相互作用與自旋

5.0引言

5.1軌道,相互作用與自旋

5.2原子和分子的磁矩

5.3晶體的磁矩

5.4晶體的磁各向異性

5.5習(xí)題5.磁性與電子態(tài)1Generalremarks

Uniaxialcases

Cubiccrystals

WhysuccesslimitedOutline2DecouplingofSpinfromOrbit

Evenforaspin-dependentExc,suchasVonBarthandHedin(1972),

Vsxc=dExc/dr=Vxc0(r)+

sVxcm

(r,m)Thespin-upandspin-downstatesaredecoupled.Totalenergydependsonlyonthemagnitudeofspinpolarization,m,butindependentofitsdirection.3Spin-orbitCouplingCausesAnisotropySpin-orbitcoupling

Hsl=[(1/4c2r)?V/?r]

l(r)×s=x(r)

l(r)×sTotalenergyvariation

Esl(s

)=E(H0+Hsl)-E(H0)4PerturbationAnalysis

Firstorderenergy

E(s)=S

x

<o|l×s

|o>because<o|l|o>=0

Second(even)orderenergy,

E(s

)=S

x2|<o|l×s

|e>|2/(e(o)-e

(e))+h.o.tMostlybetweenspin-downbands5DirectionalDependence

Duetoorbitalcharacteroftheo-epairsnearFermisurface6PerpendicularAnisotropy

D.S.Wangetal,PRB47,14932,1993Fefilm:coupling<5|lz|5*>=<xz|lz|yz>causesperpendicularanisotropySingularityoccurswhen|e(o)-e(e)|<?

x7In-planeAnisotropy

D.S.Wangetal,JMMM129,344,1994Cofilm:coupling<5|ly|1>=<xz|ly|z2><5*|lx|1>=<yz|lx|z2>causesin-planeanisotropySingularityoccurswhen

|e(o)-e(e)|<?

x8Experimentsvs

Theory

D.S.Wangetal.JMMM140,643,19959AnisotropyvsBandFilling10AnisotropyofX-Co-X

D.S.WangPRB48,15886,199311NiLayersonCuSubstrate

J.Henketal,PRB59,9332(1999)12NiLayersonCuSubstrate

J.Henketal,PRB59,9332(1999)Forfct,thebulkcontributionisnearlycorrect,butcontributionofthesub-surfacelayerseemswrong.13DistortedCubicCrystalsT.Burkertetal.PRB69,104426(2004)14CubicCrystals–EarlyEmpirical

AuthorsE(001)-E(111)inmeV/atomRemarksbccFefccCofccNiExperiments-1.41.82.7

Kondorskiietal/JETP36,188(1973)xx1.3EmpiricalFritscheetal/J.Phys.F17,943(1987)7.4x10.0

15CubicCrystals-LSDA

AuthorsE(001)-E(111)inmeV/atomRemarksbccFefccCofccNiExperiment-1.41.82.7

Daalderopetal/PRB41,11919(1990)-0.5x-0.5Strangeetal/PhysicaB172,51(1991)-9.6x10.5Tryggetal/PRL75,2871(1995)-0.50.5-0.5Razeeatal/PRB56,8082(1997)-0.950.860.11Halilovetal/PRB57,9557(1998)-0.50.30.04

-2.62.41.0xscaling

16CubicCrystals–LSDA+OP

AuthorsE(001)-E(111)inmeV/atomRemarksbccFefccCofccNiExperiment-1.41.82.7

Tryggetal/PRL75,2871(1995)-1.82.2-0.5OPYangetal./PRL87,216405(2001)U=1.2xU=1.9ineVJ=0.8xJ=1.2ineV

Xieetal/PRB69,172404(2004)U=1.15U=1.41U=2.95ineV

J=0.97J=0.83J=0.28ineV

17Ab

InitioAttempt-SummaryBulkuniaxialcasesaregoodSurface(interface)layersarefairCubiccrystalsarepoor18UniaxialCase:Two<o|*|e>

Pairs

Reconsiderthesecondorderperturbation,

E(s

)=-S

x2|<o|l×s

|e>|2/(e(e)-e

(o))Itholdsonlywhene(e)-e(o)>x,andS?1

.Foruniaxialcases,theregularpartisin2ndorder(x2

/D)!

Whene(e)-e(o)<x,degenerateperturbationapplies,

E(s

)=-S|x<o|l×s

|e>|andS?(x2/|?ke(o)×?ke(e)|).Singularatthosekpoints.Totalcontributionisin3rdorder(x3

/D2)!.19CubicCase:Two<o|*|e>

Pairs

Thesecondorderperturbation,

E(s

)=-S

x2|<o|l×s

|e>|2/(e(e)-e

(o))isisotropic.Forcubiccase,theregularpartofanisotropygoesto

E(s

)?

-S

x4|<l×s

>|4/(e(e)-e

(o))3,andS?1

.Thecontributionisinthe4thorder(x2

/D3)!

Thesingularpartwith,

E(s

)=-S|x<o|l×s

|e>|andS?(x2/|?ke(o)×?ke(e)|)Singularatthosekpoints.Totalcontributionisin3rdorder(x3

/D2)!.20ChallengeinCubicCase