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

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

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.

650 dHvA OBSERVATION OFCo IMPURITIES IN Au Vol. 21,No.7

REFERENCES

1. FROSSATIG., THOLENCEJ.L.,THOULOUZE D.& TOURNIERR.,Proc. 14thmt. ConfLow Temp.Phys.3, 370(1975).

2. TOURNIERR. & BLANDIN A., Phys.Rev.Lett. 24,397(1970).

3. BOUCAI E., LECOANETB., PILON J., THOLENCEJ.L. & TOURNIERR.,Phys.Rev.3B, 3834(1971).

4. THOLENCEJ.L. & TOURNIERR.,Phys.Rev. Lett. 25,867(1970).

5. HIRSCHKOFFE.C., SHANABARGERM.R., SYMKO O~G.& WHEATLEY J.C.,J. Low. Temp.Phys.5,545(1971).

6. THOLENCEJ.L. & TOURNIERR.,Proc. 13thmt. ConfLow Temp.Phys.749 (1972).

7. TISSIERB. & TOURNIERR.,Proc. 14thmt. Conf Low Temp.Phys.3, 378 (1975).

8. CAROLI B., J.Phys.Chem.Solids28, 1427(1967).

9. SOULETIEJ.& TOURNIERR.,J. dePhys.32,Cl-172-178(1971).

10. YUN CHUNG, UnpublishedPh.D.Dissertation,Universityof Oregon(1975).

11. YUN CHUNG & LOWNDESD.H.,SolidStateCommun.20,101 (1976).

12. COLERIDGEP.T., SCOTTG.B.& TEMPLETON I.M., Can. J.Phys.50, 1999 (1972).

13. ALLES H.G.,HIGGINS R.J.& LOWNDES D.H.,Phys.Rev.Lett. 30,705(1973).

14. SHIBA H.,Frog. Theor. Phys.50,1797 (1973).

15. HARRISR.,MUUMANI B.G. & ZUCKERMANN M.J.,Phys.Cond. Mat. 19,269(1975).

16. FENTONE.W., J.Phys.F.: Met. Phys.6,363 (1976).

17. LOWNDESD.H. & CHUNG YUN, Phys.Cond. Mat. 19,285 (1975)and(to bepublished).

18. PHILLIPS R.A. & GOLD A.V., Phys.Rev. 178,932(1969).