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Doctoral

Ferrimagnetic

romF

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Physics

Heusler

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MeinertMarkus

October

Bielefeld

2011

University

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ounds

Thiscorrworkespondingwasdonepublications,bymyself.whichTextoriginateanddirfigurectlyesfrweromethispartlywork.takenfrom

Meinert)(Markus

Reviewers:

PrProf.of.DrDr..JG¨ur¨untergenReissSchnack

Copyrightc2011MarkusMeinert
BielefeldUniversity,DepartmentofPhysics

ThinFilmsandPhysicsofNanostructures

TypefacePalladioandPazoMath10pt.
SystemLATEX2εandKOMA-Script.

”What

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Feynman

Publications

thesisthisinincludedPublications

interactions•Abinitiopredictionofferrimagnetism,exchange
CurietemperaturesinMn2TiZHeuslercompounds

CurietemperaturesinMn2TiZHeuslercompounds
M.MEINERT,J.-M.SCHMALHORST,ANDG.REISS
J.Phys.Condens.Matter23,063001(2011)

•ExchangeinteractionsandCurietemperaturesofMn2CoZ
M.MEINERT,J.-M.SCHMALHORST,ANDG.REISS
J.Phys.Condens.Matter23,116005(2011)

onicElectrestructuroffullyepitaxialCo2iSnTthinfilmsandcompoundsM.MEINERT,J.SCHMALHORST,H.WULFMEIER,G.REISS,
E.ARENHOLZ,T.GRAF,ANDC.FELSER
PhysicalReviewB83,064412(2011)

•FerrimagnetismanddisorderofepitaxialMn2−xCoxVAl
compoundHeuslerfilmsthinM.MEINERT,J.-M.SCHMALHORST,G.REISS,ANDE.ARENHOLZ
J.Phys.D:Appl.Phys.44,215003(2011)

ItinerantandlocalmagneticmomentsinrimagneticferMn2CoGathinfilmsprobedbyx-raymagneticlineardichroism:Experimentandabinitiotheory
M.MEINERT,J.-M.SCHMALHORST,C.KLEWE,G.REISS,
E.ARENHOLZ,T.BO¨HNERT,K.NIELSCH
PhysicalReviewB84,132405(2011)

5

filmsthinAlV2)xCox−1(MnepitaxialindisorderandFerrimagnetismfilmsthiniSnT2Coonic6

identifiedwithx-raymagneticlineardichroism
M.MEINERT,J.-M.SCHMALHORST,C.KLEWE,G.REISS,
E.ARENHOLZ,T.BO¨HNERT,K.NIELSCH
56thAnnualConferenceonMagnetism&MagneticMaterials,
DE-11alk,TScottsdale,

M.MEINERT,J.SCHMALHORST,G.REISS,
E.ARENHOLZ,R.LASKOWSKI
CECAMWorkshop,X-raySpectroscopy:RecentAdvances
inModellingandNewChallenges,Z¨urich,Poster

propertiesoftheHeuslercompoundCo2TiSnfromfirstprinciples
M.MEINERT,J.-M.SCHMALHORST,ANDG.REISS
Appl.Phys.Lett.97,012501(2010)

•Inuenceoftetragonaldistortiononthemagneticandelectr
propertiesoftheHeuslercompoundCo2TiSnfromfirst
principles

Publicationsnotincludedinthisthesis

M.MEINERT,J.SCHMALHORST,H.WULFMEIER,G.REISS,
E.ARENHOLZ,T.GRAF,ANDC.FELSER
DPGFr¨uhjahrstagungDresden2011,Talk,MA52.1

Conferencecontributions

•Electronicstructureoffullyepitaxial
MHCS.J,TRENIEM.MTSROHLA

M.MEINERT,J.-M.SCHMALHORST,G.REISS,ANDE.ARENHOLZ
DPGFr¨uhjahrstagungDresden2011,Poster,MA63.27

filmsnthiCoGa2MnferrimagneticinmomentsMnlocalizedandItinerantCompoundsHeuslerFerrimagneticofoscopySpectrMagnetic

Contents

1

2

3

4

5

6

7

ionuctdIntro

ExperimentalMethods
2.1SamplePreparation..........................
2.2StructuralCharacterization.....................
2.3ChemicalCompositionAnalysisbyX-RayFluorescence.....
2.4SoftX-RayAbsorptionSpectroscopy................
2.5OtherTechniques...........................

TheoreticalMethods
3.1DensityFunctionalTheory......................
3.2ImplementationsofDFTUsedinThisWork............
3.3CurieTemperaturesfromanEffectiveHeisenbergModel....
3.4X-RayAbsorptionSpectrafromElectronicStructure.......

Ferrimagnetism,exchangeandCurietemperaturesinMnTiZ
24.1Introduction..............................
4.2Results.................................

ExchangeandCurietemperaturesofMnCoZcompounds
25.1Introduction..............................
5.2Results.................................

ElectronicstructureoffullyepitaxialCoTiSnthinfilms
26.1Introduction..............................
6.2Experimentalresults.........................
6.3Electronicstructure..........................

FerrimagnetismanddisorderofepitaxialMnCoxVAl
x2−7.1Introduction..............................
7.2Methods................................
7.3Experimentalresultsanddiscussion................

9

141415182023

2424303537

434345

565657

66667081

85858687

7

Contents

8ItinerantandlocalmagneticmomentsinferrimagneticMnCoGa95
28.1Introduction..............................95
8.2Results.................................96

Concluding9rksrema

Bibliography

wledgementsAckno

8

103

106

114

onductiIntro1

TheHeuslercompoundsareaclassofintermetalliccompoundswiththegen-
greraloupchemicalelement.formuTheylaX2YZcrystallize,wherbyeX,YdefinitionareintransitiontheL21metals,structurandeZis(Fig.amain1.1),
¯aThefacecoorcenterdinatesedofcubicthesestrfouructuresites(spaceA,B,grC,oupandFmD3,m)arewithgivenabyfourA=atom(0,0,basis.0),
B=(41,41,41),C=(42,42,42),D=(43,43,43).Thestructurehasinversionsym-
metryelement.,InmakingthetwoWyckofsitesf(Anotation,andC)theAequivalent.andCcitesThesearareenamedoccupied8c,byandthetheX
othertwositesaredenotedas4a,4b.
TheprototypeoftheHeuslercompoundsisCu2MnAl,whichwasdiscovered
bywasFriedrichdeterminedHeuslerbyinBradley1903[1and].TheRodgerscompound,in1934the[2],iscrystalaferrstructuromagneteofwithwhicha
highCurietemperature,thoughnoneofitsconstituentsisferromagneticby
itself.titudeTodayof,difweferknowentprmoreoperties.thanMost1000knownHeuslerquantumcompounds[3mechanical],whichgroundhaveastatesmul-

Figure1.1:Left:Conventional(cubic)unitcelloftheL21(Heusler)structure.Right:
Conventional(cubic)unitcelloftheXA(inverseHeusler)structure.Xsitesarered,Y
andZsitesareblueandgreenrespectively.

9

oductionIntr1

ofsolidsarerepresentedwithinthisclass.Justtomentionafew,therearefer-
romagnets(Cu2MnAl[1]),ferrimagnets(Mn2VAl[4]),semiconductors(Fe2VAl
[5]),heavyfermionsystems(Cu2CeIn[6]),andsuperconductors(Ni2ZrGa[7]).
Oneparticularlyintriguingproperty,whichispredictedforanumberof
magneticHeuslercompounds,ishalf-metallicferro-/ferrimagnetism(HMF):
foreitherthemajorityorminoritydensityofstatesagapispresentaround
theFermienergy.Thus,thematerialbehavesmetallicforonespinspecies,
andsemiconductingorinsulatingfortheotherone.Thehalf-metallicferro-
magnetismofaHeuslercompoundwasfirstpredictedbyK¨ubleretal.for
Co2MnSi[8].TheCo2-basedhalf-metallicHeuslercompoundshaveagapfor
theminoritystates.Thispropertyisparticularlyinterestingforspin-electronic,
orspintronic,applications,whichmakeuseofspin-polarizedcurrents.These
includeinparticulargiantandtunnelmagnetoresistivedevices.Thefullspin
polarizationofthecurrentcarriersinaHMFgivesrisetolargemagnetoresistive
fects.efAhalf-metallicferrimagnethasadvantagesoverthewell-knownhalf-metallic
ferromagnets:duetotheinternalspincompensationithasaratherlowmag-
neticmoment,whiletheCurietemperatureremainsfairlyhigh.Alowmag-
neticmomentgivesrisetolowstrayfields,whichisdesiredforspintronics,as
isahighCurietemperatureandthusagoodthermalstabilityofthecompound
[9].ThemostprominentHeuslercompoundoutofthisclassisMn2VAl,which
hasbeenstudiedthoroughlybyexperimentandtheory[4,10,11,12,13].Sev-
eralothermaterialclasseshavebeenproposedtobehalf-metallicferrimagnets,
e.g.,Cr0.75Mn0.25SeandCr0.75Mn0.25Teinthezincblendestructure[14],orCr
antisitesinCrAs,CrSb,CrSe,andCrTe,havingthezincblendestructure[15].
Ideally,anelectrodematerialforspintronicswouldbeahalf-metalwith
zeronetmoment.Thiscannotbeachievedwithantiferromagnetsbecauseof
thespin-rotationalsymmetry(resultinginzeropolarization),butwellchosen
half-metallicferrimagnetscanbetunedtozeromoment.Thispropertyis
alsoknownashalf-metallicantiferromagnetism,andhasbeenfirstpredicted
forMnandIndopedFeVSb[16].Amongothers,La2VMnO6andrelated
doubleperovskites[17]andcertaindilutedmagneticsemiconductorshave
beenlaterpredictedtobehalf-metallicantiferromagnetsaswell[18].However,
half-metallicantiferromagnetismislimitedtozerotemperatureandasmall
macroscopicnetmomentisexpectedatelevatedtemperature—inparticular
neartheCurietemperature—becauseoftheinequivalentmagneticsublattices
[19].

10

Structureofthiswork
Amajorfocusofthisworkisputonthedirectcomparisonoftheoretical
andcompounds.experimentalPartsofprthisopertiesworkofarethinpurelyfilmstheorofetical,variousaimedferrimagnetictowardabasicHeuslerun-
derstandingofthepropertiesofferrimagneticHeuslercompounds.Otherparts
combineexperimentalworkandtheoreticalapproachestoexplainthedata
ortomethodstestarpreedictions.outlinedinTheChaptersbasicand2andmost3.usedexperimentalandtheoretical

PredictingnewferrimagneticHeuslercompounds
AveryinterestingclassofHeuslercompoundsthathasreceivedconsiderable
theoretical,butonlyfewexperimentalattentiontodate,arethehalf-metallic
ferrimagnetsMn2YZ,whereY=V,Cr,MnandZisagroupIII,IV,orVelement
[20,21,22].FollowingtheSlater-Paulingruleconnectingthemagneticmoment
mandthenumberofvalenceelectronsNVviam=NV−24inthehalf-metallic
Heuslercompounds[23],itisexpectedtofindanotherseriesofferrimagnetic
half-metalsintheMn2TiZsystemwith−3to−1µBperformulaunit(f.u.).
Thenegativemomentindicatesthatthehalf-metallicgapwouldappearfor
themajoritystates,justasinthecaseofMn2VAl.Thesecompoundscould—if
theyarehalf-metals—provideanotherseriesofpotentialelectrodesforspin-
dependentapplicationsandcouldalsobecomeastartingpointforhalf-metallic
omagnetism.antiferrChapter4discussesthepropertiesofthisnewlypredictedclassofferrimag-
compounds.Heuslernetic

ExplainingtheexchangeinteractionsofinverseHeusler
ndsoucompCloselyrelatedtotheHeuslercompoundsaretheso-calledinverseHeusler
compounds,whichhaveasimilarlatticestructure(seeFig.1.1),butmiss
theinversionsymmetry(spacegroupF4¯3m,prototypeHg2CuTi).Here,sites
BandCareoccupiedbythesameelement.ThesecompoundsareHeusler
compoundsinageneralizedsense,i.e.,aface-centeredcubicstructurewiththe
abovegivenatomicpositions.Thisoccupationispreferredwithrepecttothe
HeuslerstructureifXhaslessvalenceelectronsthanY[24,25].

11

1oductionIntr

Recently,theMn2YZinversecompoundshaveattractedconsiderabletheo-
reticalandexperimentalactivities,whereY=Fe,Co,Ni,Cu[25,26,27,28,29,
30,31,32,33,34,35,36,37].Half-metallicferrimagnetismhasbeenpredicted
fornumerouscompoundsfromthisclass[28].TheMn2YZcompoundsalso
followtheSlater-Paulingruleconnectingthemagneticmomentmandthe
numberofvalenceelectronsNVviam=NV−24inhalf-metallicHeusler
[23].compoundsThecomputedexchangeinteractionsandassociatedpropertiesoftheMn2CoZ
compoundsarediscussedinChapter5.

ferrimagnetcobalt-basedeakwAThetensivematerialstudiesclassinoftheCo2contextYZofHeuslerspintronicscompoundsduringhasthebeenlastthedecade.subjectofTheseex-
compoundsareofinterestbecausemanyofthemarepredictedashalf-metallic
ferromagnetswithfullspinpolarizationattheFermiedge.
TheHeuslercompoundCo2TiSn(CTS)ispredictedtobeahalf-metallic
ferrimagnetwithamagneticmomentof2µB/f.u.andithasahighformation
energyoftheCo-Tisite-swapdefect[38,39].Usually,disorderdestroysthe
half-metallicity.Hence,makinguseofHeuslercompoundswhichexhibitlow
disorderorhightoleranceofthegroundstatepropertiesagainstdisorderis
ed.desirhighlyInChapter6theelectronicstructureofthinfilmsoftheweakferrimagnet
Co2TiSnisdiscussed.

ensationcompmagneticfullAchievingGalanakisetal.pointedoutthatitmaybepossibletosynthesizeaHMFi
bysubstitutingCoforMnintheHeuslercompoundMn2VAl[40].Mn2VAl
isa(potentiallyhalf-metallic)ferrimagnetwithantiparallelcouplingofMn
andVmomentsandatotalmomentof-2µBperformulaunit.Thehigh
Curietemperatureof760Kmakesitinterestingforpracticalapplications.
Numerousexperimental[4,10,11,41,42]andtheoretical[12,13,20,43,44]
studiesarefoundintheliterature.FollowingtheSlater-PaulingruleforHeusler
compounds,m=NV−24[23],themagneticmomentmistobetakenas
negative,becausethenumberofvalenceelectronsNVis22.Thus,byadding
effectivelytwoelectronsperunitcell,themagnetizationshouldvanish.This
canbeachievedbysubstitutingoneMnwithoneCoatom,whichhastwo

12

additionalelectrons.Abinitiosimulationswerecarriedoutonthissystemin
theL21structurewithMnandCorandomlyspreadacrosstheWyckoff8csites
andVandAlonthe4aand4bsites.Indeed,aHMFiisfoundwithmagnetic
momentsof:-1.388(Mn),0.586(Co),0.782(V),0.019(Al)[40].Itwasshown
byLuoetal.thatthesiteoccupationpreferenceinMn2YAldependsonthe
numberofvalenceelectronsofY:ifitislowerthantheoneofMn,Ywould
preferentiallyoccupythe4a/bsites,butifitishigher,Ywouldratheroccupy
the8csitestogetherwithMn,changingthestructuretotheHg2CuTitype[45].
Accordingly,onecanexpectanoccupationasproposedbyGalanakisetal.in
Mn2−xCoxVAl(MCVA).
Chapter7focussesonthesynthesisandcharacterizationofthinfilmsofthe
ferrimagneticMn2−xCoxVAlsystem.

ThefirstthinfilmsofaninverseHeuslercompound
Todate,theinverseHeuslercompoundswerestudiedonlyinthebulk.For
manypracticalapplications,suchasintunnelorgiantmagnetoresistance
(TMR,GMR)devices,thinfilmsarenecessary.
Additionally,itcanbeverydifficulttopreparehigh-qualitysinglecrystalsof
(inverse)Heuslercompounds,sopreparationofepitaxialthinfilmsprovides
anattractivealternativeroutetostudyanisotropicpropertiesofthesematerials.
ThefinalChapter8dealswiththerelationbetweentheinverseHeusler
compoundMn2CoGaandtheHeuslercompoundsMn2VGaandCo2MnSi.

13

dsMethoerimentalExp2

Withinthiswork,thinfilmshavebeenpreparedandcharacterized.This
andchaptergivescharacterization.abriefAllintrsamplesoductionwerintoethepreparmainedbyDCtechniquesandofRFprmagnetreparationon
oftheco-sputteringfilmswasandelectrperformedonbybeamx-raydifevaporation.fractionTheandstrructuraleflectivity.Thecharacterizationchemical
resolvedcompositionmeasuranalysisementswasofthedoneelectrbyharonicdandx-rayfluormagneticescence.structureElement-oftheandsamplessite-
werecarriedoutbysoftx-rayabsorptionspectroscopy.

rationPrepaSample2.1

AllthinfilmsamplespresentedinthisworkweredepositedbyDCandRF
magnetronco-sputtering[46]onMgOsubstrateswith(001)orientation.The
apparatususedforthedepositionisacustomlydesignedmachinebuiltby
BESTEC,BERLIN.Itsultra-highvacuumrecipientisequippedwith(atthe
timeofwriting)seventhree-inchmagnetronsputtersourcesandanelectron
beamevaporator.Thesourcesareplacedinaconfocalgeometry,withthe
substratecarrierinthefocus,seeFig.2.1.FiveofthesourcesaredrivenbyDC
generators,theothertwosourcesaredrivenbyanRFgenerator,operatedat
13.56MHz.Thisallowstoco-sputtermetalsandinsulators.Thesamplecarrier
canberotatedtoobtainhomogenousthicknessandstoichiometryacrossa
diameterofabout100mm.Itcanberadiativelyheatedwithaceramicheater
withapowerofupto1000W,yieldingasamplecarriertemperatureofover
900◦C.Highpurity(6N)argonisusedasthesputtergas,typicallyatapressure
of2∙10−3mbar.Areactivegas(oxygenornitrogen)canbeaddedifdesired.
TheelectronbeamevaporatorismostlyusedtodepositaprotectiveMgO
filmontopofthesample,inordertoprotectthefilmbelowfromoxidation.
Itisusuallyoperatedat6kVandabeamcurrentof10mA(forMgO).The
depositionprocesscanbecalibratedandmonitoredwithafilmthickness
.sensor

14

CharacterizationucturalStr2.2

Figure2.1:Technicaldrawing(crosssection)oftheBESTECsputtermachine[47].

racterizationChaStructural2.2

DiffractionyX-Ra2.2.1Thediffractionofx-raysisawellknownandversatiletooltodeterminethe
structurwavelengtheofaλandcrystallinethedifsolidfraction[48].angleBragg’sθvialawrelatesthelatticespacingd,the

λ=2dsinθ.

(2.1)

hIn,ka,lcubicandthematerial,latticetheconstantlatticea,suchspacingthatcanonebecanexprexpressedesswithBragg’sMillerlaw’sasindices

λsinθhkl=2ah2+k2+l2.(2.2)
rBragg’seflection,lawbutitdescribesdoesatnotprwhichedictdifthefractionintensityangles.TheonecanintensitypossiblyI(hkl)findofananx-rayx-ray

15

MethodsExperimental2

reflectionfromthe(hkl)planeofathinfilmonasubstrateisgivenby
I(θhkl)∝|F(hkl)|2LP(θhkl)DW(θhkl)ODFhkl(ϕ,ψ).(2.3)
ThestructurefactorF(hkl)containstheinformationonthecrystalstructure.It
isderivedasaFouriertransformofthechargedensityofthesolid,giving
F(hkl)=∑fj(θhkl)e2πi(hxj+kyj+lzj),(2.4)
jwherefj(θhkl)istheatomicformfactorandxj,yj,zjarethecoordinatesofsite
jintheunitcell.fj(θhkl)equalstheatomicnumberinthelongwavelength
(λ→∞)orforwardscattering(θ→0)limit.
TheLorentz-Polarizationfactorincludesthediffractiongeometryandpo-
larizationeffectsfromthediffraction.Forapowderorpowder-likefilmitis
asgiven2LP(θhkl)=12+cos2θhkl,(2.5)
sinθhklcosθhkl
wherethenumeratordescribesthepolarizationandthedenominatorthe
diffractiongeometry(theLorentzterm).ThetemperatureorDebye-Waller
factorDW(θhkl)takesintoaccountlatticevibrations,whicharenegligibleinthe
casesdiscussedinthiswork.Finally,thepoledensityororientationdistribution
functionODFhkl(ϕ,ψ)describesthedistributionofcrystalorientationswith
respecttotheEuleranglesϕ,ψ.Itaccountsfortextureandepitaxialgrowth
anditcanbeinterpretedasasetoftwo-dimensionalrockingcurves.
Disorderisaccountedforbyappropriateweightingoffi(θhkl)withthesite
occupancies.Further,inamoregeneralexpressiontheatomicformfactor
containsanomalousscatteringcorrectionswhichdependontheenergyE:
f(θ,E)=f0(sinθ/λ)+f1(E)+if2(E),(2.6)
whereE=hc/λ.Thesecorrectionsareimportantclosetoatomicabsorption
edges.TheyaretabulatedorcomputedwiththeCromer-Libermanmethod
[49,50].Therefore,expression(2.3)ismostconvenientlyevaluatednumerically.
ForHeuslercompounds,wecandivideallpossiblex-rayreflections(those
allowedbytheextinctionrulesforthefacecenteredcubiclattice)intothree
groupswiththreedifferentstructurefactors[51]:
•h,k,lallodd((111),(311),(331),(333),(511),(531),...)
|F(111)|2=16(fA−fC)2+(fB−fD)2(2.7)


16

(2.7)

CharacterizationucturalStr2.2

•h+k+l=2(2n−1),n=1,2,...((200),(222),(420),(600),(442),...)
|F(200)|2=16[(fA+fC)−(fB+fD)]2(2.8)
•h+k+l=4n,n=1,2,...((220),(400),(422),(440),(620),(444),...)
|F(400)|2=16[fA+fB+fC+fD)]2(2.9)
Thestructurefactorsaregivenhereneglectingtheanomalouscorrectionterms.
Thethirdgroupofreflectionsisindependentofchemicaldisorderonthe
foursublattices,makingitafundamentalreflection.Theothertwogroups
dependondisorder;thefirstgroupvanishesifB-Dorderisnotpresent,i.e.,the
structureisequivalenttotheB2structure(aprimitivecubicstructurewithtwo
atomsinthebasis).Thesecondgroupvanishes,ifadditionallyA/C-B/Dorder
ismissing,suchthatthestructurebecomesequivalenttotheA2structure(a
primitivebody-centeredcubicstructure).Inthelattercase,thefoursublattices
occupied.randomlyearThewidthofthereflectionshascontributionsfromtheinstrumentitself,
fromthesizeofthecrystallitesandfromstrainwithinthecrystallites.Witha
GaussianinstrumentalpeakbroadeningandaLorentzianconvolutionofgrain
sizeandstraineffects,oneseperatesthecontributionsby
B2obs=B2inst+B2ss,(2.10)

ewherλkBss∙cosθ=D+4ε[hkl]sinθ.(2.11)
Bobsistheobservedintegralwidth,Binsttheinstrumentalwidth,Bssthesize-
strainwidth,theshapefactork=0.9,thecoherencelength(grainsize)Dand
theaveraged[hkl]componentofthestraintensorε[hkl].Thisschemeiscalled
Williamson-Hallanalysis[52].Theinstrumentusedforthiswork,aPHILIPS
X’PERTPROMPD,isequippedwithBragg-Brentanooptics,collimatorpoint-
focusoptics,andanopenEulercradle.ItisoperatedwithCuKαradiation
(λ=1.5419˚A).

ReflectionyX-Ra2.2.2Forverysmallanglesofincidence,acrystalcanbedescribedasaneffective
medium,i.e.,intermsofopticaltheory.Itisconvenienttowritetherefractive

17

MethodsExperimental2

indexinthex-rayregimeasn=1−δ+iβ,whereδ,βaresmallpositive
numbers.Therefractiveindexissmallerthanunityforx-rays,sothephase
velocityofx-raysisslightlylargerinthemediumthaninvacuum.Thisgives
risetoatotalexternalreflectionofthex-raysuptoacriticalangleθc.Neglecting
absorption(β=0)onefinds
√√θc=2δ∝naRef(0)∝naZ∝√ρ,(2.12)
withthenumberofatomspervolumena,theforwardscatteringlimitofthe
atomicformfactorf(0),thenuclearchargeZandthemassdensityρ[53].
Therefore,onecandeterminethemassdensityofafilmbydeterminingthe
criticalangle.Foracompound,thestoichiometryhastobeknownapproxi-
matelyinordertoapplytheproperanomalousscatteringcorrections.Above
thecriticalangle,thereflectivitydropsquicklyas1/θ4.
Penetrationofx-raysintoathinfilmonasubstrategivesrisetopartialreflec-
tionsattheinterfaces.Theseaddupcoherentlyandproduceaninterference
patternsimilartotheFabry-Peroteffect,theKiessingfringes.Fromthespacing
ofthemaximaorminimaθm+1−θmonecandeterminethefilmthicknessd:
1λd≈2θm+1−θm,θθc.(2.13)
Roughnessreducestheamplitudeoftheoscillationsandcancomplicatethe
determinationofthefilmdensity.Inpractice,anx-rayreflectivitymeasurement
isfitnumericallywiththeParrattformalism,whichincludesabsorptionand
roughnessandallowstofitmultiplelayers[54].

2.3ChemicalCompositionAnalysisbyX-Ray
rescenceFluoHardx-rayfluorescenceisawidelyusedtoolforchemicalcompositionanalysis
ofelementsheavierthansodium.Aphotoninteractingwithanatomcan
promoteanelectrontothecontinuumifthephoton’senergyishigherthanthe
electron’sbindingenergy.Figure2.2(left)showsthetermschemeofthelowest
absorptionedges,theK-edgeandtheL1,2,3-edges.Thecreatedvacancy(the
core-hole)isfilledbyelectronsfromhigherlevels,eitherviatheAugerprocess
emittinganotherelectron,orradiativelybyemissionofaphoton.Thelatter
processisthex-rayfluorescence,anditsprobability(thefluorescenceyield)

18

2.3

Chemical

Composition

Analysis

by

X-Ray

escenceFluor

2

Experimental

eFigur

soft

2.3:

x-ray

electron

Methods

Detection

modes

absorption:

yield

luminescence.

and

of

(total)

substrate

2.4SoftX-RayAbsorptionSpectroscopy

2.4.1X-RayAbsorptionNearEdgeStructure
Theunoccupiedstatesinasolid(orinamolecule)giverisetoresonantab-
sorption,andresultinanx-rayabsorptionnearedgestructure(XANES,also
nearedgex-rayabsorptionfinestructure(NEXAFS)).Thiscanbeusedto
extractinformationonhybridizationsororientationdependenceoforbitals.
Severaldichroiceffectscanbeobservedinx-rayabsorption,someofwhichare
associatedwithmagnetism;thesearepresentedinthefollowing.

2.4.2X-RayMagneticCircularDichroism
X-raymagneticcirculardichroism(XMCD)occursifthespin-upandspin-
downfinalstatesaredifferent,i.e.,ifthesystemisferromagnetic.Acircularly
whichpolarizedis(withmagnetizedasingleparallelphotonorhelicity)antiparallelx-raytothebeamisbeamdirabsorbedection,byseetheFig.sample,2.4.
Theresultingspectra,µ+(E)andµ−(E),canbecombinedtotheaveragex-ray
absorptionandthedifferencespectrum,
XAS(E)=1(µ+(E)+µ−(E))(2.14)
2XMCD(E)=µ+(E)−µ−(E).(2.15)
Theseorbitalspectramomentscanofbetheevalabsorberuatedwith[58].theOneXMCDdefinessumrintegralsulestop,qobtainandrasspinand
p=LdE(µ+−µ−)
3q=L3+L2dE(µ+−µ−)
µ++µ−
r=L3+L2dE2−S
Ano-free-parametertwo-step-likebackgroundfunctionSwiththresholdsset
tothepointsofinflectiononthelowenergysideoftheL3andL2resoncance
inandthesteppost-eheightsdgeofregion2/3(L3)and(”post-edge1/3(L2jump)oftheheightaverageη”)isintrabsorptionuducedcoefhere.ficientIt
accountsfortheabsorptionintodelocalized,s-likestates.
Sufficientlyfarawayfromtheabsorptionedges,interactionsamongthe
atomsinthesamplescanbeneglected[59]andthepost-edgejumpheight
ηisproportionalto∑iXiσai,whereXiistherelativeconcentrationofatomi

21

MethodsExperimental2inthesampleandσaiisitstotalatomicabsorptioncrosssection.Aspointed
outbySt¨ohr[60],thenumberofunoccupied3dstatesNhisproportional
totheintegralrviar=CNhη.TheconstantCdependsonthetransition
matrixelementsconnectingthecoreandvalencestatesinvolvedinthe2p
–3dtransitionsandhasbeenanalyzedbyScherzfordifferent3dtransition
metals(CTi=5.4eV,CV=5.3eV,CCr=5.7eV,CMn≈6.0eV,CFe=6.6eV,
CCo=7.8eV,CNi=8.1eV;theMnvalueisinterpolatedbetweentheother
data)[61].WhenneglectingthespinmagneticdipoletermTZintheXMCD
sumrules,thespinandorbitalmagneticmomentsmspinandmorbandtheir
ratioarethengivenas
41qmorb=−Phνcosθ6Cη(2.16)
1(6p−4q)
mspin=−Phνcosθ2Cη(2.17)
2qmmspinorb=9p−6q(2.18)
withtheellipticalpolarizationdegreePhνandtheangleθbetweenmagnetiza-
tionandx-raybeamdirection.
2.4.3X-RayMagneticLinearDichroism
X-raymagneticlineardichroism(XMLD)arisesasthedifferencebetween
parallelandperpendicularorientationofx-raypolarizationandmagnetization
whenusinglinearlypolarizedlight(seeFig.2.4):
XMLD(E)=µ(E)−µ⊥(E).(2.19)
BecauseXMLDisessentiallygivenasthedifferencebetweenΔm=0and
theaveragedΔm=±1transitions,itisasensitiveprobeofthelocalcrystal
field.Forsystemswithmdegeneracy,i.e.,sphericalsymmetry,itisapproxi-
matelygivenbyXMLD(E)≈0.1ΔddEXMCD(E).Δdescribesthecore-level
exchangesplittingduetothelocalmagneticfield.ΔandtheXMCDscalewith
thelocalspinmagneticmoment,whichgivesrisetoaquadraticdependence
onthelocalspinmoment[62,63,64].IncontrasttoXMCD,XMLDisonly
sensitivetothedirectionofthespinmoments,nottheirorientation.Thisal-
lowstoprobeantiferromagneticandferrimagneticmaterialswithXMLD.For
22

2.5

Other

echniquesT

3TheoreticalMethods

Thebasedtheoroneticaldensitypartsoffunctionalthisworktheoryhave(DFT).beenDiffercarriedentoutwithimplementationscomputerofcodesDFT
comewithindividualadvantagesanddisadvantages.Asauser,onehasto
decidewhichimplementationisbestsuitedfortheproblemtobeinvestigated.
Thischoicedependstoalargepartonthefeaturesetofthevariouscomputer
thecodes,butfull-potentialalsoonthelinearizedsuitabilityofaugmentedthepbasislanesetforwavesthepr(FP-LAPW)oblem.Formethod,thiswork,the
real-spacespin-polarizedrrelativisticelativisticfullmultipleKoringa-Kohn-Rostokerscatteringmethod(SPRKKR)(implementedmethod,inandFEFF9)the
used.beenhaveInthischapter,thebasicideasofDFTareoutlinedfollowingRichardMartin’s
textbook[65].Thedescriptionsofthecomputercodesinvolvedetailsofthe
basissetsandthesolutionmethods.Particularfocusisputontherelevant
Curiefeaturesprtemperaturovidedesinwithintheancodes.efTfectivewoHeisenberimportantgmodelmethods,andthethecalculationcomputationof
(ofe2=x-rayh¯=meabsorption=1)arespectra,usedarthreoughoutdiscussedthisinchapterindividual.sections.Atomicunits

3.1DensityFunctionalTheory

DensityfunctionaltheoryasformulatedbyHohenbergandKohnin1964[66]
isanexacttheoryofaninteractingelectrongasinanexternalpotential.Inthe
caseofasolidoramolecule,theexternalpotentialistheCoulombpotentialof
thenuclei,whichareassumedasfixed(Born-Oppenheimer-Approximation).
TheHamiltonianofthemany-electronsystemcanbewrittenas
111Hˆ=−2∑i2+2∑ri−rj+∑Vext(ri)+Enn(3.1)
ii=ji
inwhichthefirsttermisthekineticenergy,thesecondtermistheCoulombic
repulsionbetweenelectronpairs,andthethirdtermdescribestheenergyofthe
electronsintheexternalpotential.Ennistheclassicalinteractionofthenuclei

24

TheoryFunctionalDensity3.1

(3.3)

andalsocontainsallothercontributionstotheenergythatdonotinfluencethe
electrons.ThestationarysolutionoftheN-electronSchr¨odingerequationhas
theformΨ(r1,...,rN).Theelectrondensityn(r)isgivenbytheexpectation
valueofthedensityoperatornˆ(r)=∑i=1,Nδ(r−ri):
n(r)=Ψ|nˆ(r)|Ψ.(3.2)
ΨΨ|ThetotalenergyistheexpectationvalueoftheHamiltonian:
E=Hˆ:=Ψ|Hˆ|Ψ.(3.3)
ΨΨ|3.1.1TheHohenberg-KohnTheorems
HohenbergandKohnprovedthefollowingtheorems:
•TheoremI:Foranysystemofinteractingparticlesinanexternalpotential
Vext(r),thispotentialisdetermineduniquelyuptoanadditiveconstant
bythegroundstateparticledensityn0(r).SincetheHamiltonianisthus
fullydetermineduptoaconstantshiftoftheenergy,itfollowsthatthe
many-electronwavefunctionsaredetermined.Thereforeallpropertiesof
thesystemarecompletelydeterminedbythegroundstatedensityn0(r).
•TheoremII:AuniversalfunctionalfortheenergyE[n]intermsofthe
densityn(r)canbedefined,validforanyexternalpotentialVext(r).For
anyparticularVext(r),theexactgroundstateenergyofthesystemis
theglobalminimumvalueofthisfunctional,andthedensityn(r)that
minimizesthefunctionalistheexactgroundstatedensityn0(r).The
functionalE[n]aloneissufficienttodeterminetheexactgroundstate
.densityandgyenerInshort,theHohenberg-Kohntheoremsstatethatthereisaone-to-onecorre-
spondencebetweentheground-statedensityandtheground-statepotential,
andthattheground-statedensityistheglobalminimumoftheenergyfunc-
tionalE[n].Thus,itcanbedeterminedfromavariationalcalculation.
Inanalogytothemany-electronHamiltonian(3.1),theHohenberg-Kohn
totalenergyfunctionalEHK[n]is
EHK[n]=T[n]+Eint[n]+d3rVext(r)n(r)+Enn(3.4)

25

MethodseticalTheor3

whereT[n]istheelectronkineticenergyandEint[n]istheinteractionenergy
amongtheelectrons.Thesetermscanbegatheredinauniversalfunctionalof
thedensity,i.e.,onethatisthesameforallelectronsystems:
FHK[n]:=T[n]+Eint[n].(3.5)
Ifofthisthetotalfunctionalenergywaswithrknown,espectonetothecoulddensityfindthen(rgr).oundstatebyminimization
Ageneralizationtospin-polarizedsystemsiseasilypossible.AZeeman
termelectrisonsinaddedthetoprtheesenceofaHamiltonian,magneticwhichfield.isIndifthisferentcase,fortwospinupdensities,andonedownfor
eachspin,aredefinedandsatisfytheHohenberg-Kohn-Theoremsindividually.
Thenthedensityisn(r)=n(r,↑)+n(r,↓),andthespindensityisgivenby
s(r)=n(r,↑)−n(r,↓).

AnsatzKohn-ShamThe3.1.2Althoughitisinprinciplesufficienttofindthedensityofagivenmaterialto
understanditsproperties,thereisnowayknownhowtoextractthemfromthe
density.Further,thefunctionalFHK[n]isnotknowningeneral.Therefore,the
densityfunctionaltheoryasformulatedbyHohenbergandKohnisofminor
elevance.rpracticalKohnandShamproposedin1965toreplacethefullinteractingmany-body
problemwithasimpler,non-interactingauxiliaryproblem[67].Theiransatz
restsontheassumptionthatthegroundstatedensityoftheinteractingsys-
temcanbeexpressedbythegroundstatedensityofaproperlychosennon-
interactingsystem.Thekeyideaistore-introduceorbitalsfornon-interacting
electronsandputthemany-bodyproblemintoanexchange-correlationfunc-
tionalofthedensity.Thisway,theHohenberg-KohnfunctionalFHK[n]becomes
simplythekineticenergyofthenon-interactingficticiouselectrons.
TheauxiliaryKohn-ShamHamiltonian,replacing(3.1),isdefinedby
HσKS(r)=−12+VKSσ(r),(3.6)
2whereσdenotesthespin-index.TheN=N↑+N↓electronsoccupyorbitals
ψiσ(r)withthelowesteigenvaluesεiσdeterminedbytheSchr¨odinger-like
equationsKohn-Sham(HσKS−εiσ)ψiσ(r)=0.(3.7)

26

FunctionalDensity3.1Theory

(3.8)

ThedensityoftheKohn-Shamsystemisgivenby
σn(r)=∑N∑|ψiσ(r)|2,(3.8)
1=iσandtheKohn-Shamkineticenergyis
σN1Ts=2∑∑|ψiσ(r)|2.(3.9)
1=iσTheclassicalCoulombinteractionenergyoftheelectrondensitywithitselfis
givenbytheHartreeenergy
EHartree[n]=1d3rd3rn(r)n(r).(3.10)
2|r−r|
Withtheseingredients,theHohenberg-Kohntotalenergyfunctional(3.4)can
asewrittenrbeEKS=Ts[n]+EHartree[n]+d3rVext(r)n(r)+Enn+Exc[n](3.11)
Themany-bodyeffectsofexchangeandcorrelationareputintotheexchange-
correlationfunctionalExc[n].Now,theKohn-ShampotentialVKSσ(r)canbe
expressedintermsofvariationswithrespecttothedensityas
VKSσ(r)=Vext(r)+δδnE(rHartr,σ)ee+δnδ(Er,xcσ)=:Vext(r)+VHartree(r)+Vxcσ(r).
(3.12)As(3.12)dependsonthedensitycomputedfromthesolutionof(3.7),onehas
toiteratetheequationstoself-consistency,startingfromaninitialguess(from,
e.g.,asuperpositionofatomicdensities).

3.1.3TheExchange-CorrelationFunctional
andThetotalKohn-Shamenergy,noansatzapprisanoximationsexactwayhavetofibeenndthemadeexactyet.ground-stateUnfortunately,densitythe
exchange-correlationfunctionalisnotknown.Themajorobstacleofsolvingthe
fullmany-bodyproblemhasbeenreformulatedwiththeKohn-Shamequations,
soOnlythatamostsmallofthefractiontotalofenerthegytotaloftheenerelectrgy,onthesystemexchange-corriscalculatedelationcorrenerectlygy.,
hastobeapproximated.Twodifferentparadigmsforthederivationofthe

27

MethodseticalTheor3

approximationscanbedistinguished:empiricalandnon-empirical.While
asempiricalpossible,functionalsnon-empiricalareconstrfunctionalsuctedartoematchconstructedexperimentalbasedondatasetsknownasphysicalgood
constraints,whichthefunctionalhastoobey.

Localspindensityapproximation(LSDA)
Thesimplestapproachtotheproblemoftheexchange-correlationfunctional
istouseonlylocalquantities.Usuallyitissplitintoasumofexchangeand
correlationcontributions,whicharederivedfromthehomogeneouselectron
(HEG),gasExcLSDA[n↑,n↓]=d3rn(r)exHEG(n↑(r),n↓(r))+ecHEG(n↑(r),n↓(r)).
(3.13)Theexchangecontributionisknownanalytically,andthecorrelationterm
istypicallyaparametrizedexpressionbasedonMonte-Carlosimulations.
Variousparametrizationshavebeenproposed,namedaftertheirauthors.A
popularformisthatproposedbyPerdewandWang(PW92),whichisimproved
[68].formsearlieroverOnecanexpecttheLSDAtoworkbestinsystemsthatareclosetotheHEG,
likesimplemetallicsolids.Surprisingly,itdoesevenperformquitewellfor
molecules,thoughithasatendencytooverbind,i.e.,bindingenergiesaretoo
largeandbondlengthsaretooshort.Thusitisnotgoodenoughtobeuseful
forthermochemistry,stillitprovidesverygoodstructuralproperties.

Generalizedgradientapproximation(GGA)
Inadditiontothelocaldensity,onecanaddinformationaboutthegradient
ofthedensitytogetbetterapproximationsforsystemswithstronglyvarying
density.Functionalsthattakeintoaccountgradientsarecalledgeneralized-
gradientapproximations(GGA).Theytakethegeneralform
ExcGGA[n↑,n↓]=d3rn(r)excGGA(n↑(r),n↓(r),|n↑(r)|,|n↓(r)|),(3.14)
andaretypicallyreferredtoassemi-localfunctionals.ThestandardGGA
functionalofthenon-empiricaltypeisthePerdew-Burke-Ernzerhof(PBE)
functional,whichlargelycorrectstheoverbindingofLSDAandusuallyover-
estimatesthebondlengthsslightly[69].

28

3.1TheoryFunctionalDensity

ydensitspinrNon-collineaUsually,thespindensityhasacommonaxis;itiscollinear.Non-collinear
calculations,withaspinaxisthatvariesinspace,involveamodifiedtreatment
oftheKohn-Shamequationsandtheexchange-correlationfunctional.The
Kohn-ShamHamiltonianbecomesa2×2matrix,towhichtheexchange-
correlationpotentialcontributesoff-diagonalcomponents.Byfindingthelocal
axisofspinquantizationforeverypointinspace,theusualformoftheLSDA
canbeused.GGAexpressionshavetobemodifiedinvolvingthegradientof
axis.spinthe

3.1.4PeriodicBoundaryConditions
Periodicboundaryconditions,whicharenaturallypresentinanextended
crystal,allowtoevaluatetheKohn-Shamequationsinreciprocalspace.The
foundationforthisisgiventhroughtheBlochtheorem,
Tˆnψ(r)=ψ(r+Tn)=eik∙Tnψ(r),(3.15)
inwhichTn=n1a1+n2a2+n3a3describesatranslationalongthelattice
vectorsaiwith|ni|=0,1,2,...EigenfunctionsoftheperiodicHamiltonian
aswrittenbecanψk(r)=eik∙ruk(r),(3.16)
whereuk(r+Tn)=uk(r).TheeigenstatesoftheHamiltoniancanbefound
seperatelyforeachkintheBrillouinzone,leadingtobandsofeigenvaluesεi,k.
Onefindsintrinsicpropertiesofacrystalperunitcell–suchasthenumberof
electrons,themagnetization,thetotalenergy,etc.–byaveragingoverthek
points,whereNkisthetotalnumberofkpoints.Thedensityisgivenby
1n(r)=Nkk∑nk(r).(3.17)
Thedensityofstatesρ(E)iscalculatedfromtheenergybandsεi,kas
1ρ(E)=Nk∑δ(εi,k−E).(3.18)
k,iObviously,anadequatekpointsamplingoftheBrillouinzoneiscrucialfor
numericallyexactcalculations.Inreciprocalspacecalculations,themeshofk
pointshastobemadedenseenoughtoobtaingoodnumericalconvergenceof

29

MethodseticalTheor3

thequantitiesunderinvestigation.Symmetryoperationsareappliedtoreduce
thenumberofkpoints,suchthatonlytheirreduciblewedgeoftheBrillouin
used.iszone

3.2ImplementationsofDFTUsedinThisWork

3.2.1TheElkFP-LAPWcode
Elkisanopen-sourcefullpotentiallinearizedaugmentedplanewaves(FP-
LAPWtaneously,,FLAPW)andiscodegenerally[70].considerFLAPWedtrtheeatsmostcoreandaccuratevalencemethodelectrtoonssolvesimul-the
oblem.prKohn-ShamTheFLAPWmethodstartsfromthemuffin-tinpartitioning.Theunitcell
isinbdevidedetweeninto(thespherinterstitial).es,centerTheedbasisonthesetisnucleibuilt(thefrommufsphericalfin-tins)andharmonicsaregionin
themuffin-tinspheresandplanewavesintheinterstitial.Thisisreferredto
asanaugmentedplanewaves(APW)basis,whichwasoriginallysuggested
bySlater.Matchingconditionsonthemuffin-tinboundarycanbeimposedto
arbitraryorder.ThebasissetusedbyElkisalinearizedversionoftheAPW+lo
method[71].Itcanbeexpressedas
kG∑cGei(G+k)∙rr∈interstitial
φ(r)=∑αlkmul(r,El1)Ylm(rˆ)r∈muffin-tin(3.19)
mlwherer=|r|andrˆk=r/r.TheplanewavecoefficientscGarevariational
mufquantities,fin-tinandboundarythe.αlmarMatchingetodeterminedzerothorbyderthe(i.e.,onlymatchingthevalueconditionsoftheatwavethe
1offunction)radialSchrand¨odingerobtainingtheequationssolutionsatfixedul(rener,Elgy)E(one1ispersufficient,angulariflocalmomentumorbitalsl)
lareaddedtotheAPWset.Thelocalorbitals(lo/LO)arerepresentedbyradial
zerofunctionsontheandmuffin-tinsphericalboundaryharmonics.Theyinthedomufnotfin-tindependspheronke.sTandwoartypeseforofcedlocalto
orbitalsareaddedtothebasisset:
φllom(r)=βlmul(r,El1)+γlmul(r,El1)Ylm(rˆ)(3.20)
φlmLO(r)=δlmul(r,El1)+lmul(r,El1)+ζlmul(r,El2)Ylm(rˆ)(3.21)


30

3.2ImplementationsofDFTUsedinThisWork

Thelocalorbitalcoefficientsβlmandγlmaredeterminedbytheconditionto
havethelocalorbitalwavefunctionzeroatthemuffin-tinboundaryandits
normalization.Similarly,δlm,lm,andζlmaredeterminedbythewavefunction
anditsderivativebeingzeroatthemuffin-tinboundary,anditsnormalization.
toThedescribesecondtypesemi-corofelocalstates.orbitalsThehavelocalorbitalsatomic-likegreatlywaveimprfunctionsovetheandarflexibilityeusedof
thebasissetatverylowcomputationalcost.
Theradialfunctionsandderivativesul(r,El1),ul(r,El1),iul(r,El2)aresolu-
ationsstandarofthedlinearradialSchreigenvalue¨odingerproblem.equationTheatfixedlinearienerzationgiesEenerl,gieswhichE1resultshavetoin
bechosenapproximatelyinthecenterofthevalencebands.Thellineariza-
tionenergiesEl2areattheapproximateenergyofthesemi-corestate,andare
searchedautomatically.ThevariationalcoefficientscGareobtainedfromthe
principle.variationalRayleigh-RitzCorelevelelectronsaretreatedseparatelyinafullyrelativisticwaywith
theradialDiracequation.Spin-orbitcouplingcanbeincludedforthevalence
statesinasecond-variationalstepbyaddingaσ∙LtermtotheHamiltonian.
ThecrystalpotentialV(r)isexpandedsimilartothewavefunctions,
∑VGei(G+k)∙rr∈interstitial
V(r)=G(3.22)
ml∑Vlm(r)Ylm(rˆ)r∈muffin-tin.
aThissphericalconstitutesapprtheoximationfullpotential(usuallytrcalledeatment,atomicwhichspheristoesbeapprcontrastedoximation).withIt
correspondstoatruncationofthepotentialexpansionatl=0andG=0.
Thus,thepotentialinthemuffin-tinswouldbesphericallyaveraged,andthe
allowspotentialtointreatthetheinterstitialfullpotentialwouldbewithoutconstant.shapeTheapprpotentialoximations.expansionof(3.22)

3.2.2TheMunichSPRKKRpackage
TheMunichSPRKKRpackageisaspinpolarizedrelativisticimplementation
oftheKorringa-Kohn-RostokerGreen’sfunctionmethod.Itdeterminesthe
eletronicstructureofaperiodicsolidbymeansofmultiplescatteringtheory.
Thecode[72method].isAnotherdescribedveryinstrinuctivedetailinintraroductioneviewisarticlegivenbybytheMavrauthorsopoulosofandthe
Papanikolaou[73].Here,themainideasaresummarizedinshort.

31

3MethodseticalTheor

OnestartsfromaformalintroductionoftheGreenfunctionG(r,r,E)
throughtheSchr¨odingerequation:
(E−H)G(r,r,E)=δ(r−r).(3.23)
G(r,r,E)hasthefollowingspectralrepresentation:
G(r,r,E)=lim∑ψν(r)ψν∗(r),(3.24)
η→+0νE−Eν+iη
whereEνaretheeigenvaluesoftheHamiltonianH,andηisasmallpositive
realnumber.FromtheGreenfunction,thedensityofstatesρ(E)andthecharge
densityn(r)areobtainedas
ρ(E)=−1Imd3rG(r,r,E),(3.25)
πn(r)=−1ImEFdEG(r,r,E).(3.26)
πTheGreenfunctioncontainsallinformationwhichisgivenbytheeigenfunc-
tions,bothareequivalent.Allphysicalpropertiesofthesystemcanbefound,
iftheGreenfunctionisknown.
ThereareseveralwaysofcalculatingtheGreenfunction,themostimportant
andflexibleofwhichismultiplescatteringtheory(MST).Thesolutionofthe
electronicstructureproblemisbrokenupintwoparts,apotentialrelatedone
andageometryrelatedone.
Inthefull-potentialformulation,theunitcellisdividedintoWigner-Seitz
polyhedra,centeredonthenuclei.Thepotentialofsitenisexpandedin
sphericalharmonics,Vn(r)=∑LVLn(r)YL(rˆ),withL:=(l,m).Thepotential
ofsiteniszerooutsideitspolyhedron.IncontrasttoFLAPWthereisno
egion.rinterstitialInafirststep,thesingle-sitescatteringproblem,i.e.thescatteringofaplane
waveonthepotentialofsiten,issolvedindividuallyforallsites.Thescattering
solutionsψn(r,E)fortheisolatedpotentialwellsVn(r)areobtainedfromthe
Lippmann-Schwingerequation,anintegralformoftheSchr¨odingerequation:
ψn(r,E)=ψ0(r,E)+d3rG0(r,r,E)Vn(r)ψn(r,E),(3.27)
withthefree-electronwavefunctionψ0(r,E)=eik∙randthecorresponding
Greenfunctione−i√E|r−r|
G0(r,r,E)=−4π|r−r|.(3.28)

32

3.2ImplementationsofDFTUsedinThisWork

ThescatteringbehaviourofthepotentialVn(r)canbeexpressedintermsofa
n,-operatort

tn=Vn+VnG0tn(3.29)
=Vn(1−G0Vn)−1,(3.30)
wheretheargumentshavebeendroppedforclarity.Itisrelatedtotheradial
partofthescatteringsolutionoutsidethepolyhedronofsiten.
InsteadofworkingwiththeLippmann-Schwingerequation,onecanwritea
Dysonequation(inoperatorform)forthesingle-sitescatteringproblem:
Gˆn(E)=Gˆ0(E)+Gˆ0(E)VˆnGˆn(E)(3.31)
=Gˆ0(E)+Gˆ0(E)tˆn(E)Gˆ0(E).(3.32)
Analogousequationsarefoundinthemultiple-scatteringcase:
Gˆ(E)=Gˆ0(E)+Gˆ0(E)VˆGˆ(E)(3.33)
=Gˆ0(E)+Gˆ0(E)Tˆ(E)Gˆ0(E),(3.34)
wherethemultiple-scatteringT-matrixoperatorhasbeenintroduced.Itcanbe
asexpandedTˆ(E)=∑τˆnn(E).(3.35)
nnThescatteringpathoperatorτˆnn(E)isdefinedtotransferanelectronwave
incomingatsitenintoawaveoutgoingfromsitenwithallpossiblescattering
eventsinbetweenincorporated.Inanangularmomentumbasis(denotedby
underlines),τˆnn(E)hasthefollowingequationofmotion:
τnn(E)=tn(E)δnn+tn(E)∑G0nmτmn(E).(3.36)
n=mForafinitesystem,thisequationissolvedbymatrixinversion,
1−τ(E)=t(E)−1−G0(E).(3.37)
Thedoubleunderlinesdenotematriceswithrespecttoangularmomentum
andsites.Thematrixinsquarebracketsisknownasthereal-spaceKKR
matrix.ForaperiodicsolidwithsitesnatpositionsRn,onefindsbyFourier
transformationτnn(E)=1d3kt(E)−1−G0(k,E)−1eik∙(Rn−Rn),(3.38)
ΩΩBZBZ

33

MethodseticalTheor3

withthe(reciprocalspace)structureconstantsmatrixG0(k,E)beingtheFourier
transformedofthereal-spacestructureconstantsmatrixG0(E).
TheformalismoutlinedaboveisverygeneralwithrespecttotheHamilto-
nianH.Inpractice,theKohn-Shamequationsaresolvedintheusualiterative
.self-consistencyto,wayAmajoradvantageoftheGreen’sfunctionformalismistheconnectionofa
perturbedsystemandareferencesystemthroughtheDysonequation:
ˆG=Gˆref+GˆrefHˆpertGˆ.(3.39)
Thisequationgivesalsotheformalbackgroundfortheschemedescribed
above,inwhichthefree-electronsystemisthereferencesystem,andthepertur-
bationHamiltonianisgivenbythepotentialofthesystemunderinvestigation.
BecauseMSTseperatestheelectronicstructureproblemintoageometricanda
potentialpart,itiseasytotreatimpuritiesinaperfecthostmaterialwithout
usingsupercellsorlargeclusters,asinothermethods:
τimp=(τhost)−1−(thost)−1+(timp)−1−1.(3.40)
Similarly,disorderedsystemsaretreatedwithintheso-calledcoherentpotential
approximation(CPA).AnauxiliaryCPAmediumisintroduced,inwhichthe
concentrationaverageoftheconstituentscausesnoadditionalscattering.For
abinaryalloywithconcentrationsxA,xB,thiscanbeexpressedwiththe
matrices:operatorpathscatteringxAτAnn+xBτBnn=τCPAnn.(3.41)
Inanalogytotheimpurityproblem,thecomponentprojectedscatteringpath
operatormatricesaregivenas
ταnn=(τCPA)−1−(tCPA)−1+(tα)−1−1,α=A,B.(3.42)
Thematrixdimensionofthemultiple-scatteringproblemisNscatterers∙(lmax+
1)2.Therefore,onetriestokeeptheangularmomentumcutoffassmallas
possible,typicallylmax=3ford-electronsystems.Inprinciple,onewould
havetotaketheangularmomentumexpansiontoinfinitytoobtainthecharge
densitycorrectly.Duetothetruncation,thechargedensityissomewhat
incomplete,leadingtoaslightmiscalculationoftheFermienergy.Thisproblem
canberesolvedbyananalyticallyexactexpressiontoobtainacorrectcharge
normalization,theLloydformula[74,75].

34

3.3CurieTemperaturesfromanEffectiveHeisenbergModel

OnlyvalenceelectronsaretreatedwiththeMST.Coreelectrons,whichare
welllocalizedwithinthepolyhedra,aretreatedrelativisticallywiththeDirac
equation.TheHamiltonianforthevalenceelectronscanbechoseneitherscalar-
relativistic(neglectingspin-orbitcoupling)orfullyrelativistic,dependingon
investigated.beingoblemprthe

3.3CurieTemperaturesfromanEffective
delMoergHeisenbIntheclassicalHeisenbergmodeloflocalizedspins,theHamiltonianofthe
bygivenissystemspinH=−i,∑jeiejJij,(3.43)
withpointingtheintheHeisenberdirgectionpairoftheexchangemagneticcouplingmomentparametersonsiteJiij.,andSPRKKRunitallowsvectorsetoi
calculatetheexchangecouplingparametersbymappingthe(itinerant)system
ontoperturbativeaHeisenberreal-spacegapprHamiltonian.oachusingThetheparameterstheorybyareLiechtensteindeterminedetal.within[76a].
Inthisapproach,theenergydifferenceΔEij=Jij(1−cosθ)associatedwitha
rotationofthespinsonsitesi,jinoppositedirections±θ/2ismappedonto
viaHamiltoniangHeisenbertheJij=−1ImEFdETr(ti−↑1−ti−↓1)τ↑ij(tj−↑1−tj−↓1)τ↓ji,(3.44)
4πthewherpre↑,evious↓denotesectionthe(noteup-theandchangeddown-spinindicestandforτbetteroperatorslegibility).asdiscuThessedreal-in
spacecalculationgivesdirectaccesstothedistance-dependenceofthepair
exchangecouplingparameters.Anecessaryconditionfortheapplicability
ofthethismomentsapproachshouldisthenotlocalitychangeofonrtheotation.spinThismoments,conditioni.e.,theisnotmagnitudefulfilledinof
systems.itinerantFromtheJijtheCurietemperaturescanbecalculatedwithinthemeanfield
approximation(MFA).Forasingle-latticesystemtheCurietemperatureis
givenwithintheMFAby

32kBTCMFA=J0=∑J0j.
j

(3.45)

35

MethodseticalTheor3

Inamulti-sublatticesystem,denotedbyindicesµ,ν,(as,e.g.,theHeusler
compoundswithfoursublattices)onehastosolvethecoupledequations
3kBTCMFAeµ=∑J0µνeν(3.46)
2νJ0µν=∑J0µrν
0=rwhereeνistheaveragezcomponentoftheunitvectorerνpointinginthe
directionofthemagneticmomentatsite(ν,r).Thecoupledequationscanbe
rewrittenasaneigenvalueproblem:
(Θ−TI)E=0(3.47)
3kBΘµν=J0µν
2withaunitmatrixIandthevectorEν=eν.ThelargesteigenvalueoftheΘ
matrixgivestheCurietemperature[43,77].ToconvergetheCurietemperature
withrespecttothereal-spaceclusterradius,onehastocomputepairexchange
couplingparametersuptotypicallyrmax=3.0a,whereaisthelatticeconstant.
ToestimatetheaccuracyofourmethodfortheCurietemperaturedetermi-
nationofHeuslercompounds,wecalculatedtheCurietemperaturesofsome
compoundsattheirrespectiveexperimentallatticeparameters.Thecalculated
andexperimentalvaluesaregiveninTable3.1.Furthervalues,obtainedusing
thesamemethod,canbefoundinRef.[80].FortheCo-basedferromagnetic
periment.compounds,theHowever,incalculatedthecaseofmean-fieldthetwovaluesarferrimagneticeingoodMn-basedagreementcompounds,withex-
theMFACurietemperatureisabout25%lowerthantheexperimentalone.
Thelattercompoundsmighthavemoreitinerantcharacter,similartothecase

Ref.expt.AMFCo2MnSi1049K985K[51]
Co2TiSn383K355K[78]
Mn2VAl605K760K[11]
Mn2VGa560K783K[79]

Table3.1:CalculatedandexperimentalCurietemperaturesofsomeHeusler
compounds.

36

3.4X-RayAbsorptionSpectrafromElectronicStructure

offccNi,wheretheMFAvalueisabout380K,incontrasttotheexperimental
[76].K630ofvalue

3.4X-RayAbsorptionSpectrafromElectronic
Structure

ConsiderationsGeneral3.4.1

Inafirstapproximation,onecandescribetheabsorptionofx-raysbyamedium
asasingle-stepprocess:electronsfromanoccupiedcoreorbitalareexcitedinto
oftheunoccupiedabsorptionstatesisabovegovernedthebyFermitheenerstrgy,uctursucheofthatthetheenerunoccupiedgystates.dependenceIn
firstorderperturbationtheorywiththeelectricdipoleapproximation,wecan
expresstheenergy-dependentoptical(andx-ray)absorptionspectraµα(ω)
Rule:GoldenFermi’swith2µα(ω)∝∑ψf|pα|ψiδ(Ef−Ei−ω),(3.48)
f,iwhereαdenotesthepolarization,ωthephotonenergy,i,flabeltheinitialand
finalwavefunctions,Ei,fthecorrespondingenergylevels,andpα=−iαthe
momentumoperatorwithdirectionα.Ifonlyasingleinitialstate–asinthe
caseofx-rayabsorption–isconsideredandthemomentummatrixelements
|ψf|pα|ψi|2areassumedasenergy-independent,thisexpressionreducesto
thedensityofstates,modifiedbythedipoleselectionrules.Absorptionfroms
statesprobesthep-projecteddensityofstates,absorptionfrompstatesprobes
states.anddsFirstorderperturbationtheoryassumesaninfinitesimaldepletionofthe
initialstateduringtheabsorptionprocess.Thisapproximationis,however,
oftennotjustified.Whenaphotonisabsorbedbyacore-levelelectron,itis
promotedtothevalencestates,leavingacore-hole.Thepropagatingelectron
caninteractwiththecore-hole,aswellasallotherelectrons.Alleffectsof
thiskindarecondensedintheexpressioncore-holecorrelations.Theextent,
towhichthesecorrelationshavetobetakenintoaccountdependsonthe
absorptionedges,theabsorbingatomandthesystem,inwhichitisembedded.
Thiswillbediscussedinmoredetaillater.

37

MethodseticalTheor3

Elk3.4.2Amoregeneralformulationofopticalpropertiesisgiventhroughtheoptical
conductivitytensorσαβ(ω)[81]:
i1Πiαf,kΠβ(Πiαf,kΠβ)∗
σαβ(ω)=Ωk∑i,∑fωif,kω−ωif,kfi+,kiη+ω+ωif,kfi,+kiη,(3.49)
thewhereα,transitionβdenoteenerthegy.Thepolarization,parameterΩηtheunitsmoothscellthevolume,polesωofif,kthe=suEfm,k−withEi,ak
Lorentzianandcanbeinterpretedasaphenomenological(inverse)lifetime
broadening.ThedipolartransitionmatrixelementsΠiαf,karedeterminedby
Πfαi,k=ψf∗,k(r)pαψi,k(r)dr.(3.50)
Theopticalconductivitytensorandthedielectrictensorεαβ(ω)arerelatedby
εαβ(ω)=δαβ+4ωπiσαβ(ω)(3.51)
withponentstheKrconveroneckergeto1deltaandδαβ;theinofthef-diagonalhigh-frequencycomponentslimit,gothetozerdiagonalo.Thecom-x-
raydichroismabsorption,ofacubicx-raymaterialmagneticwithcircularmagnetizationdichroism,alongandthex-rayz-axismagnetic(whichislinearnot
necessarilyparalleltooneofthecrystalaxes)canbecalculatedas
1XAS(ω)=3Tr[Im(ε(ω))](3.52)
XMCD(ω)=Im(σxy(ω))(3.53)
XMLD(ω)=Im(εzz(ω)−εxx(ω)).(3.54)
aThisspin-orbitverycorrgeneralectiontermformulationintheisdipolaradoptedintransitiontheElkmatrixcode,andelements.alsoTheincludescode
havedoestonotbeconsiderdescribedastransitionsvalencefrombycortheelocalorbitals,orbitalssothatmethod.theorbitalsofinterest

SPRKKR3.4.3SPRKKRtreatsthex-rayabsorptiononafullyrelativisticlevel,suchthatspin-
orbiteffectsarenaturallyincluded.IntheKKRformalism,itisconvenientto

38

3.4X-RayAbsorptionSpectrafromElectronicStructure

(3.55)

(3.57)(3.58)

rewrite(3.48)usingtheidentity
−π1ImG(E)=∑|ψfψf|δ(Ef−E)(3.55)
ffortheGreen’sfunctiontoobtain
µα(ω)∝∑Φi|Xα∗ImG(Ei+ω)Xα|Φiθ(Ei+ω−EF).(3.56)
iTheΦiarethecorelevelwavefunctionsoftheinitialstates,andXα=
−c1jel∙Aαrepresentsthecouplingoftheelectroniccurrentdensitytothe
radiationvectorpotential.X-rayabsorptionandcirculardichroismarecom-
definitions:theirfollowingputedXAS(ω)=1(µ+(ω)+µ−(ω))(3.57)
2XMCD(ω)=µ+(ω)−µ−(ω).(3.58)
decoFEFF9The3.4.4TheFEFF9codeisanimplementationoftherelativisticreal-spacemultiple-
scatteringGreen’sfunctionmethodwithinthemuffin-tinapproximation[82].
Correspondigly,mostofthemathematicsdescribedin3.2.2applyhereas
well.Themuffin-tinapproximation(nottobeconfusedwiththemuffin-tin
partitioninginFLAPW)assumessphericalpotentialsinthemuffin-tinsanda
constantpotentialoutside.ThefirstversionsofFEFFweredesignedtocompute
theextendedx-rayabsorptionfinestructure(EXAFS)ofmoleculesandsolids,
whichoriginatesfrommultiplescatteringoftheexcitedphotoelectronfromthe
surroundingatoms.Therefore,itwasnaturallybasedonmultiple-scattering
theory,butemployedascatteringpathexpansionfortheGreen’sfunction:
Gsc=G¯0TG¯0+G¯0TG¯0TG¯0+...(3.59)
TheGreens’sfunctionofthesystemisgivenasthesumofthecentral(absorber)
atomandthemultiple-scatteringcontributionabove,G=Gc+Gsc.The
Green’sfunctionG¯0referstothedampedfree-electronGreen’sfunction,as
calculatedwithacomplexself-energyandcoreholelifetime.Theexpansion
isaveryefficientandfastwaytocomputeEXAFS,whicharerelevantat
energiesabout10eVabovetheabsorptionthresholduptoafewhundredeV.

39

MethodseticalTheor3

Forlowenergies,i.e.,verydistantscatteringevents,theconvergenceofthe
expansionisbad,suchthatthenear-edgeregion(x-rayabsorptionnearedge
structure,XANES)isnotdescribedcorrectly.Forthisregion,thefullmultiple-
scattering(FMS)asdescribedbyEq.(3.37)hastobeconsidered.Further,a
self-consistentpotentialisrequiredforaccurateresults.BoththeFMSaswell
astheself-consistencyareimplementedinFEFF9,allowingaccurateXANES
calculations.However,thespintreatmentisnotself-consistent.Onehasto
imposeaparticularmagneticmomentforagivensiteinthecluster,whichis
thenadjustedbyarigidshiftofspinupanddowndensities.Thecomputation
ofcirculardichroismisaccordinglylimitedtocases,wheretherigidshiftisa
gooddescriptionoftheactualbandstructure.
ThemajoradvantageoftheFEFFcodeisaself-consistenttreatmentof
coreholeeffects.Thex-rayabsorptioncanbedescribedinthefinalstate
approximation,removinganelectronfromtheinitialstateandaddingittothe
finalstates.Thisgivesrisetoaredistributionofthebands,oftenimproving
theagreementbetweenexperimentandcalculation,inparticularforKedges.
Itisdifficulttotreattheseeffectsinreciprocalspacemethods(largesupercells
havetobeconstructed),whereasthetreatmentinarealspaceclusterapproach
natural.quiteis

3.4.5MoreAdvancedTreatmentofthe
Core-Hole–PhotoelectronInteraction
Asindicatedabove,self-consistentinclusionofacoreholeimprovesagreement
betweenexperimentandcalculationinmanycases.However,thisisjust
anapproximatetreatmentoftheexcitedstate,andsomeproblemsremain.
Oneparticleoftheapprmostoximationprominent(IPA)ofexamplesx-rayofabsorpfailurtioneofastheoutlinedstandardaboveisindependenttheL3,2
absorptionof3dtransitionmetals.WithintheIPA,thebranchingratioof
thetwoabsorptionpeakscorrespondstothestatistical2:1ratio,duetothe
occupationofthe2p3/2levelwithfourelectronsandofthe2p1/2levelwith
istwocloseelectrtoons.1:1,wherHowevereas,forinNilightitis3dlargertransitionthanthemetals,statisticalsuchasScratio.orTMori,ethisrecentratio
computationschemesgobeyondthesimpleIPAandcanpartlyresolvethese
oblems.prTwomajorapproachestotreatthecore-hole–photoelectroninteractionin
asitymorefunctionalsophisticatedtheoryway(TD-DFT)haveandbeenanexplicitdeveloped:themany-bodytime-dependentperturbationden-the-

40

3.4X-RayAbsorptionSpectrafromElectronicStructure

ory(MBPT)calculationwiththeBethe-Salpeter-Equation.Neitherofthese
approacheshasbeenusedinthiswork,butforcompletenesstheyshallbe
.brieflyoutlinedIntheTD-DFTonefindsthelinearinteractingdensityresponsefunctionχ
fromaDysonequationrelatingittothenon-interactingχSvia
χ(r,r,ω)=χS(r,r,ω)
+d3xd3xχS(r,x,ω)K(x,x,ω)χ(x,r,ω).(3.60)
Here,theTD-DFTKernelKhasbeenintroduced,whichconsistsoftheCoulomb
interactionandafrequency-dependentexchange-correlationkernel:
1K(r,r,ω)=|r−r|+fxc(r,r,ω).(3.61)
SimilarlyasintheDFT,themajorproblemhereistoapproximatetheunknown
exchange-correlationkernelfxc.Differentapproximationshavebeenproposed,
withvaryingsuccess[83,84,85].Todate,nouniversalKernelisknownthatis
equallywellsuitedforallsystemsofinterest.
TheBethe-Salpeter-Equation(BSE)isderivedfrommany-bodyperturbation
theory,andiscommonlywrittenasaneigenvalueprobleminreciprocalspace
[86]:∑Hhee−kh,hekAhλek=EλAhλek.(3.62)
hek
Theelectron-holeinteractionHamiltonianconsistsofadiagonalpart,adirect
(Coulombic)termandanexchangeterm,
He−h=Hdiag+Hdir+Hx,(3.63)
whichcanbeexpressedas
diagHhek,hek=(εhk−εek)δhhδeeδkk,(3.64)
Hhedirk,hek=−d3rd3rψhk(r)ψe∗k(r)W(r,r)ψh∗k(r)ψek(r),(3.65)
Hhxek,hek=d3rd3rψhk(r)ψe∗k(r)v¯(r,r)ψh∗k(r)ψek(r),(3.66)
withtheKohn-Shameigenvaluesε(e,h),k,thescreenedCoulombpotential
W(r,r)andtheshort-rangepartofthebareCoulombpotentialv¯(r,r)[86].

41

MethodseticalTheor3

ETheλandtheimaginarycouplingpartofcoeftheficientsdielectricAhλek:functioniscalculatedfromtheeigenvalues

2Imεxx(ω)=8Ωπ∑∑Ahλekhkε|−−iεx|ek∙δ(Eλ−ω)(3.67)
λhekekhk
TheBSEgivesaphysicallytransparentpictureoftheabsorptionprocess:
excitoniceffects,i.e.excitedstatesinthebandgapofinsulators,aredueto
thethedircorecte-hole);termspectral(describingweightthetransfers,Coulombasinattractiontheofabovethementionevalencedstatescasebofy
L3,2absorption,arecausedbytheexchangeterm,whichmixesthevarious
transitionchannels[86].Theexcitoniceffectsarepartlyaccountedforbythe
oximation.apprstatefinalTheBSEiscurrentlythestate-of-the-arttreatmentoftheopticalandx-ray
absorptionprocess.However,itsuseisrestrictedtosmallsystemswithafew
atoms,becausethecalculationofthematrixelementsintheBSEHamiltonian
anditsdiagonalizationareverycumbersome.

42

4Abinitiopredictionofferrimagnetism,
CurieandinteractionsexchangetemperaturesinMn2TiZHeusler
oundscomp

ductionIntro4.1

Inthischapter,abinitiocalculationsofthepropertiesofthe(hypothetical)
Mn2TiZcompounds,crystallizedintheL21structure,arediscussed.Noexper-
imentaldataareavailableforthissystem,andonlyMn2TiAlhasbeenstudied
theoreticallybefore[87].However,itisexpectedthatpartsofthisserieswill
existintheL21structure,seeingthatMn2VAlandMn2VGa,aswellaspartsof
theCo2TiZserieshavebeenprepared[79,88,89].
Thecalculationspresentedinthisstudywereperformedwithintwodiffer-
entdensityfunctionaltheory-basedbandstructurecodes:thefull-potential
linearizedaugmentedplanewaves(FLAPW)packageElk(Chapter3.2.1)and
thefull-potentialKorringa-Kohn-RostokerMunichSPRKKRpackage(Chap-
ter3.2.2).Althoughbothmethodsareinprincipleequivalentforcrystalline
systems,therearesubtledifferencesassociatedwiththeirnumericalimplemen-
tations,andthusitisworthtocomparebothmethodsontherathercomplex
intermetallicsystemMn2TiZ.
Elkwasusedtodeterminethetheoreticallatticeparametersandthetotal
energydifferencesbetweenferrimagneticandnonmagneticstates.These
calculationswerecarriedoutona12×12×12kpointmesh(72pointsinthe
irreduciblewedgeoftheBrillouinzone).Themuffin-tinradiiofallatomswere
setto2.0a.u.toavoidoverlapsatsmalllatticeparameters.Theequilibrium
latticeparametersaweredeterminedusingathird-degreepolynomialfittothe
totalenergies.Toobtainaccuratemagneticmomentsanddensitiesofstates,
thecalculationswereperformedattheequilibriumlatticeparameterusinga
16×16×16k-mesh(145pointsintheirreduciblewedge)andnearlytouching
es.spherfin-tinmuf

43

4Ferrimagnetism,exchangeandCurietemperaturesinMn2TiZ

Figure4.1:Totalenergiesoftheinvestigatedcompoundsindependenceoftheirlat-
ticeparameters.Theresultsfortheferrimagneticandthenon-magneticstatesare
representedwith+and×,respectively.

TheSPRKKRcalculationswereperformedonthetheoreticalequilibrium
latticeparametersdeterminedwithElk.Thecalculationswerecarriedoutin
thefull-potentialmodewithanangularmomentumcutoffoflmax=3ona
22×22×22kpointmesh(289pointsintheirreduciblewedgeoftheBrillouin
zone).Boththefullpotentialaswellastheincreasedangularmomentum
cutoffarenecessarytoensureaccurateresults.TheDOSwerecalculatedona
densermeshof1145kpointswith0.5mRyaddedastheimaginaryparttothe
.gyenerTheexchange-correlationpotentialwasmodeledwithinthegeneralized
gradientapproximationofPerdew,Burke,andErnzerhofinbothschemes
[69].Thecalculationswereconvergedtoabout0.1meV.Allcalculationswere
carriedoutinthescalar-relativisticrepresentationofthevalencestates,thus
coupling.spin-orbittheneglecting

44

4.2Results

HeisenbergpairexchangecouplingparametersandMFACurietemperatures
wereobtainedasdescribedinChapter3.3.InordertoseparatethetwoMn
lattices,thecalculationswereruninF4¯3mspacegroup,inwhichtheMnatoms
arenotequivalentbysymmetry.Ther-summationinEq.(3.46)wastakentoa
radiusofRmax=3.0a,whichhasbeenshowntobesufficientforhalf-metallic
80].[90,compoundsHeusler

Results4.2

4.2.1Energyminimizationandlatticeparameters
Threetypesofmagneticstartingconfigurationsweretested:ferro-,ferri-,and
nonmagnetic.Itwasfoundforallcompoundsthattheferromagneticconfig-
urationswereunstableandconvergedintotheferrimagneticstate.Fig.4.1
displaysthetotalenergiesoftheferrimagneticandthenonmagneticconfigura-
tionsindependenceonthelatticeparametersa.Wefindthattheferrimagnetic
statehasalwayslowerenergythanthenon-magneticstate;thedifference
intotalenergyreduceswithincreasingnumberofvalenceelectrons,butit
increaseswithinthegroupswiththeatomicnumber.Thelatticeparameters
followroughlyalineardependenceontheatomicradiusoftheZelementwith

Figure4.2:(a):DependenceofthelatticeparameteraontheatomicradiusoftheZ
element.(b):NormalizedmagneticmomentsofMnandTiindependenceofthelattice
.parameter

45

4Ferrimagnetism,exchangeandCurietemperaturesinMn2TiZ

SPRKKRElkMn2TiZa(˚A)mmMnmTiP(%)mmMnmTiP(%)
Al5.962.981.83-0.57212.981.76-0.4982
Ga5.952.951.84-0.60452.971.77-0.5379
In6.233.172.17-0.8673.081.98-0.8232
Si5.781.981.16-0.31941.981.13-0.2687
Ge5.871.971.20-0.37941.971.16-0.3389
Sn6.141.971.32-0.51972.001.25-0.4893
P5.680.300.18-0.05-3————
As5.820.940.59-0.20840.970.61-0.2258
Sb6.070.970.65-0.25880.980.62-0.2479

Table4.1:ResultsofthegroundstatepropertiescalculationswithElkandSPRKKR.
ThemomentstotalaremagneticgiveninµmomentsBperareatom.giveTheninµBSPRKKRperresultsformulaforunit,Mn2TtheiAsatomicwereobtainedmagnetic
witha=5.95˚A(seetext).

thecorrelationcoefficientofr=0.92(Fig.4.2(a)).Somecompoundsshowa
strongasymmetryofthetotalenergycurveintheferrimagneticconfiguration
andevenkinksinthecurvesforverylargea.Thisiscausedbyasteepincrease
ofthemagneticmomentsforincreasingawhichcausesastrongerbinding.
However,thiseffectisneverstrongenoughtoshifttheequilibriumlattice
parametertosuchahigh-mstate.Theequilibriumlatticeparametersaresum-
marizedinTable4.2.1.Typicallywefindtheequilibriumlatticeparametersof
HeuslercompoundsobtainedwithElktobeaccuratewithin±0.5%compared
experiment.to

4.2.2Magneticmomentsanddensitiesofstates
TheresultsofthissubsectionaresummarizedinTable4.2.1andFig.4.3.

Mn2TiAl,Mn2TiGa,Mn2TiIn
Fromtherulem=NV−24weexpecttofindamagneticmomentof3µB/f.u.
forthesecompounds.TheFLAPWcalculationsshowsmalldeviationsfrom
thisrule,indicatingthatthecompoundsarenotperfecthalf-metals.Thisis

46

Results4.2

confirmedbytheDOS,whichshowspinpolarizationsattheFermilevelbelow
50%,enhancedandintop3.17µarticularB/f.u..onlyThis7%arisesforMnfr2omTiIn,thelarwhergeethelatticemagneticparametermomentandtheis
factthatallthreecompoundsdonotformagapintheDOS.TheFermilevel
for(seeMninsets2TiAlinandFig.Mn4.3),2TbutiGaisbothinofarthemegionhavewithalowveryDOSlargeforemptybothspinminoritychannelsspin
DOSrightaboveEF.Smallvariationsofthelatticeparameterwouldthuslead
tostrTheongcalculationsvariationsoftheperformedspinwithpolarization.SPRKKRreproducethemagneticmoments
largerobtainedindeviationElkisveryfoundwell.fortheAlthoughatom-rthetotalesolvedmomentsmoments.areThepracticallyFermiequalener,gya
isobservedfoundatinElkslightlyardifoundferEFentarelesspositionsprinonounced,theDOS,andespeciallythethedetaileddipinstrtheucturspin-es
downSPRKKR.statesatHoweverEF.,ThisthetrleadsendtothatMnsignificantly2TiInhashigherthelospinwestpolarizationpolarizationvalueswithinin
thisgroupisreproduced.

Mn2TiSi,Mn2TiGe,Mn2TiSn
Accordingtothe“ruleof24”atotalmagneticmomentof2µB/f.u.isexpected.
Again,smalldeviationsfromthisruleareobserved;allmomentsarelowerby
about1.5%.InElk,thethreecompoundsarefoundtoformahalf-metallicgap
inthemajorityspinstatesslightlyaboveEF.ThegaponsetaboveEF(width)
is0.16eV(0.49eV)forSi,0.24eV(0.25eV)forGe,and0.19eV(0.01eV)forSn.
Nevertheless,thespinpolarizationisabove90%inthesecalculations.The
structureoftheDOSaroundEFleadstoastablespinpolarizationandmagnetic
momentuponisotropiclatticecompressionorexpansion.Forthisseries,
havingthesamevalenceelectroncountsandnearlyhalf-metallicDOS,onecan
observeclearlyanarrowingofthebands,i.e.,theDOSarecontractedtowards
EF,whiletheFermilevelitselfmovesupwards.Thisisdirectlyassociated
withthegraduallyincreasinglatticeparameterinthisseries,whichreduces
theoverlapofthe3dorbitalsandtherebyreducestheitinerancyofthesystem.
Anincreasedlocalizationoftheelectronsprovidesalsoanexplanationforthe
increasingatomicmagneticmomentsalongthisseries.Similarbehaviorhas
beenobservedearlierforCo2MnZ,withZ=Si,Ge,Sn[91,92]andNi2MnSn
[93].InthefirstcasetheMnmomentisincreasedandtheComomentis
loweredalongtheseries,keepingthetotalmomentinteger.Calculationson
Co2MnSiwithincreasedlatticeparameterreproducedthisbehavior.Inthe

47

4Ferrimagnetism,exchangeandCurietemperaturesinMn2TiZ

Figure4.3:DensitiesofstatescalculatedwithElk.ThemajorityDOSispointingup,the
minorityDOSispointingdown.TheinsetsforAlandGashowtheregionaroundthe
.gyenerFermi

secondcase,thepressuredependenceofthemomentswasstudied.Under
increasingpressure,i.e.,withreducedlatticeparameter,boththeNiandtheMn
momentdecrease,andthusthetotalmomentdecreases.However,Ni2MnSnis
notahalf-metal,hencethetotalmomentisnotrestrictedtoanintegervalue.
Consequently,bothobservationsonquitedifferentferromagneticHeusler
compoundsareinaccordwithourcaseof(nearly)half-metallicferrimagnetic
compounds.HeuslerThemagneticmomentsandDOSfromSPRKKRareinverygoodagreement
withtheonesobtainedfromElk.However,theFermilevelisfoundatalower
position,givingrisetotheslightlyreducedpolarizationvalues.

48

Results4.2

Mn2TiP,Mn2TiAs,Mn2TiSb
Inthesecasesatotalmagneticmomentofonly1µB/f.u.isexpected.Because
oftheverysmalllatticeparameterofMn2TiP,itsspin-splittingissmallwith
only0.3µB/f.u.intheElkcalculation.ThesituationofMn2TiAsandMn2TiSb
issimilartothatofMn2TiSiandMn2TiGe.Amajorityspingapisformed
abovetheFermilevelwithonset(width)of0.29eV(0.53eV)forAsand0.19eV
(0.44eV)forSb.Thoughnotbeinghalf-metallic,bothcompoundshavespin
polarizationsofmorethan80%.
Finally,themagneticmomentsofMn2TiSbinSPRKKRagreeverywellwith
thoseobtainedwithElk.Butagain,theFermilevelislowerandthespinpolar-
izationisreduced.ForMn2TiPandMn2TiAsthesituationisquitedifferent.
Theycannotbeconvergedintoferrimagneticstatesattheequilibriumlattice
parametersdeterminedbyElk;instead,theyarefoundtobenonmagnetic.
Thisiscausedbythetinyenergydifferencebetweentheferrimagneticand
thenonmagneticconfiguration,whichleadstoanumericalinstabilityofthe
ferrimagneticstate.ByincreasingthelatticeparameterofMn2TiAsbyabout
2%to5.95˚A,theseparationisincreasedartificiallytoabout30meV/f.u.and
thecalculationconvergesintotheferrimagneticstate.Becauseofthis,theprop-
ertiesobtainedwithSPRKKRforthiscompoundhavetobetakenwithcare:in
allothercasestheindividualatomicmomentsareslightlylowerinSPRKKR
thanthosefromElk;hereinstead,largermomentsarefound.However,the
sameprocedurecannotbeappliedtoMn2TiP,withinareasonablerangeof
parameters.lattice

rksremaGeneral

ItisworthtonotethatthemagneticmomentsoftheZcomponentarealways
below0.06µBandthattheyarealwaysparalleltotheTimoment.Indetail,
thevaluesareAl0.044µB,Ga0.052µB,In0.058µB,Si0.034µB,Ge0.035µB,Sn
0.034µB,P0.0062µB,As0.018µB,andSb0.017µB.
AnotherpropertyworthnotingisthefactthattheratiosmMn/mandmTi/m
followalineardependence(withcorrelationcoefficientsofr≈0.9inbothcases
fortheElkdata)onthelatticeparameter(andhencetheinteratomicdistances)
independentlyontheZtype,seeFig.4.2(b).Asmentionedabove,with
increasinglatticeparametertheitinerantcharacterofthesystemisreduced
andlocalizesthemomentsgraduallyontheatoms.Therefore,theinfluence
oftheZcomponentinMn2TiZistwofold.First,itdeterminesthelattice

49

4Ferrimagnetism,exchangeandCurietemperaturesinMn2TiZ

parameterlocalization.oftheAndsecond,compoundtheandtotalfollowingmagneticfrommomentthat,isthedegrdeterminedeeofviaelectrtheon
rangenumberof(whichelectrisnotonsthesupplied,caseforifPtheandlatticeIn).parameterdoesnotexceedacertain

4.2.3ExchangeinteractionsandCurietemperatures
MnThe2TiSb,exchangewhichintearerractionseprareesentativeinvestigatedcompoundshereforforMntheir2TriGa,espectiveMn2TZiGe,group.and
)2(1MnFig.1(24.4)(a)andthedisplaysintertheJij-sublatticecalculatedinteractionsfortheintrMn1(2)a-sublattice-Mn2(1)andinteractionMn-TiMnofthe-
threecompounds.Allotherinteractionsareverysmallandcanbeneglected
discussion.followingtheforInallthreecasesitisclearthattheMn1(2)-Mn2(1)inter-sublatticeinteraction
providesthelargestcontributiontotheexchange.Further,thenearestneighbor
AllinteractioninteractionsofMn-Tareiismostlyalwaysconfinednegative,withinhenceaallradiusofcompounds1.5a.arApartefromferrimagnets.these
similarities,therearemanyinterestingdifferences.
Mn1(First,2)-Mnwe2(1).discussThefirsttheanddetailssecondofthenearestdominatingneighborsinterprovide-sublatticealarge,interactionpositive
exchange.ThesecondnearestneighborshavetwodifferentvaluesofJ.This
isafeaturethatisnotobservedinfrozen-magnoncalculations(see,e.g.ij[43]),
becausetheFouriertransformthatisnecessarytoobtaintheexchangeparam-
etersinvolvesasphericalaveraging.Instead,withthereal-spaceapproach
usedhereweobserveadifferenceforMnatomswithaTiatomoraZatomin
between.WefoundlargervaluesontheMnatomsmediatedviaTiandlower
valuesontheZmediatedones.ThenearestMnneighborshaveadistanceof
about2.95˚A,andtheexchangeisapparentlyindirect.Fordirectexchange,one
wouldexpectascalingwiththemagneticmoments,whichisnotobserved
herbeene.ItobtainedratherearlieroscillatesonotherwithhalftheandspfullelectronHeuslernumber.compoundsAsimilar[94].rTheesultratiohas
ofwiththeincrnearesteasingandelectrsecondonnearestconcentration,neighborandthecouplingnearisestneighborsignificantlyrinteractioeducedn
dominatesinMn2TiSb.
TheantiferromagneticMn-Tiinteractionisonlysignificantforthenearest
distanceneighbors.ofaboutAccor2.55dingly˚A,,istheessentiallyinteractiongivenbetweenbydirectMnandexchangeTi,whichcouplinghaveanda
thescalingwiththeTimomentcorroboratesthisassumption.

50

Results4.2

Figure4.4:HeisenbergexchangeparametersJijindependenceonthenormalizeddis-
tancelatticer/a.parameters.(a):Jijfor(b):MnJi2jTforiGa,MnMn2T2TiGeiGe,withMn2difTferiSbentforlatticetheirrespectiveparameters.Noteequilibriumthe
differentscalesoftheverticalaxesinthetoprow.

Theintra-sublatticeinteractionofMn1(2)-Mn1(2)exhibitsanotableoscillatory
behavior.Inthetwocaseswithoddvalenceelectronnumberitispositive
forthenearestneighbors,negativeforthesecond,andagainpositiveforthe
thirdnearestneighbors.ForMn2TiGewithitsevenelectroncountthefirst
twoneighborshavenegativeandthethirdneighborhaspositiveinteraction.
Sointhelattercase,thetotalMn-Mnintra-sublatticeinteractioniseffectively
omagnetic.antiferrInexplanationordertoforstudythethedifferencesdependenceofdiscussedJijontheabove,latticeadditionalparameterascalculationsapossibleon
Mn2TiGehavebeenperformedwithlatticeparametersof(5.87±0.2)˚A.This
compoundwaschosenbecauseofthewide(pseudo-)gapforthespin-upstates,
whichwarrantsastabletotalmagneticmomentandminimalbandstructure
effectsovertherangeofausedhere.
ThechangesresultsherefrareomratherthesesubtlecalculationsandcanarenotgiveninaccountFig.for4.4the(b).thelarObviouslygedif,ferthe-

51

4Ferrimagnetism,exchangeandCurietemperaturesinMn2TiZ

Mnences1(2)-Mndiscussed2(1)above.interactionHoweandofver,theweTinotemediatedareductionsecondofneartheestnearMnest1(2)-Mnneighbor2(1)
incrneighboreased.MnandMeanwhile,Timoments.theMn-Tiinteractionincreases,inagreementwith
Thestrongconfinementoftheexchangeinteractionstoaspherewitharadius
ofabout1.5aisreflectedintheCurietemperaturecalculatedasafunctionof
theclusterradiuswhichisnearlyMFconverAgedatr1.5a,seeFig.4.5(a).At
largerradiiaweakoscillationofTCisobserved,indicatinglong-ranged
.behaviour-likeRKKYAdeeperdiscussionoftheexchangeinteractionisbeyondthescopeofthis
work.However,itwasrecentlyshownfornumeroushalfandfullHeuslercom-
poundsthatvariousexchangemechanisms—suchasRKKY,superexchange
andAndersons-dmixing—contributetotheindirectexchangeinteractions
[94].TherelevantcontributionstotheJ0matrixinEq.(3.46)aredisplayedin
Fig.inter4.5-sublattice(b).InagrinteractioneementMnwith1(2)the-Mnpr2(1)eviousprovidesdiscussionthelaritisgestfoundcontribution,thatthe
)2(1Mn2followed(1)byinteractiontheMn-TiniMn2TinteraciIn.tion,Thewhichintra-sublatticecanbecomeinteractaslariongeMnas1(the2)Mn-Mn1(2)-
isinter-generallyandweak,intra-sublatticepositiveforAl,contributionsGa,In,arandebelownegative1formeV.Si,AGe,Sn.negativeAllintra-other
ordersublatticeonthiscontributionlatticeandmeansthusrthateducesthetheinteractionCurieactstemperaturagainste.theferromagnetic
Table4.2summarizesourcalculatedCurietemperatures.Theyarewell
aboveroomtemperatureforthecompoundswith21and22valenceelectrons,
butconsiderablylowerforMn2TiAsandMn2TiSb.TheCurietemperature
scalesroughlylinearwiththetotalmagneticmoment.Withinonegroup,the
Curietemperaturesarecomparable,thoughatrendtodecreasewithincreasing
atomicnumberoftheZcomponentisclearfor21and22valenceelectrons.
TheCurietemperaturesMFAofMn2TiAl,Mn2TiGaandMn2TiInarequitesimilar.
TheslightlyreducedTCofMn2TiIniscausedbythesteepreductionof

52

Mn2TiZAlGaInSiGeSnPAsSb
TCMFA(K)665663630424398354—132156

Table4.2:CurietemperaturesTCMFAcalculatedinthemean-fieldapproximation.

Results4.2

Figure4.5:(a):TheCurietemperatureTCMFAindependenceonthenormalizedcluster
radiusr/atakenintothesummation.(b):r-summedexchangecouplingparametersJ0.

theMn1(2)-Mn2(1)interaction.Ontheotherhand,asimultaneousincrease
oftheMn-TiinteractionstabilizesTCMFAatastillhighlevel(seeFig.4.4(b)).
IntheseriesMn2TiSi–Mn2TiGe–Mn2TiSntheMn1(2)-Mn2(1)interaction
decreases,butheretheincreaseoftheMn-Tiinteractioncannotcompensate
thisandhencetheCurietemperaturedecreases.Inanycase,theMn1(2)-Mn2(1)
interactionprovidesthedominantcontributiontoTCMFA,onlyinMn2TiInthe
Mn-Tiinteractionisdominant.ThesignificantlylowerCurietemperatureof
Mn2TiAswithrespecttoMn2TiSbcanbeattributedtotheartificiallyincreased
latticeparameterusedinthecalculation.
ThedependenceoftheexchangeparametersandTCMFAonthelatticecon-
stantwasstudiedforMn2TiGe.ThecorrespondingtermsoftheJ0matrix,the
CurietemperatureandthemagneticmomentsarepresentedinFig.4.6(a)-(c).
AdecreaseoftheMn1(2)-Mn2(1)interactionandsimultaneouslyofTCMFAwith
increasingaisobserved,althoughbothmMnandmTiincrease.Obviously,the
individualmomentsplayonlyaminorroleintheexchangeandtheinteratomic
distancesaremoreimportant.TheMn-TiaswellastheMn1(2)-Mn1(2)inter-
actionsbecomestrongerwithincreasinga,buttheynearlycompensateeach
other.Inagreementwithadirectexchangecoupling,theMn-Tiinteraction
scaleswiththemagneticmoments.ThechangesinJ0reproduceverywellthe

53

4Ferrimagnetism,exchangeandCurietemperaturesinMn2TiZ

Dependence4.6:eFigurofJ0(a),TCMFA(b)and
(c)momentsmagneticoninparameterlatticetheiGe.TMn2

changesobservedinFig.4.5(b)fortheSi–Ge–Snseries.
Putintermsofapressuredependence,weobservedTC/dp>0,i.e.,the
Curietemperatureincreaseswithincreasingpressure.Kanomataetal.pro-
posedanempiricalinteractioncurveforNi2MnZandPd2MnZfullHeusler
compoundsthatsuggestesdTC/dp>0forthesecompounds[95].Theori-
ginexchangeofthisbetweenbehaviortheisMnattributedatoms,towhichthefullyMn-Mncarrydistancetheandmagnetismtheindiroftheect
compounds.Hence,allotherinteractionscanbeneglected.Anumericalcon-
Forfirmationhalbyf-metallicfirstHeuslerprinciplesofcompoundsthisofinteractiontypeCocurveYZKwas¨ublergivenetral.ecentlyanalyzed[93].
2thedependenceofTConthevalenceelectronnumber,whichisapproximately
linear,andscalesthuswiththetotalmagneticmoment[96].Furtheritwasalso
proposedforCo2MnZcompoundstohavedTC/dp>0,althoughtheCoatom
participatessignificantlyintheexchangeinteractions[92].Experimentallythis
dependenceonthelatticeparameterwasevenobservedfortheCo2TiZseries

54

Results4.2

(withZ=Si,Ge,Sn),wheretheTiatomshavenearlyvanishingmagnetic
[88].momentInterestingly,themagneticmomentsofMnandTiinMn2TiGevarywithin
thesamerangeasthemomentsfordifferentcompoundsshowninFig.4.2(b),
whilethetotalmomentremainsfixedat2µB/f.u.Thesefindingsdemonstrate
thestronginfluenceofthelatticeparameter,whilethedetailsoftheelectronic
structureoftheZelementarelessimportant.Consequently,theZelement
influencesthepropertiesoftheMn2TiZcompoundmainlyviaitsnumberof
valenceelectronsanditsatomicradius,whichdeterminestheequilibrium
.parameterlattice

55

CurieandinteractionsExchange5temperaturesofMn2CoZcompounds

5.1tioncduIntro

IntheliteratureithasbeennotedthattheMn2YZinverseHeuslercompounds
withHg2CuTistructurearedominatedbydirectexchangebetweenthenearest
neighborMnatoms,butdirectcalculationsoftheexchangeinteractionsare
missing.Itisthescopeofthischaptertoprovidethesecalculationsforthe
Mn2CoZcompounds.Wefocusonthiscompoundseriesbecauseithasbeen
experimentallysynthesized,andbandstructurecalculationssuggestedvery
largeatomicmomentsandhalf-metallicityinmostcases.
Thehalf-metallicityofMn2CoZisconstitutedbytwoprocesses[28].First,a
broadcovalentgapofMn(B)iscreatedbycovalenthybridizationwithCoand
Mn(C),whichformthe(doubletetrahedral)nearestneighborshell.However,
thefinalsizeoftheminoritygapisdeterminedbytheeu-t1usplittinginthe
hybridizationofCoandMn(C),whichformeachother’s(octahedral)second
nearestneighborshells.Mn(B)statesdonotcontributetothishybridization
becauseofthedifferentsymmetrytransformations.Thus,thebandgapis
ad-dgap[97].ThissituationissimilartotheoneintheCo2MnZHeusler
compounds,wheretheeu-t1usplittingoftheCo-Cohybridizationgovernsthe
[23].gapminorityThecalculationswereperformedwiththespin-polarizedrelativisticKorringa-
Kohn-RostokerpackageMunichSPRKKR,seeChapter3.2.2.Thecalculations
werecarriedoutinthefull-potentialmodewithanangularmomentumcutoff
oflmax=3ona28×28×28kpointmesh(564pointsintheirreduciblewedge
oftheBrillouinzone).Inordertofurtherimprovethechargeconvergence
withrespecttolmax,weemployedLloyd’sformulaforthedeterminationof
thewithinFermitheenergygeneralized[74,75].gradientTheapprexchange-corroximationofelationPerdew,potentialBurke,wasandmodeledErnzer-
hof[69].Allcalculationswerecarriedoutinthescalar-relativisticrepresentation
ofthevalencestates,thusneglectingthespin-orbitcoupling.

56

Results5.2

Mn2CoZa(˚A)mtotalmComMn(B)mMn(C)mZ
Al5.841.990.942.69-1.59-0.05
Ga5.862.010.932.88-1.78-0.03
In6.04a1.950.993.16-2.18-0.02
Si5.702.990.842.66-0.50-0.01
Ge5.802.980.872.83-0.720.01
Sn5.96a2.980.832.96-0.81-0.01
Sb5.903.970.882.950.150.00
aexp.latticeparameters:Mn2CoIn6.14˚A,Mn2CoSn6.06A˚

Table5.1:Latticeparametersusedforthecalculationsandresultingtotalandsite
resolvedmagneticmoments.ThetotalmagneticmomentsaregiveninµBperformula
unit,theatomicmagneticmomentsaregiveninµBperatom.

HeisenbergpairexchangecouplingparametersandMFACurietemperatures
wereobtainedasdescribedinChapter3.3.Ther-summationinEq.(3.46)was
takentoaradiusofrmax=3.0a,whereaisthelatticeconstant.
ThelatticeparametersweretakenfromLiuetal.[28],whoprovideexper-
imentalvaluesforZ=Al,Ga,In,Ge,Sn,Sb.ForMn2CoSiweassumedthe
Mn2CoGeparameterreducedby0.1˚A,whichisobserved,e.g.,forCo2MnSi
–Co2MnGe[51].ThecalculationsofMn2CoInandMn2CoSnwereunstable
attheexperimentallatticeparameters,butcouldbestabilizedwithslightly
reducedvalues.AlllatticeparametersusedherearesummarizedinTable5.1.

Results5.2momentsMagnetic5.2.1Theelectronicstructurecalculationsyieldahalf-metallicgroundstateinall
caseswiththeexceptionofMn2CoGaandMn2CoIn.Ourresultsforthetotal
andsiteresolvedmagneticmomentsaresummarizedinTable5.1.Thetotal
magneticmomentscloselyfollowtheSlater-Paulingruleforhalf-metallic
Heuslercompounds,sothatwehavemagneticmomentsof2,3,or4µB/f.u.
ifZisagroupIII,IV,orVelement,respectively.Smalldeviationsfromthe
integervaluesarisefromtheangularmomentumtruncationatlmax=3,which

57

5ExchangeandCurietemperaturesofMn2CoZcompounds

givesrisetoaverysmallDOSintheminoritygap.Thisisatypicalobservation
whenusingtheKKRmethodonferromagnetichalf-metals(see,e.g.,Galanakis
etal.[23]).ThemagneticmomentoftheCoatomisnearlyconstantfor
difnearlyferentZconstantmaterials,magneticbeingmomentaboutin0.9theµB.rangeSimilarlyof2.69,tothe3.16Mn(B)µB.Inatomcontrast,hasa
themomentoftheMn(C)atomchangesconsiderablywiththevalenceelectron
numberanddeterminesfinallythetotalmoment.AllMn2CoZcompoundsare
ferrimagneticduetotheMn(C)atomwiththeexceptionofMn2CoSb,which
isaferromagnet.InallcasestheZatomisnearlyunpolarized.Onlysmall
onechangesgroup.areTheobservedincreaseforofthethesiterabsoluteesolvedvaluemomentsoftheMnwhenZismomentschangewithindwithinone
groupcanbetracedtothelatticeparameterchangeuponZchange.Theorbital
overlapisreducedwithincreasinglatticeparameter,givingrisetoweaker
ofthishybridizationsreduction(whichofisitinerancyalsothethereasonquenchingfortheofgapthewidthatomicreduction).momentsisBecauseless
effectiveandthemomentsbecomemoreatomic-like,i.e.,larger.Thissituation
issimilartotheonedescribedinChapter4.
OurresultsdifferconsiderablyfromthosegivenbyLiuetal.[28],whoused
thefullpotentiallinearizedaugmentedplanewaves(FLAPW)method.The
totalmagneticmomentsareinverygoodagreement,buttheatomicmoments
aresmallerinourcalculationsby0.3to0.7µBforMn(B)andMn(C).Incontrast,
themagneticmomentsoftheCoatomsarenearlyequal.Mostnotably,inour
calculationsMn2CoSbisferro-insteadofferrimagnetic.Therefore,wehave
resultscheckedareourconcorSPRKKRdantrwithesultsthewithSPRKKRtheFLAPWdata,leavingpackagetheElkdiscr[70].OurepanciesFLAPWwith
unexplained.Liual.etApartfromthesedifferences,theDOSareingoodagreementwith[28]and
allconclusionsabouttheelectronicstructuregiventherearetransferableto
calculations.our

5.2.2ExchangeinteractionsandCurietemperatures
Figure5.1showstheHeisenbergexchangecouplingparametersobtainedfrom
ourcalculations.Toeasethefollowingdiscussion,refertoTable5.2forthe
dinations.cooratomicWexchangeestartwithinteractionsthearediscussiontightlyoftheconfinedtoAl–Ga–Inclustersseries.ofItradiusisr≤notablea.Inthatpartic-the
ular,theinter-sublatticeinteractionshavesignificantcontributionsonlyforthe

58

eFigur

5.1:

Heisenberg

exchange

coupling

asafunctionoftheinteratomicdistance

have

been

multiplied

by

3

for

.clarity

r

parameters

.

Note

that

Jjieth

ZCointra-sublattice

fortheMn2CoZ
intra-sublattice

5.2

Results

compoundseractionsint

eractionsint

59

5ExchangeandCurietemperaturesofMn2CoZcompounds

0.500.433(r/a)distancesymmetryTOdhMn(C)/Mn(B)CoZMn(C)Mn(B)Mn(C)Mn(B)//ZCoZCo
Mn(B)Co/Mn(C)Z

Table5.2:NearestandsecondnearestneighborcoordinationsinMn2CoZ.

nearestandsecondnearestneighbors,whiletheintra-sublatticecontributions
aresignificantuptor=a.Anexponentialdampingoftheexchangeinterac-
tionsisexpectedforhalf-metals[90];inthecasesofGaandInthedampingis
alsopresent,butnotasefficientasinthehalf-metalliccaseofAl.Oneobserves
clearlythedominatingCo-Mn(B)andMn(C)-Mn(B)nearestneighborinter-
actions,wheretheMn(C)-Mn(B)interactionisclearlythestrongerone.The
Co-Mn(B)(secondnearestneighborinteraction)ismuchweakerincomparison.
InthegraphsweomittheinteractionswithZ,becausetheseareeffectivelyzero
foralldistances.CoandMn(C)coupleantiferromagenticallytoMn(B),while
CoandMn(B)coupleferromagnetically.Hence,theantiparallelalignmentof
theMn(C)momentisstablewithrespecttoMn(B)andCo.Ontheotherhand,
theintra-sublatticeinteractionsarenegative,whichleadstoadestabilization
oftheparallelalignmentofthemomentsononesublattice.Itshouldbenoted
thatintheAl–Ga–InseriestheMn(C)-Mn(C)interactionisreducedonthefirst
shell,whileitisincreasedonthesecondshellatr=1,whereitbecomeslarger
thantheCo-Mn(C)inter-sublatticeinteraction.
FortheSi–Ge–Snseriessomedifferencestothepreviousresultsarenotable.
ThemostevidentoneisthemuchlowerMn(C)-Mn(C)interaction,butalsothe
Co-Mn(B)interactionissignificantlyreduced.Inparticular,theMn(C)-Mn(B)
interactionisreducedbyafactorofabout3,inverygoodagreementwiththe
reductionoftheMn(B)moment.Thisindicatesastrongdirectexchangeinter-
action,whichisfeasiblebecauseofthesmallMn(B)-Mn(C)distanceoftypically
2.53˚A.ItisremarkablethattheCo-Mn(B)interactionsareevenslightlyin-
creasedwithrepecttotheAl–Ga–Inseries,althoughthesite-resolvedmagnetic
momentsaresystematicallylower.Theadditionallooselyboundspelectron
augmentsthedirectexchangecouplinghere.Finally,theintra-sublatticeinter-

60

Results5.2

Mn2CoZAlGaInSiGeSnSb
TCMFA(K)890886845578579536567
Table5.3:CurietemperaturesTCMFAcalculatedinthemean-fieldapproximation.

firstactionshell,ofbuttheMn(B)-Mn(B)otherisfoundintra-sublatticetobepositiveparametersinallarthreeestillnegative.compoundsonthe
Mn2CoSbisspecialinthisrespect,sinceitisaferromagnetwithasmall
andpositiveMn(C)-Mn(B)magneticmomentinteractionsonartheeMn(C)positive,site.andAccortheirdinglyvalues,theareinrCo-Mn(C)eason-
ableCo-Mn(B)agreementinteractionwithisthestillrlareductiongeandofisthewithMn(C)themoment.exceptionofInMn2contrast,CoSithethe
larnegativegestoneagainamongonallthefirstdiscussedandsecondcompounds.shells.TheSuchaMn(B)-Mn(B)periodicitywithinteractionrespectis
tothevalenceelectroncountofthesystemhasbeenpredictedby¸Sa¸sio˜glufor
somefullHeuslercompoundsandoccursinthepresenceofindirectexchange
interactionsmediatedbytheconductionelectrons[94].
Fromtheexchangecouplingparametersdescribedabovewecalculatedthe
seriesCurieAl–Ga–Intemperaturhaseswithinsurprisinglythemeanhighfieldvaluesapprofmoreoximationthan800(seeK,Tableevenr5.3).eachingThe
almost900KforMn2CoAl.FortheSi–Ge–Snserieswefoundmoderatevalues
between500and600K.TheCurietemperatureofMn2CoSbissimilarasfor
theSi–Ge–Snseries.Thisissurprisingatafirstglance,becausetheMn(C)-
Mn(B)exchangeinteractionissosmallhere.Itcanbeunderstoodµifνweneglect
allsingular3interactions×3matrixbutwithMn(C)-Mn(B)twononzerandoCo-Mn(B).eigenvalues,Inthiswhichcase,haveJ0theformbecomesofaa
rootmeansquareoftheCo-Mn(B)interactionandtheMn(C)-Mn(B)interaction.
Mn2ObviouslyCoSn),,ifthenonetheinteractioneigenvaluesiswillsignificantlybedominatedlargerbythanthethelarotherger(as,interaction.e.g.,in
ThisandtheincreasedCo-Mn(B)exchangeinteractionexplaintheunexpected
.behaviourHowever,whatismostexcitingabouttheseresultsisthefactthatMn2CoAl,
Mn2CoGa,andMn2CoInhavethepossiblyhighestCurietemperatureamong
allperatureferrimagneticdecreasesfromintermetalliconeZgroupcompoundstoranothereported,althoughtodate.theThetotalCuriemomenttem-

61

5ExchangeandCurietemperaturesofMn2CoZcompounds

increases.AbehaviourlikethisisuniquetotheMn2basedinverseHeusler
compounds.TheCo2-andMn2-basedgenuineHeuslercompoundsshowa
scalingoftheCurietemperatureroughlyproportionaltothetotalmoment
uponchangeoftheZelement,seeRef.[96]andChapter4.Nevertheless,
theMn2CoZcompoundscanberelatedtotheCo2-basedHeuslercompounds
withthesumoftheabsolutevaluesofthesiteresolvedmagneticmoments
m˜total:in,e.g.,Mn2CoAlwehave˜mtotal=5.27µB,whichisclosethevalueof
ferromagneticCo2MnSi(5µB).ThelatterhasaCurietemperatureof985K[51],
whichisclosetoTCMFAofMn2CoAl.Further,theCurietemperatureandm˜total
aredecreasedwithincreasingZelectronnumberinMn2CoZ.
Naturally,thequestionabouttheaccuracyofourCurietemperaturecalcula-
tionariseshere.FortheMn2CoZseriesonlyfewdataareavailable.Lakshmi
etal.reportedTC=605KfordisorderedMn2MFCoSnA[31].Daietal.reported
485KfordisorderedMn2CoSb[32].Hence,theTCvalueunderestimatesthe
measuredvalueinMn2CoSnandoverestimatesitforMn2CoSb,sonosystem-
atictrendcanbestatedhere.Itisapriorinotclearwhichtypeofdisordercan
increaseordecreasetheCurietemperature,sincetheexchangeinteractions
arehighlysitespecificandquitecomplex.However,thecalculatedvalues
reproducethemeasureddatawithin±100K.
InFigure5.2weshowthecalculatedCurietemperaturesindependenceon
theclusterradiustakenintothesummationinEquation(3.46).Asexpected
fromtheJijplotsinFigure5.1,TCMFAisalreadydeterminedbythenearest
neighborinteractionsinallcompounds.Onlyweakchangesareobservedwith
increasingclusterradiusandTCMFAiswellconvergedatr=1.5a.Thisplot
helpsustoidentifytheoriginofthereducedCurietemperaturesofMn2CoIn
andMn2CoSn,whichisapparentlynotthesame.ForMn2CoInwecanassign
thejumpatr=atothestrongantiferromagneticintra-sublatticeinteraction
ofMn(C)-Mn(C).InMn2CoSn,thereducedferromagneticMn(B)-Mn(B)intra-
sublatticeinteractiononthethirdneighborshellatr=0.707aisresponsible
eduction.rtheforInordertoshedsomemorelightonthecharacteroftheexchangeinteractions
andtheirdependenceonthesitespecificmagneticmoments,wecalculatedthe
groundstatesandexchangecouplingparametersforMn2CoGeintherange
ofa=5.60...5.95˚A.TherebywecanseparatetheinfluenceoftheZvalence
electroncountandthebindingenergyfromgeometriceffects.Thecompound
isaferrimagnetichalf-metaloverthewholerange,sowecanexpectminimal
bandstructureeffectsonthecalculations.
Ontheotherhand,thesiteresolvedmagneticmomentschangeconsiderably

62

5.2Results

Figure5.2:CurietemperaturesTCMFAasafunctionoftheclusterradiustakeninto
account.

withthelatticeparameter(Figure5.3(a)).Theirabsolutevaluesincreasewith
increasinglatticeparameterasalreadyexplainedabove.Allmomentsvary
approximatelylinearlysuchthatthetotalmomentremainsat3µB/f.u.The
momentofMn(C)changeswithintheinvestigatedrangebymorethanafactor
ofthree,from-0.34to-1.12µB.ThecompensationcomesmostlyfromtheMn(B)
site,andtheComomentremainsfairlyconstant.
Todisplaytheexchangeµνinteractionsinacompactform,weshowtherelevant
contributionstotheJ0matrix(Equation(3.46))inFigure5.3(b).TheCo-
Mn(B)interactionsumisnearlyconstant,althoughthemagneticmoments
increase.Thenearestneighborinteractionremainsnearlyconstant,butthe
weaklonger-ranginginteractionissignificantlydecreasedandaccountsfor
thedecreaseintheinteractionsum.Theconstantnearest-neighborinteraction

63

5ExchangeandCurietemperaturesofMn2CoZcompounds

Figure5.3:LatticeparameterdependenciesinMn2CoGe.(a):Siteresolvedmagnetic
moments.(b):J0µνcontributions.(c):CurietemperaturesTCMFAindependenceonthe
clusterradius.(d):CurietemperaturesTCMFA.

isandartheesultrofeductiontwoofopposingexchangeprefocesses,ficiencyduenamelytothelongerincreaseinteratomicofthedistances.moments
Incontrast,theinteractionsinvolvingMn(C)changeconsiderablywiththe
interatomicdistance.TheMn(C)-Mn(B)exchangeinteractionsincreasebya
factoroffour,inagreementwiththeproductmMn(B)∙mMn(C).Further,theCo-
Mn(C)interactionincreasesmorethanlinearlywiththelatticeparameter,but
AlltheintheseteractioninteractionsisprleadesumablytoanindirincrecteaseandofnothesimpleCuriedependencetemperatureiswithobvious.the
latticeparameter.Incontrast,theantiferromagneticMn(C)-Mn(C)exchange
theinteractionCurietemperaturcounteractse.theThisinfluenceferrimagneticis,ordehoweverrin,thenegligiblecompoundatandsmallreduceslattice
theparameterMn(C)-Mn(B),butibecomesnteraction.quitelarNotablygeat,thethehighestMn(C)-Mn(C)values,eveninteractioniscompensatingentirely
2governedbythenearestneighborinteractionanddependsapproximatelyon
mMn(C).

64

Results5.2

Figure5.3(c)displaystheCurietemperatureindependenceonthecluster
radius.Thegeneralfeaturesoftheexchangeinteractionsarethesameforall
latticeparametersconsidered.However,therearesomesubtledifferences
onthesecondandthirdshellsatr=0.5aandr=0.707a,respectively.The
changeinteraction.ontheRelatisecondvetoshellthecansecondbeshell,tracedthebacktocontributiontheincrofeasedthethirdCo-Mn(C)shell
isreduced.ThisarisesfromtheincreasedantiferromagneticMn(C)-Mn(C)
interactiondiscussedabove.Forclarity,Figure5.3(d)showsthattheresulting
CurietemperatureTCMFAincreasesfrom526Kto631Kwithincreasinglattice
.parameterIntermsofapressuredependence,theCurietemperatureofMn2CoGeis
thuspredictedtodecreaseuponhydrostaticpressure,i.e.,dTC/dp<0.This
situationisverydifferentfromthatinHeuslercompounds,whereusually
dTC/dp>0isfound.However,itisinagreementwithCastelliz’[98]and
coefKanomata’sficientof[T95]forempiricalshortMn-Mninteractiondistancescurves.asinTheyprhexagonaloposeaMnAsnegativeorprMnSb,essurbute
CapositivecoefficientatlargerdistancesasintheHeuslercompoundsX2MnZ.
AbinitiocalculationsbyYamadaetal.onhexagonalMnAs[99]andby¸Sa¸sio˜glu
etal.ontheHeuslercompoundNi2MnSn[93]areinagreementwiththe
experimentallyobservedpressuredependencies.AsshowninChapter4,we
havealsocalculatedapositivepressurecoefficientofTCinthe(hypothetical)
Mn2TiZHeuslercompounds.TheMn-MndistanceintheMn2CoZcompounds
isnegativeevenprsmalleressurethanindependencethehexagonalofTCisinMnAsgoodoragrMnSbeementcompounds,withthesoastravailableong
data.experimentalSincethelatticeparameterdependenceoftheCurietemperatureispositive,
thereductionofTCMFAinMn2CoInandMn2CoSn(whichhavethelargestlattice
parameterswithintheirgroups)canbeascribedtoabindingenergyeffectdue
tothehigh-lyingvalencestatesinInandSn.

65

6Electronicstructureoffullyepitaxial
thinTiSnCofilms2

tioncduIntro6.1

Co2TiSnhasbeenthesubjectofmanyexperimentalandtheoreticalstudies.The
groundstatepropertiesobtainedbydensityfunctionaltheory(DFT)depend
sensitivelyonthechoiceoftheDFTmethod[23,38,39,100,101,102,103,
104].Thepotentialhasstrongnon-sphericalcomponentsandthusonlyafull-
potentialtreatmentinconnectionwiththegeneralizedgradientapproximation
(GGA)tothedensityfunctionalyieldsahalf-metallicgroundstate[38,101].
ExperimentsconductedonbulkCTSfindalatticeparameterof6.07˚A,a
355magneticK[38,88,moment105].ofFurtherabout,it1.95isµBfound/f.u.toandhaveaaCuriestronglytemperaturanomalouse(TC)artempera-ound
turedependenceofresistivity,thetemperaturecoefficientbecomesnegative
abovetheCurietemperature.Alargenegativemagnetoresistancerevealsthe
importanceofspinfluctuationsinthecompound[78].
ArathernewdevelopmentaimsatthethermoelectricpropertiesofCo2TiSn,
whichhasalargeandconstantSeebeckcoefficientof−50µV/KaboveTCinthe
bulk[88].Therehavebeensomeeffortstounderstandtheunusualtransport
[pr88,operties106].ofTheseCTSprbyabopertiesinitiomakebandstrCTSucturintereandestingforasemi-classicalpossibletransportapplicationtheoryin
spincaloritronics,whichattempttomakeuseoftheinteractionsbetweenheat
andspin.Animplementationintothinfilmsisofparticularimportancefor
applications.suchOnlytwostudiesonthinfilmsofCTSareavailableasfarasweknow.Gupta
etal.appliedpulsedlaserablationtogrowCTSonSi(001)substratesfroma
stoichiometrictargetatgrowthtemperaturesupto200◦C[107].Theauthors
foundoff-stoichiometric,polycrystallinefilmswith(011)texture.Suharyadiet
al.utilizedanatomicallycontrolledalternatedepositiontechniquebasedon
electronbeamevaporation[108].Theyhavegrown(001)oriented,L21ordered
filmsonCrbufferedMgO(001)substratesatgrowthtemperaturesupto600◦C

66

oductionIntr6.1

andinvestigatedthembynuclearresonantscattering.
Inthischapterwepresentasuccessfulpreparationtechniquebasedon
DCneticprmagnetropertiesonofourco-sputtering.films.FurtherWe,prweesentdatacharacterizeonthethestrelectructuralonicandtransportmag-
propertieswhichmakeCTSaparticularlyinterestingcompound.Finallywe
discusstheelectronicstructureofourCTSfilmsbasedonsoftx-rayabsorption
spectroscopyandabinitioelectronicstructurecalculations.

detailserimentalExp6.1.1ThesamplesweredepositedusingtheBESTECUHVsputteringsystem,see
Chapter2.1.Withthequartzsensorandx-rayreflectometry(XRR),thefilm
stoichiometryofacompoundcanbesetupwitharelativeaccuracyofabout
±was10%.usedtoInductivelyfine-tunethecoupledsputterplasmaoparameters.pticalForemissionthesamplesspectroscopydepositedat(ICP-OES)high
temperaturewecheckedthestoichiometrybyenergydispersivex-rayanalysis
(EDX)inanelectronmicroscopeandfoundnodeviationfromthestoichiometry
ofroomtemperaturedepositedfilmsofsamethickness.Thesputteringpower
ratioswere1:1.67:0.34(Co:Ti:Sn).Thevoltageswereconstantlymonitored
duringthedeposition,whichremainedconstantthroughoutalldeposition
prtheSnocesses,targetensuringconstitutedtherepraoseriousducibilityprofoblemthefortmethod.heCrdepositionoss-talkprefocessfectsdueon
tothelowsputteringpowerappliedtothesource.Thiswassuppressedby
achimney-likecylinderputaroundthesource,suchthattherewasnoline-
ofof-sight1.5˚A/s.fromSamplethistarrgetotationtowasanotherset.toThe28rpm,compoundmakingwassuredepositedthatwithataeachrate
turnonlyoneprimitivecellwasdeposited.Allelementaltargetshad4N
purity.Thesputteringpressurewassetto2∙10−3mbar.Withthistechnique
wehavefabricatedthinfilmsampleswithapreciselysetupstoichiometryof
Co2.0Ti1.0Sn1.0,witherrorsof<3%fortheindividualconstituents.
Allsamplesusedinthisstudyhadthefollowingstacksequence:MgO(001)
/RFMgOsputtering5nm/atCTS2.3∙1810−nm2/mbarMgOto2ensurnm.eThegoodlowercrystallinityMgOofwasthebufdepositedfer.Theby
upperMgOwasdepositedbye-beamevaporationfromsinglecrystalMgO
slabsaftercoolingthesamplestolessthan100◦C.Thebase−pr8essureduring
depositionwiththeheatedsubstratewasalwaysbelow5∙10mbar.
closed-cycleResistivityHewascryostatmeasurandedainvacuumstandardfurnace.in-linefourTher-presistiviobeDCtyρisgeometrycalculatedina

67

6ElectronicstructureoffullyepitaxialCo2TiSnthinfilms

fromthefilmthicknessd,thevoltageUandthecurrentIasρ=d∙(π/ln2)∙
(U/I).Magnetoresistancewasmeasuredwithavariablepermanentmagnet
(coaxialHalbachcylinderconfiguration,MagneticSolutionsMultimag)with
amaximumfieldstrengthof10kOeinthecryostat.Thedataweretakenby
drivingfullmagnetizationloopsandthenaveragingthepointsforeachfield
magnitude.TheSeebeckcoefficientwasdeterminedinahomebuiltsetupinair.The
samplewascontactedwithplatinumtips.Itwasmeasuredatanaverage
temperatureofT¯=310KwithatemperaturegradientofΔT=10K.
Magneticmeasurementsweretakenusingasuperconductingquantuminter-
ferencedevice(SQUID)attemperaturesintherangeof5Kto400Kinmagnetic
fieldsofupto50kOe.
X-raydiffraction(XRD)andreflectometry(XRR)havebeenperformedinthe
PhilipsX’PertProMPDinBragg-Brentanoconfiguration.Texturecharacteri-
zationwasadditionallyperformedwithcollimatorpointfocusopticsonthe
cradle.EulerianopenTemperaturedependentx-rayabsorptionspectroscopy(XAS),x-raymag-
neticcirculardichroism(XMCD)andx-raymagneticlineardichroism(XMLD)
wasperformedatBL6.3.1andBL4.0.2oftheAdvancedLightSourceinBerke-
ley,USA.Theelement-specificmagneticpropertieswereinvestigatedatthe
Co-andTi-L3,2edgesinsurface-sensitivetotalelectronyieldmode(TEY)[56]
fortemperaturesbetween20Kand370K.
ForXMCD,thesamplewassaturatedbyapplyingamagneticfieldofmax.
±20kOealongthex-raybeamdirectionusingellipticallypolarizedradiation
withapolarizationofPhν=±60%(BL6.3.1)andPhν=±90%(BL4.0.2),
respectively.Thex-raysangleofincidencewithrespecttothesamplesurface
wasα=30◦(BL6.3.1)andα=90◦(BL4.0.2),respectively.I+andI−denote
theabsorptionspectra(normalizedtothex-rayfluxmeasuredbythetotal
electronyieldofaAugridinfrontofthesample)forparallelandanti-parallel
orientationofthephotonspinandthemagnetizationofthesample.The
XASandXMCDspectraaredefinedasXASc=(I++I−)/2andXMCD=
(I+−I−),respectively.Tocalculatetheelement-specificspinandorbital
magneticmomentsfromthedataweappliedsum-ruleanalysis,seeChapter
2.4.2.AnisotropicXMLDspectraweretakenatBL4.0.2with100%linearlypolar-
izedlightinnormalincidenceusingtheeight-poleelectromagnetendstation
[109].Themagneticfieldforswitchingthemagnetizationofthesamplewas
appliedparallelandorthogonaltothepolarizationvectoroftheincominglight,

68

6.1oductionIntr

theaccordingabsorptionspectranormalizedtothex-rayfluxaredenotedasI
andI⊥.TheXASandXMLDspectraarethendefinedasXASl=(I+I⊥)/2
andXMLD=(I−I⊥),respectively.Spectraweretakenwithmagneticfields
alignedappliedalongmagneticthefield[100]ofand4.5thekOe[110]wasdircantedectionsoutofofththeeCosurface2TiSnplanefilms.by10The◦
toimprovetheelectronyieldsignal.However,theXMLDresultsarenearly
unaffectedbythisbecausethedemagnetizingfieldperpendiculartothefilm
planeissostrongthatthemagnetizationistiltedout-of-planebylessthan5◦
(measuredbyanalyzingtheXMCDasymmetryfordifferenttiltingangles).
atTheeachXMCDenergyandpoint.XMLDTorspectraemovewerenon-dichrtakenoicbyartifactsswitchingwetheperformedmagneticmea-field
surementsforpositiveandnegativepolarization(XMCD)ordifferentspatial
orientationsofthepolarizationvector(XMLD)andaveragedthecorresponding
spectra.

roachappreticalTheo6.1.2Theelectronicstructureprobedbyx-rayabsorptionspectroscopyhasbeen
investigatedindirectcomparisonwithabinitioelectronicstructurecalculations.
Weusedtwodifferentapproachestothisend.First,electronicstructurecalcu-
lationswereperformedwiththeMunichSPRKKRpackage,seeChapter3.2.2.
Andsecond,inordertotakecareoftheexcitedstatebandstructure,whichis
actuallyprobedinXAS,spectrumsimulationswerecarriedoutinFEFF9,see
3.4.4.ChapterInSPRKKR,thebandstructureandthegroundstatepropertieswerecalcu-
latedinthefully-relativisticrepresentationofthevalencestates,thusincluding
spin-orbitcoupling.Theangularmomentumcutoffwassettolmax=3(spdf-
basis)andthefullpotentialwastakenintoaccount.Thebulklatticeparameter
ofa=6.07A˚wasused.Theexchange-correlationpotentialwasmodeledwith

mCospinmorbCoNhComTispinmTorbiNhTi
0.960.042.06-0.030.017.65

Table6.1:ResultsofbandstructurecalculationswithSPRKKR.Themagneticmoments
aregiveninµB/atom.

69

6ElectronicstructureoffullyepitaxialCo2TiSnthinfilms

thegeneralizedgradientapproximation(GGA)inthePerdew-Burke-Ernzerhof
parametrization.Theresultingatomicmagneticmomentswerethenusedasinputparame-
terstoFEFF9,whichisnotspinself-consistent.Theself-consistentpotential
wasobtainedonaclusterof59atomsandthex-rayabsorptionnearedge
spectrosopy(XANES)wascalculatedonaclusterof229atoms.Thecomplex
Hedin-Lundqvistself-energywasappliedandthecalculationsweredonewith
thefinalstaterule,includingafullscreenedcoreholeontheabsorber.The
angularmomentumforthefullmultiplescatteringwastakentolmax=3.
ThegroundstatedescribedbytheSPRKKRcalculationisnothalf-metallic
withtheexperimentallatticeparameter,incontrasttocalculationswithfull
potentiallinearizedaugmentedplane-wavescodes[38,104].TheFermienergy
isslightlyabovetheminorityspingap;asmallincreaseofthelatticeparameter
wouldmoveEFintothegap.Thisisduetotheangularmomentumtruncation
atlmax=3,whichisinsufficienttocapturethenon-sphericalcomponents
ofthedensity.Fortechnicalreasons,itcannotbetakentohighervalues.
However,thisdoesnotsignificantlychangetheshapeofthecalculatedXAS
spectrum.Thetotalspinmomentis1.9µB/f.u.andthetotalorbitalmomentis
0.09µB/f.u..Theatom-resolvedmagneticmomentsandthenumbersofholes
forCoandTiaregiveninTab.6.1.ThenegativeTispinmomentindicatesa
CTS.ofbehaviorferrimagneticweakly

6.2Experimentalresults

Structure6.2.1XRDandXRRwereutilizedtoinvestigatethestructureofthefilms.Figure6.1
displaysasetofdatathatwereextractedfromthemeasurements.Asisclearly
visibleinFigure6.1(a),thefilmsshowLaueoscillationsonthe(002)reflection
thatbecomemorepronouncedwithincreasingdepositiontemperature.Laue
oscillationsareonlyobservedifthecrystallinecoherenceisverygoodand
theroughnessissmall.Whilethetwofilmsdepositedatlowertemperatures
showonlyweakoscillations,thetwofilmsdepositedathighertemperature
exhibitpronouncedfringes.Onlyweakasymmetryofthefringesisobserved
forTS=700◦C,indicatingnearlyhomogeneous(orno)strainalongthegrowth
ection.dirFourintense(111)reflectionshavebeenobservedinpolefigureanalysisat

70

esultsrExperimental6.2

Figure6.1:(a):X-raydiffractionscansofthe(002)reflectionsshowingLaueoscillations.
(b)X-rayreflectometry(XRR)scans.Thedashedlinerepresentsthebestfittothe
experimentalcurvewithTS=700◦C.(c):Full-widthsathalf-maximum(FWHM)of
therockingcurvesandeffectivedensitydeterminedbyXRR.(d):Out-of-planelattice
.cparameter

◦theconsiderablyexpectedtiltwithangleincrofeasingΨ=54.74depositionforalltemperatursamples.e.TheTheintensityepitaxialrincrelation-eases
shipisCo2TiSn[100]MgO[110],whichiscommonlyobservedforHeusler
substrates.(001)MgOoncompoundsTheout-of-planelatticeparametercmeasuredonthe(004)reflection,dis-
playedperatureinandFigureconver6.1ges(d),isforthefoundtohighestincreasedepositionwithincrtemperatureasinges.depositionFor700◦tem-C,
wefindalatticeparameterofc=6.105˚A.
Thefull-widthathalf-maximum(FWHM)oftherockingcurvesmeasuredon
the(004)reflectionsaredisplayedtogetherwiththedensitydeterminedbyXRR
inFigure6.1(c).ForhighdepositiontemperaturetherockingcurveFWHM
isfoundtobeaslowas0.6◦,whichdemonstratesthenarroworientation
distributionoftheindividualfilmgrains.
XRRdeterminedprovidesbyXRRindirhasecttobeinformationseenasanonefthefectivefilmdensity,morphologywhich.onlyTherdensityeflects

71

6ElectronicstructureoffullyepitaxialCo2TiSnthinfilms

onthearealsmallfilmlateraldensityscalife.theInFig.surface6.1r(b)weoughnesspreissentlowthewithXRRacurvesGaussianofourdistributionsamples.
Theroughnessishighforthetwosampleswithlowergrowthtemperature,
coverwhichislayerdoesidentifiednotbyshowaupquickasanvanishingindividualoftheresonance.KiessingWefringes.findanTheincrMgOease
intheXRRdensityfordepositionwith600◦Candmore,whiletheroughness
isgreatlyreducedandtheMgOcoverlayerbecomesvisible(seearrow).The
XRRroughnessofthefilmwithTS=700◦Cis0.3nm.Thescansforthetwo
lowerdepositiontemperaturescannotbefitwiththeParrattalgorithm[54].
TheyshowtwomainFouriercomponentsat18nmand23nm,andadifference
componentat5nm.Acolumnargrowthwithhighandlowgrainsthathave
18gr±owth5nmchangesthicknesstoacanmodebewithinferrlaredgefrandomthis.smoothAtgrainshigherofequaltemperaturheight.es,Thisthe
behaviorhasbeenconfirmedbyatomicforcemicroscopy.
toFrfindomaThornton’stransitionfrmodel[oma110]offine-grainedfilmgrowthcolumnarforstrsputteructuredefilmstoaitrisegimeexpectedwith
larmeltinggegrainstemperaturgovernede,TbyS/Tbulkm≈dif0.5.fusionInandfact,rthemeltingecrystallizationpointatofaboutCo2ThalfiSntheis
◦W1720(20)iththeK,[111]i.e.,experimentalthisbulktransitionlatticeisexpectedparameterara=ound6.07600˚A,C.thedensityofthe
3epitaxialcompoundgrisowthoncalculatedthetoMgObe8.446substrate,g/cmthe.Iflatticeonewillbeassumesadistortedperfect,tetragonallystrained,
withanin-planelatticeparametera=√2∙4.21A˚=5.95A˚andaccordingly
expandedout-of-plane.Ifthevolumeremainedconstant,theout-of-plane
latticeparameterwouldbe6.32˚A.Forthefilmdepositedat700◦C,wemea-
suredc=6.105˚A.Recalculatingthedensityforthistetragonalconfiguration
givesρ=8.74g/cm3,whichisincloseagreementwiththemeasureddensity
ofρ=8.7g/cm3.Thisresultsupportsthegrowthmodeldiscussedabove.
Furtherdistortion,weofCohaveTiSnshowncaninaeasilyrecentoccurpaperbecausebyabofinitiothelowtheoryenerthatgyatetragonalassociated
2withactivatedtheduringdistortionthe[gr104].owth.ItisofHowevertheor,deratoflower50meVtemperatur/f.u.,esandthisisthusconstituteseasily
state.metastablea

72

esultsrExperimental6.2

Magnetism6.2.2SQUIDmeasurementstakenonthesamplewithTS=700◦Cgiveamagnetic
momentofm=1.6(1)µB/f.u.andaCurietemperatureofTC=375(5)K(Fig.
6.2).TheCurietemperatureishigherthaninbulksamples,whereithasbeen
reportedtobeabout355K.Thecoercivefieldis160Oeat20Kand150Oeat
roomtemperature.SincethemagnetizationdeclinessharplyatTC,wecan
concludethatthefilmsconsistofasinglemagneticphase.

rtotranspElectronic6.2.3Resistivityandmagnetoresistancehavebeenmeasuredonasampledeposited
atTD=700◦C;thedataareshowninFig.6.3.Theresistivityshowsclearly
thecusp-typeresistivityanomalythatisalsoobservedforbulksamplesof
Co2TiSnatTC.Detailsofthetransitioncanbefoundbyanalyzingthefirstand
secondderivativesoftheresistivitycurve.Wedefinetheonsetoftheasthefirst
inflectionpointoftheresistivity;itisfoundat350(5)K.Themaximumofthe
resistivityisat395(5)K,i.e.,20KaboveTC.Theoffsetofthetransition,given
bythesecondinflectionpoint,isat440(5)K.AtTC=375(5)Kwefindthe
maximalchangerateoftheresistivity’sslope,identifiedbyaclearminimum
derivative.secondtheofByplottingthelogarithmoftheresistivityagainst1/Tforthedatapoints
abovethesecondinflectionpoint,wefindtheeffectivegapwidthoftheparam-

Figure6.2:Magnetizationindependenceofthesampletemperature(markers).Itwas
takenasatemperaturesweepwithaconstantfieldof100Oe.Thesolidlineisaguide
eye.theto

73

6ElectronicstructureoffullyepitaxialCo2TiSnthinfilms

Figure6.3:Top:ResistivityofaCo2TiSnfilmdepositedatTS=700◦ConaMgOsingle
crystal.Theinsetshowstheregionaroundtheferrimagnet-paramagnettransition.Bot-
tom:Correspondingmagnetoresistanceforfieldsof1kOeto10kOewiththemagnetic
fieldHinthesampleplaneandthecurrentj⊥H.

widthagneticofstate12.7to±1beEmeVg=r6.5eported±0.5formeVbul.kThisissamples.considerablyHowever,itsmallerhasbeenthanartheguedgap
byfoundBarthetsignificantal.thatdifanferactualencesfortransitionthetocalculateadsemiconductorconductivityisimprtensorsobable.betweenThey
byaspin-polarizedmolecularandfieldunapprpolarizedoximationforcalculations.theBymagnetization,mixingthetheystatescouldweightedpartly
explaintheanomalousbehavioroftheresistivity[88].
Comparedwithbulksamples,wealsofindanotablylowerresidualresis-
216tivityµΩρ(cm,20K)=compar89µedΩtocm310andandatotal205µrΩcmesistivity[78],oramplitude245and(ρ135maxµ−Ωρcmmin[)88=],
rdensityespectively,i.e.,.Therdislocations,esidualrdisoresistivityder,ofimpuritiesametalandisgrainmainlygivenboundaries.byitsInadefectthin
havefilm,oneveryhaslowtortakeesidualtheresistivityinterfacialcomparscatterinedgtointobulkaccount.samples,Ourwhichthinmightfilms
Weindicateattributethatthistheirtolarge,crystallineflatprgrainsopertiesandaregoodsuperiorchemicaltoorthoseder.ofbulksamples.

74

6.2esultsrExperimental

ThetemperaturedependenceoftheresistivityiswelldescribedbyaT2
termupto180K,whichismainlyattributedtoelectron-electronscattering.
Above180KuptothefirstinflectionpointthecurveisbetterfitbyaT3law.
Inbulksamples,theparabolicshapeoftheresistivitycurveatintermediate
temperaturesislesspronouncedthaninourfilms.However,theoverallshape
isinagreementwiththecurvesfoundbyotherauthors.

Themagnetoresistance(MR)ofthefilm,definedbyMR(H,T)=(ρ(H,T)−
ρ(0,T))/ρ(0,T),showsstronglynonlinearbehavior.Atlowtemperatureonly
weakMRisfound.WithincreasingtemperatureanincreasingMRisobserved,
whichisnegativeoverthewholetemperaturerange,i.e.,theresistivityislower
ifamagneticfieldisapplied.Ithasapronounced,nonlineardependenceon
theappliedmagneticfield.Withanavailablemagneticfieldof10kOetheMR
wasbyfarnotsaturated.Adistinctextremumisobservedatlargefieldsright
belowTC,beingtheglobalminimumofthecurveatfieldslargerthan7kOe.
AboveTCtheMRvanishes.Theappearanceoftheextremumanditsamplitude
areinagreementwiththedatapublishedbyMajumdaretal.[78].TheMRcan
beexplainedintermsofspinfluctuationsandassociatedspin-flipscattering:
atlowtemperature,thefluctuationsarenearlyzeroandasmallmagneticfield
issufficienttosaturatethefilm.Withincreasingtemperature,fluctuations
becomemoreimportant,butcanbesuppressedbyenforcingaparticularspin
orientationinastrongfield.Thispictureissupportedbytheshiftofthefirst
minimumwithincreasingmagneticfield,denotedbythedashedlineinFig.
6.3.TheMRisenhancedatTCbecausethespinfluctuationsarestrongestatthe
transitiontemperatureandtheferrimagneticstateisstabilizedinalargefield.
Furthermore,theMRhasnotraceableanisotropicMR(AMR)contribution:
thetypicalinversionoftheMRatzerofieldforj⊥HcomparedtojHis
missing.

TheSeebeckeffecthasbeenmeasuredonthesamesampleastheresistivity.
ItwasS=−14±2µV/Kat310K,whichisabout2.6timeslowerthanin
thebulk(−37±2µV/K)[88].Thisisinagreementwiththemuchlower
resistivityofourfilmscomparedtobulksamples.Barthetal.pointoutthat
theSeebeckcoefficientcanbeenhancedbyscatteringongrainboundariesor
impurities,[88]whichappeartoberarerinthefilms.Ontheotherhand,the
Seebeckcoefficientisproportionaltoν/σ,withtheelectricalconductivityσ
andthethermalconductivityν.Thus,thelowerSmayalsoindicatealower
film.theofconductivityheat

75

6ElectronicstructureoffullyepitaxialCo2TiSnthinfilms

chemistryInterfacial6.2.4XMCDandXAScmeasurementswereperformedatBL6.3.1at20K◦andat◦RT
for◦films◦depositedonMgOsinglecrystallinesubstrates(TS=400C,500C,
600C,700C,andpost-annealedsamples).
tableThetrCoends:XMCDtheCosignalsmagneticfordifferentmoment,depositionmeasuredattemperatur20K,esandshowthetworationo-of
theCoXMCDsignalsmeasuredatRTandat20Kincreasewithincreasing
TS.Thisimpliesthatthechemicalorderimproveswithincreasingsubstrate
temperature,resultinginhighersaturationmagnetizationandhigherCurie
temperature.ThatisinagreementwithSQUIDmeasurementsonthesame
samples.AtTS=400,500◦CwefoundmultipletstructuresontheTiL3,2edges,which
indicateformationofinterfacialTiO[112].Thesestructuresalmostvanishat
TS=600◦Candarenottraceableat2TS=700◦Canymore.Thespectralshapes
oftheXMCDsignalsonCoandTidonotchangeontheotherhand,onlythe
amplitudeisreducedatlowerdepositiontemperature.Thelargeroughnessof
thewithfilmstheprdepositedotectiveatMgOthelowlayerer.ThetemperaturCTSescompoundleadstoisanthusincompleteoxidizedincoveringair,
whichisparticularlyobservedassurfacialTiO2,whichisnotmagnetic.
Invacuumpost-annealedsampleshavebeenadditionallyinvestigatedfor
theirinterfacialchemistry.Annealingattemperaturesabove350◦Cresulted
ininterfaceformationtotheofMgOinterfacialsubstrate.TiO2.BecauseNaturallyof,thethishighwillgralsoowthhappentemperaturatthees,lowerwe
forcantheexpectlowanaverageoxidethicknessmagnetizationofseveralmeasuredinnanometers.theSQUID.ThisAneffectoxidizedmayaccountbottom
layerof3nmthicknesscanaccountforthedeviationfromthenearly2µB/f.u.
measuredinthebulkandpredictedtheoretically.
Usingtheresultsfromthissystematicanalysiswechosetwosamplesfor
in-detailinvestigationsdescribedinthenextsection.

6.2.5Elementspecificmagnetization
HighlyresolvedXMCDandXMLDspectraweretakenatBL4.0.2at20Kfor
thesamplesdepositedat400◦Cand700◦C,respectively(seeFig.6.4and6.5).
WhereastheXAScspectrashowsignificantdifferencesforthetwodeposition
temperaturesforCoandTi,theshapeoftheXMCDspectradoesnotdependon
thedepositionconditions.ForCothedepositionathighertemperatureresults

76

esultsrExperimental6.2

cFigursamplese6.4:depositedNormalizedonMgOXASsingleandcrystalsXMCDatspectra400◦CofandTi700and◦C,Cormeasurespectivelyed.at20Kfor

rinaesonancemoreprandaonouncedshoulderfineaboutstr4uctureVe,aboveconsistingtherofesonance.adoubleThesestrpeakucturatesthearLe3
alsoreflectedintheL2resonance,butlesspronounced.Klaeretal.investigated
[Co1132].TiSnTheybulkalsosamplesobserved(inasitudoublefracturpeakedstrinucturUHVeatfortheL3XMCDresonance,investigation)butless
◦prpeakstronounceducturecomparattheedL2toedgeourwassamplenotfounddepositedinatthese700C.bulkMorsamples.eover,Ytheamasakidouble
etXMCDal.[114]haveinvestigation),alsobutinvestigatedincontrastbulktothesamplesresults(insitubyKlaerscrapedetal.inandvacuumustheyfor
observedthreeseparatedpeaksattheL3edgeandonlyonebroadpeakatthe
L2resonance.Obviously,theirsampleshadadifferentelectronicstructure.
OurCoXMCDspectraalsoshowthedoublepeakstructureattheL3edge,
whileXMCDatthespectraL2areedgesharperonlyathanshoulderthoseisgivenvisible.byKlaerAgain,etal.theandstrYucturamasakiesinetoural..
YOuramasakiTiXMCDetal.dospectranotprshownovideindataFig.on6.4bthearTeisimilarL-edges.totheHoweverdata,bytheKlaershapeetal.is;

77

6ElectronicstructureoffullyepitaxialCo2TiSnthinfilms

verydifferentcomparedtodatacollectedbyScherzetal.[115]onthesystem
Fe/Ti/Fe(110).ThereforetherelativealignmentoftheCoandTimagnetic
momentsisnotobviousfromacomparisonwiththeirreferencedata.
Inordertogetfurtherinsightintotheelementspecificmagneticproperties,
weappliedtheXMCDsumrules(Chapter2.4.2).Theresultsofthesum-rule
analysisfortheCoXMCDspectraaresummarizedinTab.6.2.
TheCospinmomentiscloseto1µBforadepositiontemperatureof700◦C.
Forthedepositionat400◦CtheCospinmomentisafactoroftwosmaller,
buttemperaturtheorbitales;thetospinorbitalmomentmomentratioisisparallelnearlytotheidenticalspinformoment.bothBothdepositionthe
spinandorbitalmomentsareinverygoodagreementwiththetheoretical
results.Thenumberofd-holesislowerthanforpureCometal(1.75and1.5for
Co2TiSndepositedat700◦Cand400◦C,respectively,and2.4forpureCo[61]),
whichindicatesaratherlargechargetransfertotheCodstatesinCo2TiSn.It
isactuallyevenabitlowerthanthetheoreticalvalueof2.06.
WhilethesumrulesworkwellforCo,core-hole-photoelectroninterac-
tionanddynamicalscreeningeffectsofthex-rayfieldprohibittheirdirect
applicationtotheearly3dtransitionmetals[84].Theinteractionleadstoan
intermixingoftheL3andL2resonances,whichisthereasonforthedeviation
fromthestatisticalbranchingratioof2:1forthetwoedges.Theintermixing,
alsowhentheknownsumasrulesjj-mixingareofappliedthe2pto1/2theandearly2p3/3d2levels,transitionleadstometals.wrItonghasresultsbeen
suggestedbyScherzthatonecanestimatetheTispinmomentbymultiplying
theresultfromthesumruleanalysisbyafactorof4[61].Thisresulthasbeen
obtainedontheFe/Ti/Fe(110)trilayersystem.Ontheotherhand,itmust
beexpectedthatthiscorrectionfactoritselfdependsontheactualelectronic
structureandthusthescreeningstrength.Thedirectresultfromthesumrule

Ttheable6.2:samplesResultsdepositedoftheatsum400r◦Culeandanal700ysis◦C,ofrtheCoespectivelyXMCD.spectrameasuredat20Kfor

78

TSmspinmorbmorb/mspinNh
400◦C0.48µB0.025µB5.2%1.50
700◦C0.98µB0.055µB5.6%1.75

esultsrExperimental6.2

Figure6.5:NormalizedXASlandXMLDspectraofComeasuredat20Kinthe[100](a
andb)and[110](candd)directionsforsamplesdepositedonMgOsinglecrystalsat
400◦Cand700◦C,respectively.

◦inanalysisgoodisagrmspieementn=−with0.038theµBtheorfortheeticalrsampleesult.Indepositedparticularat,an700C,anti-parallelwhichis
alignmentwiththeCospinmomentisfound.Itisworthtomention,thattheTi
orbitalmoment(theapparentvalueismorb=0.022µB)isalignedanti-parallel
totheTispinmoment.ThelatterisinaccordancewithHund’srules,which
expectananti-parallelalignmentofthespinandorbitalmoment,becausethe
Tthei3dXMCDshellisdatalesscanthannothalfbequantfilled.ifiedBecauseforTof=the400◦C.formationHoweverof,allinterfacialqualitativeTiO2
SconclusionswithrespecttothealignmentoftheCoandTiorbitalandspin
ofthemomentsCoarandeprTieservedXMCDforspectralowerdonotdepositiondependontemperaturTS.Ines,summarybecause,thetheXMCDshapes
resultsareinverygoodagreementwiththeoreticalexpectations.
Ingeneralitisexpected,thattheXMLDsignalisproportionaltothesquare
ofthetotalmagneticmomentoftheindividualatoms(XMLD=βl∙mtotal2),
whereastheXMCDsignalshouldbedirectlyproportionaltothemagnetic
moment(normalized(XMCDtothe=βc∙post-edgemtotal)[jump62].heightComparingη,becausethetheXMCDandnumberXMLDof3d-holessignals

79

6ElectronicstructureoffullyepitaxialCo2TiSnthinfilms

squarFigureeof6.6:theNormalizednormalizedCoXMCDXMLDsignal.signalThefordatathepo[110]intsdircorrectionespondasatomeasurfunctionofementsthe
takenat20K,300Kand370K.ThesamplewasdepositedonMgOsinglecrystalsat
C.◦700

Nhisdifferentforthesamplesdepositedat400◦Cand700◦C,respectively)for
Co,itisinterestingtonotethatXMLD/XMCD2isabout65%largerforthe
sampledepositedat400◦Cthanforthe700◦Csample.Inthesimplepicture
thattheproportionalityfactorsβcandβlarethesameforbothdeposition
temperatures,thismeansthatinthedisordered400◦CsamplesomeoftheCo
atomsareanti-ferromagneticallycoupledtotheotherCoatoms.Ontheother
handitisknown,thattheXMLDeffectcanbecomequitelargeinsystems
withlocalizedelectrons.ThemagnitudeoftheXMLDisgivenessentially
bythemagneticmomentandthe2plevelexchangesplitting,whichitself
isproportionaltothemagneticmoment.Actually,withouttheexchange
splittingofthe2plevels,theXMLDwouldvanish.Localized3delectronstates
increasethe2p-3dexchangeinteraction,givingrisetoanenhancedXMLD[63].
Therefore,thedecreaseofXMLD/XMCD2withthedepositiontemperature
could◦alsohinttoahigherdegreeoflocalizationoftheComomentsforthe
400Csample.Thisisinagreementwithanoxidizedsurface,inwhichthe
electronsshouldbemorelocalized.However,thefinestructureattheCo-L
edgesbecomesmorepronouncedforhigherdepositiontemperature(seeFig.
6.4a,6.5aand6.5c)whichmightindicateahigherdegreeoflocalizationfor
higherdepositiontemperatures.TheelectronlocalizationwouldgivetheCoa
moreatomiccharacter,andatomicmultipletswouldbecomeimportant,giving
risetoafinestructureonthex-rayabsorptionspectrum.Ontheotherhand,
thiswouldcontradicttheXMLDresult.ThemaximumamplitudeoftheXMLD

80

6.3Electronicstructure

forTS=700◦Cis5.7%attheCoL3edgeinthe[110]direction.Thus,theCo
3dstatestakeanintermediatepositionbetweentheelementalferromagnets
Co(Ga,andMn)AsFe,,thatwhichhavehasaroundabout212%,%[and63].stronglyObviously,tlocalizedhisdiscrsystemsepancylikeneedsMninto
beinvestigatedbydirectabinitiocalculationsoftheabsorptionspectra,which
willbediscussedinSec.6.3.
Forthesampledepositedat700◦CtheXMCDandXMLDeffectwasstudied
alsohaveattheelevatedsameshapetemperaturat20K,es.300TheKandnormalized370K.XMCDFurthermorsignalse,theofTCoiandXMCDTi
andasymmetry370K.TherchangeseforethebytheratiosamebetweenfactortheasTiathendCoComagneticasymmetrymomentsbetweenis20Knot
significantlychangedatelevatedtemperatures.Thetemperaturedependence
of6.6,thetheXMLDXMLDsignalsignalwasscalesmeasurwelledfowithrtheXMCD[110]2,dirwhichection.wasAsalsoshownfoundinFig.for
othermaterialslike(Ga,Mn)As[63]inaccordancewiththeabovementioned
expectation.

structureElectronic6.3Asdiscussedabove,thefinestructureobservedattheCoL3,2edgescanhave
itsorigininatomicmultipleteffectsrelatedtoelectronlocalizationorsimply
intheparticular(itinerant)electronicstructureofCo2TiSn.Theexperimental
XASandXMCDspectraarecomparedtocalculationswithSPRKKRandFEFF9
6.7.Fig.inTheSPRKKRspectrashowbroadedgesandsomeweakshouldersonthe
highenergysideofthewhitelines.Further,theratiooftheL3andL2XMCD
signalsisincorrect,theL3XMCDistoosmall.
Bekenovetal.havecalculatedtheXAS/XMCDspectraofCTSabinitio
usingthespinpolarizedrelativisticlinear-muffin-tin-orbital(SPRLMTO)
method.[116]Theirsimulationsdonotreproducethedouble-peakstructures
andarerathersimilartoourSPRKKRspectra.
InFEFF9,theSPRKKRspectrumcanbeprincipallyreproducedwhenthe
groundstatedensityisused.Instead,ifthedensityinthepresenceofa
screenedcoreholeiscalculated,wefindastructurethatisverysimilartothe
experimentalspectrum.Becausetheself-consistencyalgorithmofFEFF9is
onlyaccuratewithin1eVinitsdeterminationoftheFermienergy,onecan
useasmallenergyshiftforfitting,therebymovingEFwithinthedensityof

81

6ElectronicstructureoffullyepitaxialCo2TiSnthinfilms

Figure6.7:ComparisonofthecalculatedCoL3,2XASandXMCDspectracarriedout
inFEFF9andSPRKKRtoexperimentalspectra.TheXMCDsignalshavebeenscaled
to90%toaccountfortheexperimentalpolarizationdegree.Theexperimentalandthe
FEFF9spectraarescaledto1inthepost-edgeregion.TheSPRKKRspectraarescaled
tomatchtheexperimentalL3resonance.Thetheoreticalspectraarealignedinenergy
um.spectrexperimentalthewith

states(DOS).Withashiftof-0.2eVweobtainedthespectrumshowninFig.
6.7.Obviously,boththedouble-peakstructureofthewhitelineaswellasthe
peaksmallstructurshouldereof4eVtheaboveXMCDthesignalwhiteiswelllinerareprereproduced.oduced.NotablyAlso,nottheonlydouble-the
shapeofthespectrumisbasicallycorrect,butalsotheintensitiesmatchthe
experimentaldataverywell.However,thedouble-peaksplittingoftheL3line
iscalculatedas1.3eV,comparedtoameasuredsplittingof1.5eV.
SinceFEFF9isbasedonthelocaldensityapproximationwithintheden-
sityfunctionaltheory—andthusreliesonsingle-particletheory—itdoesnot
accountforatomicmultipleteffects,whichnaturallyaremany-bodyeffects
arisingfromwave-functioncoupling.Consequently,weconcludethatthefea-
turesobservedinourexperimentalspectradonotarisefrommultipleteffects
andelectronlocalization.Instead,theyarefeaturesarisingfromtheexcited
statebandstructureduetothepresenceofacore-hole.Thisisconsistentwith

82

6.3Electronicstructure

Figure6.8:ComparisonofthecalculatedCositeprojecteddDOSfromSPRKKR(shaded
bluearea)andFEFF9inthegroundstate(solidredline)andwithanL3corehole
line).black(dotted

theXMLDmeasurementsdiscussedabove,whichindicateratheritinerant
moments.OurconclusionisfurthersupportedbytheanalysisgivenbyKlaeretal.,
whofoundthattheobservedstructurescannotbeexplainedbycharge-transfer
multiplettheory[113].Theystatethatthesplittingarisesfromanearlypure
CoegstateaboveEFgivingrisetothefirstpeak,andfromaCo-Tihybridstate
oft2gcharacter,whichresultsinthesecondpeak.Sincethet2gstateshavemore
itinerantcharacter,thecoreholeismorescreenedbythesurroundingatoms,
whiletheegstatesaresignificantlyloweredinenergy.Thiscore-holecorrelation
energyΔECwasassumedtobe0.5eVandconfirmedbyameasurementon
Co2TiSi.Neglectingthe(onlyweak)energydependenceofthetransition
matrixelements,andusingthiscore-holecorrelationenergyandspectral
deconvolution,theyfinallyfoundthattheFermilevelofCo2TiSnisattheedge
oftheminorityvalenceband,i.e.,Co2TiSnwouldbeonthevergeofbeinga
half-metal.Withthesamemethod,theyfoundthatCo2MnSihashalf-metallic
characterfortheunoccupieddensityofstates.
UsingtheFEFF9calculations,wecaninvertthisprocedure.Fromabinitio
calculationswefoundtheFermienergybyfittingtheexperimentalspectrum.
NowwecanusethesameFermienergyandinvestigatethegroundstateDOS
calculatedbyFEFF9.TheCositeprojecteddDOSareshowntogetherwiththe

83

6ElectronicstructureoffullyepitaxialCo2TiSnthinfilms

SPRKKRcalculationinFig.6.8.First,weshallnotethatthegroundstateDOS
fromFEFF9andtheSPRKKRcalculationproduceprincipallythesamefeatures,
butFEFF9underestimatesthesplittingbetweenthebondingandtheanti-
bondingstates.Thisisbecauseofthesphericalpotentialapproximationand
theuseofthevonBarth-Hedinexchangecorrelationpotential.Theunoccupied
DOSarehoweveringoodagreement.Becauseoffiniteclustersizeeffects,the
DOSfromFEFF9isbroadened.Theminoritystatesgapcanbeidentifiedjust
belowthecalculatedFermilevel.WhencomparingtheDOSinpresenceofthe
coreholetothegroundstate,wefindthatthecurveismainlyshiftedtolower
energiesbyΔEC≈0.3eV.IntheunoccupiedDOS,thisisbestseenforthe
peakminorityat1.4egeVrpeak,emainswhichshiftsessentiallybelowunalthetered.calculatedThatisinEFr.Instead,emarkabletheagrCo-Teementit2g
withtheproceduregivenbyKlaeretal..WhenthesameFermilevelisapplied
tothegroundstatedensityastotheexcitedstatedensity,wecanconclude
fromourdatathatCo2TiSnhashalf-metalliccharacterwithEFrightbelowthe
minorityvalenceband(seedottedenergylevelinFig.6.8).
Finally,weshalldiscussthelimitationsofourmodel.Asmentionedabove,
theabinitiocalculationunderestimatesthedouble-peaksplittingoftheXASby
about0.2eV.ThisintroducesanuncertaintyintheFermienergydetermination
byspectralfittingoftheorderofthecorrectionitself.Withthecurrentlyavail-
ablelevelofabinitiotheorythisissuecannotberesolvedanditremainsunclear
ifCo2TiSnisahalf-metallicferrimagnet.Atleast,afullpotentialtreatment
wouldbedesirable,andspinself-consistencywithmoreadvancedexchange
correlationfunctionalsmayhelptoresolveproblemswiththeexchangesplit-
ting.Ontheotherhand,theSPRKKRcalculationfindsthet2gpeakatslightly
lowerenergythanFEFF9.Thusitispossiblethatamoreaccuratecalculation
oftheXASrequiresapproachesgoingbeyondDFT.

84

7Ferrimagnetismanddisorderofepitaxial
Mn2−xCoxVAlHeuslercompoundthin
films

ductionIntro7.1

Inthischapter,weattempttotestGalanakis’predictionofafullmagnetic
compensationintheMn2−xCoxVAl(MCVA)system[40].Formanypractical
applicationsitisnecessarytopreparehighqualitythinfilmsofthemagnetic
materials.Thereforeonehastofindsuitabledepositiontechniquesandopti-
mizetheparameters.TheparentcompoundsMn2VAlandCo2VAl[117,118]
havebeensuccessfullysynthesizedinthebulkandepitaxialgrowthofMn2VAl
filmswithL21orderingonMgO(001)singlecrystalswasalsodemonstrated
[119,120].Experimentalresultsonthestructuralandmagneticpropertiesof
epitaxialMn2−xCoxVAlthinfilmsarepresentedhere.
Disorderisamajorconcernwhendealingwithhalf-metallicHeuslercom-
pounds.ThepresenceofdisorderhasbeenrepeatedlydemonstratedforCo2-
basedHeuslerbulkandthinfilms(see,e.g.,[121,122,123,124]).Theoretical
studieshaveinvestigatedtheimpactofdisorderonthemagneticproperties
andthehalf-metallicityofthecompounds[39,125,126,127,128,129].For
somecompoundsmajorimpactofdisorderonthehalf-metallicityisobserved,
whichalsodependsonthetypeofdisorder.Particularly,Picozzietal.[125]
foundthataMnatomsubstitutingaCoatominCo2MnSi,whichhasMnas
nearestneighbors,wouldcoupleantiparalleltothesurroundingMnatoms,
andthusreducethetotalmagnetizationdrastically.Thestrongdependenceof
themagneticmomentofMnonitschemicalandmagneticenvironmenthas
beendemonstratedby,e.g.,Raderetal.[130].Hence,disorderbringingMn
intonearest-neighborpositionshastobecontrolled.

85

7FerrimagnetismanddisorderofepitaxialMn2−xCoxVAl

sdMetho7.2

detailserimentalExp7.2.1

ThesamplesweredepositedwiththeBESTECsputtersystem(Chapter2.1).
teringElementalpressurtaregetswasofsetMn,to2Co,∙10V,−3andmbarAl.ofThe99.95corr%ectpuritysputterwerepowerused.ratiosThewersput-e
setupusingacombinedx-rayreflectivityandx-rayfluorescencetechnique.
Allsamplesusedinthisstudyhadthefollowingstacksequence:MgO
x(001)=0/single0.5/0.9crystal/1.0//Mn1.12−/x1.5Co/xV2.AlThe18nmupper/MgMgO0.5wasnmde/MgOposited1.5bynme-beamwith
temperaturevaporation.esDifrevealedfractionthatmeasurasubstrateementsoncarrierMn2VAltemperaturfilmseofdepositedatleastat600various◦C
wasnecessarytoobtaingoodorder,buttemperaturesabove700◦Cleadto
strongMnsublimation,whichcannotbereliablycompensatedbyhigher
sputteringpower(comparewith[119]).Thereforeallsamplesdiscussedinthis
paperweredepositedatacarriertemperatureof700◦C.TheprotectiveMg/
MgObilayerwasdepositedaftercoolingthesamplestopreventoxidationand
fusion.difinterX-raydiffraction(XRD),reflectometry(XRR),andfluorescence(XRF)were
performedinthePhilipsX’PertProMPDdiffractometerwithBragg-Brentano
andcollimatorpointfocusoptics,theopenEulercradleandtheAmptek
fluorescencedetectorinaHeenclosure.
X-raymagneticcirculardichroism(XMCD)wasmeasuredatbeamline6.3.1
of1.6TtheparallelAdvancedtotheLightincomingSourcex-ray(Berkeleybeam,CA,wasUSA).applied,Athemagneticsamplefieldsurfacesof±
wereinclinedby30◦withrespecttotheincomingbeam.Elementspecific
Themagneticmagnetichysterfieldesiswasloopsswitchedwerefortakeneverywitheneragymagneticpointtofieldobtainofupthetodi±chr2oicT.
wersignal.etakenDataatwerleastetakentwice,at20withK,cir150cularK,200K,polarizationand300degrK.AlleesofXMCD+60%spectraand-
60%,respectively.Systematicmeasurementswereperformedinthesurface
sensitivesubstratetotalwaselectrdetectedonyibyeldamode,photoanddiodethebehindvisiblethelightsamplefluor(seeescenceofChaptertheMgO2.4).
Thus,bulkinformationofthefilmscouldbeobtainedinx-raytransmission.

86

7.3Experimentalresultsanddiscussion

calculationsstructureElectronic7.2.2Electronicstructurecalculationsofdisorderedcompoundswereperformed
withtheMunichSPRKKRpackage,seeChapter3.2.2.Thegroundstateself-
consistentpotentialcalculationswereperformedon834kpointsintheirre-
duciblewedgeoftheBrillouinzone.Theexchange-correlationpotentialwas
approximatedwiththePerdew-Burke-Ernzerhofimplementationofthegener-
alizedgradientapproximation[69],theFermienergywasdeterminedusing
Lloyd’sformula[74,75].Theangularmomentumexpansionwastakenupto
lmax=3.Ascalarrelativisticrepresentationofthevalencestateswasused
inallcases,thusneglectingthespin-orbitcoupling.ForMn2VAltheatomic
spheresapproximationwasappliedandCo2VAlwastreatedwithfullpoten-
tialcalculations.Half-metallicgroundstateswereobtainedforMn2VAland
Co2VAlwiththeirrespectivebulklatticeparameters.Toaccountfordisorder,
thecoherentpotentialapproximation(CPA)wasused.Inourcalculationswith
theideallyorderedL21structure,Mn2VAlhasatotalmomentof2.01µB/f.u.,
with1.54µBonMnand-1.03µBonV.Co2VAlhasatotalmomentof1.99µB/f.u.,
with0.87µBonCoand0.28µBonV.Thesevaluesareingoodagreementwith
calculationspresentedbyotherauthors[131].

7.3Experimentalresultsanddiscussion

structureLattice7.3.1AllMCVAfilmswerefoundtobehighlyepitaxialwithMCVA[001]MgO
[001],rockingcurvewidthsof0.6◦to1.5◦,andanMCVA[100]MgO[110]in-
planerelation.Laueoscillationsobservedatthe(002)reflectionsdemonstrate
thelatticeandinterfacecoherenceofthefilmsinthetwolimitingcasesof
Mn2VAlandCo2VAl(Fig.7.3.1(a)).Forx=1,however,theoscillationsare
onounced.prlessFigure7.3.1(b)displaystheout-of-planelatticeparametercasafunctionof
x.AccordingtoVegard’slaw[132],alineardecreaseofthelatticeparameter
withincreasingxcanbeexpectedforasimplesubstitutionalmodel.However,
awewillsignificantseeindetaildeviationlaterfr,aomstrthisucturallawisandobservedmagneticatorx=der1.-disorThisderindicates,transition.as
˚hasForalsoMn2aVAl,slightlycisrslightlyeducedclowercomparthanedthetothebulkbulkvaluevalueofof5.8755.77A[A˚42[];117Co].2VThisAl
iscompatiblewithatetragonaldistortioncausedbytheepitaxialmatching

87

7FerrimagnetismanddisorderofepitaxialMn2−xCoxVAl

Figure7.1:(a):θ-2θscansofthe(002)reflectionsofMn2VAl(x=0),Mn1Co1VAl(x=1),
andCo2VAl(x=2).ClearLaueoscillationsarevisibleinbothcases.(b):out-of-plane
latticeparametercasfunctionofx.(c):OrderparametersSB2andSL21asfunctionsof
x.(d):Microstrainε[001]and(e):coherencelengthDandasfunctionsofx.Thedashed
linein(e)denotesthefilmthickness.

without-of-planethesubstrate:direction.theForlatticetheiscaseofexpandedCo2TiniSnthewehaveplanerandecentlyshrinksperformedinthe
firstdistortion.principlesInthiscalculationscaseitisofofthethechangeorderinoftotal25−ener50gymeVfor,andthisistypethusoflatticeeasily
activatedduringthefilmgrowth[104].Forthecompoundspresentedhere,we
expectasimilarenergyrange.
pliedTtoakamura’sobtaintheextendedorderorderparametersmodelSforB2andHeuslerSL21fromcompoundsthe[measur133]edwasXRDap-

88

7.3Experimentalresultsanddiscussion

(7.1)

peakintensities.Theorderparametersdescribetherelativeoccupationofthe
individualsublatticesofthestructurewiththe”right”and”wrong”atoms.
ThedegreeofB2order(i.e.,thedegreeoforderingbetweentheXandtheY/Z
asdefinedissublattices)randomS=nMn/CoonX-sites−nMn/CoonX-sites.(7.1)
B2nfullMn/CoorderonX-sites−nMn/CorandomonX-sites
Correspondingly,thedegreeofL21orderingisdefinedby
S=nVonY-sites−nVonrandomY-sites.(7.2)
L21nVfullonorYder-sites−nVonrandomY-sites
Therefore,SB2/L21is1ifthecompoundisfullyorderedandisreducedwithin-
creasingdisorder.SB2/L21=0meansrandomoccupationofthesublattices.The
orderparameterscanbeobtainedfromx-raydiffractionmeasurements,by
comparingtheexperimentallyobservedintensityratioswithcalculatedideal
values,seeChapter2.2.1andRef.[133]fordetails.UnlikeWebster’smodel
[51],Takamura’smodeltakesthedependenceofSL21onSB2intoaccount.
Thestructurefactorswereobtainedfromthemeasuredintensitiesbycor-
rectingfortheLorentz-Polarizationtermandthetemperaturefactorwithan
effectiveDebye-WallerfactorofBeff=0.4.SB2iscalculatedfromthefour
structurefactorratiosof(002)and(222)versus(022)and(004),respectively.
SL21iscalculatedastheaverageofthe(111)structurefactorversus(022)and
(004).Thefullatomicscatteringfactorsincludingangulardependenceand
anomalouscorrectionswereusedinthenumericalmodelcalculations.As
showninFig.7.3.1(c),theMn2VAlfilmsareorderedintheL21structurewith
significantV-Aldisorder(SL21≈0.4).WithincreasingCocontent,theL21
orderdisappearsinthealloysystem;Co2VAldoesnotshowanysignofL21
ordering.Ontheotherhand,thedegreeofB2orderincreasesslightlywith
increasingCocontent,fromSB2=0.7toSB2=0.8,i.e.,85%to90%oftheCo
atomsareonthe8csites.However,wenoteherethatdisorderbetweenCo,
Mn,andVcannotbeidentifiedwiththismethod,becausetheatomicform
factorsaretoosimilar.
AWilliamson-Hallanalysis(Chapter2.2.1)oftheintegralpeakwidthsof
the(002),(004),and(006)reflectionswasperformed.Theanalysisresultsare
displayedinFig.7.3.1(d)and(e).Themeasuredcoherencelengthmatchesthe
filmthicknessesquitewellwithintheaccuracyofthemeasuringandfitting
procedure.Acleartrendofincreasingstraincanbeobserved,from0.18%to

89

7FerrimagnetismanddisorderofepitaxialMn2−xCoxVAl

Figure7.2:ExperimentalXMCDspectraforV,Mn,andCoat20K.Thecorresponding
XASspectrawerenormalizedtoapost-edgejumpheightof1.Thespectraforx=
0.9,1.1aresimilartox=1andareomittedforclarity.

0.47%.ThelatticemismatchofCo2VAl(3.1%)isabout2.4timesaslargeasthe
mismatchofMn2VAl(1.3%)withMgO.Thesamefactorappliestothestrain
cohervalues,ence,whichtheverifiesdeviationthefrhighomVqualityegard’soflawtheandepitaxythe.Theratherlowerlowdegrstraineeinofspitefilm
ofthelargelatticemismatchindicateanincreaseddensityoflatticedefectsin
Mn1Co1VAl.Thedefectsallowforrelaxationofthefilm,whichcanreducethe
microstrainatalossofcoherence.
ZiebeckandWebsterfoundthatCo2VAlcrystallizesintheL21phase,but
exhibitssomeprefer◦entialV-Aldisorder[117].Thesamplesmeasuredbythem
atwerupetoannealed1200◦C,at800andCstillfor24h.exhibitedTheasamplescomplexbygrainKanomatastretuctural.ewereconsistingannealedof
L21andB2orderedfractions.Depositionat700◦Cmaythusbeinsufficientto
promoteL21orderinCo2VAl.However,asstatedinitially,ahigherdeposition
temperaturewasnotusablebecauseofMnsublimation.

7.3.2Magneticandelectronicstructure
WebeginwithadiscussionoftheXMCDspectraindependenceonx,which
areshowninFig.7.2.Forx=0,i.e.,forpureMn2VAl,wefindanantiparallel
alignmentoftheMnandVmoments,whichwasverifiedwithelementspecific

90

7.3Experimentalresultsanddiscussion

Figure7.3:NormalizedXMCDspectraofMnandVinelectronyieldandluminescence
detection.

hysteresisloops(notshown).Thisispreserveduptox=0.5,goingalongwith
anantiparallelcouplingofCotoMn.Here,wefindthepredictedferrimagnetic
orderwiththeCoandVmomentspointingoppositetotheMnmoments.
Withfurtherincreasingx,allmagneticmomentspointinthesamedirection;
thealloysbecomeferromagnets.Thistransitioniscloselyrelatedtochemical
disorderwhichisindicatedbythedeviationofthelatticeparameterfrom
Vegard’slaw.Acrossthestoichiometryseriestheshapeofthespectrachanges
significantly.Mostprominently,thesplittingoftheVandMnlinesvanishesat
x=0.9andabove.Theappearanceofthissplittingisdirectlycorrelatedwith
theappearanceofferrimagnetism.ThelineshapeoftheMnXMCDforx=1.5
isverysimilartotheMnlineshapeinCo2MnAlorCo2MnSi[134].Forthe
ferrimagneticcouplingofCoandMn,theyhavetobesecondnearestneighbors
onoctahedralpositions.CoandMnontetrahedralnearest-neighborpositions

91

7FerrimagnetismanddisorderofepitaxialMn2−xCoxVAl

coupleferromagnetically,asinCo2MnGe[51]andtheotherCo2Mn-based
compounds.HeuslerToassertthatthecomplexshapeoftheMnandVspectraisnotasurface
effect,wehavemeasuredthetransmittedx-rayintensityinluminescence
detectionatroomtemperatureforMn2VAl.TheXMCDspectraarealmost
equalintotalelectronyieldandintransmission(seeFigure7.3.2),although
inbothcasestheL3pre-peakismorepronouncedintransmission.However,
comparedtothetotalareaofthepeaks,thisdeviationissmall.Thefine
structureofthespectraisconsequentlyrelatedtotheelectronicstructureofthe
filmsratherthantoasurfaceeffect.
Usingthesumruleanalysis(Chapter2.4.2)weextractedthespinandorbital
magneticmomentsfromtheXMCDspectra.Table7.1summarizesthetotal
magneticmomentsobtainedfromsumruleanalysisandprovidesestimatesof
theCurietemperaturesobtainedfromtemperaturedependentXMCDforx=
0,1,2(thespectraarenotshownhere).Figure7.3.2displaystheelementspecific
totalmomentsindependenceonx.Becauseofcore-hole–photoelectron
interactions,thesumrulesfailfortheearly3dtransitionmetals[84].To
compensatetheresultingspectralmixingeffects,theapparentspinmagnetic
momentscanbemultipliedwithcorrectionfactorsassuggestedbyD¨urret
al.andScherzetal.,i.e.1.5forMn[135]and5forV[136].Actually,the
appliedcorrectionfactorsdependontheactualelectronicstructureandcan
notbesimplytransferredtodifferentsystems.However,weassumethatthis
influenceisrathersmall,sothatquantitativeresultscanbeobtained.
InMn2VAlwefindaloweredMnmoment(1µB)andanenhancedVmoment
(−1.1µB),resultinginatotalmagnetizationof0.88µB/f.u.Nochangeofthe

TmCtotMn2VAl0.88RT
Mn1.5Co0.5VAl0.1-
Mn1.0Co1.0VAl1.09≈350K
Mn0.5Co1.5VAl2.29-
Co2VAl1.66≈210K
Table7.1:Experimentaltotalmagneticmomentsat20K(giveninµB/f.u.)andesti-
matedCurietemperaturesderivedfromtemperature-dependentXMCD.

92

7.3Experimentalresultsanddiscussion

Figure7.4:Elementspecificmagneticmomentsasfunctionsofx.Ferrimagnetic(FiM)
orderisobservedforx≤0.5,ferromagnetic(FM)orderisobservedforx≥0.9.

magneticmomentswasobservedatRTascomparedto20K,hencetheCurie
temperatureismuchhigherthanRT.Thefilmisnotwelldescribedbyapure
L21ordermodel.Asdiscussedearlier,thefilmhassomedisorderbetween
Mnand(V,Al).Inthiscase,Mnatomsresideonsitessurroundedbyother
Mnatoms,whichcoupleantiferromagneticallyatshortdistance.Indeed,by
calculatingtheself-consistentpotentialinSPRKKRwith20%Mn-AlorMn-V
swap,wefindantiparallelcouplingoftheantisites,similartothefindings
byPicozzietal.forCo2MnSi[125].ForMn-Alswap,theMn(8a)momentis
reducedto1.22µBandtheMnontheAlsitehas−2.48µB.TheVmomentis
reducedto−0.83µB.Thisresultsinatotalmagnetizationof0.85µB/f.u.,and
theaverageMnmomentisconsequently0.85µB.InthecaseofMn-Vswap,
theMn(8a)momentremainsat1.58µBandtheMnontheVsitehas−2.63µB.
TheVmomentonthe4bsiteis−0.87µBand+0.84µBonthe8asite.Inthis
casethetotalmomentis1.78µB/f.u.,withanaverageMnmomentof1.16µB.
Further,thecaseofMn-Alswapisenergeticallypreferredwithrespecttothe
Mn-Vswap.SeeingthelowtotalandMnmomentsandthehighVmoment,
apreferentialMn-AlswapinMn2VAlisthusingoodagreementwiththe
structuralandthemagneticdata.Ourcalculationsshowthatthe20%Mn-Al
disorderandB2disorderbarelyinfluencethehalf-metallicgapofMn2VAl.For

93

7FerrimagnetismanddisorderofepitaxialMn2−xCoxVAl

B2disorder,thetotalmagneticmomentalsoremainsunaffected.Incontrast,
20%Mn-Vdisorderdestroythegap.ThisisincontrasttothefindingsbyLuo
etal.,obtainedwithasupercellapproachinapseudopotentialcode.Theystate
thatthegapispreservedunder25%Mn-Vdisorder[24].
Co2VAlhasareducedComoment(0.69µB)andaVmomentof0.28µB,
givingatotalmagnetizationof1.66µB/f.u.ThefilmhasB2order,whichis
expectedtoreducethemagnetizationfromthehighlyorderedL21case.We
findmagneticmomentsof0.75µBforCoand0.4µBforVinaB2ordered
SPRKKRcalculation,withatotalmomentof1.86µB/f.u.,ingoodagreement
withourmeasurements.SomeadditionaldisorderinvolvingCoandVcould
explainthefurtherreducedmoments.TheCurietemperatureisabout210K
(seeTable7.1),whichissignificantlylowerthanthevalueforbulksamples
(310K[117]).AcalculationoftheCurietemperaturewithSPRKKRwithin
themeanfieldapproximation(Chapter3.3)yields352KintheL21caseand
165KintheB2orderedcase.TheobservedsignificantreductionoftheCurie
temperatureinthedisorderedalloyisthusinagreementwiththeory.The
half-metallicgapofCo2VAlvanishesintheB2structure.
Atx=0.5,anearlycompletemagneticcompensationwithatotalmomentof
only0.1µB/f.u.isobserved.Remarkably,atx=1.5thetotalmagneticmoment
becomeslargerthan2µB/f.u.,causedbythehighMnmomentof1.67µB.This
isinagreementwiththedifferentMnlineshape:in,e.g.,Co2MnAl,inwhich
Mnhasasimilarlineshape,Mnhasamomentofabout3µB[51].Thus,the
mechanismmainlyresponsiblefortheferromagneticcouplingofallmoments
isthepreferentiallytetrahedral(insteadofoctahedral)coordinationofMn
Co.withatoms

94

8Itinerantandlocalmagneticmomentsin
ferrimagneticMn2CoGathinfilms
probedbyx-raymagneticlinear
dichroism:experimentandabinitio
rytheo

ductionIntro8.1

Inthischapter,weinvestigatethepropertiesofepitaxialthinfilmsofthe
inverseHeuslercompoundMn2CoGa,whichisinvestigatedtheoreticallyin
5.ChapterEpitaxialthinfilmsofMn2CoGawith(001)orientationwerepreparedwith
theBESTECsputtermachineonMgO(001)substrates.AMn50Ga50targetand
anelementalCotargetwereusedforthedeposition.TheresultingMn:Garatio
inthefilmswas2.2:1,asdeterminedbyx-rayfluorescence.Cowasaddedto
matchtheGacontent,i.e.,thestoichiometryoftheunitcellcanbewrittenas
Mn2.1Co0.95Ga0.95.
Amongthevariousheattreatmentstested,depositionat200◦Candinsitu
post-annealingat550◦Cwasfoundtoprovideoptimalfilmquality.Thelattice
parameterperpendiculartothesurfacewas5.81˚A,whichisslightysmaller
thanthebulkvalueof5.86A˚[28].Asmalltetragonaldistortionofthefilm
isinducedbythelatticemismatchwiththesubstrate,hencethelatticeis
expandedinthefilmplaneandcompressedperpendiculartotheplane.The
bulkmagnetizationmeasuredbyasuperconductingquantuminterference
device(SQUID)correspondsto1.95(5)µB/unitcell,whichisverycloseto
thebulkvalue.Nosignificantchangeofthemagnetizationbetween5Kand
roomtemperaturewasobserved,whichisconsistentwithaCurietemperature
K.600thanhigherX-rayabsorption(XAS)measurementswereperformedatBL4.0.2ofthe
AdvancedLightSourceinBerkeley,CA,USA.X-raymagneticcircular(XMCD)

95

8ItinerantandlocalmagneticmomentsinferrimagneticMn2CoGa

inandx-raylineardichrtransmissionoismthr(XMLD)oughthemeasurfilmbyementscollectingwerethetakenatvisibleroomandtemperaturultraviolete
lightsaturatedfluorwithescenceafrmagneticomthefieldsubstrateof0.6Twithandathecirphotodiode.cularor[57]linearThesamplepolarizationwas
degreewas90%and100%,respectively.
3.4.2).WeThecomputedexperimentaltheXAS,bulkXMCDlatticeandparameterXMLDwasusingchosentheforElkthecodecalculations;(Chapter
thesmalldistortionandoff-stoichiometryhavenegligibleinfluence.The
irrBrillouineduciblezonewedge,integrationthePerwaspdew-Burke-Ernzererformedonahof16×16functional×16[69k-point]wasmeshchoseninthefor
exchangeandcorrelation,andspin-orbitcouplingwasincludedinasecond-
spinvariationalmagneticscheme.momentAof2µBhalf-metallic/f.u.,andgrsiteound-stateresolvewasdspinobtained(orbital)withamomentstotal
(as−0.019follows:µB).CoA1.03detailedµB(0.046µdiscussionB),Mn(B)ofthe2.91eµlectrB(0.011onicµstrB),ucturandeisMn(C)given−in1.93Ref.µB
[28].

Results8.2

Theexperimentalx-rayabsorptionandcirculardichroismspectraareshown
inFig.8.1(a)and(b).Bothx-rayabsorptionspectrahavethetypicalshape
ofametallicsystemwithoutpronouncedmultiplets.However,theXMCD
spectrumofMnshowssomeuncommonfeatures(seearrowsinFig.8.1a).The
CoXASexhibitsfinestructuresattheL3andL2resonances.Thereisaweak
shoulderabout2.6eVabovethresholdandamorepronouncedoneat5eV
abovethreshold.TheCoXMCDspectrumreflectstheshoulderintheXAS.The
Coand(effective)Mnmomentsareparallel.Allthesefeaturesarereproduced
bytheabinitiocalculations(Fig.8.1(c)and(d)),whicharebroadenedwitha
Lorentzianof0.3eVwidthtoaccountforlifetimeeffects.Wecanthusidentify
thefeaturesinthespectraasbandstructureeffects.The5eVfeatureinthe
CoXASresultsfromtransitionsintoans-dhybridizedstateofCoandGa.
ItiscommonlyobservedforCoinCo2YZtypeHeuslercompounds,butits
positiondependsontheZelement.Theasymmetriclineshapeandthebroad
tailsoftheresonancesareaconsequenceof2p-3de-ecorrelation[137],which
isneglectedinoursimulations.Electron-holecorrelationscansignificantly
altertheshapeoftheXASorXMCDspectraof3dtransitionelements,even
inametallicenvironment(seeChapter6).Thus,thegoodagreementofour

96

Results8.2

Figure8.1:Top:experimentalXASandXMCDspectraof(a):Mnand(b):Coin
Mn2CoGa.Middle:theoreticalXASandXMCDspectraofMn2CoGa.(c):MnXASand
XMCD.(d):CoXASandXMCD.Bottom:decompositionoftheMnXAS(e)andXMCD
(f)forthetwoinequivalentMnsites.Thetheoreticalspectraarenormalizedto1about
40eVabovetheL3edgeandareshiftedtomatchtheexperimentalabsorptiononsetat
.L3

calculationswiththeexperimentalspectraindicatesaneffectivescreeningof
e-hole.cor2thepInFig.8.1(e)and(f)weshowthedecompositionofthecalculatedXASand
Mn(B)XMCDandintotheMn(C)Mn(B)areandslightlyMn(C)shiftedcomponents.(about0.15WeeV)findagainstthattheeachcoreotherlevels.Theof
shapesbranchingoftheratioisspectraassignificantlywellaslarthegerthanbranchingtheoneofratiosarMn(C).edifTheferent,thedecompositionMn(B)
oftheXMCDspectrumshowstwodifferentsignalswithoppositesigns.The

97

8ItinerantandlocalmagneticmomentsinferrimagneticMn2CoGa

antiparallelMn(C)contributionisresponsibleforthefeaturesmarkedinthe
experimentalspectrum.Thesefeaturesarelesspronouncedintheexperimental
spectrum,whichindicatesasmallercore-levelshiftthanthecalculatedone.
Asumruleanalysiswasperformedtoobtainthespinandorbitalmagnetic
momentsfromtheXMCDdata(Chapter2.4.2).Theresultingmagneticmoment
ratiosare:mMnspin/mCospin=0.48,morbMn/mMnspin=−0.013,morbCo/mCospin=0.055.
Usingthebulkmagnetizationwederivetheelementspecificmoments.The
averageMnspinmomentis0.47µBperatomandtheCospinmomentis
0.98µBperatom.TheaverageorbitalmomentofMnis-0.006µBperatom,
beingantiparalleltothespinmagneticmoment.ForCowefind0.055µBfor
theorbitalmoment.InthisanalysistheapparentMnspinmomenthasbeen
multipliedby1.5tocompensatethe2p1/2-2p3/2channelmixing,assuggested
byD¨urretal[135].Thesevaluesmatchthetheoreticalvalueswithintheerrors.
BoththepositiveCoorbitalmomentaswellasthesmallnegativeMnorbital
momentareinagreementwiththecalculation.Theorbitalmomentsofall
atomsareparalleltotherespectivespinmoments,buttheorbitalmomentof
Mn(C)islargerthantheoneofMn(B),resultingintheeffectivelyantiparallel
alignment.Thesinglecrystallinecharacterofepitaxialfilmsallowstomakeuseofthe
anisotropicx-raymagneticlineardichroism,whichisasensitiveprobeofthe
localcrystalfield.Bycomparisonwithreferencesystem,XMLDprovides
informationonthelocalityofmagneticmoments,seeChapter2.4.3fordetails.
ItwasshownthattheMnmomenthasalocalcharacterintheHeusler
compoundsCo2MnSi(CMS)andCo2MnAl(CMA).[134]K¨ubleretal.proposed
anexclusionofminoritydelectronsfromtheenvironmentofMn,givingrise
toalocalmomentcomposedofitinerantelectrons[8].Asimilarmechanism
cangiverisetoalocalMn(B)momentinMn2CoGa[28].Therefore,wechose
CMSasareferencesystemwithsimilarcrystalstructureforlocalmoments.
Mn2VGa(MVG),alsocrystallizingintheHeuslerstructure,ispostulatedtobe
itinerant,andischosenasareferencesystemforitinerantMnmoments.
Asimpletheoreticaltestforthe(non-)localityofspinmomentsisbasedon
non-collinearspinconfigurations.Weperformedself-consistentcalculations
fornon-collinearconfigurations(withoutspin-orbitcoupling)inwhichthe
magneticmomentofinterestwastiltedbyanangleϑoutofthecommon
magnetizationaxis.Onlythedirectionswerefixed,andthemagnitudeswere
determinedself-consistently.Alocalmomentwouldnotchangeinmagnitude
whentilted.InFig.8.2therelativechangesofthemagneticmomentsfor
Mn2CoGaandthereferencesystemsCMSandMVGareshown.InMn2CoGa,

98

Results8.2

Figurfigurations.e8.2:TheCalculatedspinrmomentelativeunderchangeoftheinvestigationmagneticistiltedoutmomentsoftheforcommonnon-collinearaxisbycon-ϑ.

Mn(B)hasaweakdependenceonϑ,whereasMn(C)andCochangesignifi-
cantlyontilting:Mn(B)haslocalcharacter,whereasMn(C)andCoarerather
itinerant.BoththeCoandtheMnmomentinCMShaveweakornodepen-
denceonthetiltangle,showingclearlythelocalityofbothmoments.MVG
incontrast,isanitinerantsystem;boththeMnandtheVmomentdepend
stronglyonϑ.Mn2CoGahasamorecomplexmagneticstructurethantherefer-
encecompounds,beingahybridbetweenitinerantandlocalmagnetism.Local
momentsystemscanbedescribedwithintheHeisenbergmodel.Thishasbeen
successfullyappliedtoexplaintheCurietemperaturesinCMSandrelated
compounds[80].ForMVG,thismodelunderestimatestheCurietemperature,
similartofccNi(Chapter3.3).Thiscanbeseenasexperimentalevidence
fortheitinerancyofMVG.Consequently,weexpectsignificantdeviationof
experimentalCurietemperaturesfromtheoreticalvaluesforMn2CoGa.
WehaveperformedXMLDmeasurementsforCoandMnalongthe[110]
directionofthefilm.InFig.8.3weshowtheexperimentalandtheoretical
spectraofMn2CoGaandthereferencecompounds.AllXMLDdataweretaken
atthesamebeamlineandaredirectlycomparableintermsofenergyresolution.
TheCoXMLDofMn2CoGaisverysimilarinshapetothesignalofCMS,all
finedetailsarereproduced.ThecomputedspectrumofMn2CoGaresembles
thegeneralshapeoftheexperimentaldata,althoughthenegativecontributions
areoverestimated.Theseareinthetailsoftheresonances,inwhiche-ecorre-
lationplaysarole,whichweneglectasstatedabove.Thelocalcrystalfields
areconsequentlysimilarinMn2CoGaandCMS,andtheabinitiocalculationis
abletodescribethesereasonablywell.

99

8ItinerantandlocalmagneticmomentsinferrimagneticMn2CoGa

CoFigur2eMnSi8.3:Left:(experimentalExperimentalspectrumandfromtheorRef.etical[134Co]).XMLDRight:spectraExperimentalofMn2(blackCoGasoliandd
Colines)2MnSiandtheor(experimentaletical(thinspectrlines)umfrMnomRef.XMLD[134]).spectraMn(B)ofMntype2VGa,spectraMnar2esolCoGaidranded,
height.Mn(C)Alltypetheorspectraeticalarespectradottedareblue.shiftedTheandXMLDisexpandednormalizedtomatchtothetheL3rexperimentalesonance
absorptiononsetatL3andtheL3,2spin-orbitsplitting.Theyarescaledtomatchthe
intensities.experimental

L3.ForAtLMn,2wehoweverfind,thattheytheareMn2somewhatCoGaanddiftheferent.CMSMn2signalsCoGaarehasanvirtuallyoverallequallessat
pronouncedstructureandlessintensityhere.TheMVGsignalismuchweaker
andhasanentirelydifferentshape,whichindicatesdifferentcrystalfields
actingonMnonaBorCposition.ThecomputedspectraofMn(B)inMn2CoGa
anddeviationforCMSisresemblobserved,etheparticularlyexperimentalfordataCMS.atL3Theverymainwell.peakAtLat2,LinsignificantCMS
2stemssurvivesfromtheabandfeatureinformationtheXASandthatcorrwasoboratesassignedthetolocalityanofatomicthemomentmultiplet,[134that].
InMn2CoGathisfeatureislesspronounced,leadingtoabetteragreement
ofexperimentandtheory.LesslocalityoftheMn(B)momentincomparison
toCMScanbeinferredfromthat.TheinfluenceoftheMn(C)spectrumin
Mn2CoGacannotbetracedintheexperimentaldata.ThecalculatedMn(C)
turn,spectragrumeesis,onlyhowever,modestlyverywithsimilartoexperiment.thecomputedBecauseofXMLDtheofsimilarityMVG.This,ofthein
similarcomputedshapespectra,asthewemeasurassumeedthatMVGthespectractualum.Mn(C)TheMn2contributionCoGaXMLDwouldis,havein

100

Results8.2

Figure8.4:XMLDvs.ms2forvariousMnandCocontaining(inverse)Heuslercom-
pounds:Co2MnSi(CMS),Co2MnAl(CMA),Co2TiSn(CTS),Mn2VGa(MVG),and
(MCG).CoGaMn2

conclusion,clearlydominatedbytheMn(B)signal.
NowweturntotheobservedintensitiesoftheXMLDsignals.Fig.8.4shows
acomparisonofthemaximumXMLDsignals(definedas(I||−I⊥)|max/[(I||+
I⊥)/2]|max)attheL3edgesversusthesquaredspinmagneticmomentsofCo
andMnforCMS,CMA,Co2TiSn(CTS),MVG,andMn2CoGa.TheCTSdata
weretakenfromChapter6.TheCoXMLDamplitudesareclosetoacommon
lineforCMS,CMA,andCTS.CMSisabitabovethough,indicatingastronger
localityoftheComomentinCMSthaninCMAorCTS.TheMn2CoGasignal
isaboutafactorof2.5smallerthanexpectedfromthereferences.Inagreement
withthelocalitytestdescribedabove,thisshowstheitinerancyoftheCo
momentinMn2CoGa.BecauseoftheantiparallelMnmoments,theMnXMLD
ofMn2CoGaisverystrongcomparedtotheMnspinmoment,anditisfaroff
thelinegivenbyCMSandCMA.
WiththelinearfitsthroughtheCMAandCMSpointsasaguideforlocalMn
momentsandthroughtheMVGpointforanitinerantsystemwecanpredict
theMnXMLDamplitudeofMn2CoGa.WetreattheMnXMLDofMn2CoGa
asasuperpositionofthespectrafromCMA/CMSandMVG.OurFLAPWcal-
culationgivesaMn(B)/Mn(C)spinmomentratioof−1.5.Withthisvalueand
themeasuredsummsMn(B)+msMn(C)≈0.94µBweobtainmsMn(B)=2.82µBand
msMn(C)=−1.88µB.Accordingtotheerrorsofthemagneticmomentsoftheref-
erencedata,weexpectanXMLDof(2.7±0.5)%forMn2CoGa.Themeasured
valueof1.53%isclearlybelowthisrange;theratiodetermineddirectlyfrom

101

8ItinerantandlocalmagneticmomentsinferrimagneticMn2CoGa

theXMLDis−1.7,whichleadstomsMn(B)=2.28µBandmsMn(C)=−1.34µB.
inThoughMnCoGathisisstillindicatesraeasonable,loweritdegrseemseeofmuchMn(B)morespinlikelymomentthatthelocalitylowerthanXMLDin
2CMS.However,theMn(B)momentisclearlynotpurelyitinerant.

102

9rksremaConcluding

AbGa,In,initioSi,Ge,computationsSn,P,As,forSb,theMnsuggest2TiZthatHeuslerthesecompoundcompoundsseriescanwithexhibitZ=ferri-Al,
performedmagnetisminwithaccortwodifdanceferent,withtherulestate-of-the-artm=NV−density24.Thesefunctionalcalculationstheorywermeth-e
andods:thethe(ffull-potentialull-potential)linearizedspin-polarizedaugmentedrelativisticplanewavesmethodKorringa-Kohn-Rostocker(FLAPW)
method(SPRKKR).Theresultsareingoodagreementwitheachother.Mostof
thecompoundshavelargespinpolarizationandaspin-upgapformsabove
theFermienergy.TheCurietemperaturescalculatedwithinthemean-field
approximationindicatethatthecompoundswith21and22valenceelectrons
willbeferrimagneticatroomtemperature.Athoroughunderstandingofthe
influenceoftheZcomponentonthepropertiesofthecompoundshasbeen
establishedonthebasisofabinitiobandstructureandexchangecoupling
calculations.ItwasfoundthatthepressuredependenceofTCispositive,in
agrandeementstablewitspinhferrpolarizationsomagenticandfulltheirHeuslerhighCuriecompounds.temperaturBecauseesofwetheirprlaroposege
inparticularMn2TiSi,Mn2TiGe,andMn2TiSnascandidatesforspintronic
applications.WehaveperformedabinitiobandstructurecalculationswiththeSPRKKR
methodontheMn2CoZinverseHeuslercompoundswiththeHg2CuTistruc-
ture.Theexchangeinteractionparametersobtainedfromthecalculationsare
foundtobegovernedbytheCo-Mn(C)exchange,whichisofdirectnature.In
thecaseofZ=Al,Ga,andIn,theMn(C)-Mn(C)interactionisthedominating
one,whichisdirectaswell.Theindirect,long-rangedinteractionsareexponen-
tiallydampedandthusweak,andtheintra-sublatticeinteractionsaremostly
antiferromagnetic.Curietemperaturescalculatedwithinthemean-fieldap-
proximationareinreasonableagreementwithexperimentaldataforMn2CoSn
theandtotalMn2CoSb.moment,ThewhichCurieisdiffertemperaturentfresomshowtheanfullHeusleranomalouscompounds.dependenceForon
Mnthough2CoAlthewetotalpredictmomentanoftheexceptionallycompoundhighisonlyCurie2µB/temperaturf.u.Theeof890dependenceK,al-

103

emarksrConcluding9

oftheexchangeparametersonthelatticeparameterinMn2CoGesuggests
anegativepressuredependenceofTCintheMn2CoZcompounds,which
originatesfromtheexchangeinteractionsofMn(C)-Mn(B)andCo-Mn(C).
WehavegrownthinfilmsoftheHeuslercompoundCo2TiSnbyDCmag-
netronco-sputtering.Structuralinvestigationsrevealedhighlyordered,fully
epitaxialgrowthofCo2TiSnthinfilmsonMgO(001)substratesatgrowthtem-
peraturesabove600◦C.Alowresidualresistivitysupportstheconclusionof
wellorderedfilms.Theresistivityhasapronouncedcusp-typeanomalyatTC.
Alargemagnetoresistancehasbeenobservedandcanbeexplainedinterms
ofspinfluctuations.FromtheXMCDmeasurementswefindatotalmagneti-
zationof1.98±0.05µB/f.u.,wheretheuncertaintyarisesfromtheunknown
systematicerrorintheestimateoftheTispinmoment;thereducedaverage
saturationmagnetizationofthebestfilm(TS=700◦C,m=1.6(1)µB/f.u.)
canbeeasilyexplainedbyanoxidizedbottominterfacelayerof3nmthickness.
Theresultsfortheelementspecificspinandorbitalmagneticmomentsarein
quantitativeagreementwithabinitiobandstructuretheory.Thefinestructures
observedfortheCoL3,2edgeswereexplainedbydirectcalculationsoftheXAS
usingFEFF9.Inclusionofthecore-holepotentialwasfoundtoreproducethe
splitwhitelines,assessingthemasbandstructureeffects.Formationofatomic
multipletscanberuledout,inagreementwithXMLDresults.However,dueto
shortcomingsofthetheoreticalmodeling,itremainsunclearwhetherCo2TiSn
isahalf-metallicferrimagnetornot.
EpitaxialthinfilmsofMn2−xCoxVAlhavebeensynthesizedonMgO(001)
substratesbyDCandRFmagnetronco-sputtering.Itwasintendedtoobserve
aferrimagneticcompensationofthemagnetizationatx=1.Thefilmshave
significantchemicaldisorder,dependingonthedegreeofMn-Cosubstitution.
Mn2VAlwasfoundtobeL21ordered,withapreferentialMn-Aldisorderand
additionalV-Aldisorder.TheMn-Aldisorderreducesthetotalmomentconsid-
erably,becausethenearest-neighborMnatomscoupleantiferromagneticallyin
thisconfiguration.Accordingly,themagnetizationofMn2VAlisverysensitive
todisorderinvolvingMn.However,thebandstructurecalculationssuggest
thatonlyMn-Vdisorderhasaninfluenceonthehalf-metallicgap.Because
ofthedisorder,anearlycompletemagneticcompensationwasobservedfor
Mn1.5Co0.5VAl.WithfurtherCosubstitution,theelectronicstructurechanges
considerably,andaparallelcouplingofCo,Mn,andVwasobserved.We
supposethatCoandMnbecomepreferentiallynearest-neighbors,whichleads
toaparallelcouplingoftheirmagneticmoments.TheCo2VAlfilms,beingthe
secondextremumofthesubstitutionalseries,hadB2order.Thebandstructure

104

calculationswithB2ordersuggestreducedmoments,buttheexperimentally
determinedmomentsareevenlower,whichindicatesadditionaldisorderin-
volvingCo.TheCurietemperaturewassignificantlyreduced,whichisin
agreementwiththetrendobservedinthemeanfieldcalculation.Itisinprinci-
plepossibletoobtainahighdegreeofL21orderinbulkCo2VAlbyappropriate
thermaltreatment,butourmaximumsubstratetemperaturewaslimitedby
Mnevaporation.Whileitmaybepossibletoobtainthecorrectoccupationfor
theferrimagneticcompensationinthebulk,itseemsnotpossibletoobtain
filmswithahighdegreeoforder.
WehavepreparedepitaxialfilmsoftheferrimagneticinverseHeuslercom-
poundMn2CoGabyco-sputteringandobtainedgoodfilmqualitybydeposi-
tionat200◦Candinsitupost-annealingat550◦C.Wefoundgoodagreement
oftheexperimentalL3,2x-rayabsorptionanddichroismspectrawithabinitio
calculationswithinindependentparticletheory.Thetotalandelementre-
solvedmagneticmomentsareclosetotheoreticalvalues.X-raymagneticlinear
dichroismspectraweretakentoprovideinformationonthelocalityoftheCo
andMnmoments.Non-collinearelectronicstructurecalculationsprovidedthe
footingfortheinterpretationoftheobservedXMLDamplitudes.Thelocality
oftheMn(B)momentisnotaspronouncedasinCo2MnSi,theCoandMn(C)
momentshaveclearlyitinerantcharacter.Becauseofthesefindings,weexpect
significantdeviationofexperimentalCurietemperaturesfromthepredicted
onesintheMn2CoZcompounds.
Inallexperimentalpartsofthiswork,densityfunctionaltheoryhasproven
tobeanindispensabletoolfortheinterpretationoftheobtaineddata.Di-
rectcomparisonofexperimentandabinitiotheoryprovidesamuchdeeper
understandingoftheunderlyingphysicsthanempiricalworkalone.

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stwledgemenAckno

MysupervisorsPROF.DR.GU¨NTERREISSandDR.JANSCHMALHORST
deservemyfirstanddeepestgratitude.Itwasthemwhoofferedmethe
opportunitytoworkwithanew,state-of-the-artthinfilmdepositiontool,and
itwasthemwhogavemethefreedomtobecomewhatIcalla“computationist”.
Iinamme.IthankfulthankforJANmanyforXMCDdiscussions,sumradvices,uleforanalysistheirandsupport,fortheandmanytheirgoodfaith
beamtimesattheAdvancedLightSource.
TRIUCamTUREobligedS-Dto2.myIncolleaguesparticular,frIomthankTHIDNRF.ILKMASRASTNEDNPRHOYTSITCSforOFhisNANmanyOS-
lessonsinvacuumtechnologyandforkeepingthelabequipmentalive,DR.
DAadviceNIELandEBKElessonsformanyonnoise,helpfulandAGdiscussions,GIWINPDETMEARNHNEDforWIGhelpingformetypographicmany
timesstudentswithHEtheNDRIKwheelsWUofLFburMEIER,eaucracyCH.RIISTOwouldPHKlikeLEWtoE,exprCHRessISTmyIANSthanksTERtoWERmyF,
andtothecolleaguesinmyoffice,forjusthavingagoodtimewiththemand
support.theirforIamdesministeriumindeptedf¨urfortheBildungfinancialundsupportForschungofmy(BMBF)workandbythetheDeutscheGermanBun-For-
(DFG).schungsgemeinschaftIthankfortheopportunitytoworkatBL6.3.1andBL4.0.2oftheAdvanced
LightSource,Berkeley,USA.Inparticular,Iamverythankfulforthemany
occasionstoworkanddiscusswithDR.ELKEARENHOLZ.
Further,IthankDR.TANJAGRAFandTIMBO¨HNERTforconductingthe
SQUIDmeasurementsonvarioussamples.
work.SpecialWithoutthankstheirgotefofortthetomakedevelopersmodernoftheDFTmethodscodesofIDFThaveaccessibleusedfortomyus
experimentalists,aworklikethiswouldnothavebeenpossible.Inparticular,
IthankwouldthemlikefortointrexpressoducingmymethankstotothethegrSPRKKRoupofPpackageROF.HandUBtoERTDFTEBEinRT,generalandI
withdeveloperstheiroftheworkshopElkincode,MunichDR.JinOHNJuneKAY2009.DEVWHeryURSTspecialandDthanksR.SAgoNGtoEETtheA
SHARMA.Ithankthemforbringingmeclosertoitsmathsandalgorithmswith
thehands-oncourseinLausanneinJuly2011.
JuliaAndBullikfinally,,whoIamtrsupportedulyindebtedmeduringandallthankfulthosetoyearsmyofgirlfriendmywork.andbestfriend

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