Theory of Magnetic Materials

James M. MacLarenTulane University

Topics covered• Introduction• Magnetism as a quantum phenomenon• Spin and orbital moments• Paramagnetism/Diamagnetism• Spin-orbit coupling• Hundt’s rules and magnetic ions• Magnetic interactions, dipolar, exchange• Direct, indirect, super, and double exchange• The Heisenberg Hamiltonian and ferro/antiferro magnetism• Weiss model of ferromagnet• Exchange coupling in metallic superlattices• Domain walls• Micromagnetics• Kerr effect• Magneto-crystalline anisotropy

Introduction• Magnetic Moment

– Interactions, ordering, temperature dependence

• Sources – intrinsic (spin), orbitalOrbital moment m = γLEinstein-de Haas experiment

dm = IdS

m = I

m

dS

• Consider electron charge e, mass m moving in a circular orbit radius r

• Angular momentum is quantized

• Leading to µ = π r2I = -eu/2m = -µB ~9.274 × 10-24Am2

• γ=-e/2m

The Bohr Magneton (µB)

v

e

I = −e

ττ =

2πr

m

m mr= u

Magnetism is a Quantum Phenomenun

• Magnetism is inherently a quantum phenomenon• Von Leeuwen theorem

"At any finite temperature, and in all finite applied electrical or thermal fields, the net magnetization of a collection of electrons in thermal equilibrium vanishes identically."

• From classical mechanics

(Canonical momentum)

• Z=∫∫…∫exp (-β Eri,pi) dr1…drN dp1…dpN

• Limits of integration are -∞ to ∞• Simple change of variables shows partition function Z is independent of A

T =1

2m m2

p = m v + qA

=(p − qA )2

2m

Susceptibility/Magnetization and Field

• free space magnetic induction B=µ0 H, where H is the magnetic field

• In a magnetic material B= µ0 (H+M), M is the magnetization, or ,magnetic dipole moment per unit volume

• In the special case of a linear material M=χ H, χ is the susceptibility B= µ0 (1+ χ )H

• Consider a region of space with fields B0 and H0, B0 = µ0 H0,

• Now add in magnetic material. Fields inside the material Bi and Hi differ markedly from B0 and H0, and can be complicated to calculate except in simple cases e.g.slab, sphere… in which case Hi=H0-NM, N is the demagnetization factor =1/3 for a sphere.

• More complex shape need to satisfy Maxwell’s equations with M=0 outside the sample

Quantum Mechanics of Spin• orbital angular momentum

– state defined by l and ml

• Spin angular momentum– state defined by s and ms

• Energy of a spin in a magnetic field E = gµBmsB

• g~2, leads to Zeeman splitting

N¥(¥+ 1)u,m ¥u

Ns(s+ 1)u,m su

Electrons in a Magnetic Field

• Now since , we can choose gauge

condition

• Leads to

H =Q

S

([pS+ eA S]2

2m+ VS

u

+ gµB µ ·S

µ = ∇ × A

A =µ × r

2

Q

S

p2S

2m+ VS+ µB (· + gS)·µ +

e2

8m

Q

S

(µ × rS)2

H 0paramagnetism diamagnetism

Paramagnetism J = 1/2• Partition function

• Free energy

• Magnetization

• For high T, M = gµBB/kBT, χ ∝ 1/T or Curie’s Law

• Can be generalized to other values of J (Brillouin function)

= 2Ioqh(gµB /[B T)

Z = exp(gµB /[B T)+ exp(−gµB /[B T)

x = −0o

0B= tanh(gµB /[B T)

o = n[B T l nZ

Diamagnetism• Diamagnetic term in Hamiltonian

• For B along the z axis

• For a solid with N ions in a volume V, we get

• where Zeff is the # of outer shell electrons and r the ionic radius• - sign similar to Lenz’s law in induction

H d =e2

8m

Q

S

(µ × rS)2

=e2B 2

12m

Q

S

0gr2

Sg0^

∆ E =e2B 2

8m

Q

S

0gx2

S + y 2Sg0

^

− = −N

V

e2µ0

mmZeffr2

Spin Orbit Coupling• Total angular momentum is the sum of both spin and orbital contributions

• The magnetic moment is given by

• The spin and orbital moments are weakly coupled. This coupling falls out of the Dirac equation.

• Lande g-factor

J = · + S

µ = gjj(j+ 1)µB

gj =3

2+

S(S + 1)− L(L + 1)

2J(J + 1)

µ = µB (· + gsS)= gJµB J

Classical View of Spin-Orbit

• Orbiting nucleus produces a magnetic field

• where

• ½ is the relativistic Thomas factor

E = −3 V (r)

H s+ = −1

2µQµ =

eu2

2m c2r

dV

dr·QS

µ =E × v

c2

Magnetism of ions/Hundt’s rules

• Hundt’s rules used to determine total angular momentum. Minimize Coulomb energy taking into account the Pauli principle

1. Maximise S (Pauli principle, minimizes U)2. Maximise L (keeps electrons I same orbit apart)3. attempts to minimize spin-orbit interaction.

• shell less than half full J=|L-S|• shell greater than half full J=|L+S|

• 3rd rule not always applicable in a solid since surrounding ions in the crystal provide an electrostatic field

Taken from “Magnetism in Condensed Matter, by Stephen Blundell, Oxford, 2001.

Magnetic Ions in a solid• Relative importance of crystal field terms and spin-orbit coupling determines

the moment.

• Rare earth f-shell electrons– Angular momentum barrier confines electrons close to nucleus,

moments localized & atomic– Spin-orbit important

• Transition metal d-shell electrons– Crystal field larger than spin orbit – Orbital moments quenched

Taken from “Magnetism in Condensed Matter, by Stephen Blundell, Oxford, 2001.

Magnetic Interactions• Dipole-Dipole forces (classical)

• Size estimate for two atomic moments separated by 1 Angstrom E ~ 1K

E =µ0

4πr3

p

µ1µ2 −3

r2(µ1 ·r)(µ2 ·r)

N

Exchange• The exchange interaction is a result of the Pauli principle and the Coulomb

Interaction.• Form antisymmetric wavefunctions as product of spatial and spin functions.

– Singlet state has S=0

– Triplet state has S=1

ψs =1π

2[ψa(r1)ψb(r2)+ ψa(r2)ψb(r2)]−s

ψN =1π

2[ψa(r1)ψb(r2)− ψa(r2)ψb(r2)]−N

Heisenberg Hamiltonian• Energy of singlet & triplet states

• After some algebra we can show

• Where we have used

– For a singlet

– For a triplet

Es = ψ∗sH ψs EN = ψ∗

NH ψN

H =1

4(E s + 3E N)− (E s − EN)S1 ·S2

S1 ·S2 = −3

4

S1 ·S2 =1

4

Heisenberg Hamiltonian

• Two electrons on same atom, J usually positive. Spatial part of wavefunction antisymmetric lowers Coulomb Energy. (Hundt’s 1st rule)

• Two electrons on different atoms is more complex. Some saving in kinetic energy to have a symmetric spatial wavefunction thus J can be negative

H = −Q

Sj

JSjSSSj

Direct Exchange

• Exchange interactions between electrons on neighboring sites• In rare earths f-orbitals interact with each other weakly due to localization• In transition metals d-d overlap is larger• In these metals s-d and s-f interactions important for bonding and

communicating exchange (indirect between d or f orbitals) important for RKKY like interactions in multilayers

Superexchange• Found in magnetic ionic materials, eg. MnF2 and

usually antiferromagnetic• Cannot be a direct exchange as the magnetic ions

are too far apart• Exchange is mediated through the non-magnetic ions

orbitals• Electron from non-metal hops onto metal ion and is

coupled strongly, resulting single p-electron has direct exchange with the adjacent metal ion

• Coupling may be determined via Hundt’s rulesparallel for ions with less than half filled d-shell antiparallel for ions with more than half filled d shell.

• Chromium telluride ferromagnetic, manganese telluride antiferromagnetic

internal coupling and exchange coupling couplingspin alignment of p-electron with other atom is

parallel ferromagnetic antiferromagnetparallel antiferromagnet ferromagneticantiparallel ferromagnetic ferromagneticantiparallel antiferromagnet antiferromagnet

P.W. Anderson, Phys. Rev 79, 350 (1950)H.A. Kramers, Physica 1, 182 (1934)

Double Exchange• Occurs in some oxides where the magnetic ion can have mixed valence• Leads to ferromagnetic coupling• Examples include Fe2O3

Electron hopping

Fe2+ (3d6) Fe3+ (3d5)

Ferromagnetism

• Mean field solution

• Assuming S=1/2

H = gµB

Q

S

SS·(µ + µ m f)

µ m f = ∗M

H = −Q

Sj

JSjSi·Sj+ gµB

Q

j

Sj ·µ

µ m u = −2

gµB

Q

j

JSj < Sj >

x

x s= tanh

pgµB S(B + ∗x )

[B T

N

Curie temperature• Solve the two equations for appearance of non zero M

• Leads to

– z is # of nearest neighbors

• susceptibility

x

x s= tanh(y )

x

x s=

2[B T

gµB ∗x sy

Tc =gµb∗x s

2[B=

nµ2eff

3[B=

JJ

2[B

− ∝1

T − TC

Properties of some common ferromagnets

M aterial T c (p ) m om entQe 1043 2.2No 1394 1.c2ui m31 0.m1G H 289 c.TM nSb T8c 3.TEuO c0 m.9EuS 1m.T m.9

Moment in Bohr magnetons

Stoner Criterion & Band magnetism• Consider a density of electron states g(E)• Move spin down electrons from to spin up band at• Change in KE

• Change in PE

• Where we used the molecular field = and• Change in total energy is

• Spontaneous ferromagnetism (Stoner criterion) is possible if•

Eo − ffE E o + ffE

ffE p E = g(E o )ffE /2× ffE = 12g(Eo )ffE 2

ffEP E = −ix

0 µ0(∗x )dx = − 12µ0∗x 2 = − 1

2µ0µ2B ∗(n↑ − nt)2

∗x n↑ − nt = g(Eo )ffE

∆ E = ∆ E p E + ∆ EPE = 12g(E o )(ffE )2(1− U g(E o ))

U g(Eo )≥ 1

Antiferromagnetism• If the exchange interaction J<0 then nearest neighbor moments will align

antiparallel.• Often occurs for crystals with interpenetrating lattices• Can be treated via a Weiss mean field theory using the two sublattices• Leads to a susceptibility

• TN is known as the Neel temperature shows this form but the extrapolated Weiss temperature differs from simple mean field Neel temperature. Need to include 2nd neighbor interactions (both sublattices)

− ∝1

T + TN

Other magnetic ordering• Ferrimagnetism – two sublattice moments with different values are coupled

antiferromagnetically e.g. Fe3O4

• Helical ordering seen in some rare earth magnets e.g Dy

– Minimizing wrt

– Leads to or

• Frustrated systems such as antiferromagnetism on a triangular lattice, spin glasses

E= -2uS2(J1 Ioqθ+ J2 Ioq2θ)

θ

(J1 + 4J − 2Ioqθ)qinθ= 0

θ= 0,i θ= −J1

4J2

Exchange coupling in Magnetic Superlattices

• Model proposed by Stiles (PRB 48, 7238, 1993)• Based upon a quantum well model and the reflection coefficient for waves

incident upon the well (see below) with R<<1• Change in energy

• In 1-d for free electrons leads to

• Reason for the oscillations is the step function in occupation numbers at kF

R T

t

∆ E (N)Mu

i

mo

2

1

Nqin(2[o N)

∆ E (N)=

m E o

µdE (E − Eo )ffn(E ,N)

Exchange coupling• Ferromagnetic state

• Antiferromagnetic state

• Exchange coupling

+M M M M

M MM

M+

J(N)=u

i

mo

2

UgG ↑↑g2 + gG ↑tg2 − 2gG ↑↑G ↑tg

E×

1

Nqin(2[o N+ φ0)

3-d and realistic Fermi surfaces• Moving from 1-d to 3-d for free electrons

– Fermi surface spherical– Contributions from all k//

– Contributions to integrand oscillate except when parts of Fermi surface are parallel (approximate with stationary phase integration)

– Spanning vector of 2kF determines oscillations just like 1-d model• Realistic Fermi surface oscillations determined by spanning vectors in the

growth direction that link parallel sheets of the Fermi surface

Stiles, PRB 48, 7238 (1993)

Fe/Au/Fe wedgesData from Unguris et al. PRL 79, 2734 (1997)

SEMPA

MOKE

Theory: M.D. Stiles, J. Appl. Phys. 79(8), 5805 (1996)

Period and coupling strength in good agreement with measurement

http://physics.nist.gov/Divisions/Div841/Gp3/Projects/MagNano/sempa_exchange_proj.html

# of layers

Domain Walls• Wall is region between two magnetic

domains

– Bloch wall (a) - spins rotate in plane parallel to the wall

– Neel wall (b) - spins rotate in plane perpendicular to the wall

• Competition between exchange and anisotropy.

Exchange wants a slowly varying wall since the energy rises as Si.Sj

Anisotropy wants to lock spins to a particular crystal direction

Wall size (Bloch)• Suppose wall extends over N layers• Anisotropy contributions – assuming uniaxial anisotropy

• Exchange

for small angles

adding contribution from the N layers

• Total energy is sum of exchange and anisotropy

NQ

S

K qin2 θS MN

i

m i

0K qin2 θdθ=

N K

2

E = −2JS1 ·S2

M JS2θ2

= JS2i2/N

E = JS2 i2

N a2+

N K a

2

Wall size (cont.)• Minimizing wrt N leads to

• And

• Where is known as the exchange stiffness

ff= N a = iSN

2J/K a

N = iSN

2J/K a3

A ex = JexS2/a

σB f = iSN

2JK /a = iN

A exK

Domain Formation• Creation of domain walls costs energy• Dipolar energy reduced• Domains are formed to balance these two effects

Micromagnetics• Micromagnetics minimizes the energy of the system that includes exchange,

anisotropy, external fields, and demagnetization (dipolar)• Typically solved by breaking the sample into small elements and calculating the

torque on the moment from an effective field determined from the energy function

• Damping is included to drive the solution to equilibrium• In principle exchange, anisotropy can be found from electronic structure theory• Several public domain codes oommf (NIST), magpar (Vienna)

– http://magnet.atp.tuwien.ac.at/scholz/magpar/– http://math.nist.gov/oommf/

Etot =

m

Ω(l exIh + l ani+ l ext + l Hem ag)dm

=

m

Ω

s

A

(

(3 jx)2 + (3 j y )2 + (3 jJ)2

u

+

K 1(1− (a ·j)2)− J ·H ext−1

2J ·H Hem ag

V

dm

Calculation of hysteresis from 1st principles

HRTEM image of grain boundary in Co1Pd5

Idealized model of the interface

Calculated boundary exchange

R.H. Victora, S.D. Willoughby, J.M. MacLaren, and Jianhua Xue, IEEE Trans. Mag., 39, 710 (2003).

Calculated hysteresis loop• Anisotropy only perturbed right at boundary plane• Exchange coupling falls off quickly• Modeled distribution of grains and used calculated anisotropies and

exchanges

Secondary Magnetic Properties• magneto optics

– Rotation of the plane of polarized light on either reflection/transmission• magneto crystalline anisotropy

– Energy cost as magnetization rotated wrt crystal axes• magnetostriction

– Change in shape of a sample as it is magnetized• magnetoelastic effect

– Influence of stress on magnetization

• all dependent on spin-orbit coupling

Kerr & Faraday effects• Kerr effect – rotation of the plane of polarization of reflected light• Faraday effect – rotation of the plane of polarization of transmitted light• Key is the symmetry of the dielectric tensor• Same as symmetry in Lorentz force

Example of Polar Kerr Effect• M perpendicular to film, dieletric tensor

• Maxwell’s equations

• Linear media

T=

S

].1 .2 0

−.2 .1 00 0 .3

H

J

D = TE, J = σE, µ = µH

3 ·D = 4iρ 3 × H =4i

cJ +

1

c

0D

0N

3 ·µ = 0 3 × E = −1

c

0µ

0N

Polar Kerr Effect (cont)• Consider incident plane waves

• Leads to modified Maxwell equation

• and

• At normal incidence for transverse waves

• Eigenvalue problem (circularly polarized light)

3 × H =1

cT(≥)

0E

0N, T(≥)= 1+

S4iσ(≥)

≥with

[20D = [2E − (, ·E),,

(T− n2)D = D

n2 = .1 ± S.2 D = x ± Sy

E(r,N) = E 0exp(S(, ·r− ≥N))

H (r,N) = H 0exp(S(, ·r− ≥N))

Polar Kerr effect (cont)• Incident plane wave equals sum of left and right circularly polarized beams• Each reflected from the surface (see Jackson) for reflection coefficients

• After some simplifications we arrive at the standard formula

• Formulae for different geometries more complicated (see chapter by J.M MacLaren in Magnetic Interactions and Spin Transport , eds Chtchelkanova, Wolf, Idzerda, pub. Kluwer, 2003.

E r =1− n+

1+ n+E + +

1− n−1+ n−

E − = 2E 0

;(1− n+ n− )x + S(n− − n+ )>

[

(1+ n+ )(1+ n− )

D =.2

i.1(1− .1)

=Tx y

iTxx(1− Txx)

=−σx y

σxxi

1+ S4i;σxx

Density Functional Theory• Based on two theorems by Hohenberg & Kohn

– Ground state energy is a functional of the electron density– Energy minimized for the true density

• Made practical by Kohn-Sham– Find density from an effective single particle Schrödinger equation– Potential depends on electron density– Density determined from solutions (wavefunctions)– Potential includes electron-ion, electron-electron, and exchange-correlation

potential.– Many accurate approximations to exchange & correlation– Solve self-consistently

• Extended to magnetic systems as spin-density functional theory• Despite being a ground state theory, wavefunctions and eigenvalues of

excited single particle states can explain properties like the Kerr effect

Kubo formula• Linear response leads to the Kubo formula for the optical conductivity

• This can be evaluated using single particle wavefunctions and operators using the momentum rather than current operators

σ×β = Sne2

m ;ff×β +

1

u;Ω

m µ

0dNeS;Neψg

;j× (−,N),jβ(,0)

[gψs

σ×β = Sn0e2

(; + S/τ4 )[m ∗]− 1

+Se2

m 2uΩ

Q

,

HQ

T σ

HHQ

T HσH

1

; T HσH,T σ

|− ×

T σ T HσH−βT HσHT σ

; − ; T HσH,T σ + Sff−

−βT σ T HσH− ×

T HσHT σ

; + ; T HσH,T σ + Sff,

r

Kubo formula (cont)• Matrix elements include spin-orbit coupling

− ×T σ,T HσH(,)=

m

Ωdrψ∗

T σ,(r)

(

−Su0

0x×+

u

4m c2(σ × 3 V (r))×

u

ψ T HσH,(r)

σxx(;) =e2

m 2u

m

d3[

HQ

T σ

HHQ

T HσH

g− +T σ,T HσHg2 + g−−

T σ,T HσHg2

2; T HσH,T σ

×

(1

(; − ; T HσH,T σ(,)+ Sff)+

1

(; + ; T HσH,T σ(,)+ Sff)

u

σx y (;) =e2

m 2u

m

d3[

HQ

T σ

HHQ

T HσH

g− +T σ,T HσHg2 − g−−

T σ,T HσHg2

2; T HσH,T σ

×

(1

(; − ; T HσH,T σ(,)+ Sff)+

1

(; + ; T HσH,T σ(,)+ Sff)

u

Example of Fe

Anisotropy• Magnetization prefers to align with certain crystallographic directions – easy

and hard axes• Origin is spin orbit coupling• Can be calculated using modern electronic structure methods• Larger in technologically interesting superlattices where the symmetry is

lower• Often seen in superlattices is a sum of interface/surface and bulk

contributions

• Neel model is a useful framework for understanding anisotropy – symmetry based expansion

K j = 2K s/N+ K m

E = E0 + L(r)Ioq2 θ+ H (r)Ioq4 θ+ QQQ

Anisotropy for magnetic transition metals

Column 1:Daalderop et al., PRB 41, 11919, 1990 (LMTO s,p,d partial waves)

Column 2:Daalderop et al., PRB 41, 11919, 1990 (LMTO s,p,d,f partial waves)

Column 3: Fritsche et al., J. Phys. F 17, 943 (1987)

Column 4: Trygg et al., PRL 75, 2871 (1995)

Column 5: Trygg et al., PRL 75, 2871 (1995) (adding in orbital polarization)

Agreement ok, Ni LSD gives poor Fermi surface

S> qtem theor> (µenAatom ) exp. eaq> axiqbII Qe -0.T -0.4 c.4 -0.T -1.8 -1.4 (100)fII ui -0.T -0.m 10 -0.T -0.T 2.c (111)hIp No 1m -29 -29 -110 -mT (0001)fII No 0.T 2.2 1.8 (111)

Anisotropies for Superlattices

• Theory Victora and MacLaren, PRB, 47 11919 (1993)• Experiment Engel et al. PRL, 67, 1910 (1991)• Interface term well described (larger)• Volume term less accurate (smaller)

NoAPH 2K s (ergqAIm 2) K m (ergqAIm 3)theor> experim ent theor> experim ent

(111) 1.32 1.3 -1.8 × 10c -0.m × 10c

(100) 1.14 1.2 -m.4 × 10c -4.T × 10c

(011) 1.3 -1.9 × 10c

Other results - Co/Pd

(+) Draaisma et al., JMMM 66, 351 (1987)(♦) den Broeder et al., Appl. Phys. A 49, 507 (1989)Theory Daalderop et al. in “Ultrathin magnetic films”, Springer Verlag, 1994

Neel Model & Disorder• Disorder at the interface is seen to reduce the anisotropy• Disorder – Pj = probability of a magnetic atom on a particular site.

• For pure materials K=0, so

hence

mjj

mjj

fjj

mjj

fn

jji

WPPWPP

WPPWPPWPK

1111

11001

2

)1()1(

)1(22

++

+=

−+−

++−+=∑

mmff WWWW 1010 ; −=−=

21

1)(2 +

=

−= ∑ j

n

jjpi PPKK

Application to FePd• Hirotsu’s data for L10 FePd (private communication)• Kp=2.6x107 ergs/cc • Ki=1.0x107 ergs/cc • LRO = 0.65

• Neel theory for Ki ~ 0.652 x 2.6x107 ergs/cc ~ 1.1x107ergs/cc

• Ki/Kp can also be derived from XRD diffraction intensities (MacLaren,Victora, Sellmyer, unpublished)

• Calculated Kp for a=.386 nm ,c/a=0.968 1.2x107 ergs/cc