Detailed transport investigation of the magnetic anisotropy of (Ga,Mn)As

KPappert,CGould,

PhysikalischesInstitut(EP3),Universit¨atW¨urzburg,AmHubland,D-97074W¨urzburg,Germany

MSawicki,

InstituteofPhysics,PolishAcademyofSciences,al.Lotnik´ow32/46,PL-02668,Warszawa,Poland

JWenisch,KBrunner,GSchmidt,LWMolenkamp

PhysikalischesInstitut(EP3),Universit¨atW¨urzburg,AmHubland,D-97074W¨urzburg,GermanyE-mail:gould@physik.uni-wuerzburg.de

Abstract.

Thispaperdiscussestransportmethodsfortheinvestigationofthe(Ga,Mn)Asmagneticanisotropy.Typicalmagnetoresistancebehaviourfordifferentanisotropytypesisdiscussed,focusingonanindepthdiscussionoftheanisotropyfingerprinttechniqueandextendingittolayerswithprimarilyuniaxialmagneticanisotropy.

Wefindthatinall(Ga,Mn)Asfilmsstudied,threeanisotropycomponentsarealwayspresent;Theprimarybiaxialalong([100]and[010])alongwithbothuniaxialcomponentsalongthe[110]and[010]crystaldirectionwhichareoftenreportedseparately.Variousfingerprintsoftypical(Ga,Mn)Astransportsamplesat4Kareincludedtoillustratethevariationoftherelativestrengthoftheseanisotropyterms.Wefurtherinvestigatethetemperaturedependenceofthemagneticanisotropyandthedomainwallnucleationenergywiththehelpofthefingerprintmethod.

PACSnumbers:75.50.Pp,75.30.Gw

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As2

Asthesophisticationofspintronicdeviceinvestigationscontinuestorapidlygrow,adeeperandmoredetailedcharacterizationoftheferromagneticsemiconductormaterialusedintheelaborationofmanyofthesestructuresisbecomingevermoreessentialtoproperlyunderstandingtheoperationanddesignofdeviceelements.Thespin-orbitmediatedcouplingofmagneticandsemiconductorpropertiesinthesematerialsgivesrisetomanynoveltransport-relatedphenomenawhichcanbeharnessedfordeviceapplications.Fortheunderstandingandreliablefunctioningofsuchdevicesitisimportanttounderstandandbeabletodeterminethemagneticanisotropyoftheparentlayerandofreadilystructuredsamples.WhileFMR[1]andSQUID[2]caneffectivelymeasurethemainmagneticanisotropyoftheparentlayer,theyarenotpracticalforanisotropystudiesonindividualsmallstructureswhichhavetoolittlemagneticmomenttobedetected.Transportmeasurements,ontheotherhandprovideaveryeffectivemeansofextensivelystudyingtheanisotropyatafixedtemperature.Usingavectorfieldmagnet,manymagneticfieldscansindifferentin-plane(orevenspace-)directionscanberecordedwithinashorttimeframewithoutremountingthesample.Anisotropictransportpropertiesallowforelectricalmonitoringofthemagnetization.Thisprovidesdetailedinformationontheangulardependenceofthemagneticbehaviour.

Atechniqueforextractingthemagneticanisotropybytransportmeanswasintroducedin[3].Inthistreatisewediscussinvestigationsofthemagnetizationbehaviourbytranportmeansingeneralandinparticulartheanisotropyfingerprinttechniqueinmuchgreaterdetail.Wepresentavarietyoffingerprintsofdifferent(Ga,Mn)Aslayersat4Kanddiscussthealwayspresentthreesymmetrycomponentsofthemagneticanisotropyat4K.Wethenextendthemethodtothecaseofauniaxialmaterial,whichisnecessarytodescribe(Ga,Mn)Aslayersathighertemperaturesorstructuredsubmicrondevices.Weinvestigatethetemperaturebehaviourofthe(Ga,Mn)Asanisotropyusingthefingerprintmethod.ItshowsthetypicaltransitionfromamainlybiaxialsystematlowtemperaturetoauniaxialsystemclosetoTC.Fromthesefingerprintswecanalsoextractthetemperaturedependenceofthedomainwallnucleationandpropagationenergy.

1.AnisotropicTransportandMagneticAnisotropyin(Ga,Mn)As

Theferromagneticsemiconductor(Ga,Mn)Asisstronglyanisotropicbothintransportandinitsmagneticproperties.Itshowsastronganisotropicmagnetoresistanceeffect(AMR):Theresistivityforacurrentflowperpendiculartothemagnetizationρ⊥islargerthanρ||paralleltothemagnetization[4].Ohm’slawisbestexpressedwiththeelectricfieldEbrokenupincomponentsparallelandperpendiculartothemagnetizationM[5,6]

E=ρ||J||+ρ⊥J⊥

(1)

withJthecurrentdensity.Theprojectionontothecurrentpathgivesthelongitudinalresistivityρxx(longitudinalAMReffect):

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As

ρxx=ρ⊥−(ρ⊥−ρ||)cos2(ϑ),

(2)

3

whereϑistheanglebetweenMandJ.ThedependenceoftheHallresistivityρxy(transverseAMRorPlanarHalleffectPHE)onthemagnetizationdirectionfollowsdirectlyfromtheelectricfieldcomponentperpendiculartothecurrentpath:

ρxy=−

ρ⊥−ρ||

sin(2ϑ),2

(3)

WiththehelpoflongitudinalAMRandPHEmeasurementsitisthuspossibletomonitorthemagnetizationdirectionϑandconcludeonthemagneticanisotropyofthematerial.

Thecubicanisotropyofthecrystalstructureisreducedbygrowthstrain.Herewediscusshighlydopednotannealed(Ga,Mn)AslayersgrownundercompressivestrainonGaAs(001)substrates.Forstandardthicknessandexperimentallyrelevantholedensities,thegrowthstrainresultsinanadditionalstronghardaxisingrowthdirectionthatconfinestheeasyaxestothelayerplane.Thisin-planeanisotropyisstronglytemperaturedependentaswillbediscussedinsection4.At4Kthematerialshowsamainbiaxialmagneticanisotropywitheasyaxesalongthe[100]and[010]crystaldirection.Theaboveiswellunderstood,however,inadditiontothistwouniaxialanisotropytermshavebeenobservedtheoriginofwhichisnotclear.Oneadditionaluniaxialanisotropytermwitheasyaxisalong[110]or[110]istypicallypresentandhasbeenseeninmanylaboratories.Amuchsmalleradditionaluniaxialanisotropycomponentwitheasyaxisalong[010]or[100][7]hasoftenbeenoverlooked,becauseitistypicallytoosmalltobevisibleinstandardSQUIDmeasurements.Recently,theanisotropyfingerprinttechnique[3]allowedustoshowthatallthree,themainbiaxialandthetwouniaxial,anisotropycomponentsaresimultaneouslypresentintypical(Ga,Mn)Aslayersat4K.Section2.1explainsthedetailsofthemethod.Fingerprintsoftypical(Ga,Mn)Aslayersareshowninsection2.4todiscussthetypicalrelativestrengthoftheanisotropycomponentsandtheirvariationfromlayertolayer.

Inthiscontextitishelpfultonotethatforthepurposeofcalculatingthemagnetostaticenergyinthesingledomainmodel,anylinearcombinationofuniaxialanisotropycomponentswithdifferenteasyaxescanbeexpressedasalinearcombinationofa[110]anda[010]uniaxialanisotropyterm.Itisknownthat:

√

asinα+bcosα=a2+b2·sin(α+β),

(4)

whereβisgivenbyarctan(b/a)andarctan(b/a)±πifa≥0anda<0respectively.Thisrelatestwosinefunctionsofthesameperiodbutwithdifferentphasetoathirdsinefunctionwiththesameperiodandanewphase.Consequently,wecanexpressanycombinationoftwouniaxialanisotropycomponentsina(Ga,Mn)Aslayerbyanequivalentlinearcombinationofthe[110]andthe[010]uniaxialanisotropyterm.The

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As

E(a)

E(b)H=0

H=5% K /Mcryst4

ε0

45

90

135

180

225

270

315

360

ϑ

04590135180225270315360

ϑ

E(c)

H=20% K /Mcryst

E(d)

H=50% K /Mcryst

04590135180225270315360

ϑ

04590135180225270315360

ϑ

Figure1.Energylandscapeatzerofield(a).Thesymmetrycomponentsofthe

anisotropyareshownwiththinlines(biaxialred;uniaxialalong[110]blue;uniaxialalong[010]black).Theenergysurfaceevolveswithincreasingfieldalong45◦(b-d)causingmagnetizationreversalthroughdomainwallnucleationandpropagation(b)orthroughStoner-Wohlfarthrotation(candd).

choiceofonlythesetwodirectionsisthusfullygeneralanddoesnotexcludeotheruniaxialanisotropycomponents,e.g.duetospecificstrainconditionsinaspecificsample.

Summingupthethreeanisotropytermsofdifferentsymmetry,wecanexpressthemagnetostaticenergyEofamagneticdomainwithmagnetizationorientationϑwithrespecttothe[100]-crystaldirection:

E=

Kcryst

sin2(2ϑ)+Kuni[110]sin2(ϑ−135◦)+Kuni[010]sin2(ϑ−90◦)−MHcos(ϑ−ϕ),(5)4

wherethelasttermistheZeemanenergy.TheanisotropyconstantsKcrystinthebiaxialanisotropytermandKuni[110]andKuni[010]inthetwouniaxialtermsdependdifferentlyonthemagnetizationMandthusontemperature[2].Thisresultsinacharacteristictemperaturedependenceoftheoverallmagneticanisotropyofthelayer.Thistypicaltransitionfrommainlybiaxialbehaviourat4KtouniaxialbehaviourclosetotheCurietemperatureisinvestigatedwiththeanisotropyfingerprinttechniqueinsection2.1.

Fig.1showstheenergylandscape,aplotoftheenergyofamagneticdomainasafunctionofthemagnetizationangle,andhowitevolveswithmagneticfield.Underanappliedmagneticfield,themagnetizationcanreversethroughtwomechanisms.Onemechanismiscalledcoherent(Stoner-Wohlfarth[8])rotation:Withincreasingmagneticfieldthemagnetization(markedwithareddotinFig.1)followsthelocalminimumoftheenergysurfaceuntiltheminimumdisappearsasillustratedinpanel

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As5

Figure2.Hallbarstructuretypicalofthoseusedinthisstudy,processedbyopticallithographyanddryetching.Thecontactsareestablishedbygolddeposition.Thisbaris40µmwideand0µmlong.

btod.Thedashedarrowinpanelb,ontheotherhand,illustratesthereversalbyDWnucleationandpropagation.ItappearsiftheenergygainedbyreorientingthemagnetizationdirectiontoanotherlocalminimumoftheenergysurfaceislargerthantheDWnucleation/propagationenergyε.ADWisnucleatedandanewdomainwiththenewmagnetizationorientationgrowsuntilitextendsoverthewholestructure.Aswillbecomeevident,theinherentbehaviourof(Ga,Mn)AsisgenerallydominatedbyStoner-WohlfarthrotationathighmagneticfieldsandbyDW-nucleation/propagationrelatedeventsatlowfields.

2.MonitoringtheMagnetizationBehaviourinTransport

Thedescribedmagnetizationbehaviourcanbeobservedindirectorindirectmagnetizationmeasurements,andleadstoaverycharacteristictwo-stepreversalprocessinSQUIDandmagnetoresistancemeasurements.Three-jumpmagneticswitchingisalsopossibleinveryspecificsituations[9].

HerewewilldiscussthecharacterizationofthemagneticanisotropyoftypicalGa1−xMnxAstransportlayers.Thelayersweregrownbylow-temperaturemolecularbeamepitaxy(LT-MBE,270◦C)onahigh-qualityGaAsbufferonanepireadysemiinsulatingGaAs(001)substrate.Theycontainbetweenx=2%and5%Mnandshowanas-grownTCaround50Korabove.Alllayerswerepatternedinto40to60µmwideHallbarstructuresasshowninFig.2byopticallithographyandchlorineassisteddryetching.Contactsareestablishedthroughmetalevaporationandliftoff.Duringtheprocessingcareistakentonotexposethesamplestoanyannealingtreatment.

Assumeabiaxialmagneticanisotropywitheasyaxesalongthein-plane󰀆100󰀇crystaldirections(coordinateaxesinFig.3a)asafirstapproximationofthe4Kanisotropyof(Ga,Mn)As.AssumefurtherthatthelongitudinalresistanceofaHallbarwithitscurrentalongthe[100]axisismeasuredwhiletheexternalmagneticfieldissweptfromahighnegativetoahighpositivevaluealongadirection30◦awayfrom

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As6

Rxy+ a)

2easy[010]b)

3Rxxϕ = 30°Hc1Hc241B[100]easy23Rxy= 0 1Rxx||-2-10142Rxy- Magnetic Field [arb. u]Figure3.Two-stepmagnetizationreversal.a)Sketchshowingthemagnetization

behaviourinhard(blue)andeasy(red)axisHallbars.b)ThecorrespondingcalculatedAMRscanfortheeasyaxisHallbar(leftscale),whichisequivalenttoaPlanarHallscanonthehardaxisbar(rightscale).

the[100]axis.Usingeq.5and2wecannowcalculatethecorrespondingAMRsignalshowninFig.3b(lefty-axisscale).Athighnegativefields,themagnetizationisforcedalongthefielddirection(notshown).(1)AsthefieldisdecreasedMgraduallyrelaxesthroughStoner-Wohlfarthrotationuntilitisalignedwithitsclosesteasyaxis.AtzerofieldMisthusparallelto[100]andtothecurrent,yieldingthesmallestresistancevalueR||.(2)AtasmallpositivefieldHc1a90◦-DWisnucleatedandpropagatesthroughthestructureresultinginanabruptchangeofthemagnetizationdirectiontothe[010]direction.Misnowperpendiculartothecurrent,yieldingthemaximumresistancevalueR⊥.(3)AtthesecondswitchingfieldHc2,another90◦-DWisnucleatedandthemagnetizationjumpsclosetothe[100]easyaxis.(4)WithincreasingmagneticfieldsMrotatesagaintowardsthemagneticfielddirection.Theentireprocessisofcoursehystereticallysymmetric(notshown).

IfanotherHallbarisorientedalongthe[110]crystaldirection(blueinFig.3a)theeasyaxes[100]and[010]haveanangleof±45◦withthecurrentpath.Anabruptswitchofmagnetizationfromoneeasyaxistotheothercorrespondsaccordingtoeq.3toasharpswitchingeventbetweentwoextremaofthetransverseresistance.ThecalculatedPlanarHallsignalisthusuptoaconstantidenticalwiththepreviouslydiscussedcurveinFig.3b,inthiscasecenteredaroundzerotransverseresistance(blue/righty-axis).Becauseofthis,transverseresistancemeasurementsarethemethodofchoiceforHallbarsorientedalongacrystallinehardaxis.ForHallbarsalonganeasyaxis,longitudinalresistancemeasurementsaretheonlyusefultechnique.Indeed,ifthecurrentdirectionisrotatedby45◦,Eq.3transformsintoEq.2(plusanuninterestingoffset).

Fig.4showsAMR(middle)andPlanarHalleffect(right)curvesforfieldsweepsalongdifferentanglesϕintheplanecalculatedusingEq.5incombinationwithEq.2and3respectively.Thedomainwallnucleationenergyεwasexaggeratedinthesecalculations(30%ofKcrystinsteadof5-10%aswouldbetypicalfor(Ga,Mn)As))toillustratebothStoner-WohlfarthrotationandDW-relatedmagnetizationswitchingin

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As

Anisotropic MR

[010]7

Planar Hall Effect

90°60°

a)0°[100]R+

ΔR xy2

45.1°60°30°,

45.1°

0ΔR xy

2ΔR xy2

0°,90°

30°

R||

[010]0°

-

b)20°[100]R

90°

+

60°45.1°90°30°

60°

0ΔR xy2

45.1°

R||

0°30°

-

0°

c)[010]R+

ΔR xy290°60°

45°[100]0°,90°

0ΔR xy-24-4-3-2-10

1

2

45.1°

30°,60°

30°0°

R||

-4-3-2-101245.1°

334

Magnetic Field [in units of K /M ]crystMagnetic Field [in units of K /M]cryst

Figure4.CalculatedAnisotropicMagnetoresistance(middle)andPlanarHalleffect

(right)curvesformagneticfieldsweepsalongseveralin-planeangles(ϕ=0◦,30◦,45.1◦,60◦and90◦)forHallbarorientationsasindicatedinthesketchesontheleft,withcurrentalonga)0◦b)20◦c)45◦.Theunderlyingmagneticanisotropyisbiaxialwitheasyaxesalong[100]and[010].Allangleswithrespecttothe[100]crystaldirection.Thedomainwallnucleation/propagationenergyεisexaggeratedwith30%ofKcryst.

thesamefigure.ThemiddlepanelofFig.4a,showsMRcurvesforaHallbaralongabiaxialeasyaxis.Iftheexternalmagneticfieldissweptalongthe[100]easyaxis(0◦),themagnetizationisalwaysparalleltothecurrentdirection.Theresistance(blackline)thustakesitslowestvalueR||throughoutthewholefieldrange.Ifthefieldissweptalongthe[010]easyaxis(90◦),themagnetizationisalwaysperpendiculartothecurrentresultinginahighresistancevalueR⊥throughoutthewholecurve(thincyan).Forintermediatemagneticfieldangles,themagnetizationisparalleltothefieldathighpositiveandnegativefields,yieldingintermediateresistancevalues.Atzerofieldthemagnetizationrelaxestotheclosesteasyaxis,whichis[100]forthe30◦scanand[010]forthe60◦and45.1◦scans,correspondingtothelowestandhighestresistancevaluerespectively.The45.1◦-scan(greenline)canbeusedtomeasurethestrengthofthemagneticanisotropy.Wecanreadouttheanisotropyfield(-2Kcryst/M),atthepointwherethemagnetizationstartstoturnawayfromthemagneticfielddirection.A

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As8

measurementwithtwopossibleresistancestatesatzerofieldalwayssuggestsabiaxialmagneticanisotropy.However,notethatthesetwostatescancorrespondtothesameresistancevalueas,e.g.,iftheeasyaxisandthecurrentincludeanangleof45◦(leftpanelofFig.4c),wherethe0◦(black)and90◦(cyan)curvefallontopofeachother.ThepanelsontherightshowthecalculatedPlanarHallresistancecurvesintherespectiveconfigurations.Note,that,theAMRsignalinFig.4aisidenticaltothePHEsignalinFig.4c,asdiscussedabove.Theeasyaxisshowingconstantresistancethroughoutthewholescancaneasilybeidentifiedinanyoftheconfigurations,evenifcurrentandeasyaxisincludeanobliqueangleasinFig.4b.

Theswitchingfields(Hc1andHc2inFig.3b)canbederivedanalyticallyfromeq.5[10](hereforapurebiaxialanisotropy;Kuni[110]=Kuni[010]=0).TypicallyDWnucleationandpropagationdominatesthemagnetizationreversalprocess,i.e.εismuchsmallerthanthecrystallineanisotropy.Thatiswhyitcanbeassumedthatthemagnetostaticenergyminimaremaintoagoodapproximationalongthebiaxialeasyaxesduringthedouble-stepswitchingprocess.TheenergydifferencebetweenstablemagnetizationstatesisthusgivenbytherespectivedifferenceinZeemanenergy(Eq.5).Whentheenergygainedthrougha90◦magnetizationreorientationislargerthanε90◦,thenucleationandpropagationenergyofa90◦-DW,athermallyactivatedswitchingeventbecomespossible.This,onthetimescaleofourmeasurement,resultsinanimmediateswitchingevent.Forexample,tocalculatethefieldrequiredforthemagnetizationtojumpfrom0◦to90◦,thedifferenceintheZeemantermsisequatedwithε

∆E0◦→90◦=−MH[cos(0◦−ϕ)−cos(90◦−ϕ)]=ε90◦>0.

ReorganizinggivestheswitchingfieldHcasafunctionofϕ.

−ε90◦

Hc=

M[cosϕ−sinϕ]

(7)(6)

Thisequationisthesameforotherpairsofangles,exceptforthesignsinfrontofthesineandcosinefunctionsinthedenominator,inthefollowingmarkedwith±.TheswitchingfieldequationabovedescribesstraightlinesifplottedinapolarcoordinatesystemusingHasradialandϕasangularcoordinate.ThepolarplotinFig.5showstheresultingcharacteristicsquarepattern[10].Weexpresstheswitchingfieldpositionsinthisplot(thicklines)incartesiancoordinatesusingx=Hccosϕandy=Hcsinϕtogetabetterfeelingfortheswitchingfieldbehaviourandtoextractimportantparameters.

Hc·M[±cosϕ±sinϕ]=−ε90◦

M[±x±y]=−ε90◦

y=±x±

(8)

ε90◦MThecharacteristicpolar-plot-patternforabiaxialmaterialisthusasquarewithdiagonalsalongtheeasyaxes(thecoordinateaxesinFig.5).Thefirstswitching

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As9

[010]Hc2Hc1ε/M[100]xyFigure5.Switchingfieldpositions(thicksolidlines)inapolarplotforabiaxialmaterialwitheasyaxesalong[100]and[010](coordinateaxes).Themagnetizationdirectionineachregionoftheplotisindicatedbyarrows(red/black:high/lowresistance)andthehardaxesbydashedlines.

field(thickbluelines)islargestalongtheeasyaxes,whereHc1=ε/M.TheDWnucleation/propagationenergycanbeextractedfromthediagonalofthesquare,whose

90◦

lengthisequalto2ε.AllswitchingfieldlinesinFig.5haveanangleof45◦totheMcoordinateaxes.Thedashedlinesrepresentthehardmagneticaxes.ArrowsillustratethedirectionofthemagnetizationandtheircolorthecorrespondingresistancestateoftherespectivesectioninanAMRmeasurementwithcurrentalongoneoftheeasyaxes.

NeglectingcoherentrotationistypicallyagoodmodelforthefirstswitchingfieldsHc1,whereasHc2isinfluencedbymagnetizationrotationespeciallyclosetothehardaxes.PairsofparallellinesinFig.5donotextendtoinfinityinpractice,theymoveclosertothehardaxes(seethefiguresandthediscussioninsection2.4).ThemagneticfieldneededtoforcethemagnetizationparalleltotheexternalfieldinthehardaxisdirectioniscalledtheanisotropyfieldHa.ItisameasureoftheanisotropystrengthandcanbecalculatedfromEq.5usingthedefinitionoftheanisotropyfield:Haisthestrengthofafieldalongthehardaxis(here45◦)neededtosuppressthelocalminimaalongtheeasyaxes.

Ha=

2KcrystM

(9)

2.1.TheAnisotropyFingerprintTechnique

TraditionallythemagneticanisotropyisinvestigatedbydirectmeasurementoftheprojectionofthemagnetizationontocharacteristicdirectionsusingSQUIDorVSM.

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As

[010](a)0.100.05Rx [ Ω ]y10 mT0.00(b)10Magnetic Field [ mT ]10

Hc2J0

[100]-0.05-0.100Hϕ=80°Hc124681012Magnetic Field [ mT ]14-10

-100Magnetic Field [ mT ]

10Figure6.a)PlanarHallEffectmeasurementalongϕ=80◦withmarkedfirstandsecondswitchingfield,colorscaleandthecorrespondingsectionofacolorcodedresistancepolarplot(inset).b)ResistancepolarplotfromafullsetofPlanarHallmeasurementsalongevery3◦.The80◦-sectioncorrespondingto(a)ismarkedbyadashedline.

Theadventofvectorfieldmagnetshasrecentlyopeneduppossibilitiesforacquiringadetailedmappingoftheanisotropy.Weintroducedsuchamethod,whichbuildsontheabovediscussedangulardependenceofthemagnetizationswitchingfields,inRef.[3]andexpanduponithere.Itisbasedonsummarizingtheresultsoftransportmeasurementsintocolorcodedresistancepolarplots(RPP)whichactasfingerprintsfortheanisotropyofagivenstructure.Notonlyisthismethodfasterthanthetraditionalalternatives,butitisalsomoresensitivetocertainsecondarycomponentsoftheanisotropy,inparticularthosewitheasyaxescollineartotheprimarybiaxialanisotropycomponent[10].Thetechniquethusoftenrevealstheexistenceofcomponentswhichwouldbemissedusingstandardtechniques.Moreover,thetechniquecanbeappliedtostudytheanisotropyoflayersbyusingmacroscopictransportstructures,orapplieddirectlytodeviceelements.Itcanrevealeffectsofprocessingortheinfluenceofsmallstrainfieldsdueto,forexample,contacting.

InthepresentcasetheplanarHalleffectisusedtomonitorthemagnetizationbehaviourinastandardHallbarorientedalongthe󰀆110󰀇crystaldirection.Fig.6ashowsaplanarHallscanalongϕ=80◦.Aftermagnetizingthesampleat-300mTalong80◦,thefieldisbroughtdowntozero.Thefigureshowsthetypicaldouble-stepswitchingbehaviourasdiscussedpreviouslyinconnectionwithFig.3b.ThearrowsindicatethemagnetizationdirectionintherespectivefieldregionswithrespecttothecrystaldirectionsgiveninFig.6b.AbruptjumpsinresistancemarkthefirstandsecondswitchingfieldHc1andHc2.ThenormalizedresistancevalueiscolorcodedaccordingtothescaleinFig.6a.Itisplottedinapolarcoordinatesystemalongthemagneticfielddirectionϕandwiththemagneticfieldasradialscale.TheinsetofFig.6ashowsthepolarplotsectioncorrespondingtothe80◦-scaninthisfigure.SuchPlanarHallscansarerecordedalongmanydifferentin-planefielddirectionsandsummarizedinthe

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As11

ε/ΜH [ ε/M ]00-1BeUhε/Μ2Κ /Μu1ε/Μ90°−δ1

(a)

(b)1

10-1

1(c)

ε/Μ-1

-1

0H [ ε/M ]

1

BeUe-10H [ ε/M ]

-1

0H [ ε/M ]

1

Figure7.Calculatedresistancepolarplotsforabiaxialmaterialwitheasyaxesalongthe[100](0◦)and[010](90◦)crystaldirections(a)andthesamematerialwithanadditionaluniaxialanisotropyalong[010](b)or[110](c).ColorscaleoftheresistanceasinFig.6.εdenotesthe90◦-DWnucleation/propagationenergy.

resistancepolarplot(RPP)inFig.6b.The80◦-segmentismarkedbyadottedwhiteline.

WecannowcomparetheobservedswitchingfieldpatterninFig.6bwiththecalculatedshapeinFig.5.WhileacursoryexaminationsuggestsasimilarHc1-pattern,amoredetailedcomparisonrevealssignificantdifferences:Focussingontheinnermostswitchingevent,thepatternisindeedstronglysquare-like,confirmingthatthe(Ga,Mn)Ashasamainlybiaxialmagneticanisotropyat4K.Thediagonalsofthissquare-likeHc1-patternareclosetothe[100]andthe[010]crystaldirection,theeasyaxesofthebiaxialanisotropyterm.However,theinnerregioniselongated(arectangleandnotasquare)-thesignatureofanadditionaluniaxialanisotropytermwithaneasyaxisbisectingthebiaxialeasyaxes(Fig.7c),aswillbediscussedinsection2.3.Additionallyweobserveadiscontinuityinthemiddleoftherectanglesidesanddark”open”cornersclosetothe[010]direction.Thisischaracteristicofauniaxialmagneticanisotropytermcollinearwithoneofthebiaxialeasyaxes(Fig.7b)andwillbediscussedindetailinsection2.2.

ThesequalitativechangesinthebehaviourofHc1arekeysignaturesofthedifferentanisotropytermsofthe(Ga,Mn)Aslayer.Asetofhighresolutiontransportmeasurementscompiledintoacolorcodedresistancepolarplotthusconstitutesafingerprintofthesymmetrycomponentsoftheanisotropy.Itallowsforthequalitativeandquantitativedeterminationofthedifferentanisotropyterms.Itcanprovetheirexistenceandvisualizetheirrespectiveeffectsonthemagnetizationreversal.2.2.Signatureofa󰀆010󰀇UniaxialTerm

ThefingerprintofamagneticallybiaxialmaterialinFig.7aisequivalenttotheswitchingfieldpatterninFig.5.IfanadditionalsmalluniaxialanisotropyKuni[010]alongoneofthebiaxialeasyaxes(herealong90◦)ispresent,thesquarepatternisalteredasshown

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As12

inFig.7b.Thefour-foldsymmetryisbroken,andthebiaxialeasyaxescorrespondtoenergyminimaofslightlydifferentdepth,becauseoneofthemisparallel(biaxialeasy,uniaxialeasy;BeUe)andoneperpendicular(biaxialeasy,uniaxialhard;BeUh)totheeasyaxisoftheuniaxialanisotropycomponent.

Theangledependentswitchingfieldcanbederivedasdiscussedabovefollowing[10]:Againitisassumed,thattheminimaoftheenergysurfaceremainattheirzerofieldanglesthroughouttheswitchingprocess.Inthepresentcasehowever,theenergyminimumalongthe[010]directionisfavored.Itsenergyis∆E=Kuni[010]smallercomparedwiththe[100]direction,whichresultsin

Hcy

uni[010]90

=±M

[cosϕ±sinϕ]

ε◦±K

(10)

=±x±

ε90◦M±

Kuni[010]

MMagnetizationreorientationstowardstheeasierbiaxialeasyaxisBeUeoccurnowatlowermagneticfieldscomparedtothepurebiaxialanisotropy;switchesawayfromBeUeathigherfields.Thesignsineq.10arechosenappropriately.Asaresult,theHc1-patternchangesasdisplayedinFig.7b.Characteristicfeaturesarethestepsalongthebiaxialhardaxes,forexamplealong45◦,andthetypical”opencorners”alongtheBeUeaxis.Theseopencorners(inblackalong90◦inFig.7b)arisebecausea180◦-magnetizationreorientationthroughthenucleationofa180◦-DWisenergeticallyfavoredinasmallangularregionaroundtheBeUeaxis[10].

Sincetheisotropicmagnetoresistance[11]oftypicalsamplesisrelativelysmallcomparedtotheAMR,twomagnetizationdirectionsdifferingby180◦arenotdistinguishableonthescaleconsideredhere,andhavenearlythesamecolorintheRPP,creatingthecharacteristic”opencorner”.Thestrengthoftheuniaxialanisotropy

u1

componentcanbedeterminedfromtheseparation2KbetweenHc1andHc2alongtheMBeUhaxis.

2.3.Signatureofa󰀆110󰀇UniaxialTerm

InthissectionwedescribetheeffectsofauniaxialanisotropytermKuni[110]withitseasyaxis(along135◦)bisectingthebiaxialeasyaxes.Thisuniaxialanisotropycomponentflattenstheenergysurface(eq.5)andshiftsthepositionsoftheenergyminimaby(seeFig.8a)

Kuni[110]δ1

=arcsin()22Kcryst

(11)

towardstheuniaxialeasyaxis[12].Allfourminimahavethesameenergyvalue.To

derivetheswitchingfieldsweequatetheDWnucleation/propagationenergyεwiththedifferenceinZeemanenergybetweentheinitialandfinalmagnetizationangleintherespectiveswitchingevent.AsillustratedinFig.8a,themagnetizationdirectioncan

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As

E

(a)

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K = 30% Kuni [110]cryst

90°+δ/2

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uniaxialeasy

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x [ε/M]

ε90°+δ0°45°90°135°180°225°270°315°360°

ϑ

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-2

Figure8.a)Auniaxial[110]anisotropycomponentflattenstheenergysurface(eq.5)andshiftsthepositionsoftheenergyminima.b)Energylandscapewithmagneticfieldappliedalongthe−δ/2-globaleasyaxisdirection.Aclockwiseandcounterclockwisejumpofthemagnetization(withtherespectiveε)areequallypossible.c)Switchingfieldpositionsinthepolarplot(thickblacklines),globaleasyaxes(orange)andeasyandharddirectionofthe[110]anisotropycomponent(blue).

changeby90◦+δor90◦−δdependingonwhetherthemagnetizationrotatesclockwiseorcounterclockwise.Following[13]weusedifferentDWnucleation/propagationenergiesε90◦+δandε90◦−δrespectively.Theswitchingfieldpositionsinthepolarplotgivenincartesiancoordinatesare

y90◦+δy90◦−δ

=x±

ε90◦+δ

M2[cos(45◦−δ/2)]√(12)

=−x±

ε90◦−δ

M2[cos(45◦+δ/2)]√Equation12describestwoparallelsetsoflines,asshowninFig.8c(thickblacklines),whosedistancefromtheoriginisdeterminedbytherespectiveε.MOKEexperimentsonepitaxialironfilmsgrownonGaAs(withsimilaranisotropytermsas(Ga,Mn)As)confirmthatasexpectedthesenseofthemagnetizationrotationchangeswhencrossingaglobaleasyaxis[12].Thetwolinesetsofeq.12representtheclockwiseandcounterclockwisesenseofmagnetizationrotation.Ifthefieldisappliedalongaglobaleasyaxes(minimaofFig.8a)bothrotationdirectionsareenergeticallyequivalent.Consequentlythelinesmustintersectalongglobaleasyaxesdirections.Fig.8bshowstheenergylandscapeofFig.8awhenamagneticfieldisappliedalongthe−δ/2-globaleasyaxisdirection.Forbothrotationdirections,theZeemantermatthefirstswitchingfieldHc1isequaltotherespectiveε.Wecanthuscalculatethedependenceofεontheangle∆ϑbetweeninitialandfinalmagnetizationdirection:

ε90◦±δε∆ϑ

=Hc1M(1−cos(90◦±δ))

(13)

=ε90◦(1−cos(∆ϑ))

whichisintuitivelyreasonable.Atthesametimewefind,thatHc1alongaglobaleasyaxisisε90◦/M.Notethatthiscarefultreatmentofεisnecessary,thesimplified

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As

2014

(a)[110](b)

20.019.819.6Magnetic Field [mT]100R(kΩ)19.419.2-10

19.0 0 mT 20 mT 50 mT 100 mT04590135180Angle ° 225270315360-20

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Figure9.a)Fingerprintofatypical20nmthick(Ga,Mn)AsHallbarandb)angle-dependentlongitudinalresistanceatdifferentfieldsaftermagnetizingalongϕ.

modelofaconstantεindependentoftheDWangle∆ϑ,wouldleadtotheincorrectconclusion,thattherectangleinthepolarplotwouldhaveitslongaxisperpendiculartotheuniaxialeasydirection.

AsummaryoftheaboveisshowninFig.7c.Thecharacteristicpatternofamainlybiaxialanisotropywithabisectinguniaxialanisotropycomponentisrectangular.Thediagonalsoftherectanglearethe”globaleasyaxes”,theirlengthis2ε90◦/M.Theanglebetweentheglobaleasyaxesgivestherelativestrengthofthetwoanisotropycomponents(usingeq.11).Theeasyaxisoftheuniaxialtermisalongthemedianlineofthelongeredge,forexamplealong135◦inFig.7c.

Thepresenceandsignofthe󰀆110󰀇anisotropytermcanbeverifiedwiththehelpofAMRorPHEmeasurementsatmagneticfieldsofmediumamplitude.Forcomparison,longitudinalresistancemeasurementsonaHallbarsampleorientedalonga(Ga,Mn)Aseasyaxis(0◦)areconvertedintotheRPPdisplayedinFig.9a.Thisfingerprintshowsanoverallbiaxialanisotropywitheasyaxescloseto0◦and90◦.Thecentralpatterniselongatedalong135◦,suggestingthatauniaxialanisotropycomponentwitheasyaxisalongthisdirection(the[110]crystaldirection)ispresent.

ThisisconfirmedbythemeasurementsinFig.9b.HeretheHallbarsampleisfirstmagnetizedinahighmagneticfieldof300mTalonganangleϕ.Thelongitudinalresistanceisthenmeasured,whilethefieldisslowlysteppeddowntozero.Fig.9bshowstheresistancevaluesat100mT,50mT,20mTand0mTasafunctionofthefieldangle.Fortheinterpretationofthesecurves,imagineforexampleanenergylandscapeasshowninFig.8,wherethestrengthofthe[110]uniaxialanisotropytermisexaggerated.Thistermdescribesthewidthandtheheightofthe”hills”intheenergysurface.The”hill”intheuniaxialeasyaxisdirection(here135◦)islowerthantheenergybarrierperpendiculartothisdirection,whichissteeperandcoincideswiththe

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As

300

[010]15

200

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[100]0

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-300-200-1000100200300Magnetic Field [mT]

Figure10.TypicalAMRfingerprintmeasurementofa100nmthick(Ga,Mn)AsHallbar.Thecurrentdirectionisalong0◦.

hardmagneticaxisofthe[110]uniaxialterm.Atzerofieldthemagnetizationisalignedwithoneofthebiaxialeasyaxes(blackcurveinFig.9b).Thestepsinthiscurvemarkthepeakpositionsofthe”hills”intheenergylandscape-thebiaxialhardaxes.Atmediumfields(e.g.50mTinFig.9b),themagnetizationisrotatedawayfromtheglobaleasyaxes,causingdeviationsfromthestep-likebehaviouratzerofield.Thesedeviationsoccuratsmallerfieldvaluesalongtheuniaxialeasydirection[110]comparedwiththeuniaxialhardaxis[110].Thedirection(meaningthesignofKuni[110])ofthe[110]uniaxialanisotropyisthusthesameasinFig.9a:theabruptresistancechangeat45◦marksthehardandthesmootherbehaviourat135◦theeasyuniaxialaxisdirection.2.4.(Ga,Mn)Asat4K-TypicalFingerprints

Intheabovesectionswedescribeamethodwhichissensitiveenoughtodetectboth,the[110]andthe[010]uniaxialanisotropyterm.Hereweapplythemethodtoourtypical(Ga,Mn)Aslayersandfindthatallthreeanisotropycomponents,thebiaxialandthetwouniaxialones,arepresentineverysample.Variousfingerprintsshowthetypicalvariationoftherelativeanisotropytermsandthecharacteristicsatlowandhighfields.

ThefingerprintsinFigs.9a,10and11werecompiledfromlongitudinalAMRmeasurementsontypical(Ga,Mn)Aslayersofdifferentthickness.AlltheseplotsincludingFig.6bshowthesamegeneralpattern,resemblingthefour-foldswitchingfieldpatterninFig.7.Themainanisotropycomponentinalltheselayersat4Kisthusbiaxialwitheasyaxesalongthe[100]and[010]direction.ThestrengthofthisbiaxialtermismeasuredbytakingseparatehighresolutionAMRcurvesalongthehardmagneticaxesandconcludingtheanisotropyconstantfromtheanisotropyfields.Typical2K/Mvaluesareoftheorderof100mT...200mT,ascanbeseene.g.inthehighfieldfingerprintsinFigs.10and11aandc,whereinthesectionsalongthehard

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Figure11.Highangularresolutionfingerprintmeasurements(a,b)andcloseupsofthecentralregion(c,d)fortwoHallbarsmadeofthesame70nmthickmaterialbutorientedalongorthogonalcrystaldirections.Thecurrentflowsalong0◦inaandbandalong90◦incandd.

axesthemagnetizationisalignedwiththeexternalfieldatthesefieldvalues.

Theanisotropycomponentsandεdifferofcoursefromwafertowafer.Thegeneralpatternontheotherhandisverysimilar.AlloftheRPPshowclearlyanelongationoftheHc1-patternintoarectangle,thesignatureofthe[110]anisotropycomponent.Stepsalongthehardaxesandthetypicalopencornerarealsoalwayspresent,thetypicalfeatureofthe[010]anisotropycomponent.Bothuniaxialanisotropycomponentsarethusclearlypresentinallinvestigatedsamples.

Thecrystaldirectionsareindicatedinallfingerprintsinyellow.Wefind,thattheelongationofthecentralpattern,andthustheeasyaxisofthe[110]uniaxialcomponent,pointsalongthe[110]crystaldirectioninallshowntypicaltransportsamples.Atthispoint,wewouldliketonote,thatthesignofthe[110]uniaxialcomponent,i.e.whethertheeasyaxispointsalong[110]or[110],dependsoncarrierconcentrationandMn

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As

10017

(a)Magnetic Field [mT]10(b)Magnetic Field [mT]5050[100]0[100]-50[010]-5[010]-10-100-100

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Magnetic Field [mT]

Figure12.FingerprintmeasurementandhighresolutionRPPatlowmagneticfieldfora70nmthick(Ga,Mn)Aslayerwithstronglyvisible[010]uniaxialanisotropycomponentbutarelativelysmall[110]term.

dopingasshownby[14].Theelongationcouldthusalsobealongthe[110]directiondependingongrowthconditionsandapossibleannealingtreatment.Intheasgrownsamplesinvestigatedhereat4K,thetypicalstrengthofthe[110]uniaxialanisotropyisoftheorderof10%ofthebiaxialanisotropyconstant.Asexamplesoftherangeofvaluestypicalforthisratiovarieswenoteavalueof10%fromFig.12,15%fromFig.6band20%fromFig.11.

TheopencornersoftheHc1-patternindicatethedirectionoftheeasyaxisofthe[010]uniaxialterm.Thiseasyaxisdirectionissampledependentandcanbealongeitherofthebiaxialeasyaxesofthesample.Intheshownpolarplotsweseethiseasyaxisalong[010]inFig.6bandalong[100]inFigs.9,10,11and12.Alsothestrengthofthe[010]termissampledependent.ItcandominatethelowfieldswitchingbehaviourasforexampleinFig.12orbebarelyvisibleasinFig.9a.Inanycase,thestrengthofthisanisotropycomponentisextremelysmallcomparedtothemainbiaxialanisotropy.EveninFig.12,wherethepresenceofthe[010]uniaxialcomponenthasastronginfluenceonthemagnetizationbehaviouratlowfields,itsanisotropyfield2Kuni[010]/Misonly1.6mT,only1%ofthetypicalbiaxialanisotropyconstant.

Figures11aandcshowsimilarAMRfingerprintsontwoHallbarsmadefromthesamewafer,butorientedalongorthogonalcrystaldirections.Panelscanddshowacloseupofthecentralregion.Bothfingerprintsshowvirtuallythesameswitchingpatternwithinvertedcolorsbecauseoftheorthogonalcurrentdirections.Thisshowsthehighhomogeneityofthewaferandtherobustnessofthemethod.Evenonseveralcooldowns,weseevirtuallythesameswitchingpattern(notshown),althoughtheresistanceofthesamplechangesslightlyuponrecooling.NotethatshapeanisotropyinthesestructuresisnegligiblecomparedwiththecrystallinemagneticanisotropycontributionasdiscussedinRef.[15].Itistoosmalltoplayanysignificantroleinthesemeasurements.

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As18

Wehaveneglectedcoherentrotationtoderivesimpleformulasfortheswitchingfieldsinthepolarplots.ThisistypicallyagoodmodelforthefirstswitchingfieldsHc1,whereasHc2isinfluencedbymagnetizationrotationespeciallyalongthehardaxes,asmentionedabove.Theeffectsofcoherentrotationareofcoursetakenintoaccountinthenumericalmodellingthatisbasedontheenergyequation5.

Asdiscussedpreviously,theextentoftheHc1-patternisdeterminedbyε,whiletheextentoftheHc2-featuresismainlygivenbythebiaxialanisotropyconstantthroughtheanisotropyfield(seeeq.9).Iftheratioofbiaxialanisotropytoεisverylarge,thecentralpatterniswelldescribedbyDW-nucleation/propagation-relatedswitchingeventsalone.Inthehigherfieldregion,theminimaoftheenergysurfacemoveconsiderablyandtheHc2-switchingeventsapproachthehardaxesdirections.

ThetypicalsituationisforexampleseeninFig.10,whereAMRcurvesalongevery10◦weretakenonaHallbaralong0◦.Thecentralregionshowsarectangularpattern(signatureofthe[110]uniaxialterm)withopencornersandstepsalongthehardaxes(signatureofthe[010]uniaxialterm).Thereisalmostnocoherentrotationattheselowfields.MagnetizationreorientationsoccurthroughDWnucleationandpropagationasseenfromtheabruptcolorchanges(betweenredandblue).Thesecondswitchingfieldsalongthehardaxes(e.g.along45◦at50mT)aremarkedbysmoothcolortransitionsprovingthatcoherentrotationisatplay.Smoothcolortransitionsatevenhigherfields(greentoblackaround0◦andredtogreenaround90◦)finallyarecausedbytheisotropicMReffect[11].

ThefingerprintinFig.9showsaslightlydifferentsituation.Theratiooftheanisotropyenergytoεcannotbetreatedasinfinite.ForthisreasonalsotheHc1-patternshowsaconsiderableinfluenceofcoherentrotation.Thesidesoftherectanglearenolongerparalleltoeachotherandthecornersdonotdrawanangleof90◦.Still,theelongationisobviousandthedifferencebetweenswitchingeventstowardsthetwoeasyaxesisobservable.

Insummarywehaveshownavarietyoffingerprintsoftypical(Ga,Mn)Astransportlayersat4K.Thefingerprintmethodallowedustoidentifythesimultaneouspresenceofthebiaxialandtwouniaxialanisotropycomponents.Indeedall(Ga,Mn)Aslayersinvestigatedshowboththeseuniaxialcomponents,includinglayerswherethe[010]componentcouldnotbeidentifiedinSQUIDmeasurements.Asaruleofthumb,thetypicalrelativestrengthoftheanisotropytermsisoftheorderofKcyst:Kuni[110]:Kuni[010]∼100:10:1.

3.UniaxialMagneticAnisotropy

Thissectiondealswiththemagnetizationbehaviourofmagneticallyuniaxialmaterialsandhowitmanifestsitselfintransportmeasurements.Thelatterwithtwospecificapplicationsinmind:

•The(Ga,Mn)Asmagneticanisotropyisstronglytemperaturedependentwiththe󰀆110󰀇uniaxialanisotropytermbeingdominantclosetotheCurietemperature

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As

easyaxis0°19

a)

1.00.8(R-Rmin)/ΔRxx0.60.40.20.0-400b)20°-2000200H (% of K /M)cryst400-400-2000200H (% of K /M)cryst400c) 45°d)90°1.00.8(R-Rmin)/ΔRxx0.60.40.20.0-400-2000200H (% of K /M)cryst400-400-2000200H (% of K /M)cryst400 0° 5° 10° 15° 20° 25° 30° 35° 40° 45° 50° 55° 60° 65° 70° 75° 80° 85° 90° 95° 100° 105° 110° 115° 120° 125° 130° 135° 140° 145° 150° 155° 160° 165° 170° 175°Figure13.Calculatedanisotropicmagnetoresistancecurvesinamagneticallyuniaxialmaterialformagneticfieldsweepsalongmanyin-planedirections(0◦..85◦thinsolid,90◦thick,95◦..175◦dashed)forHallbarorientationsasinthesketcheswithcurrentalong(a)0◦,(b)20◦,(c)45◦,and(d)90◦.Allangleswithrespecttotheuniaxialeasyaxis.Thefieldissweptfromlefttoright.

(section4).

•Thefingerprintmethodcanalsobeusedtocharacterizeindividualtransportstructuresorevendevicecomponents.Uniaxialmagneticbehaviourwasrecentlyachievedbysubmicronpatterningof(Ga,Mn)Asandthecorrespondinganisotropicstrainrelaxation[15].

WeagaintrackthemagnetizationangleusingAMRmeasurementsandfinallydiscussthecolor-codedRPP,theanisotropyfingerprint,expectedforamaterialwithuniaxialmagneticanisotropy.

Fig.13showsAMRcurvescalculatedforamagneticallyuniaxialmaterialusingEq.5withε=30%Kuni.TheindividualpanelsillustratehowthecurrentdirectionwithrespecttotheeasyaxismodifiestheoverallpictureofasetofAMRcurves.Inallfourpanelsasinglezerofieldresistancestatecanbeidentified,correspondingtothe

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As

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(c)

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Figure14.CalculatedAMRfingerprintsofamagneticallyuniaxialmaterialwitheasyaxisalong135◦andcurrentalong0◦.Magnetizationreversalthrough(a)coherentrotationonly(Stoner-astroid)and(b)DWnucleationandpropagationwithεaccordingtoEq.13(c)simplifiedmodelassumingaconstantε=2ε90◦.(ε90◦=30%Kuni)

easyaxismagnetizationorientation.TheresistancevalueisgivenbytheanglebetweencurrentandeasyaxisthroughEq.2.Iftheexternalfieldissweptalongthiseasyaxisdirection(0◦,thinblackline),themagnetizationisalignedwiththeeasyaxisthroughoutthewholescan,yieldingahorizontallinethroughthefocalpointatzerofield.Thehardaxisscan(thickline)revealstheanisotropyfield;(thesameasinthebiaxialcase,Eq.9)

Ha=

2KuniM

(14)

theexternalmagneticfieldperpendiculartotheeasyaxis,wherethemagnetizationstartstodeviatefromthefielddirection.Themagnetizationrotationinpanels(a)and(d)yieldsaparabolicdependenceoftheresistanceonthefieldamplitude[16].InallotherMRscansthemagnetizationrelaxestotheclosesteasyaxisdirectionwhilethefieldisdecreasedfromhighnegativevalues,reachingthefocalpointatzerofield.Afterzero,themagnetizationdirectionreversesbycirca180◦throughDWnucleationandpropagation,whichisvisibleasabruptresistancechangesinFig.13,forexamplethespikesaround100%Kuniinpanel(a).Abacksweepresultsinahystereticallysymmetriccurvewiththeswitchingeventsatnegativefields(notshown).

Fig.14showstheresultsofsimilarcalculationswithhighangularresolutionplottedinRPPfashion.Heretheeasyaxisisorientedalong135◦andthecurrentflowalong0◦.Thecolorsareafunctionofthethecurrentdirection,forexampledarkcolorathighmagneticfieldsalongthecurrent,whiletheswitchingeventpatternisdefinedbythemagneticpropertiesalone.

Ifastructureissmallerthanthesingle-domainlimit[17,18]itisenergeticallyunfavourabletonucleateaDW.Insteadthemagnetizationrotatescoherently(Stoner-Wohlfarthmodel[8]).Fig.14ashowsthewellknownStoner-Wohlfarthastroid[8,19]whichdescribestheswitchingpositionsofauniaxialparticleundercoherentrotation.Itsextentinboththeeasyandthehardaxisdirectionisgivenbytheanisotropyfield.

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As21

AllowingforDWnucleationwithεaccordingtoeq.13truncatestheeasyaxiscornersoftheastroidasshowninFig.14b.Theextentε90◦/MintheeasyaxisdirectionisameasurefortheDWnucleation/propagationenergy.Afieldsweepalongthehardmagneticaxis,isstillfullydescribedbyStoner-Wohlfarthrotationandtheextentinthisdirectionisgivenbytheanisotropyfield.Thedetailedshapeoftheswitchingfieldpatterndependsonthemodelusedfortheε-dependenceontheDWangle.Fig.14cshowstheRPPcalculatedassumingaconstantε∆ϑ=2ε90◦independentofthemagnetizationdirectionsofthedomainsseparatedbytheDW.WhiletheeasyandhardaxisextentarethesameasinFig.14b,thebettercorrespondenceoftheshapeofthefeaturesin(b)then(c)totheexperimentaldataisfurtherevidenceinsupportfortheabovedescribeddescriptionoftheDWenergies.

4.TemperatureDependenceofthe(Ga,Mn)AsAnisotropy

Thefingerprintmethodprovidesuswiththeopportunitytoinvestigatethetemperaturedependenceofthemagneticanisotropy.Figures15and16showAMRfingerprintsatvarioustemperaturesforthelayerinvestigatedinFig.11at4.2K.TheleftcolumnshowsresultsonaHallbarpatternedalong90◦(the[100]crystaldirection).IntherightcolumntheHallbarisorientedalong0◦.Thelayeris,astypical,veryhomogeneousandtheswitchingpatternsinthetwocolumnsarevirtuallyidenticalatalltemperatures(exceptforatrivialinversionofthecolorscales).

Themainlybiaxialanisotropyistheoriginofthenearlyfour-foldsymmetryinthelowtemperaturefingerprints.Theuniaxialanisotropytermwitheasyaxisalongthe[110]crystaldirectiontakesoverwithincreasingtemperatureandbecomesthedominanttermclosetoTC:alreadythefingerprintsat30Kexhibitthetypical2-foldsymmetryofauniaxialanisotropy.Theshortaxisofthepatternmarkstheuniaxialeasyaxis;theextendedfeatureperpendiculartoitthemagnetichardaxis(seeSec.3fordetails).TheAMRamplitudeandtheswitchingfields,i.e.thesizeofthefingerprintpattern,decreasesignificantlywithtemperature(notethedifferentmagneticfieldscales).

ThisisconsistentwithdetailedSQUIDstudies[2,20].TheretheanisotropyconstantsKcrystandK[110]wereextractedfromhardaxismagnetizationmeasurementsvsmagneticfield.Thetwotermsexhibiteddifferenttemperaturedependence.Inparticularitwasobservedthatthetemperaturedependenceoftheanisotropyconstantsoriginatesintheirpower-lawdependenceonthevolumemagnetizationM.WhiletheuniaxialanisotropyconstantisproportionaltothesquareofM,thebiaxialtermdependsonM4.Asaresult,thebiaxialanisotropyterm,whichdominatesthemagneticbehaviourat4K,decreasesmuchfasterwithincreasingtemperaturethantheuniaxialterm.Thisisthereasonwhythemagneticanisotropyundergoesatransitionfrommainlybiaxialtomainlyuniaxialwhenthetemperatureincreasesfrom4KtoTC.

Fig.17ashowsSQUIDmeasurementsonthesampleofFigs.15and16.Aftermagnetizingthesamplealongagivendirection,wemeasuretheprojectionoftheremanentmagnetizationontherespectiveaxisanditsevolutionwithincreasing

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Figure15.TemperaturedependentAMRfingerprintmeasurementsofthesampleinFig.11aandb(rightcolumn)withcurrentalong0◦andFig.11candd(leftcolumn)withcurrentalong90◦.

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As

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Figure16.Fig.15.

HightemperatureAMRfingerprintmeasurements,continuationof

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As

20Longitudinal Moment [10A/m]Long Moment [kA/m]24

a)320b)DW Nucleation Energy ε [J/m]3015100-20-1000100Magnetic Field [mT]ε / M [ mT ]10151050020405 <010> [110] [-110] |M| Fingerprints10T [ K ]180051015202530Temperature [K]3045001020304050Temperature [K]6070Figure17.a)MeasurementoftheprojectionoftheremanentmagneticmomentofthesampleofFigs.15and16ontodifferentcrystalaxesbySQUIDmagnetometry.VerticalgraylinesindicatethetemperaturesofthefingerprintmeasurementsinFigs.15and16.b)Domainwallnucleationenergyε90◦(symbols)versustemperature,derivedfromε90◦/M(inset)extractedfromthefingerprints.

temperature.Displayedaremeasurementsalongthetwo4Khardmagneticaxes[110]and[110]andoneofthebiaxialeasyaxes󰀆010󰀇.Theyshowthesameanisotropytransitionasthefingerprintsabove.At4K,the󰀆010󰀇crystaldirectionisclosetoaglobalmagneticeasyaxisandthusshowsthelargestprojectionoftheremanentmagneticmoment.The[110]directioncoincideswiththeeasyaxisoftheuniaxialKuni[110]anisotropyterm.Thatiswhyitisclosertoaglobaleasyaxisthanthe[110]direction[21]andinconsequenceshowsalargerprojectionoftheremanentmoment.Astemperatureincreases,themagnetizationdecreasesandtherelativeamplitudeoftheanisotropytermschanges,asdescribedabove.Thisresultsinagradualreorientationoftheglobaleasyaxeswithtemperature,changingtheanglebetweenremanentmagnetization(alongtheglobaleasyaxisclosesttothesweepdirectioninFig.17a)andtherespectivesweepdirection.Theresultofboththedecreasingvolumemagnetizationandchangingrelativeprojectionsontothedifferentsweepdirections,canbeseeninFig.17a.Thegreen[110]remanence,e.g.,gainsrelativeweightwithincreasingtemperature.Thissupportstheobservationsofthefingerprintmeasurements,wherethe[110]anisotropytermgainsininfluenceathighertemperatures.Giventhespecificanisotropybehaviour,knownfromthetransportmeasurements,wecanestimatetheabsolutevalueoftheremanentmagnetizationfromthesquarerootofthesumofthesquaresofthetwomagnetizationprojectionsalong[110]and[110](Pythagoreantheorem)[2].TheresultisdisplayedingrayinFig.17a.SuchamagnetizationmeasurementwithSQUIDiscomplementarytotransportinvestigations,sincethosecanonlygiveenergyscalesinfieldunits,i.e.normalizedtothevolumemagnetizationlikeK/Morε/M.

Thequantitativedeterminationoftheanisotropyconstantsathighertemperaturesismorecomplexthanat4Kandworkisongoingtofindasetofstraightforwardrulesasforthemainlybiaxialsystemat4K.Determiningthedomainwall

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As

a)

4

25

b)

, , <100> <110>|M| Magnetic Field [mT] Fingerprint in b)10

-6Remanent Moment [10emu]5

3

0

2

1

-5

00

10

20

30

40

50

60

70

80

Temperature [ K]

-10

-10

-5

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Figure18.a)Projectionoftheremanentmagneticmomentofa20nmthick(Ga,Mn)AslayermeasuredwithSQUIDalongdifferentcrystalaxes.35K,thetemperatureoftheanisotropyfingerprintmeasurement[28]inb)isindicatedbyaverticalgrayline.

nucleation/propagationenergyε,however,ispossiblewiththedescribedtechniques.BlacksymbolsinFig.17bshowpreliminaryresultsdeterminedfromthefingerprintsinFigs.15and16.Thelineisaguidetotheeye.Themethodfortheextractionbuildsonthetechniquesdescribedinsection2:2ε/Misbasicallygivenbythediagonaloftherectangularfirstswitchingfieldpatternformainlybiaxialsamplesandbytheeasyaxisdirectiondiameterforpurelyuniaxialsamples.Thestrengthofthismethodisthatwecanextractε90◦easilyfromtheplots,becausetheglobaleasyaxesdirectionsareobviousfromsymmetryconsiderations.Itisnotnecessarytoassumeaconstant(orknown)globaleasyaxisdirectionandwecanthusfullyaccountforthecomplextemperaturedependenceoftheeasyaxisbehaviourwithoutfittingthedatatoacomplicatedmodel.BoththedeterminationofMandofε/Marenotasaccurateinthetransitionregion,wheretheenergysurfaceatzerofieldisalmostflatoverawideangularrange.ThisisaprobablecauseofthedeviationfromperfectexponentialbehaviourforthedatainFig.17batintermediatetemperatures.

Thesquarehysteresisloopwithabruptswitchingevents,shownintheinsetofFig.17a,pointstoaDWnucleationdominatedprocess,asopposedtoaprocess,wheretheenergyneededforDWpropagationisthelimitingparameter[22].AlsothetemperaturedependenceoftheDWnucleationenergyinFig.17bfitstothestandardexponentialbehaviourexpectedforthetemperaturedependenceofthecoercivity[23,24].Wesuggestthattheabovemethodisonetoolthat,incombinationwith,e.g.,timedependentandopticalinvestigations[25],canclarifytheDWnucleationprocessin(Ga,Mn)As.Itcancomplementrecentopticalstudies,thatidentifythenatureofpinningcentersandvisualizetheprocessofDW-relatedmagnetizationswitchingin(Ga,Mn)As[26,27].

DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As26

SQUIDstudiesonanothersampleareshowninFig.18a.AsinFig.17aweplottheprojectionoftheremanentmagneticmomentvstemperature.Shownaremeasurementsalongthetwobiaxialeasyaxes(redandpurple)andalongthetwobisectingdirections(greenandblue).Theabsolutevalueofthevolumemagnetizationisestimatedasdiscussedabove(gray).Thelargedifferencebetweenthetwobiaxialeasyaxesdirections(redandpurple)atintermediatetemperatures(25to60K)pointstoasymmetrybreakingcausedbyarelativelystronguniaxial[010]anisotropycomponent.Forthisreasonweinvestigatethissampleat35K,wherethe[010]componentshouldbestronglyvisibleinthesymmetryofthefingerprintpattern,andwherethetransportsignalisstilllargeenoughtogetcleanmeasurements.Fig.18bshowstheresultingfingerprint.Thesymmetrybreakingbetweenthetwobiaxialeasyaxes(herealong0◦and90◦)isapparentfromthepicture.Therelativelystronguniaxial[010]termcausesapreferenceforthemagnetizationorientationalong90◦.Theresistancepolarplotinturnresemblesinpartsatypicalbiaxialfingerprintpattern(between45◦and135◦andthepointsymmetricregion)andintheotherquadrantsatypicaluniaxialfingerprintpattern(between135◦and225◦).[28]Wecanthusconclude,thattherelativelysmalluniaxialtermgainsinimportanceatintermediatetemperaturesinthissample.Thisiswherethetwostrongeranisotropytermshaveapproximatelyequalweight,compensatingeachotherinspecificangularregions.Asmallextratermintheenergyequationthenplaysahugerole:itcreatesanadditionallocalminimumintheenergysurface,causingverydifferentswitchingbehaviourindifferentquadrantsofthepolarplot.

Insummary,wehaveshownthattheextendedanisotropyfingerprinttechniqueisapowerfulmethodtoaccessthefinedetailsofcomplexanisotropiesinferromagneticsemiconductors.Weusedthismethodtoshowthatalltransportlayersinvestigatedshowedthreesymmetrycomponentsofthemagneticanisotropy;themainbiaxialtermandtwouniaxialtermsalongthe[110]andthe[010]crystaldirections.Therelativestrengthoftheseanisotropytermsisroughlyspeaking,oftheorderofKcyst:Kuni[110]:Kuni[010]∼100:10:1at4K.Athighertemperaturestherelativestrengthofthe[110]anisotropycomponentincreases.TheoverallbehaviouroftheanisotropytermsisconsistentwithSQUIDinvestigations,showingthetypicaltransitionfromamainlybiaxialtoamainlyuniaxialmaterialwithincreasingtemperature.Anextractionofthe90◦-DWnucleationenergyanditstemperaturedependenceisalsopossible.Measurementshaveshown,thatthe[010]uniaxialanisotropyterm,whoseexistenceissometimesquestioned,canbeclearlyobserved.Weshowthatitcanhaveaparticularlystrongimpactontheswitchingbehaviourforcaseswherethecooperativeeffectofthebiaxialandthe[110]uniaxialanisotropytermleadtoaflattenedenergysurface.Acknowledgements

TheauthorswishtothankS.H¨umpfnerandV.HockforsamplepreparationandC.ChappertandW.VanRoyforusefuldiscussions,andacknowledgethefinancialsupportfromtheEU(NANOSPINFP6-IST-015728)andtheGermanDFG(BR1960/2-2).

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DetailedTransportInvestigationoftheMagneticAnisotropyof(Ga,Mn)As28

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