

⎣

r −r '

ij

^{Try now a spin-polarized ground state:}k_F^↑≠ k_F^↓

Kinetic energy loss can be compensated by the potential energy gain!
Fock (exchange)
Hartree (direct)

interaction interaction. Only exists if spin projections are the same

in the states i and j

5

1

For

- spin-polarized ground state (ferromagnetism!!)

k_Fa_B<

2π 2^1/3+1

a_B= =²/ me²

Never occurs in real life.
- Bohr radius

Antiferromagnetic ordering

2

A different situation: a pair of magnetic atoms in an insulating matrix Consider d-electrons, 5 electrons per atom
2^ndorder corrections to the ground state: virtual states

M_iν
∆E = −

∑

E − E_i

νν

Virtual

Ground

U – ionization energy; t – overlap between the atoms

t

ꢀ

U

∆E = −25t²/U ꢀ t,U

Unperturbed ground state: either parallel or antiparallel spins.
Ferromagnetic state: no second-order correction (forbidden by Pauli principle)

Antiferromagnetic state preferential!

Spin Hamiltonian

Mean field approach

Let us single out one particular spin at i.
Does not work for many atoms – but still represents a good model to treat

G

i

G

ˆ

magnetism

ˆ
ˆ

H = −JS S +(all other spins)

G G

i

∑

j

ˆ ˆ
ˆ

H = − J S S

i and j – lattice sites

j

∑

ij

j

ij

G

ˆ

GG

Approximation: this spin sees the average field (Weiss field)

ˆ
ˆ

h = −J
S

Common approximation: only nearest neighbours interact; the same exchange integrals for all bonds

∑

H = hS_i

j j

G G

i

Now we need to calculate the average field self-consistently.

ˆ ˆ
ˆ

Heisenberg model:

H = −J S S

∑

j

Each site has

N

nearest neighbours. For each neighbour,

ij

P ∝ exp(−h/ 2k_BT)

↓

chance to be “up” (parallel to the field)

P^↑∝ exp(+h/ 2k_BT)

J > 0 (J < 0)

favors ferromagnetism (antiferromagnetism) chance to be “down” (antiparallel to the field)
Exact solution: only known for 1D chain (Bethe 1931) – no magnetism!

JN

h

Can also be treated for high spins (classical) Let us see what we can do with approximate solutions.
Equation for the field:

h = −

P − P

=

tanh

(

)

↑

↓

2