Sep. 11. 2017

Spintronics Basic II
Hisazumi Akai
The Ins(tute for Solid State Phisucs
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 Kanamori-Goodenough rulen Double exchangen Magnetic anisotropyn Anisotropic exchange
n Dzialshinsky-Moriya interactionn Heisenberg model2

Interaction between electrons
n two electrons –Pauli’s exclusion principle
possibility that two electrons come closer→ increase in coulomb energy

less probability of two electrons meet
→ suppress increase in coulomb energy

parallel spin is preferable

3

Interaction between electrons
n two electrons –Pauli’s exclusion principle
possibility that two electrons come closer→ increase in coulomb energy

less probability that two electro meet
→ suppress increase in coulomb energy
reason why electrons do not come closer is notcoulomb repulsion but Pauli’s exclusion principle

parallel spin is preferable

4

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

r p
antiparallel spin is preferable

5

Exchange interaction between electronsn energy difference between parallel and antiparallel
couples of electrons

EX = E– E6

Single atomn electrons are bounded in a potential
n neutral atom with an odd number of electrons
n nonzero total electron spinn nonzero angular momentum
n shows magnetism (paramagnetic)LS7

Tow atoms
n interaction between atomic magnetic momentsn parallel or antiparallel couplingn exchange interaction between atoms

EX = E– E

parallel if EX>0, antiparallel otherwise
8

Parallel or antiparallel?
n Simple model I Heitler–London limit
n two hydrogen atoms (hydrogen atom1 and hydrogen atom2)R)n potential of each atom v(ratomic orbital

atomic hamiltonianhydrogen atom1

H1 =2
1 + v(r1R1 )H1 1 =1hydrogen atom2

H2 =2
2 + v(r2R2 )H2 2 =2（we use Slater’s atomic unit:= 1, m = 1/2, e2 = 2 ）9Parallel or antiparallel?
n Simple 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 wave
functions φ’s of two hydrogen atoms
hamiltonian of hydrogen molecule

coulombic interaction

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

molecular energy

Er2 )R2 ) + v(r2(r1 , r2 )HR1 ) atractive potential due tothe other atom

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

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

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

Parallel or antiparallel?
n Simple 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

E1

12

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

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

even 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 ) = 00 (r1 , r2 )Vif

r1 = r20 (r1 , r2 )dr1 dr2 >1 (r2 ) 2 (r1 )}small interaction V

1 (r1 , r2 )V

1 (r1 , r2 )dr1 dr2
13

Parallel or antiparallel?
n Simple 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 (r1 )v(rR1 ) 1 (r1 )dr1 <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 > 0n triplet state realized in Heitler-London limit
n not corresponding to most two-atom moleculesn two-atom molecules are mostly spin singlet

n molecular bonding is caused by H12n ε’ < ε ：effect of attractive potential of neighbor
15

Parallel or antiparallel?
n Simple 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?
n Simple model 2: molecular orbital limit
antibonding statea (r) = A{ 1 (r)2 (r)}

most stable if
coulombic interaction
is missing

single electron energy

bonding stateb (r) = A{ 1 (r) +2 (r)}17Parallel or antiparallel?
n Simple 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?
n Simple model 2: molecular orbital limit
more stablewithout electroninteraction

E0electron interaction stabilizes
parallel coupling

E1
E0

E1orE0E1 ?

19Hitler–London vs. molecular orbital limitsn Heitler–London limit excludes ionic statesn strong correlation limit
n electron interaction energy

kinetic energyn equal probability of ionic and neutral states inmolecular orbital limit
n weak correlation limit
n electron interaction energy
ionic state

atom1

kinetic energy
neutralneutralcoulomb
atom2 interaction U atom1

atom2

ionic state

20Analysis using simple modeln 2-site Hubbard model

H=t†(a1a2†+ a2a1 ) + U

ni ni
i=1,2

coulombic
interaction U

site 1site 2electron transfer t
21

Exact solutionn ground state is spin singlet state for any U
energysingletsingletsinglet in mean
field approximation
triplet0molecular orbital limit

singletelectroninteraction U

Heitler-London limit

22

Exact solutionn ground state at U=0 andenergy+–
singlet+singlet
singlet in MFA
triplet0singlet tripletmolecular orbital limit

singletelectroninteraction U–

Heitler-London limit

23

Mean field approximation (MFA)n nonmagnetic to ferromagnetic transition occurs
energynonmagnetic

nonmagnetic

nonmagnetic
0electroninteraction U

ferromagnetic
singlet exact solution

molecular orbital limit

Heitler-London limit

24Mean field approximation (MFA)n nonmagnetic to ferromagnetic transition occurs
energyferromagnetic
0electroninteraction U

singlet exact solution
nonmagnetic

molecular orbital limit

Heitler-London limit25

In reality
n somewhere in between Heitler-London andmolecular orbitaln star8ng from Heitler-London
nionic states gradually mixed into neutral states
nalthough ionic states have higher energies, energy gain due toelectron transfer is expectedn star8ng from molecular orbital limit
nionic states are suppress owing to their higher energy
nbonding–an8bonding spliAng decreases, which reduce energy
gain due to electron transfer26

real atomic exchange couplingsn between two atomic spins S 1and S 2 of atom 1 andatom 2, there exists an interaction of the form

Ex = E0
whereE1 =J12 > 0,2(J12 + W12 )S 1 · S 2

W12 < 0is always satisfied and in many cases

J12 + W12 < 027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.

28

If many atoms exist
n when interaction is strong enough, spontaneous
symmetry breaking occurs
n magnetic moment arises at each siten stable magnetic structure realized
n in this situation MFA gives reasonable descriptions
n energy of systems depends of magnetic structuresn lowest energy structure is realized at ground states

29

Magnetic ions in solids
n consider a situation where interactions are large
enough to realize a magnetic staten each atom carries a magnetic momentn consider at T=0n which magnetic structure is most stable?
n what determines the magnetic structure?

30

Kanamori–Goodenough rulen exchange interactions of oxides and haleids
180o configuration

anion

anion

12

1

2

J12 < 0J12 > 0anion

anion

90o configuration
11J12 > 02J12 < 0231

Analysis using Hubbard modeln simple but qualitatively correct
H=t(a†1 a2 + a†2 a1 ) + U
i=1,2superexchange
energy gain due to electron transfer

antiferromagnetic
atom1
atom2

ni ni

t2
U

no energy gain because electrontransfer is prohibitedX

ferromagnetic
atom1
atom232

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

ni ni
i=1,2superexchange

double-exchange

2tUenergy gain due to electron transfer

energy gain due to electron transfer

nt

when holes exist

antiferromagnetic
atom1
atom2

ferromagnetic
atom1
atom2

33

Ferromagnetism appears if carriers existn at half-filled only superexchange works
n antiferromagnetic (effects in second order of t)n if carriers exist, double-exchange works inproportion to carrier concentration (effects in firstorder in t)n when n exceeds some value, ferromagnetism appears

n for small t’s, even a small n realizes ferromagnetismt2< nt
Ufor smallt34In terms of band electronn two mechanisms in band picture
ferrimagnetism stabilizedband broadening by 2tEF

double-exchange

antiferromagnetism stabilizedt2
widening of split by 2U

EFspuperexchange35

Spin-orbit interactionn a relativistic effectn in view of electrons turning around a nucleus, thenucleus is turning around the electrons
n turning nucleus thus produces electric current

LS

BZeL
mr3S
Hso = L · S

36

Effects of spin-orbit interactionn magnetic anisotropy
n without spin-orbit interaction, spins do not see latticesn because of spin-orbit interactions, correlation between
directions of spins and lattice orientations arises
n spins prefer special direction relative to a latticen anisotropic exchange interaction
n anti-symmetric exchange interaction(Dzyaloshinsky–Moriya interaction: DMI)n orbital magnetic moment in crystals
37

Magnetic anisotropy
n there exists a special direction to whichmagnetization is apt to aligns
MJ(100)
(110)

0.06 Τ

μ0H

38

Anisotropic exchangen orbitals are distorted as a result of spin-orbitinteraction
n exchange interaction thus depends on the directionof spins

JAJB

JA = JBThis interaction can be expressed as

S A JS B using symmetric tensor J39

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

J = 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
40of two spinsθ as sinθ.

Heisenberg modeln phenomenological model describing magnetism of mattern 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 j41

Jij obtained from electronic structure calculationn example of calculated exchange coupling constants

42

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 j
43Summaryn magnetism of insulators in mind, we discussed
n atomic magnetism
n molecular magnetism and exchange interactions
n mechanisms of exchange interactions
n exchange interactions between magnetic ions in crystals
n effects of spin-orbit interactionsn Heisenberg modeln for further study:n J. Kanamori, “Magnetism” (Japanese) (Baifukan, PhysicsSeries 7)44