PhysicsLettersAl60(1991)315—318 PHYSICSLETTERS ANorth-Holland

Nuclearmagneticresonancein the Condondomainstate

A. Gordon’Max-PlanckInst itUt für Festkorperforschung,W-7000Stuttgart80, Germany

B. GrushkoInstit Ut fürFestkorperforschung,Forschungszentrum,Julich GmbH,Postfach 1913, W-5170 Jülich, Germany

I.D. VagnerDepartmentofPhysicsandAstronomy,TheJohnsHopkinsUniversity,Baltimore, MD 21218,USAandSolidStateInstitute,Technion— IsraelInstituteofTechnology,TechnionCity, Haifa 32000,Israel

and

P. WyderMax-Planck-InstitutflkFestkorperforschung,Hochfeld—Magnetlabor,B.P.166X,F-38042,GrenobleCedex,France

Received30 September1991; acceptedfor publication9 October1991Communicatedby J. Flouquet

Calculationsof splittingsof nuclearmagneticresonance(NMR) lines atdiamagneticphasetransitionsarecarriedOut for athree-dimensionalelectrongas.Thereis a good agreementbetweenthetheoryandtheexperimentalresultsin silver.ThesizeoftheCondondomainsandthewidth of thedomainwalls areestimated.Experimentalresultson thedeHaas—vanAlpheneffectinsilverareanalyzedwith thehelpof thetemperature—magnetic-fielddiagram.Theresultsoftheanalysisarein agreementwith theNMR data.TheNMR splitting in atwo-dimensionalelectrongasdueto Condondomainsis calculated.

As is known, normal metalsunderthe condition netic interactionbetweenelectronswhich becomesof the strongde Haas—vanAlphen (dHvA) effect importantwhenthedHvA effectis strongenoughtomay undergoa diamagneticphasetransition to a make the internal field in the samplesignificantlyCondon-domainstate [1,2]. Thereforetherecanex- different from the appliedfield [41.The differenceist foreachdHvA cycle aregionof appliedmagnetic- betweenthesetwo fields becomessignificant onlyfield strengthin which no portion of the field-in- when the amplitude of the magnetizationoscilla-duction isothermis thermodynamicallystable [3]. tionsis comparedwith theirperiod.TheinternalfieldThenthesampleis divided into regionswith differ- responsiblefor determiningthe energylevels insideent valuesof the magnetizationanda domainstruc- the sampleis the macroscopicmagneticinduction,tureresults.Thesedomainsare regionsin which the B. Theexistenceof magneticdomainsin silver wasmagnetizationis uniform; in the domain walls the provedby Condonand Walstedt [2]. The appear-magnetizationchangessmoothlyfromitsvalueinone anceof domainsin silverplatewasdemonstratedbydomaintoits valuein theneighbouringdomain.The the simultaneousoccurrenceof two nuclearmag-reasonof theexistenceof Condondomainsisa mag- netic resonance(NMR) frequenciesover just the

appropriatepart ofthedHvA cycle,while only asin-Permanentaddress:DepartmentofMathematicsandPhysics, gle NMR frequencywasobservedover the part ofOranim,Haifa University,36910Tivon, Israel. thecyclewithout domains.Theveryexistenceoftwo

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well-separatedNMR linesof 109Ag indicatesthat the thermodynamicpoint of view, coexisting domainsdomain-wallwidth is much smallerthanthe domain representdifferent phasesof a metal with differentthickness, magneticinductions.The mostimportantcaseis that

Accordingto the analysisof the NMR datamade in which the characteristiclinear dimensionsof ain ref. [21, thereis only a qualitativeagreementbe- samplearelargecomparedwith the cyclotronradiustweenthe experimentandtheavailabletheoryof the r~so that the domainsized is considerablygreaterdHvA effect. TheseNMR experimentsshowed:(i) thanthethicknessof thedomainwall A. Sinced>> A,an existenceof two magneticdomains;(ii) the fact the domainwall may be regardedas planarandthethat theparameterconnectedbetweentheamplitude problemas one-dimensional[141. Wechoosethe xof the magneticoscillationsandthe periodof dHvA axis as perpendicularto the domain wall. A mm-oscillationsis larger thanunity in the domainstate. imization of the thermodynamicpotential (1), (2)

In spiteof the fact that in recentyears the dia- includingthegradientterm (DteO)givesthe knownmagneticphasetransitionshave been extensively result [6,8,14,151studied in the three-dimensionaland the two-di- — 2

B=±(a/b)” tanh(x/A), (5)mensionalelectrongas [5—11],calculationsof theNMR splitting havenot beencarriedout and the where thewidth of the domain wall A is givenbyquantitativerelationbetweenthe magneticdomains 4—2 D ‘a 1/2 6and the splitting of the NMR lines has not been — 1

established, for the following boundaryconditions,In this communicationwe explain the NMR ex- — . —

Fm B—+ ‘a’b”2 lim dB’dx—O ‘7periments[2] on the basisof the theory of the dia- ~ — — / / ‘ I —

magneticphasetransitions[12,7,11].Our thermodynamicpotentialis As a result of (4), thedifferencebetweenthevalues

of the magneticinductionin two differentdomains

~=~LKs(Bo)+ ~‘-~ + J~dv, (1) EtB is equaltoB

2—B1_—Ei.B=2(a/b)”2. (8)

where— — To calculate~.Bwe usethefollowing expressionsfor

~=~aB2+~bB4+D(VB)2, (2) coefficients a and b: ~a=(4~t~—1)/4it [15] and

whereweconsiderthespatiallyvaryingmagneticin-4b= (Ef/ilQ~Bo) [12], whereE~is the Fermi en-

ductionB as ergy,Q~is the cyclotronfrequency,x= ÔM/0B is the— maximal differential susceptibility,where M is the

B(r) —B0+B(r) , IB(r) I << I Bo I (3) magnetization.Accordingto ref. [3] 4m~=A,where

Herefl is the nonuniform part of the magneticin- A is a measureofthe magneticinteraction.ForA> Iductionwhich is small in comparisonwith the uni- the transition into the domainstateoccurs.form part B0. bLKS is the thermodynamicpotential Substitutingexpressionsfor aandb into (8) wefor thecaseof auniform magneticfield B0 calculated obtainby Lifshitz andKosevich [13], taking into account ~ l\1~~2hQ~B0the Shoenbergcorrection [4] replacingH by B. The ~ = ~ 2~ ) E, ‘ (9)coefficient b andthe coefficient D of the gradientterm are positive;D = ~r~[141; r~is the cyclotron Let us comparethe valueof i~s.Bcalculatedby (9)radius. with that determinedin the experimentin ref. [2].

A minimizationof the functional (1), (2) in the Theexperimentwascarriedoutin a plate-shapedsil-caseD=0 gives two domainswith magneticinduc- ver sampleorientedperpendicularto the magnetictions B1 andB2: field H of 90 kG with the [1001directionparallelto

B —B ‘b’~2 B —B + ‘b”2 (4\ the field. For this samplegeometrythe demagneti-— o— (a, / , 2— 0 (a, / “ / zationfactorn= 1 andthusB=H. TheFermienergy

while in the statewithoutdomainsB=B0. Fromthe for silver is Ef= 5.5 eV [4], the cyclotronfrequency

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calculatedfor this field Q~=l.57x 1012 Hz. The val- 3ues A or x~dependingon the appliedfield, temper-ature and sample orientation for each material,shouldbe determinedin independentexperiments. ~ 2

period:~=H~/F and M0 is the magnetizationam- 1 -In thefollowing weshallfind theparameterA whichis equal to A=8it2M

0/Aff [3,4] (All is the dHvAplitude) is 3.3 for theseconditionsat T= 1.4K. Thenthecalculationaccordingto (9) givesAB= 10 0. Theexperimentalvaluefound by CondonandWalstedt 0 ‘ ‘ I

—it —2—1 01 2 itin silver [2] is close to this magnitude: AB=11 Gat 2it FT= 1.4 K (AB is the difference between the magneticinductionsin two domainscorrespondingto the ob- hservedfrequencydifferenceoverthe domainregion(see (9)). Theaccuracyof a determinationof the Fig. 1. Temperature—magnetic-fielddiagramfor onecycleof the

dHvA oscillation in thesilver sampleorientedwith the [100]magneticinductionB in ref. [2] is to within 0.5 0. directionparallelto themagneticfield of 90kG. Thediagramis

Theaboveresultcanbe illustratedby meansof the calculatedfor TD = 0.8K. HereH0 is thestrengthof themagnetic

magnetic-field—temperaturediagrambuilt on theba- field at thecentreof thedHvA cycleandh = H—H0 is thedevia-

sisof the Lifshitz—Kosevichformula.Neglectingthe tion fromthis value.

higherharmonicsin the first approximationwe canpresentthe Lifshitz—Kosevichformula [13] for the whereH0 is the strengthof the magneticfield at theamplitudeM0 of the magnetizationoscillationsat the centreof thedHvA cycle.Thush is themeasuredex-fundamentalfrequencyasfollows [4], ternalappliedmagneticfield. Thevertexofthecurve

QFGCTB”2RD in fig. 1 is at T=2.55 K (A=l). It is very closeto

M0 = sinh(aT/B) (10) the temperatureof the domain appearancein the

NMR measurements[2]: T=2.5 K. Fromthis dia-where T is the temperature,Q= (1/it) (2e/~/cit)1/2

gramwe candirectlyestimatea possibleNMR split-RD is the Dingle reductionfactor:

ting connectedwith thedomainexistence.At T= 1.4RD=exp(—aTD/B), (11) K we obtain AB=A110=l2.5 G, whereAll0 is the

field differencein two domains(n= 1), while thewhere TD is the Dingle temperatureand a= 1.469X 10

5(m”/m) G/K, m* is the effectivemassof the measuredsplitting AB=l 1 G. The slight differenceelectron,m is the electronmass,F is the frequency maybecausedby the fact that the demagnetization

factorn for therealplate-shapedsample(sizesoftheof themagnetizationoscillations,C is thecurvaturefactor of the Fermi surface,G is the spin splitting plateare 8 mmx 8 mmx0.8 mm) isnotequalto un-factor. ity but n=0.84 [4].

If aT/B is largeenoughfor sinh to bereplacedby We canestimatethe typical domainsize andthe~exp,we obtain domain-wall width. The domain size is found by

minimizing thesumof thefield andthedomain-wallM

0=(2FQCT/B”2) exp[—a(T+TD)/B] . (12) energies [16]: d=(Lr~/4it~—l)”2~37~.tm,where

If we use the following experimental data for silver L is the sample size (about8 mm in the experimentthen for belly oscillations, for B

0=H=90 kG (the [2]). This value is very close to that mentioned indemagnetizationfactorn = 1 fora plate),TD = 0.8K, ref. [2]: d~30 j.tm. Theestimateof thedomain-wallm*/m=0.936[17],F=4.77xl0SG[18],C=0.354 width gives 4=r~/(4it~—1)”

2~0.4jsm (for[19], G~1 [20], wehavethediagramfor the (100) r~=lx l0~cm [4]). Theratio d/Ashouldbeclosedirectionof theappliedmagneticfield shown in fig. to the ratio AB/r, where~ is the naturallinewidth1. The details of the diagramconstructionare de- of the resonance[2]. Since the linewidth is of thescribedin ref. [21]. In fig. 1 hisequalto h=H—H

0, orderof a tenth of a gausswe havevery close the-

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oretical and experimental results:d/A 0.9 x 102 and Israel Foundationfor Scientific Researchand De-AB/a~lx102. velopment, Grant No. 0-112-279, 7/88. One of us

It was pointed out in ref. [5] that the magnetic (A.G.) is indebtedto the Minerva foundation forinteraction should be strikingly different in a two-di- the support.mensionalelectrongas. In particular,domainspinthe Fermilevel betweenLandaulevels — a conditionnecessaryfor the observationof the quantumHall Referenceseffect. Since,within eachdomain,the Landaulevelsare either empty or completelyfilled, the longitu- [1] J.H.Condon,Phys.Rev.45 (1966)526.

dinal resistancevanishes.In the two-dimensional [2] J.H.CondonandR.E.Walstedt,Phys.Rev.21(1968)612.

electron gas the magnetic-inductiondifferencebe- [3] A.B. Pippard,Proc. R. Soc. A 272 (1963) 192.tweentwo domainscorrespondingto the NMR-fre- [4] D.Shoenberg,Magneticoscillationsin metals(CambridgeUniv. Press,Cambridge,1984).quencysplitting canbe obtainedasfollows [5,8] [5] I.D. Vagner,T. ManivandE. Ehrenfreund,Phys.Rev.Lett.

till (2~T—T~I\h/2 hQCBO 51(1983)1700.(13) [6] R.S.Markiewicz, Phys.Rev.B 34(1986)4172,4177.— T U Ef [7] A. Gordon, I.D. Vagnerand P. Wyder, Phys. Rev. B 41

(1990)658.In this equationthe critical temperatureT~is equal [8] T.ManivandID. Yagner,Phys.Rev.B 41(1990)2661.

to [9] A. Gordon,I.D. VagnerandP.Wyder,Solid StateCommun.74 (1990)401.

T~= (hQC/4kB)(l —n~/n~), (14) [lOlA. Gordonandl.D.Vagner,J. Phys.C2 (1990) 3787.[111 A. Gordon,T.Salditt, I.D. VagnerandP. Wyder,Phys.Rev.

where n,, is thecriticalvalueandnf=Ef/hQC.Thelat- B 43 (1991) 3775.ter is determinedas follows, [12] S.C. Ying, B.J. McIntyre andJ.J. Quinn, Phys.Rev. B 6

(1970) 1801.n~=(m~~1/2mr

0)’’2, (15) [13]I.M. Lifshitz andA.M. Kosevich,Zh. Eksp. Teor. Fiz. 29

wheremis the free-electronmass,r0=e

2/mc2 is the (1955) 730 [Soy.Phys.J~TP2(1956)636].[14] l.A. Privorotskii,Zh. Eksp. Teor. Fiz. 52 (1967) 1755 [Soy.

classicalradiusof the electron,~ is the electron Phys.JETP25(l967)1167].

effectivemassreflectingthat the systemconsistsof [15] S.C.Ying andJ.J.Quinn, Phys.Rev.Lett. 22 (1969)231.

ananisotropicfree-electrongasin whichtheeasyaxes [161C. Kittel andJ.K.Gait,SolidStatePhys. 3 (1956)437.

(x, y) are perpendicularto a uniform static mag- [17]B. Lengeler, W.R. Wamper, R.R. Bourassa,K. Mika, K.WingerathandW. Uelhoff,Phys.Rev. B 15 (1977)5493.

neticfield H~,1 isthe distancebetweenneighbouring[18] A.S. Josephand A.C. Thorsen, Phys.Rev. 138 (1965)

planes. Aii59.

Eq. (13) showsthat the NMR splitting displays [19] W.M. Bibby, P.T. Coleridge,N.C. Cooper,C.M.M. Nex and

thecritical-temperaturedependenceof a square-root D. Shoenberg,J.Low Temp.Phys.34 (1979)681.type (I (T—T~)/ TI)”2 at the diamagneticphase [20] D. ShoenbergandJ. Yandercooy,J. Low Temp. Phys. 2

(1970)483.transition.

[21] B. Grushko,A. Gordon, ID. VagnerandP. Wyder,to bepublished.

This researchwas supportedby the German—

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