Feb. 17. 2020

Spintronics Basic II
Hisazumi Akai
The Institute for Solid State Physics
The University of Tokyo1

Contentsn Origin of magnetism and magnetic structuresn Exchange interactionn Exchange interaction between two atoms
n Heitler–London limit and molecular orbital limitn Direct exchange and kinetic exchangen Atoms in a crystaln A bit more detailsn Kanamori–Goodenough rulen Double-exchangen Magnetic anisotropyn Anisotropic exchange
n Dzialshinsky-Moriya interactionn Heisenberg model2

Interaction between electrons
ntwo electrons–Pauli’s exclusion principle
possibility of two electrons coming closer→ increase in coulomb energy

less probability of two electrons coming closer→ decrease in coulomb energy
reason why electrons do not come closer is notcoulomb repulsion but Pauli’s exclusion principle

parallel spin is preferable

3

Kinetic energy of electrons
ntwo electrons –Pauli’s exclusion principle
no limitation to the volume that electrons
can move freely→ no increase in the kinetic energy
the volume that electrons can move is halved→ increase in zero-point energy

r p
antiparallel spin is preferable

4

Exchange interaction between electrons
nenergy difference between parallel and antiparallel
coupled electrons, positive or negative?

EX = E

– E

≷0?

5

Single atom
nelectrons are bound in a potential
na neutral atom with an odd number of electrons
nnonzero total electron spinnnonzero angular momentum in generalnshows magnetism (paramagnetic)LS6

Tow atoms
ninteraction between atomic magnetic moments
nparallel or antiparallel couplingnexchange interaction between atoms

EX = E– E

parallel if EX>0, antiparallel otherwise
7

Parallel or antiparallel?
nSimple model I: Heitler–London limit
n two hydrogen atoms (hydrogen atom1 and hydrogen atom2)
n potential of each atom v(rR)atomic orbital

atomic hamiltonianhydrogen atom1

H1 =21 + v(r1R1 )H1 1 =1hydrogen atom2H2 =22 + v(r2R2 )H2 2 =2（we use Slater’s atomic unit:= 1, m = 1/2, e2 = 2 ）8Parallel or antiparallel?
nSimple model I: Heitler–London limit
n two hydrogen atoms (hydrogen atom1 and hydrogen atom2)
n potential of each atom
hamiltonian of hydrogen molecule

coulombic interaction

H =H1 + H2 + H12 + V (r1H12 = v(r1R2 ) + v(r2r2 )R1 ) atractive potential due tothe other atom

9

Parallel or antiparallel?
nSimple model I: Heitler–London limit
n assume that each atom keeps charge neutralityn assume that the wave function of hydrogen molecule is a product of 1s wavefunctions φ’s of two hydrogen atoms
hamiltonian of hydrogen molecule

coulombic interaction

H =H1 + H2 + H12 + V (r1H12 = v(r1molecular wavefunction

molecular energyEr2 )R2 ) + v(r2(r1 , r2 )HR1 ) atractive potential due tothe other atom

1 (r1 ) 2 (r2 )d3 r10Parallel or antiparallel?
nSimple model I: Heitler–London limit
n antisymmetrization（sign change accompanied by exchange of two electrons)n two cases depending on spin configuration

s=0 state（spin singlet）
0 (r1 , r2 ) = A0 { 1 (r1 ) 2 (r2 ) +−1 (r2 ) 2 (r1 )}(symmetric orbital, antisymmetric spin)＋

s=1 state（spin triplet）
1 (r1 , r2 ) = A1 { 1 (r1 ) 2 (r2 )1 (r2 ) 2 (r1 )}（antisymmetric orbital, symmetric spin)
11

Parallel or antiparallel?
nSimple model I: Heitler–London limit
expectation value of energy
spin singlet

0 (r1 , r2 )(H1 + H2 + H12 + V )0 (r1 , r2 )dr1 dr2 = E0

spin triplet

1 (r1 , r2 )(H1 + H2 + H12 + V )Ex = E0

1 (r1 , r2 )dr1 dr2 = E1

E112

Parallel or antiparallel?
nSimple model I: Heitler–London limit
s=0 state（spin singlet）

0 (r1 , r2 ) = A0 { 1 (r1 ) 2 (r2 ) +0 (r1 , r2 ) = 0even if

r1 = r2

1 (r2 ) 2 (r1 )}large interaction V

s=1 state（spin triplet）

1 (r1 , r2 ) = A1 { 1 (r1 ) 2 (r2 )1 (r1 , r2 ) = 0

0 (r1 , r2 )Vif

r1 = r2

0 (r1 , r2 )dr1 dr2 >1 (r2 ) 2 (r1 )}small interaction V

1 (r1 , r2 )V

1 (r1 , r2 )dr1 dr2
13

Parallel or antiparallel?
nSimple model I: Heitler–London limit
s=0 state（spin singlet）

0 (r1 , r2 ) = A0 { 1 (r1 ) 2 (r2 ) +0 (r1 , r2 )(H1 + H2 + H12 )s=1 state（spin triplet）

1 (r1 , r2 ) = A1 { 1 (r1 ) 2 (r2 )1 (r1 , r2 )(H1 + H2 + H12 )1 (r2 ) 2 (r1 )}0 (r1 , r2 )dr1 dr2 = 2

1 (r2 ) 2 (r1 )}
1 (r1 , r2 )dr1 dr2 = 2

the same energy except coulomb interaction energy= +1 (r)v(rR2 ) 1 (r)dr <14

Heitler–London limit
therefore
0 (r1 , r2 )H0 (r1 , r2 )dr1 dr2 >Ex = E0

1 (r1 , r2 )H

1 (r1 , r2 )dr1 dr2

E1 > 0ntriplet state realized in Heitler-London limit
nnot corresponding to most two-atom molecules
ntwo-atom molecules are mostly spin singlet

nmolecular bonding is caused by H12nε’ < ε ：effect of attractive potential of a neighbor
15

Parallel or antiparallel?
nSimple model 2: molecular orbital limit
n two hydrogen atoms (hydrogen atom1 and hydrogen atom2)n wavefunctions are molecular orbitals constructed from two atomic orbitals
n two molecular orbitals: bonding and antibonding statesn how to place two electrons in these molecular orbitals?
bonding stateb (r) = A{ 1 (r) +2 (r)}antibonding statea (r) = A{ 1 (r)(2 (r)}i (r)(rRi ))

16

Parallel or antiparallel?
nSimple model 2: molecular orbital limit
antibonding statea (r) = A{ 1 (r)2 (r)}single electron energy

most stable if
coulombic interaction
is missing

bonding stateb (r) = A{ 1 (r) +2 (r)}17Parallel or antiparallel?
nSimple model 2: molecular orbital limit
antibonding statea (r) = A{ 1 (r)single electron energy

2 (r)}
exchange interaction
stabilizes the parallel
coupling

bonding stateb (r) = A{ 1 (r) +2 (r)}18

Parallel or antiparallel?
nSimple model 2: molecular orbital limit
more stablewithout electroninteraction

E0electron interaction stabilizes
parallel coupling

E1E0

E1or

E0

E1 ?

19

Hitler–London vs. molecular orbital limits
nHeitler–London limit excludes ionic states
nstrong correlation limit
nelectron interaction energy

kinetic energy

nequal probability of ionic and neutral states inmolecular orbital limit
nweak correlation limit
nelectron interaction energy
ionic state

atom1

kinetic energy
neutralneutralcoulomb
atom2 interaction U atom1

atom2

ionic state

20Analysis using simple modeln2-site Hubbard model

H=t†(a1a2†+ a2a1 ) + U

ni ni
i=1,2

coulombic
interaction U

site 1

site 2electron transfer t
21

Exact solution
nground state at U=0 andenergy+–
singlet+singletsinglet in MFA
triplet0singlet triplet

molecular orbital limit

singletelectroninteraction U–

Heitler-London limit

22Mean field approximation (MFA)
nnonmagnetic to ferromagnetic transition occurs
energynonmagnetic

nonmagnetic

nonmagnetic
0electroninteraction U

ferromagnetic
singlet exact solution

molecular orbital limit

Heitler-London limit

23

Mean field approximation (MFA)
nnonmagnetic to ferromagnetic transition occurs
energyferromagnetic
0electroninteraction U

singlet exact solution
nonmagnetic

molecular orbital limit

Heitler-London limit24In reality
nsomewhere in between Heitler-London andmolecular orbital
nstarting from Heitler-London
nionic states gradually mixed into neutral states
nalthough ionic states have higher energies, energy gain due toelectron transfer is expected

nstarting from molecular orbital limit
nionic states are suppress owing to their higher energy
nbonding–antibonding splitting decreases, which reduce energy
gain due to electron transfer25

Real atomic exchange couplings
nbetween two atomic spins S 1 and S 2 of atom 1
and atom 2, there exists an interaction of the form

Ex = E0
where

E1 =

J12 > 0,2(J12 + W12 )S 1 · S 2

W12 < 0is always satisfied and in many cases

J12 + W12 < 026Direct and kinetic exchanges
n J: direct exchangen W: kinetic exchange
n in many cases, W determines magnetic structures

Hereafter, we use J to indicate a sum of J and W

J12 + W12 ⇒ J12
Therefore J12 can be either positive or negative.
A positive J is ferromagnetic and vice versa.

27

If many atoms exist
nwhen interaction is strong enough, spontaneous
symmetry breaking occurs
nmagnetic moment arises at each site
nstable magnetic structure realized
nin this situation MFA gives reasonable descriptions
nenergy of systems depends on magnetic structures
nlowest energy structure is realized at ground states

28

Magnetic ions in solids
nconsider a situation where interactions are large
enough to realize a magnetic stateneach atom carries a magnetic momentnconsider at T=0
nwhich magnetic structure is most stable?
nwhat determines the magnetic structure?

29

Kanamori–Goodenough rulenexchange interactions of oxides and haleids
180o configuration

anion

anion

12

1

2

J12 < 0J12 > 0

anion

anion

90o configuration
11J12 > 02

J12 < 0230

Analysis using Hubbard modelnsimple but qualitatively correct
H=t(a†1 a2 + a†2 a1 ) + U
i=1,2

superexchange
energy gain due to electron transfer

antiferromagnetic
atom1
atom2

ni ni

t2
U

no energy gain because electrontransfer is prohibitedX

ferromagnetic
atom1
atom2

31

Analysis using Hubbard model
nsimple but qualitatively correct
H=t(a†1 a2 + a†2 a1 ) + U

ni ni
i=1,2

superexchange

double-exchange

energy gain due to electron transfer

t2
Uenergy gain due to electron transfer

nt

when holes exist

antiferromagnetic
atom1
atom2ferromagnetic
atom1
atom2

32

Ferromagnetism appears if carriers exist
nat half-filled cases only superexchange works
nantiferromagnetic (effects in second order of t)

nif carriers exist, double-exchange works inproportion to carrier concentration (effects in firstorder in t)n when n exceeds some value, ferromagnetism appears

nfor small t’s, even a small n realizes ferromagnetism

t2
< ntU

for smallt33

In terms of band electron
ntwo mechanisms in band picture
ferromagnetism stabilized

antiferromagnetism stabilizedband broadening by 2tt2
widening of split by 2UEFdouble-exchange

EFspuperexchange34

Spin-orbit interaction
na relativistic effectnin view of electrons turning around a nucleus, thenucleus is turning around the electrons
nturning nucleus thus produces electric current

LS

B

ZeL
mr3S
Hso = L · S

35

Effects of spin-orbit interaction
nmagnetic anisotropynwithout spin-orbit interaction, spins do not see lattices
nbecause of spin-orbit interactions, correlation between
directions of spins and lattice orientations arises
nspins prefer special direction relative to a latticenanisotropic exchange interaction
nanti-symmetric exchange interaction(Dzyaloshinsky–Moriya interaction: DMI)
norbital magnetic moment in crystals
36Magnetic anisotropy
nthere exists a special direction to whichmagnetization is apt to aligns
MJ(100)
(110)

0.06 Τ

μ0H

37

Anisotropic exchange
norbitals are distorted as a result of spin-orbit
interaction
nexchange interaction thus depends on the directionof spins

JAJB

JA = JBThis interaction can be expressed asS A JS B using symmetric tensor J38

Dzyaloshinsky–Moriya interactionnbecause of spin-orbit interaction, J depends ondirection (along or opposite way)
■ from A to B, there is C to the right of interaction lineB■ from B to A, there is C to the left of interaction line
As a result JAB = JBA
In general J is expressed as
CJ = Js ± Ja

This can either be expressed using spin as

A
Due to sin θdependence, thetwo spins cant

2JS A · S B + D · (S ASB )The second terms is called anti-symmetric exchangeinteraction (Dzyalonshinsky–Moriya interaction: DMI).
The interaction depends on the angle between direction
39of two spinsθ as sinθ.

Heisenberg modeln phenomenological model describing magnetism of materialsn describes magnetic behavior of magnetic insulators rather
well
n sometimes can be used for phenomenological descriptionof metallic magnetism as well (there also are many cases
where this does not work at all)H=<ij>

2Jij S i · S j40

J ij obtained from electronic structure calculation
nexample of calculated exchange coupling constants
Nd4g
Fe4c

B4g

Nd4f

C

Fe16k2

Fe8j2

Fe8j1
Fe16k1
Fe4e

B

OANd2Fe14B 68 atoms/u.c. P42/mnm
41

Heisenberg modeln magnetic anisotropy, anisotropic exchange, antisymmetricexchange, etc. can be also treated in the frameworkn variations such as Ising model, XY modeln typical playground for statistical physics is used to discuss
finite temperature properties and phase transitions ofmagnetic materials.

H=<ij>

2Jij S i · S j42

Summary
nmagnetism of insulators in mind, we discussed
natomic magnetism
nmolecular magnetism and exchange interactions
nmechanisms of exchange interactions
nexchange interactions between magnetic ions in crystals
neffects of spin-orbit interactions
nHeisenberg modelnfor further study:
nJ. Kanamori, “Magnetism” (Japanese) (Baifukan, Physics
Series 7)43