Ferrimagnetic Heusler compounds [Elektronische Ressource] : from first principles to thin films / Markus Meinert. Fakultät für Physik

Publié par

DoctoralThesisinPhysicsFerrimagnetic Heusler CompoundsFrom rst principles to thin lmsMarkusMeinertOctober 2011Bielefeld UniversityDepartment of PhysicsThis work was done by myself. Text and figures were partly taken fromcorresponding publications, which originate directly from this work.(Markus Meinert)Reviewers:Prof. Dr. Gunter¨ ReissProf. Dr. Jur¨ gen SchnackCopyrightc 2011 Markus MeinertBielefeld University, Department of PhysicsThin Films and Physics of NanostructuresTypeface Palladio and Pazo Math 10 pt.ASystem LT X 2 andKOMA-Script.#E”What I cannot compute, I do not understand.”- adapted from Richard P. FeynmanPublications included in this thesis Ab initio prediction of ferrimagnetism, exchange interactions andCurie temperatures in Mn TiZ Heusler compounds2M. MEINERT, J.-M. SCHMALHORST, AND G. REISSJ. Phys. Condens. Matter23, 063001 (2011) Exchange interactions and Curie temperatures of Mn CoZ compounds2M. MEINERT, J.-M. SCHMALHORST, AND G. REISSJ. Phys. Condens. Matter23, 116005 (2011) Electronic structure of fully epitaxial Co TiSn thin films2M. MEINERT, J. SCHMALHORST, H. WULFMEIER, G. REISS,E. ARENHOLZ, T. GRAF, AND C. FELSERPhysical Review B83, 064412 (2011) Ferrimagnetism and disorder of epitaxial Mn Co VAl2 x xHeusler compound thin filmsM. MEINERT, J.-M. SCHMALHORST, G. REISS, AND E. ARENHOLZJ. Phys. D: Appl. Phys.
Publié le : dimanche 1 janvier 2012
Lecture(s) : 65
Tags :
Source : D-NB.INFO/1018631674/34
Nombre de pages : 114
Voir plus Voir moins

Doctoral

Ferrimagnetic

romF

Thesis

in

Physics

Heusler

Comp

thintorinciplespfirst

MeinertMarkus

October

Bielefeld

2011

University

Department

of

films

Physics

ounds

Thiscorrworkespondingwasdonepublications,bymyself.whichTextoriginateanddirfigurectlyesfrweromethispartlywork.takenfrom

Meinert)(Markus

Reviewers:

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

Copyrightc2011MarkusMeinert
BielefeldUniversity,DepartmentofPhysics

ThinFilmsandPhysicsofNanostructures

TypefacePalladioandPazoMath10pt.
SystemLATEX2εandKOMA-Script.

”What

I

cannot

-

compute,

adapted

omfr

I

do

dRichar

.P

not

understand.”

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.

105

Bibliography

[1]F.Heusler,Verh.dt.phys.Ges.5,219(1903).
[2]A.J.BradleyandJ.W.Rodgers,Proc.R.Soc.57,115(1934).
[3]T.Graf,S.S.P.Parkin,andC.Felser,IEEETrans.Magn.47,367(2011).
[4]H.Itoh,T.Nakamichi,Y.Yamaguchi,andN.Kazama,Trans.JapanInst.Met.24,
(1983).265[5]Y.Nishino,M.Kato,S.Asano,K.Soda,M.Hayasaki,andU.Mizutani,Phys.Rev.
(1997).1909,79Lett.[6]H.Nakamura,Y.Kitaoka,K.Asayama,Y.Onuki,andT.Komatsubara,J.Magn.
Magn.Mater.76,467(1988).
[7]J.Winterlik,G.H.Fecher,C.Felser,M.Jourdan,K.Grube,F.Hardy,H.von
L¨ohneysen,K.L.Holman,andR.J.Cava,Phys.Rev.B78,184506(2008).
[8]J.K¨ubler,A.R.Williams,andC.B.Sommers,Phys.Rev.B28,1745(1983).
[9]W.E.PickettandJ.S.Moodera,PhysicsToday54,39(2001).
[10]Y.Yoshida,M.Kawakami,andT.Nakamichi,J.Phys.Soc.Japan50,2203(1981).
[11]C.Jiang,M.Venkatesan,andJ.M.D.Coey,SolidStateCommun.118,513(2001).
[12]S.Ishida,S.Asano,andJ.Ishida,J.Phys.Soc.Japan53,2718(1984).
[13]R.WehtandW.E.Pickett,Phys.Rev.60,13006(1999).
[14]K.Nakamura,T.Ito,andA.J.FreemanPhys.Rev.B72,064449(2005).
[15]I.Galanakis,K.¨Ozdo˜gan,E.¸Sa¸sio˜glu,andB.Akta¸s,Phys.Rev.B74,140408(2006)
[16]H.vanLeukenandR.A.deGrootPhys.Rev.Lett.74,1171(1995).
[17]W.E.Pickett,Phys.Rev.B57,10613(1998).
[18]H.AkaiandM.Ogura,Phys.Rev.Lett.97,026401(2006).
[19]E.¸Sas¸io˜glu,Phys.Rev.B79,100406(R)(2009).

106

Bibliography

[20]K.¨Ozdo˜gan,I.Galanakis,E.¸Sa¸siogluandB.Akta¸s,J.Phys.:Condens.Matter18
(2006).2905[21]S.Fujii,M.Okada,S.Ishida,andS.Asano,J.Phys.Soc.Jpn77,074702(2008).
[22]S.Wurmehl,H.C.Kandpal,G.H.Fecher,andC.Felser,J.Phys.:Condens.Matter
(2006).6171,18[23]I.Galanakis,P.H.Dederichs,andN.Papanikolaou,Phys.Rev.B66,174429(2002).
[24]H.Luo,Z.Zhu,L.Ma,S.Xu,X.Zhu,C.Jiang,H.Xu,andG.Wu,J.Phys.D:Appl.
(2008).055010,41Phys.[25]H.Luo,G.Liu,Z.Feng,Y.Li,L.Ma,G.Wu,X.Zhu,C.Jiang,andH.Xu,J.Magn.
Magn.Mater.321,4063(2009).
[26]H.Luo,Z.Zhu,G.Liu,S.Xu,G.Wu,H.Liu,J.Qu,andY.Li,J.Magn.Magn.
Mater.320,421(2008).
[27]H.Z.Luo,H.W.Zhang,Z.Y.Zhu,L.Ma,S.F.Xu,G.H.Wu,X.X.Zhu,C.B.
Jiang,andH.B.Xu,J.Appl.Phys.103,083908(2008).
[28]G.D.Liu,X.F.Dai,J.L.Chen,Y.X.Li,G.Xiao,andG.H.Wu,Phys.Rev.B77,
(2008).014424[29]N.Xing,H.Li,J.Dong,R.Long,andC.Zhang,ComputationalMaterialsScience
(2008).600,42[30]X.P.Wei,X.R.Hu,G.Y.Mao,S.B.Chu,T.Lei,L.B.Hu,andJ.B.Deng,J.Magn.
Magn.Mater.322,3204(2010).
[31]N.Lakshmi,R.K.Sharma,andK.Venugopalan,HyperfineInteractions160,227
(2005).[32]X.Dai,G.Liu,L.Chen,J.Chen,andG.Wu,SolidStateComm.140,533(2006).
[33]R.B.HelmholdtandK.H.J.Buschow,J.Less-CommonMet.128,167(1987).
[34]H.Luoetal.,J.Appl.Phys.105,103903(2009).
[35]S.T.Li,Z.Ren,X.H.Zhang,andC.M.Cao,PhysicaB:Cond.Matter404,1965
(2009).[36]J.Winterliketal.,Phys.Rev.B83,174448(2011).
[37]P.Klaeretal.,Appl.Phys.Lett.98,212510(2011).

107

Bibliography

[38]H.C.Kandpal,V.Ksenofontov,M.Wojcik,R.Seshadri,andC.Felser,J.Phys.D:
Appl.Phys.40,1507(2007)
[39]Y.Miura,M.Shirai,K.Nagao,J.Appl.Phys.99,08J112(2006)
[40]I.Galanakis,K.¨Ozdo˜gan,E.¸Sa¸sio˜glu,andB.Akta¸s,Phys.Rev.B75,092407
(2007).[41]M.Kawakami,Y.Yoshida,T.Nakamichi,S.Ishida,andH.Enokiya,J.Phys.Soc.
(1981).1041,50Jpn.[42]T.NakamichiandC.V.StagerJ.Magn.Magn.Mater.31,85(1983).
[43]E.¸Sa¸sio˜glu,L.M.Sandratskii,andP.BrunoJ.Phys.:Condens.Matter17,995
(2005).[44]L.Chioncel,E.Arrigoni,M.I.Katsnelson,andA.I.Lichtenstein,Phys.Rev.B79,
(2009).125123[45]H.Luo,Z.Zhu,L.Ma,S.Xu,X.Zhu,C.Jiang,H.Xu,andG.Wu,J.Phys.D:Appl.
(2008).055010,41Phys.[46]S.Swann,Phys.Technol19,67(1988).
[47]BESTEC,BERLIN(2008).
[48]B.D.CullityandS.R.Stock,ElementsofX-RayDiffraction,ThirdEdition,
PrenticeHall,UpperSaddleRiver2001.
[49]B.L.Henke,E.M.Gullikson,andJ.C.Davis.X-rayinteractions:photoabsorption,
scattering,transmission,andreflectionatE=50-30000eV,Z=1-92,AtomicData
andNuclearDataTablesVol.54(no.2),181-342(July1993).
[50]D.T.CromerandD.Liberman,ActaCryst.A37,267-268(1981).
[51]P.J.Webster,J.Phys.Chem.Solids32,1221(1971).
[52]G.K.WilliamsonandW.H.Hall,ActaMetall.1,22(1953).
[53]D.Attwood,SoftX-RaysandExtremeUltravioletRadiation,CambridgeUniver-
1999.Cambridgeess,Prsity[54]L.G.Parratt,Phys.Rev.95,359(1954).
[55]H.Ebel,X-RaySpectrom.28,255(1999).
[56]Y.U.Idzerda,C.T.Chen,H.J.Lin,G.Meigs,G.H.Ho,andC.C.Kao,Nucl.
Instrum.MethodsPhys.Res.A347,134(1994).

108

Bibliography

[57]M.Kallmayer,H.Schneider,G.Jakob,H.J.Elmers,B.Balke,andS.Cramm,J.
Phys.D:Appl.Phys.40,1552(2007).
[58]C.T.Chen,Y.U.Idzerda,H.J.Lin,N.V.Smith,G.Meigs,E.Chaban,G.H.Ho,E.
Pellegrin,andF.Sette,Phys.Rev.Lett.75,152(1995).
[59]A.Thomsonetal.:X-rayDataBooklet,LawrenceBerkeleyNationalLaboratory,
http://xdb.lbl.gov2009,[60]J.St¨ohr,J.Elect.Spect.Rel.Phen.75,253(1995).
[61]A.Scherz,PhDthesis,FreieUniversit¨atBerlin(2004).
[62]J.KunesˇandP.M.Oppeneer,Phys.Rev.B67,024431(2003)
[63]A.A.Freeman,K.W.Edmonds,G.vanderLaan,N.R.S.Farley,T.K.Johal,
E.Arenholz,R.P.Campion,C.T.Foxon,andB.L.Gallagher,Phys.Rev.B73,
(2006)233303[64]E.Arenholz,G.vanderLaan,A.McClure,andY.Idzerda,Phys.Rev.B82,
(2010).180405(R)[65]RichardM.Martin,ElectronicStructure,CambridgeUniversityPress,Cambridge
2004.[66]P.HohenbergandW.Kohn,Phys.Rev.136,B864(1964).
[67]W.KohnandL.J.Sham,Phys.Rev.140,A1133(1965).
[68]J.P.PerdewandY.Wang,Phys.Rev.B,45,13244(1992).
[69]J.P.Perdew,K.Burke,andM.Ernzerhof,Phys.Rev.Lett.77,3865(1996).
ge.netceforp://elk.sourhtt[70][71]G.K.H.Madsen,P.Blaha,K.Schwarz,E.Sj¨ostedt,andL.Nordstr¨omPhys.Rev.
(2001).195134,64B[72]H.Ebert,D.K¨odderitzsch,andJ.Minar,Rep.Prog.Phys.74,096501(2011).
[73]P.MavropoulosandN.Papanikolaou,TheKorringa-Kohn-Rostoker(KKR)Green
FunctionMethodI.ElectronicstructureofPeriodicSystems,inComputational
Nanoscience:DoItYourself!,J.Grotendorst,S.Bl¨ugel,D.Marx(Eds.),Johnvon
NeumannInstituteforComputing,J¨ulich,NICSeries,Vol.31,pp.131-158,2006.
[74]P.LloydandP.V.Smith,Adv.Phys.21,69(1972).
[75]R.Zeller,J.Phys.:Condens.Matter20,035220(2008).

109

Bibliography

[76]A.I.Liechtenstein,M.I.Katsnelson,V.P.Antropov,andV.A.Gubanov,J.Magn.
Magn.Mater.67,65(1987).
[77]P.W.Anderson,Theoryofmagneticexchangeinteractions:exchangeininsu-
latorsandsemiconductorsSolidStatePhysicsvol14,edFSeitzandDTurnbull
(NewYork:Academic),1963,pp99–214.
[78]S.Majumdar,M.K.Chattopadhyay,V.K.Sharma,K.J.S.Sokhey,S.B.Roy,and
P.Chaddah,Phys.Rev.B72,012417(2005).
[79]K.R.Kumar,N.H.Kumar,G.Markandeyulu,J.A.Chelvane,V.Neu,andP.D.
Babu,J.Magn.Magn.Mater.320,2737(2008).
[80]J.Thoene,S.Chadov,G.Fecher,C.Felser,andJ.K¨ubler,J.Phys.D:Appl.Phys.42,
(2009).084013[81]H.RathgenandM.I.Katsnelson,PhysicaScriptaT109,170(2004).
[82]J.J.Rehr,J.J.Kas,F.D.Vila,M.P.Prange,andK.Jorissen,Phys.Chem.Chem.
(2010).5503,12Phys.[83]J.SchwitallaandH.Ebert,Phys.Rev.Lett.80,4586(1998).
[84]A.L.Ankudinov,A.I.Nesvizhskii,andJ.J.Rehr,Phys.Rev.B67,115120(2003).
[85]H.Wende,A.Scherz,C.Sorg,K.Baberschke,E.K.U.Gross,H.Appel,K.Burke,
J.Minar,H.Ebert,A.L.Ankudinov,andJ.J.Rehr,AIPConf.Proc.882,78(2007).
[86]R.LaskowskiandP.Blaha,Phys.Rev.B82,205104(2010).
[87]H.Luo,Z.Zhu,L.Ma,S.Xu,X.Zhu,C.Jiang,H.Xu,andG.Wu,J.Phys.D:Appl.
(2008).055010,41Phys.[88]J.Barthetal.,Phys.Rev.B81,064404(2010).
[89]T.Graf,J.Barth,B.Balke,S.Populoh,A.Weidenkaff,andC.Felser,Scripta
(2010).925,63Materialia[90]J.Rusz,L.Bergqvist,J.Kudrnovsk´y,andI.TurekPhys.Rev.B73,214412(2006).
[91]S.Picozzi,A.Continenza,andA.J.Freeman,Phys.Rev.B66,094421(2002).
[92]Y.Kurtulus,R.Dronskowski,G.D.Samolyuk,andV.P.Antropov,Phys.Rev.B
(2005).014425,71[93]E.¸Sas¸io˜glu,L.M.Sandratskii,andP.Bruno,Phys.Rev.B71,214412(2005).
[94]E.¸Sas¸io˜glu,L.M.Sandratskii,andP.Bruno,Phys.Rev.B77,064417(2008).

110

Bibliography

[95]T.Kanomata,K.Shirakawa,andT.KanekoJ.Magn.Magn.Mater.65,76(1987).
[96]J.K¨ubler,G.H.Fecher,andC.Felser,Phys.Rev.B76,024414(2007).
[97]C.M.Fang,G.A.deWijs,andR.A.deGroot,J.Appl.Phys.91,8340(2002).
[98]L.Castelliz,Z.Metallkde.46,198(1955).
[99]H.Yamada,K.Terao,K.Kondo,andT.Goto,J.Phys.:Condens.Matter2002,11785
(2002).[100]S.Ishida,S.Akazawa,Y.Jubo,J.Ishida,J.Phys.F:Meth.Phys.12,1111(1982).
[101]P.Mohn,P.Blaha,K.Schwarz,J.Magn.Magn.Mater:140-144,1,183-184(1995).
[102]S.C.Lee,T.D.Lee,P.Blaha,K.Schwarz,J.Appl.Phys.97,10C307(2005).
[103]M.C.Hickey,A.Husmann,S.N.Holmes,G.A.C.Jones,J.Phys.:Condens.
(2006).2897-2903,19Matter[104]M.Meinert,J.M.Schmalhorst,andG.Reiss,Appl.Phys.Lett.97,012501(2010).
[105]P.J.WebsterandK.R.A.Ziebeck,J.Phys.Chem.Solids34,1647(1973).
[106]V.Sharma,A.K.Solanki,andA.Kashyap,J.Magn.Magn.Mater.322,2922
(2010).[107]P.Gupta,K.J.S.Sokhey,S.Rai,R.J.Choudhary,D.M.Phase,andG.S.Lodha,
ThinSolidFilms517,3650(2009).
[108]E.Suharyadi,T.Hori,K.Mibu,M.Seto,S.Kitao,T.Mitsui,andY.Yoda,J.Magn.
Magn.Mater.322,158(2010).
[109]E.ArenholzandS.Prestemon,Rev.Sci.Instrum.76,083908(2005).
[110]J.A.Thornton,J.Vac.Sci.Technol.11,666(1974).
communication.privateBalke,B.[111][112]R.Brydson,H.Sauer,W.Engel,J.M.Thomass,E.Zeitler,N.Kosugi,andH.
Kuroda,J.Phys.:Cond.Matt.1,797(1989).
[113]P.Klaer,M.Kallmayer,C.G.F.Blum,T.Graf,J.Barth,B.Balke,G.H.Fecher,C.
Felser,andH.J.Elmers,Phys.Rev.B80,144405(2009).
[114]A.Yamasaki,S.Imada,R.Arai,H.Utsunomiya,S.Suga,T.Muro,Y.Saitoh,T.
Kanomata,andS.Ishida,Phys.Rev.B65,104410(2002).

111

Bibliography

[115]A.Scherz,H.Wende,andK.Baberschke,Appl.Phys.A78,843(2004).
[116]L.V.Bekenov,V.N.Antonov,A.P.Shpak,andA.N.Yaresko,Condens.Matter
(2005).565,8Phys.[117]K.R.A.ZiebeckandP.J.Webster,J.Phys.Chem.Solids35,1(1974).
[118]T.Kanomata,Y.Chieda,K.Endo,H.Okada,M.Nagasako,K.Kobayashi,R.
Kainuma,R.Y.Umetsu,H.Takahashi,Y.Furutani,H.Nishihara,K.Abe,Y.
Miura,andM.Shirai,Phys.Rev.B82,144415(2010).
[119]T.Kubota,K.Kodama,T.Nakamura,Y.Sakuraba,M.Oogane,K.Takanashi,and
Y.Ando,Appl.Phys.Lett95,222503(2009).
[120]P.Klaer,E.A.Jorge,M.Jourdan,W.H.Wang,H.Sukegawa,K.Inomata,andH.
J.Elmers,Phys.Rev.B82,024418(2010).
[121]M.P.Raphael,B.Ravel,Q.Huang,M.A.Willard,S.F.Cheng,B.N.Das,R.M.
Stroud,K.M.Bussmann,J.H.Claassen,andV.G.Harris,Phys.Rev.B66,104429
(2002).[122]B.Ravel,J.O.Cross,M.P.Raphael,V.G.Harris,R.Ramesh,andV.Saraf,Appl.
Phys.Lett.81,2812(2002).
[123]T.M.Nakatani,A.Rajanikanth,Z.Gercsi,Y.K.Takahashi,K.Inomata,andK.
Hono,J.Appl.Phys.102,033916(2007).
[124]K.Inomata,M.Wojcik,E.Jedryka,N.Ikeda,N.Tezuka,Phys.Rev.B77,214425
(2008).[125]S.Picozzi,A.Continenza,andA.J.Freeman,Phys.Rev.B69,094423(2004).
[126]K.¨Ozdo˜gan,E.¸Sa¸sioglu,B.Akta¸s,andI.Galanakis,Phys.Rev.B74,172412
(2006).[127]I.Galanakis,K.¨Ozdo˜gan,B.Akta¸s,andE.¸Sa¸sioglu,Appl.Phys.Lett.89,042502
(2006).[128]M.J.Carey,T.Block,andB.A.Gurney,Appl.Phys.Lett.85,4442(2004).
[129]H.C.Kandpal,V.Ksenofontov,M.Wojcik,R.Seshadri,andC.Felser,J.Phys.D:
Appl.Phys.40,1587(2007).
[130]O.Rader,C.Pampuch,W.Gudat,A.Dallmeyer,C.Carbone,andW.Eberhardt,
Europhys.Lett.46,231(1999).
[131]H.C.Kandpal,G.H.Fecher,andC.Felser,J.Phys.D:Appl.Phys.40,1507(2007).

112

[132]

[133]

[134]

[135]

[136]

[137]

L.Vegard,Zeitschriftf¨urPhysik5,17(1921).

Bibliography

Y.Takamura,R.Nakane,andS.Sugahara,J.Appl.Phys.105,07B109(2009).

N.D.Tellingetal.,Phys.Rev.B78,184438(2008);thecorrespondingspin
CMACMSCMSmCMAmagnetic=1.91momenµ.ErrtsarorseofmCo±10%=wer0.89eµB,assumedmMnfor=these2.42µB,values.mCoN.=D.T0.59elling,µB,
BMncommunication.private

H.A.D¨urr,G.vanderLaan,D.Spanke,F.U.Hillebrecht,andN.B.Brookes,
Phys.Rev.B56,8156(1997).

Phys..RevBA.66,Scherz,184401H.W(2002).ende,K.Baberschke,J.Minar,D.Benea,andH.Ebert,

(2011).215601,23Matterondens.CPhys.:J.Manghi,.FandBellini,.Vdini,ParL.

113

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

114

Soyez le premier à déposer un commentaire !

17/1000 caractères maximum.