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Solid StateCommunications,Vol. 21,pp. 647—650,1977. PergamonPress. Printedin GreatBritain
dHvA OBSERVATIONOF SPIN-DEPENDENTSCATTERINGBY MAGNETIC PAIRSOF Co IMPURITIES IN Au*
Yun ChungandD.H. Lowndest
Departmentof Physics,University of Oregon,Eugene,OR97403,U.S.A.
(Received16 July 1976by H. Suhi)
Wereportthe first dHvA observationsof interactioneffectsbetweenimpurities,via measurementsof the spin-dependentscatteringof con-ductionelectronsby magneticpairs of Co impuritiesin Au. Thelocalmomentassociatedwith Co pairsproducesthe onlyspin-dependentscatteringin thesedilutealloysbecauseisolatedCo impurities(whicharealsopresent)carryno localmomentat low temperatures.Theresultsareusedto estimatethe concentration(or averageseparation)of magneticpairs.
THE MAGNETISM associatedwith interactingpairs, the conductionelectron-impurity(o~relectron-pairortripletsandevenlargerclustersof transitionmetal electron-triplet)interaction,if possible.Suchobser-impuritiesin non-magnetichostshasrecentlybecomean vationscouldprovidealink betweenthemagnetismareaof considerableexperimentaland theoretical occurringsuccessivelyat isolatedimpurities,pairs,trip-activity. Forexample,the low temperaturemagnetiz- lets(or othersmallclusters)and theeventualoccurrenceationsof Au(Fe),1 Au(Co),2’3 Cu(Fe)46andCu(Co)2’7 of long-range-orderingof thespin-glasstype,if the samedilutealloy serieshavebeenshownto containcontri- longrangeinteractionvia theconductionelectronsis in-butionsproportionalto c2 andc3 (c = soluteconcen- deedresponsiblefor both.tration),which areinterpretedasarising from pairsand During thepastyearwehavedevelopedanew tech-tripletsof soluteatoms,respectively.Suchexperiments niquefor obtainingatomiclevelinformation abouttheareof fundamentalinterestin demonstratingthe interactionsof conductionelectronswith localmagneticimportantrole of local environmentin determining momentsin non-magnetichosts.10’~Thetechniqueuseswhethera localmomentwill appearon a given impurity the information containedin thefirst threeharmonicsofsite.8TheFriedel—Andersonconditionfor magnetism the dHvA effect(“Third HarmonicWaveshapeAnalysis”,at a soluteatomsite dependsdirectlyon the local den- THWA). Informationis obtainedregardingboththesity of d-statesat theFermilevel,which maybe locally spin-dependentscatteringof conductionelectronsbymodifiedby interactions,eitherwithnearestneighbour local moments{[r’(k)JT * [f’(k)] ~} andabouttheatoms(perhapsof d—doverlaptype)or by alonger exchangeenergyshift, ~e, in theelectronicLandaurangeinteraction(perhapsof the RKKY type).9 energylevels,dueto their interactionwith localEvidencethatpairs arenotrestrictedto nearneighbours moments.In comparisonwithearliertechniques~”3forcomesfrom thelargefractionof ferromagneticpairs obtainingthis typeof information from the dHvA effect,estimatedfor Au(Fe)(c* = pair concentration~ theTHWA techniquehas theadvantagethat changesin1080c2), which cannotbe explainedwithout long- thespin-dependentscatteringratesandexchangeenergyrangeinteractionsbetweenimpurities.1 Similar (though shifts canbemappedoutasfunctionsof H and T, andsmaller)estimateshavebeenmadefor Au(Co), Cu(Fe) that thetechniqueitself is somewhatmoregenerallyandCu(Co),with thecritical radiusfor pair formation applicableto a varietyof extremalorbits, on theFermiestimatedto lie in the range5—11 A.3’6 surfacesof typicalmetals.U
Since theconductionelectrons(at the Fermilevel) A local magneticmoment(either at anisolatedwould directly mediatean RKKY.type interactionbe- impurity site, or thatassociatedwith a pairof interactingtweenimpurities (or betweenmagneticpairs,triplets, impuritieswhich behavesasan independentmagneticetc.),it is of considerableinterestto directly observe entity) canaffect conductionelectronenergylevelsin a
magneticfield in two ways: by producingan exchange* Researchcarriedout at theUniversityof Oregonwith energyshift, ~aC !IBHeX, in therelativeseparationof
supportfrom NationalScienceFoundationGrantNo. thespin-upandspin-downelectronicLandauenergyM 7 -07652A02 levels (aneffecton therealpartof the electronicself-
t Addressfor 1976—77:FysischLaboratorium, energy)andby producingunequalscatteringratesforKatholiekeUniversiteit,Nijmegen,TheNetherlands. thespin-upandspin-downelectronsin amagneticfield
647
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648 dHvA OBSERVATIONOF Co IMPURITIES IN Au Vol. 21,No.7
(aneffecton the imaginarypart of the electronic wherer is theharmonicindex, X = 146.9 (KG/K), ~‘ =
self-energy).’4’6In dHvA measurementsonedeter- 1.304x 1o—~(Oe1~‘2/K),F is the dHvA frequency,~ isminesan orbitally-averagedelectronicscatteringrate(or theOnsagerphasefactor,p is thereducedcyclotronDingle temperature),X = (~‘21rkB)((1/r)orb),for an effectivemass,C is the Fermi surfacecurvaturefactor,extremalorbit on the Fermi surface,sothat in the g is thecyclotronorbitally-averagedg-factor,andlandpresenceof spin-dependentscattering(SDS)thereare ~5Xwere definedabove.separateDingle temperatures,X~andX~,for theup. The effectof SDSentersthroughboth theamplitudespinanddown-spinelectrons, factor,A., andthroughthephaseshift, L~Or,which is
Shiba,14Harrisetal.15 andFenton’6haveshown inducedin therth dHvA harmonicby the combinationthatexplicit expressionsfor! = (Xt + I ~)/2, 1JX= of SDS andtheexchangeenergyshift &. In theabsence(xt ~Xt)/
2, and~e, andtheir effecton the of SDSno phaseshift occursin anyof the resultant,dHvA amplitudeandphase,maybe determinedby observableharmonicoscillations.In practice,theobser-calculatingthecomplexelectronicself-energy(orthe vabledHvA waveformusuallycontainsoneadditionalt-matrix) in variousapproximations.Frompertur- sourceof harmonicdistortion: that due to the Magneticbationtheorycalculations(valid for [(~SpBH)
2+ Interaction(MI, or Shoenberg)Effect. However,the(ITkBT)2]”2 ~ lc~T~,whereTK in our situationrefers effectsof MI canbe convenientlyincludedin theTHWAto the carriersof local moments)onemay obtain technique,beginningfrom theiterativeexpansionpro-
cedureof PhillipsandGold.’8As we haveshownelse-
[(~exY~— GCex)~] PBHex = 2c*JCd(Sz), (1) where,it is usuallynecessaryto considertheeffectsof— * 2 MI in carryingoutTHWA, sinceMI andSDSeffectskB6X — 2c C~p(S~)J, (2) . . , . 1117
mix nonlinearlyat thehigher dHvA harmonics.
The experimentalproblemposedby (3) is to obtainwherec~’is the fractionalconcentrationofcarriers of uniquevaluesfor ~X, I andH~as functionsof H andTlocal moments,J is the full exchangeintegralincluding from measurementsof the amplitudesandphasesof thetheeffectsof orbital degeneracy,Cd is thefractional dHvA harmonics.In theTHWA techniquethis isd-wavechargedensityfor theextremalorbit used(an accomplishedby Fourier transformof a shortblock oforbitalaverage),p is theconductionelectrondensityof dHvA data,obtainedusingthe field modulationtech-statesfor a singlespinand(Se)is thespinvalueassoci- nique;this yieldsresultantharmonicvoltageamplitudes,atedwith the local moments.As (1) and(2) show,both Rr, andphaseangles,Or, (thelattermeasuredrelativeto~X andH~,are expectedto be proportionalto <Se> and the left edgeof a datawindow) for r = 1, 2, 3. Fromto theconcentrationof local moments,c*. thesewe define4 “observables”:two ratiosof harmonic
The effect of ~X * 0 andHex *0 is to alterboth amplitudes,R2/R1 andR3/R ~, andtwo relativehar-
therelativeamplitudesof differentharmonicsin the monic phasedifferences,(20k — 02) and(3O~— 03).dHvA magnetizationoscillationsandto altertheir Thesefour “observables”are independentof the elec-phases.Consideringtheeffectsof SDS andthe Landau tronic gainof the signaldetectionsystemandof theenergylevel splittingsdue to both theZeemaneffect arbitraryinitial phaseof a shortdatablock
13 and,usingandthe local momentexchangeenergyshift, the (3) and(4), canbe expressedin termsof the 3 micro-equationfor theoscifiatorymagnetizationis modified scopicparameters~X, I andHex which governthemag-from thatof the pureLifshitz—Kosevitchtheory, and netizationwaveformitself.’°”1 The equationsconnectingbecomes’1”7 the4 “observables”to the 3 microscopicparametersare
/ transcendental,andsothey aremostconvenientlysolved— 00 , IF \ ~ 1 usinga laboratorycomputerinterfacedto a plotter.
M = — ~1ArCr!Y sin 2irr I~j—3/ + P~-+ Mrj~ THWA experimentswere carriedout for 5 different
singlecrystalsamplesof Au(Co), using theNeckorbit
where (3) with themagneticfield along(111). The ~JXvaluesfoundfor bothhigh (— 500 at.ppm)and low (150 at.
A = [E2” + E2’~+ 2 cos (2irrS’)]~2/2, ppm)concentrationsamplesare shownin Fig. 1. For
L~Or= tan’ [tan (irrS’) tanh(r?~p~X/H)], the low concentrationsamplesthemeanvalueis ~X0.022 K and for thehighconcentrationsamplebX
D = exp (— XpX/H), E = exp( ?~z~X/H), 0.20K, a ratio of 9: 1. Since the concentrationratio isvTF only — 3.3:1, it seemsveryclearthat the spin-depen-
Cr = (CrH)Lll 2 sinh(rXpT/I-f) S = [g — (Hex/H)], dent part of theconductionelectronscatteringis notlinearly dependenton concentration,but scalesnearly
(4) proportionalto theCo concentrationsquared.We
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Vol. 21,No.7 dHvA OBSERVATION OFCo IMPURITIES IN Au 649
.~ theexchangecouplingbetweenconductionelectronsandthelocal momentassociatedwith a Copair, then
20 ~ (usingp=0.15. ~ U states/eV-at.andCd= 0.5for the Neck orbit). Taking. U <Se) 2S 3.0(assumingsaturatedmomentsfor
.15 H= SOkOe)3givesc* 100c2,seeminglyin goodagree-
ment with the resultfor pairs formedvia a long-range(RKKY-like) interaction.(Taking(SZ) <25 givesstilllargervaluesfor c*).
.05 However,equation(1)then predictsanexchangeenergyshiftof magnitudeHex~7.SkOe;in fact, we
~ ,. •, observedno suchshift, thoughwe probablywould not
40 ~ H~G) 70 havedetecteda shift of 2.5kOeor less.Equations(1)and(2) canbothbesimultaneouslysatisfied,with
Fig. 1. The difference,61, in thespin-upandspin-down H~< 2.SkOe,if IJ~‘~‘4eV. In this casec* 10c2 ifconductionelectronDingle temperatures(orbitally (SZ)hasits saturationvalue butc* is againlargerthanaveragedscatteringrates)for the (111)Neck orbit in thenear-neighborpairlimiting valueif <Se> islessthanAu(Co) at T = 1.1 K. The datawereobtainedfrom 3differentsinglecrystalsamplesfor the 150 at.ppm saturated.Thus,onlyif IJI has the unreasonablylargealloy and2 differentsamplesfor the 500at.ppmalloy, valueof 4eVcanthelocal momentconcentrationThe errorbarsrepresenttheprobablerandomuncertain- approachthelow valuedescribingnearestneighbortiesinSI resultingfrom the estimatedexperimental pairs.errorsin the relativedHvA harmonicphaseangle Webelieve,instead,thatourmeasurementsof 61,measuremen via equation(2), dosupportthe ideaof a muchhigher
interpretthis resultasclearevidencefor spin-dependent concentrationof local moments(anda reasonableIJIscatteringof conductionelectronsby interactingpairsof value)andthatequation(1), givingtheconductionelec-Co impurities(whoseconcentrationc* ~ c2), which tron exchangeenergyshift, maybeinvalid in thecaseapparentlyact asindependentmagneticentitiesand of a spatiallyextendedpairmomentformedby a longwhich carryaninducedlocal momentin an applied rangeinteraction.Forexample,anRKKY interactionmagneticfield. ThemeanNeck electronscatteringrates, shouldgive riseto both ferromagneticandantiferro-1, for the two concentrations,wereI = 0.93(±0.04)K magneticpairs,perhapsproducingsomecancellationand!= 2.97(±0.15)K, linearlyproportionalto the in the resultantexchangeshift for the conduction(non-magnetic)Co concentrationwithin the (small) electrons,but withonly the ferromagneticpairsexperimentalerrors.Thus,(Xt — It) = 261is only 4% producingspin-dependentscattering,via their localand13% of Ifor thelow andhigh concentration moments.samples,respectively.(That61 c2 while~ x c is con- In summary,theseexperimentsshowthat the dHvAvincingly demonstratedby attemptingplotsof SIand effectcanprovidea “window” throughwhich to viewIvs bothc andc2,with the constraintthatSI= I = 0 the “onsetof magnetism”dueto interactionsbetweenforc = 0). impurities, if theseimpuritiesare non-magneticwhen
Accordingto equations(2) and(1) theseresults isolatedsufficiently from eachother. Our interpretationcanbe usedto estimatethe concentration,c*, of local that theSDS whichwe seeis due to the inducedmoments,hence,theeffectiverangeof theinteraction momenton interactingCopairs completesthepictureproducingmagneticpairsof Co impurities. Forrandom proposedby Boucai et aL,3 in which isolatedConearestneighborpairs the pair concentrationexpected impuritiescarrynolocal momentbut groupsof 3 oris c* = 6c2,while for randompairswith aneffective moreCo atomsare spontaneouslymagnetic.And, it isinteractionrangeof 10A3’6 the pair concentrationis amusingthatwedetectthis inducedlocal momentvia
= 1 25c2.ForsingleFe momentsin Au11 our recent the quantumoscifiationsin the susceptibffityof thoseTHWA measurementsshowedthat IJI 1.3eV using sameconductionelectronswhich arepresumedtobe theequations(1) and(2). If we assumea similar value for mediatorsof the pairinginteractionbetweenimpurities.
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650 dHvA OBSERVATION OFCo IMPURITIES IN Au Vol. 21,No.7
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