Solid StateCommunications,Vol. 27,Pp. 1061—1063. 0038—1098/78/0915—1061$02.00/0© PergamonPressLtd. 1978.Printedin GreatBritain.

INCOMPLETEMEISSNEREFFECTOFTRIPLET SUPERCONDUCTIVITY

K. Machida*andR.A. Klemm

Ames Laboratory—USDOEandDepartmentof Physics,IowaStateUniversity,Ames,IA 50011,U.S.A.

(Received11 April 1978 by R. Barrie)

The magneticpropertiesof a triplet superconductorareinvestigatedusinga phenomenologicalGinzburg—Landautheory.Due to the presenceof aparamagnetictermin thefree energyarisingfrom the energyrequiredtoflip the spinsof a triplet pair by themagneticfield, the systemdoesnotexhibit a completeMeissnereffect belowT~.Thisparamagneticcontri-bution to themagnetizationis stabilizedby thenon-lineartermsin thefreeenergy,andfor certainvaluesof the parameters,canevencancelthediamagneticterm.Theresultsarediscussedin termsof the Anderson—Morel andBalian—Werthamerstates.

ALTHOUGH thepossibility of triplet superconductivity requiredto flip thespinsof a pair. Thisresulting para.in certainmaterialshasbeendemonstratedtheoretically magnetictermpreventstheMeissnereffect from becom-[1—31,the fundamentalmagneticpropertiesof sucha ing complete,andfor certainvaluesof theparameters,systemhavenot beenfully investigated.Thus,if sucha caneveneliminatetheMeissnereffect altogether.systemwere to exist in nature,it would be difficult to Weassumea freeenergy functionalof the form:distinguithit from an ordinarysinglet superconductor,

F = ~ FO+FH+FN, (1)basedon our currentunderstanding.Forexample,BalianandWerthamer[3] (BW) havedemonstratedtheexist-enceof the Meissnereffectin a triplet superconductor, whereFN is thenormal statefreeenergy,but theydid not investigateto what extentthemagnetic ~ h2(r)field canbe expelled.Thisquestionmay beof particular FH = f d rrelevanceto thelinearchaincompoundHg

2.82(AsF6)0~~,which for T� 1 K doesnotexhibit a completeMeissner is themagneticfree energy,andeffect evenfor magneticfields as weakas 15 mG,butshowsanincreasingdegreeof magneticexclusionas the F0 = $ d

3r — [a + aj3h(r)] I ~II~12+ ~b I ‘I’d I~field is loweredin thea*/c* plane [4]. Since it hasbeensuggestedthat this compoundmight be exhibitingsome + ~cI~’

0~2~’J’_

0J2+ —1—i I(V— 2ieA)~PaI2~,(2)

featuresof non-s-wavepairing,it isessentialthat at least 2m jthe qualitativefeaturesof triplet superconductivitybe whereh = V x A is the local field,H

0 is theappliedinvestigated,in order to determineif this mechanism magneticfield, andm* is theeffectivepair mass,whichmight be at leastpartially responsiblefor the observed may takeon any value.In equation(2) we haveeffects. assumeda two-componentorder parameter,neglectinga

In this paper,we presenta phenomenological possiblethird componentarisingfrom pairingof twoGinzburg—Landaumodelof the fundamentalmagnetic particleswith oppositespin the inclusionof whichpropertiesof a triplet superconductor(welimit our con- would give riseto a correctionto thebulk susceptibilitysiderationto thep-wavepairingcase)in a weakexternal x of higherorderin H0. We haveassumedtheordermagneticfield. We includeonly thetermsin the free parameter‘I’17(r) to obey theusualelectromagneticgaugeenergythat contributeto themagneticsusceptibilityin invariance.The parametera —~a’(T—T0), wherea’ >0,the limit of zero field. We find that in additionto the asusual for Ginzburg—Landautheories,and~3= 13(7)orbital diamagnetism,which in a singletsuperconductor j3~+ 131(T~— T) dependsweaklyon temperatureasis responsiblefor completeexpulsionof themagnetic T -+ T~.The parametersb andc are takento befield in the weakfield limit (i.e. belowHe,),thereis a temperature-independent.In this paper,we takeparamagneticterm arisingprimarily from the free energy ii = c = kB = 1.

____________ Since ‘I’d may be complex,we define‘4’~= p~e’°G,* Permanentaddress:Departmentof Physics,Kyoto wherep0 andO~arerealfunctionsof r. The supercon-

University,Kyoto, Japan. ducting currentj = ~,j0where

1061

1062 INCOMPLETE MEISSNEREFFECTOF TRIPLET SUPERCONDUCTIVITY Vol. 27, No. 11

e 4e2 wherex is the distancefrom the surface.The external.io ~ (‘I’~’1’a — ‘I’

0V’I’~) — ‘1’012A field is parallelto zaxis. The termindependentofx is

— (3~ thusthebulk valueof h, andthe exponentiallydecaying— 2p

0v~0e, ~ ~ part ofh gives rise to the magneticsurfaceenergy.

wherev~0= (VU0 — 2eA)/m* is thevelocity of the The Gibbsfree energyis relatedto F bysuperfluid.We may now write thefree energyas

G = F—~ H0• h(r) d3r. (9)

= Jd3r —a(pt+p~)—13h(pt—p4)

Gb~k= V{—ap÷—H013p—2irf32p~

+ ~b(p~ + p~)+ Cptpt + ~~(Pfv9

2t + p~v~) + ~b(p~ + p~)+ ~c(p2+— p~)}, (10)

1 / 1 where V is the volumeof thesystem.+ h2/8rr + —--~ — (Vp~)2+ —(Vp~)2 . We now minimizethe bulk Gibbsfreeenergywith8m ~Pt P~ I respectto variatipnsin p.. andp. obtaining

(4)Wenowletp~=p~±p~,v+=~(v

8t±v~~).Usingthe aG/V = o = —a+~(b+c)p~ (11)Maxwell equationV x h = 4irj~,we obtain:

F—F~=J d3r[_aP+_13hP+~b(P2++P2) ~V = 0 = —H013—4~132p+~(b—c)p, (12)

which canbe easily solved to obtain~ = 2a/(b+c) (13)

~ P- = 210u31(b—8iri3

2). (14)2 1 6m Substitutingtheseequationsin the expressionfor Gb~k,

x [((V(p÷ + p4)2/(p÷+ p-) we haveGb~k/V = —a2/(b + c)— 2~ (15)

+ (V(p+ —p-))2/(p+—p-)] 1 (5) b—c —

Now the normal stateGibbsfree energyis givenbywhereX

1, = (m*/4ire2p+)~~2is the Londonpenetration 82 BH

depth. GN = FN(B=0)+~~. (16)For A = 0, we may find a uniform solution with ~ iT

‘1’I = ‘I’~= ‘I’ = p”2 e~,wherep = a/(h + c) is obtained The magnetizationis thusgiven by M =

by minimizingF with respectto p, and B is arbitrary. GN) I B =ii~’ which impliesThis solution is stablefor T < T~if b + c>0. For 2132Ha H~

A � 0, we may find an equationfor h by minimizingF M = - -~—— (17)h—c--8ir/3 4ii

with respectto variationsin h:

—X~V2h(r)+h(r)= 4iT13p. (6) ~—8~ (18)

Similarly, varyingF with respectto v_ gives iT iTf3

F Sinceb — c— 81r132 >0 for p positive,the secondterm— = rn*v(p2+ — p2_)/p+ = 0 (7) is positive,and is thusa paramagneticterm,which for

non~vanishing13 preventstheexpulsionof themagneticwhichimplies v_ = 0, andhencethat VOt = VU ~. If we field from beingcomplete.assumethat the spatialvariationsin p±takeplaceover a We remark that in generalwe expect13 to be non-distanceE±~ XL from thesurface,we mayneglectthe vanishingat T~,sothe temperaturedependenceof 13 istermsinvolving spatialderivativesof p±in equation(5), 13(T) f3~+ 13~(T~— T). We also note that i3 is relatedtoand treatp_ asa constantin equation(6). With the thespin susceptibilityof the system.In theAnderson-boundaryconditionh(0) = H

0, for a semi-infinite Morel (AM) state(equalspin state)13(T) is a constantsystemwe have belowT~, becausethe spin susceptibilityof this stateis

temperatureindependent.The AM stateis basedon ah2(x) = {H0—4irIlp..}e L+4irj3p_two-componentorderparameter,aswasour calculation.

= (41rXL)~{H0 — 4irflp_} ~ (8) On the otherhand,the BW state,which hasbeenshown [3] to be lower in free energythan the AM state,

Vol. 27,No. 11 INCOMPLETEMEISSNEREFFECTOFTRIPLET SUPERCONDUCTIVITY 1063

containsthethird componentof theorder parameter. componentin thesimplestpossibleway doesnot changeSince thespinsusceptibilityof this stateisdescribedby our result.thetemperaturedependentYosidafunction,$3 may be Wenote that themagnitudeof the paramagneticwritten as$3(7) $3~+ $3~(T~— 7); theorderof magni- term maybe eitherlarge orsmall,dependinguponthetudeof $3,/$3ois 1/Ta. The resulting magneticsuscepti- parameters.It is simplestto think of this paramagneticbility x is temperaturedependentandmay be written as termasarising solely from thePauli-like termin the freex(T) Xo + x~(T0— 7). Wehavealso consideredthe energy;in fact as$3 -÷ 0, this paramagneticterm indeedeffect of thethird componentof the order parameterby vanishes.However,the magnitudeof theeffect alsowriting the freeenergyanalogousto equation(2) as dependsuponb andc, theparameterscharacterizingthe

quartictermsin the free energy.Thus,themagnitudeofF = FH + FN + — (a+ u$3h)1W01

2+ ~bI~’0I

4 the paramagnetictermdependsstronglyupon,andcano=±i.0 in factbe enhancedby thenon-linearorderparameter

+ ~ C ~ I ‘I1~~j2 I ‘I’~’12 + —~—~ I (V — 2ieA)’I’0 2 fluctuations.

2m(19) Acknowledgement— This work wassupportedby the

Minimizing this free energyin thesamemanneraswe U.S. Departmentof Energy,Division of BasicEnergyhavealreadydone givesa magneticsusceptibilityequal Sciences.The authorsaremuchindebtedto J.R.Clemto that in equation(18). Thusinclusionof the third and E.H. Brandt for their helpful discussions.

REFERENCES

1. PALS J.A.,VAN HAERINGENW. & VAN MAAREN M.H.,Phys.Rev.B15, 2592 (1977).

2. ANDERSONP.W. & MOREL P.,Phys.Rev. 123, 1911(1961).

3. BALIAN R. & WERTHAMER N.R.,Phys.Rev. 131, 1553(1963).

4. SPAL R.,CHIANG C.K., DENENSTEIN A., HEEGERA.J.,MIRO N.D. & MACDIARMID A.G.,Phys.Rev.Lett. 39,650 (1977).