Magnetism Magnets

Zero external field Finite external field • Types of magnetic systems

• Pauli paramagnetism in metals Paramagnets • Landau diamagnetism in metals

• Larmor diamagnetism in insulators Diamagnets • Ferromagnetism of electron gas • Spin Hamiltonian Ferromagnets • Mean field approach

• Curie transition Antiferromagnets

Ferrimagnets …… …

Pauli paramagnetism Pauli paramagnetism

Let us first look at magnetic properties of a free electron gas. ε =−p2 /2meBmc= /2 ε =+p2 /2meBmc= /2 ↑ ↓G Electron are spin-1/2 particles #of majority spins: dp3 NV= f()ε In external magnetic field B – Zeeman splitting #of minority spins: ↑,,↓ ∫ (2π= )3 ↑ ↓ 2 = 2 = ε↑ =−p /2meBmc /2 ε↓ =+p /2meBmc /2 Magnetization (magnetic moment per unit volume):

- minority spins e= M =−()NN : aligned along the field and proportional Fermi level ↑ ↓ 2Vmc to B in low fields

χ - magnetic succeptibility - majority spins M = χB χ > 0 - paramagnetism

Pauli succeptibility Landau quantization

G 2 2 A free electron in magnetic field: B & zˆ ε↑ =−p /2meBmc= /2 ε↓ =+p /2meBmc= /2 G 2 µ+eB= /2 mc 2 G VgeB= Schrödinger equation: = ⎛⎞ieA ABxAA===;0 NN−= gd()εε ≈ V −∇+=⎜⎟ψ εψ yxz ↑↓ ∫ 2mc= 22µ−eB= /2 mc mc ⎝⎠

B=1T corresponds toeBmc= /1=× K k provided m is free electrons’s mass Solutions: labeled by two indices nk, B G z For any fields, eBmc=/ µ ψ nk(r )= exp( ik y y+− ik z z )ϕ n ( x= ck y / eB )

Magnetic succeptibility: ϕn - wave functions of a harmonic oscillator 22 2 Energies: ε =+==kmeBmcn/2 ( / )( + 1/2) - strongly degenerate!! ⎛⎞e= nk z χP = ⎜⎟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 We know that there are diamagnetic metals. How can we explain their and create magnetic field themselves. existence? This magnetic field is directed antiparallel to the external one Different spectra of electrons in metals and free electrons. * Example: renormalized electron mass m Diamagnetism dk This only affects orbital motion, not Zeeman splitting Energy: Ef= 2 ε ε s ∑∫ nk() nk n 2π ** *2 −1 χχχχ==;(/)mm Magnetization: M =−VEB ∂/ ∂ PPLL 2 Result for the succeptibility: 11⎛⎞e= We can explain paramagnetic and diamagnetic metals! χL =−⎜⎟g =− χP 32⎝⎠mc 3 But there is no way we can explain ferromagnetic and antiferromagnetic metals in non-interacting electron model. Total succeptibility: χ =+=χχPL2/30 χ P > : paramagnetic! Try insulators?

Larmor diamagnetism Larmor diamagnetism

Consider an ionic insulator with filled shells. Electrons are localized at the ions. 22 eeZN22 22 G 2 ∆=EBrBr = Electron Hamiltonian: 2 gi22∑ g ==⎛⎞GGGieA e GG G 12mci 12 mc Hˆ =−⎜⎟ ∇+ +BS; A = B × r g 2mcmc⎝⎠= 22 Succeptibility: 1 ∂ EeZ 2 -ionic charge r small: consider terms with the field as a perturbation. χ =− =− cr Z 22A g VB∂ 6 mc cA- #of atoms per unit volume Correction to the energy in the ground state: ˆ ∆=EgsHgsg .. Diamagnetism! Total spin and total momentum of electrons in a filled shell are zero; 2 only the term with r contributes. If the ionic shells are not filled – can get paramagnetic contribution due to other terms. ee22 EBxyBr222 22 ∆=giii22∑∑() + = Can explain paramagnetic and diamagnetic insulators, but not 812mciig mc g ferromagnetic and antiferromagnetic ones!

Ferromagnetism for localized Exchange interaction electrons

Two electrons: antisymmetric wave function! Try to visualize for localized electrons (atoms; artificial atoms – quantum dots; defects etc) Take an electron level ε The simplest model: Wave function of two electrons: Discrete electron states; spacing δ δ 1 (each level is doubly degenerate) ψ (,rr12 )=±[]ϕϕ 11 () r 22 () r ϕϕ 21 () r 12 () r for S = 0,1 2 To put an electron into the system costs electrostatic energy U and exchange Energy splitting due to interaction: spin-dependent! Same spin: UJ+δ − energy -J ε →+ε CJ ±/2 Energy to pay: 22 Opposite spin: U CdrdrUrrr=−()()()ϕϕ r ∫ 12 1 2 11 22 J < δ Non-magnetic state J > δ Ferromagnetic state J =−2dr dr U ( r r )()()()()ϕϕ r** r ϕϕ r r ∫ 12 1 2 1112 21 22 GG ˆ Spin Hamiltonian H=− const JS12 S

2 Itinerant ferromagnetism Itinerant ferromagnetism

If we can not get ferromagnetism with free electrons, try interacting Working the interaction terms out for free electrons (see Advanced electrons. Quantum Mechanics, lecture 4) 33N =22k Hartree-Fock approximation: Kinetic energy: ENE== F kin552 F m EgVgint= int 32 2 Potential energy: 3 2 kF NB: NVk= /(3π ) 1 GGe G2 G2 G G G G ENe=− F ≈−drdr'⎡⎤ψψ ( r ) ( r ') ψψψψ** ( r ) ( r ') ( r ') ( r ) int ∑∫ GG⎢⎥i j iji j 4 π 2'ij rr− ⎣⎦ Try now a spin-polarized ground state: ↑↓ kkF ≠ F Kinetic energy loss can be compensated by the potential energy gain! Fock (exchange) Hartree (direct) interaction. 51 interaction For - spin-polarized ground state (ferromagnetism!!) Only exists if spin kaFB< 1/3 projections are the same 22π + 1 in the states i and j 22 Never occurs in real life. ameB = = / - Bohr radius

Antiferromagnetic ordering Antiferromagnetic ordering

A different situation: a pair of magnetic atoms in an insulating matrix 2nd order corrections to the ground state: virtual states 2 Miν Consider d-electrons, 5 electrons per atom ∆=−E ∑ Ground Virtual ν Eν − Ei

U – ionization energy; t – overlap between the atoms tU

2 Unperturbed ground state: either parallel or antiparallel spins. ∆=−E 25tU / tU ,

Ferromagnetic state: no second-order correction (forbidden by Pauli principle)

Antiferromagnetic state preferential!

Spin Hamiltonian Mean field approach

Does not work for many atoms – but still represents a good model to treat Let us single out one particular spin at i. magnetism GˆˆG HJSSˆ =− +(all other spins) GGˆˆ ij∑ ˆ i and j – lattice sites j HJSS=−∑ ij i j ij Approximation: this spin sees G Gˆ GG Common approximation: only nearest neighbours interact; the same hJ=− S ˆ ˆ the average field (Weiss field) ∑ j HhS= i exchange integrals for all bonds j GGˆˆ Now we need to calculate the average field self-consistently. Heisenberg model: HJSSˆ =− ∑ ij Each site has N nearest neighbours. For each neighbour, ij

chance to be “up” (parallel to the field) P ∝−exp(hkT / 2B ) JJ><0( 0) favors ferromagnetism (antiferromagnetism) ↑ chance to be “down” (antiparallel to the field) P↓ ∝+exp(hkT / 2B ) Exact solution: only known for 1D chain (Bethe 1931) – no magnetism! JNJNh Can also be treated for high spins (classical) Equation for the field: hPP=−() − = tanh Let us see what we can do with approximate solutions. ↑↓ 222kTB

3 Mean field approach Curie transition

JNh Magnetization close to transition temperature JN h = tanh Tc = 3 4k 22kTB tanhxxx≈− / 3 B

Solution:

2 Square-root singularity hkTTT= 12Bc ( c− ) Depending on the temperature, either one solution h=0 (no magnetism) or three solutions (ferromagnetism) Numerical solutions: give power laws, but not the square root. Three solutions at: JNJN1 >⇒1 kTB < Curie’s temperature Obviously, these are problems of the mean-field approximation. 22kTB 4 To describe paramagnetic state: Can add the external field h0

A phase transition between a ferromagnetic and paramagnetic state! JNh hh=+0 tanh 22kTB

4