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Vortices in small superconducting disks E. Akkermans and K. Mallick † Department of Physics, Technion, 32000 Haifa, Israel 0 0 andA~ areconstantoverthethickness,thecovariantNeu- We study the Ginzburg-Landau equations in order to de- 0 mannboundarycondition,statedabove,isautomatically scribe a two-dimensional superconductor in a bounded do- 2 satisfied on the upper and lower surface of the disk. main. Usingtheproperties of a particular integrability point TheGinzburg-Landauequationsarenonlinear,second n (κ = 1/√2) of these nonlinear equations which allows vor- a order differential equations whose solutions are usually J tex solutions, we obtain a closed expression for theenergy of unknown. However, for the special value κ= 1 known the superconductor. The presence of the boundary provides √2 6 a selection mechanism for thenumberof vortices. as the dual point7, the equations for ψ and A~ reduce 1 A perturbation analysis around κ = 1/√2 enables us to to first order differential equations and the minimal free include the effects of the vortex interactions and to describe energy can be calculated exactly for an infinite plane. ] n quantitativelythemagnetizationcurvesrecentlymeasuredon This relies on the identity true for two dimensional sys- o small superconducting disks1. We also calculate the optimal tems (~ iA~)ψ 2 = ψ 2 + ~ ~ + B ψ 2 where ~ c vortex configuration and obtain an expression for theconfin- is the|c∇urr−ent den|sity a|nDd |the o∇pe×rator |is|defined as - r ing potential away from theLondon limit. = ∂ + i∂ i(A + iA ). At the dDual point, the p D x y − x y expression (1) for is rewritten using this identity as u PACS:74.25.Ha, 74.60.Ec, 74.80.-g F s follows . at ThedimensionlessGinzburg-Landauenergyfunctional = 1 B 1+ ψ 2 2+ ψ 2+ (~+A~).d~l (2) m of a superconductor depends on only one parameter2, F ZΩ2| − | | | |D | I∂Ω - F d the ratioκbetweenthe Londonpenetrationdepthλand where the last integral over the boundary ∂Ω of the sys- n the coherence length ξ tem results from Stokes theorem. o For aninfinite plane, we impose that the systemis su- c = 1 B 2+κ2 1 ψ 2 2+ (~ iA~)ψ 2. (1) perconducting at large distance, i.e. ψ 1 and ~ 0 [ F ZΩ2| | | −| | | | ∇− | at infinity so that the boundary ter|m|i→n (2) coinc→ides 1 with the London fluxoid. It is quantized and equal to The order parameter ψ is dimensionless as well as the 9v magneticfieldBmeasuredinunitsof 4πφλ02,withφ0 = h2ec. rHa∂mΩ∇e~teχr..d~lT=h2eπinn,tewghererne χisisthtehewpinhdaisnegofnuthmeboerrdoefr pthae- 1 Lengths are measured in units of λ√2. The expression 2 (1)assumesthatboththeorderparameterandthevector order parameter ψ and as such is a topological charac- 1 teristic of the system8. The extremal values of , are potential have a slow spatial variation. The integral is F 00 overthe volumeΩ=πR2dofathindiskofradiusRand vFan=is2hπesn,idaenndtiacraellyobgtiavininegdrwisheentothtweobufilrkstinoterdgrearldiniff(e2r)- thickness d. 0 ential equations. These two equations can be decoupled / Outsidethesuperconductingsample,theorderparam- t to give for ψ a second order nonlinear equation which a eter vanishes and the magnetic field is solution of the admitsfami|lie|sofvortexsolutions9. However,forthein- m Maxwell equation. The boundary condition between a finite plane, there is no mechanism to select the value of d- superconductor and an insulator is (∇~ − iA~)ψ|~n = 0 n, which only plays the role of a classifying parameter. where ~n is the unit vector normal to the surface of the n The extension of these results to finite size systems, disk. The presence of a boundary precludes a complete o namelytheexistenceandstabilityofvortexsolutionsand analyticalsolutionofthe 3dGinzburg-Landauequations c theirbehaviourasafunctionoftheappliedfieldreceived : forathindisk. Wearethenledtomakesomesimplifying iv assumptions, based upon numerical results4. aGpinazrbtiuarlgn-Luamnedraicualeqanusawtieorn.sN10usmheorwictahlesiemxuisltaetniocnesooffstthae- X The thickness d of the sample considered in the tionary vortex solutions whose number depends on the ar ethxepemriamgennettsic1flfuulxfilllisndes≪,giξveanndbydR≤/λλ.2eI(fwthheerecuλrev(adt,uRre,λo)f acaptpelitehdamttahgenpethiycsfiicealdl.piMctourreeodveerri,vtehdesfoersκim=ula1tiornesmianidnis- stands for the effective screening length), is smaller than √2 1/λ ,i.e. ifR λ ,thenbothψandthevectorpotential qualitatively valid for quite a large range of values of κ, e ≪ e with a small corresponding change of free energy11 . A~ canbe consideredto be constantaccrossthe thickness We consider finite size systems at the dual point i.e. and the disk is effectively two-dimensional. The expres- 1 for κ= . There, the edge currents screenthe external sion of the effective screening length λ (d,R,λ) is not √2 e known, except for the case R where5,6 λ λ2/d. magnetic field therefore producing a magnetic moment e In the London limit (i.e. κ → ∞), such a syst≃em has opposite to the direction of the field, whereas vortices in been described using Pearl’s →solu∞tion5. Finally, since ψ thebulkofthesystemproduceamagneticmomentalong 1 3 the direction of the applied field. Assuming cylindrical 1 2 a 4 2 2 (n,φ )=n+a(n φ ) (n φ ) a φ e e e e symmetry, the current density ~ has only an azimuthal 2πG − − 2 − − component, with opposite signs in the bulk and on the (5) edgeofthesystem. Thus,thereexistsacircleΓonwhich ~vanishes13. ThisallowsustoseparatethedomainΩinto where we have defined a = λ√2. The relation (5) con- two concentric subdomains Ω1 and Ω2 whose boundary sists in a set of quartic functioRns indexed by the integer is the circle Γ. Therefore one can extend to the subdo- n. The minimum of the Gibbs potential is the envelop main Ω1 the results obtained for the infinite case. The curve defined by the equation ∂ = 0, i.e. the sys- existence of vortices in a finite domain such as Ω1 was tem chooses its winding number∂Gnn|φien order to minimize checkednumerically14. Itwasshownthat ψ vanishesas . This provides a relation between the number n of | | a power law at the center of the disk, hence there is a G vorticesin the system and the applied magnetic field φ . e (multi-)vortex in the center whose multiplicity is deter- We consider the limit of large enough R, such that mined by the exponent of the power law. The magnetic λ the quartic term is negligible. The Gibbs potential then flux Φ(Ω1)=n is quantized and the free energy in Ω1 is reduces to a set of parabolas. The vortex number n is (Ω1)=2πn. then given by the integer part F The contribution of (Ω2) to the free energy can be F written, using the phase and the modulus of the order R 1 n=[φ + ] (6) parameter ψ, as e− 2√2λ 2 ( ψ )2+ ψ(~χ A~)2+ B2+(1−|ψ|2)2 (3) while the magnetization M =−∂∂φGe, is given by ZΩ2 ∇| | | ∇ − | 2 2 M =2a(φ n) 2a φ (7) e e Weknow,fromtheLondonequation,thatboththemag- − − − neticfieldandthevectorpotentialdecreaserapidlyaway For φ smaller that R , we have n = 0 and ( M) in- e 2√2λ − from the boundary ∂Ω of the system over a distance of creases linearly with the external flux. This corresponds orderλ√2. Overthesamedistance,atthedualpoint,|ψ| totheLondonregime. ThefieldH1atwhichthefirstvor- saturatestounity. Onecanthusestimatetheintegral(3) tex enters the disk corresponds to (n =0)= (n =1), usingasaddle-pointmethod. Weassumecylindricalsym- i.e. to G G metry,andweneglectthe term( ψ )2 ontheboundary because of the boundary conditio∇n|s,|so that the relation φ0 φ0 (3) is now given by an integral over the boundary of the H1 = 2π√2Rλ + 2πR2 (8) system. To go further, we need to implement boundary The subsequent vortices enter one by one for each cross- conditions for the magnetic field B(R) and the vector ing (n + 1) = (n); this happens periodically in the potential A(R). The choice B(R)=Be, where Be is the applGied field, witGh a period equal to ∆H = φ0 and a external imposed field, corresponds to the geometry of πR2 an infinitely long cylinder, where the flux lines are not discontinuity of the magnetization ∆M = 2√2λ. R distorted outside the system. A more suitable choice for There is a qualitative similarity between the results a flat thin disk is provided by demanding φ = φ . This we derived using the properties of the dual point and e boundary condition implies that the vector potential is thoseobtainedfromalinearisedversionoftheGinzburg- identified by continuity to its external applied value A~ . Landaufunctional16. However,thetwoapproachesdiffer e It should be noticed that the magnetic field B~ has then in their quantitative predictions due to the importance a non monotonous variation: it is low in the bulk, larger of the nonlinear term. than B near the edge of the system, because of the dis- Within the previous approximations, the expression e tortion of flux lines, and eventually equal to its applied (5) captures the main features observed experimentally value far outside the system15. i.e. the behaviour of the magnetization at low fields Finally, the minimization of the free energy with re- (before the first discontinuity), the periodicity and the spect to ψ gives 1 ψ 2 = ~χ A~ 2, such that, per- linear behaviour between the successive jumps. From | | −| | |∇ − | the experimental parameters1 namely R = 1.2µm and formingtheintegralovertheboundaryofthesystem,we λ(T) = 84nm at T = 0.4K, we compute from our ex- obtain pressionsH1 =25Gand∆H =4.6G.Thesevaluesagree 1 λ√2 1 λ√2 with the experimental results4 to within a few percent. 2 3 4 2πF(Ω2)= R (n−φe) − 2( R ) (n−φe) (4) We emphasize that H1 scales like R1, whereas ∆H scales like 1 in accordance with the experimental data1. We WehaveneglectedthecontributionoftheB2term,which calcuRla2te the ratio of the magnetization jumps to the 2 is smaller by a factor of the order (λ/R) . maximum value of M to be 0.20 as compared to a mea- The thermodynamic Gibbs potential of the system sured value of 0.22. The total number of jumps scales G is then like R2 and the upper critical field is independent of R in our theory in agreement with the experimental data. 2 At the duality point κ = 1/√2, the contribution of extreme type II limit18 using the London equation. It is the vortices to the free energy is topological and does importantto emphasize thataroundthe dualpoint, vor- not depend neither on the precise shape of the vortices ticesarenotpoint-likeandthereforetheusualexpression nor on the form of their interaction. This property does of the Bean-Livingston energy barrier does not hold. not hold for other values of κ. For a 2d film with an The possible equilibrium configurations of vortices re- infinitesimal current sheet, the vortex configuration has sult from the competition between the bulk and edge been computed by Pearl5, using the London equation, contributions to the free energy derived above. It is ei- anddiffers qualitativelyfromthe presentmodel. Indeed, ther a giant vortex at the center of the disk, a situation away from the dual point, both the shape of the vor- whichpreservesthecylindricalsymmetry,orapolygonal tices and their interaction modify the free energy and patternofsmallvortices. Inordertoevaluatetheenergy the magnetization. For instance, in the London limit oftheseconfigurations,wegeneralizethe relation(11)to (κ ), radially symmetric solutions become unstable the case of a polygonal configuration of vortices placed → ∞ and different geometrical configurations of the vortices atadistancexfromthecenterofthedisk. Theresulting have different energies. This results from two contribu- energy17 is n-times the barrier contribution f(a,y,φ ) e tions to the energy, arising from the interaction between obtainedin (11)providedthe following substitutions are thevorticesthemselvesandbetweenthe vorticesandthe made: a na, y yn and φ φe. → → e → n edge currents. We have performed a perturbative analy- In conclusion, we have investigated the question of sisaroundthe dualpoint, andweobtainedthatthe bulk the existence and stability of vortices in small two- free energy (Ω1) is given by dimensionalboundedsuperconductingsystems. Wehave F shown12 that starting from the exact solution of the 1 1 (Ω1)=n(1+ (κ√2 1))+β(κ√2 1) (rij). Ginzburg-Landau equations for an infinite plane and for 2πF 2 − − XU the special value κ = 1/√2, it is possible to derive an i<j analytical expression for the free energy in a bounded (9) system. The resulting expression provides a satisfactory quantitative description of the magnetization measured The part which is linear in n is independent of the po- on small superconducting aluminium disks in the low sition of the vortices. It has been evaluated using a magnetic field regime. For larger fields, we cannot ne- variationalansatz11. The two-bodyinteractionpotential glect anymore the interaction effects due to the vortices near the dual point is well approximated by a function andtheedgecurrents. Perturbationtheory17 aroundthe 1 (r )= (r) where (0)=1, ( )=0withβ = . In U ij U U U ∞ 4 value κ = 1/√2, has allowed us to derive an expres- particular, for a configuration where all the vortices are sion for both the confining potential barrier of the vor- close to the center of the disk, the bulk free energy is ticesandthestrengthoftheinteractionbetweenvortices. Thisprovidesamorerefineddescriptionofthemeasured 1 5 3√2 1 (Ω1)=n( +κ )+ n2(κ√2 1) (10) magnetization at larger fields. 2πF 8 8 8 − AcknowledgmentK.M.acknowledgessupportbythe Thus, away from the dual point, the linear term in the LadyDaviesfoundationandE.A.theverykindhospital- bulk free energy is not topologicalanymore and is modi- ity of the Laboratoire de Physique des Solides and the fied by the interaction. The attractive or repulsive char- LPTMS at the university of Paris (Orsay). acter of the interaction between vortices depends on the sign of (κ√2 1). − Toobtainthe edgecontributionto the energy,wecon- sider first the case of a single vortex placed at a distance x (in units of λ√2) from the center of the disk. Then, the phase of the order parameter is given by17 tanχ = sinθ , where y = ax = x/R (whereas one has † Permanentaddress: ServicedePhysiqueThéorique,Centre cosθ y d’EtudesNucléairesdeSaclay,91191GifsurYvettecedex, χ = nθ for t−he symetrical case). Starting from the ex- France pression (3), we obtain 1A.K.Geim,I.V.Grigorieva,S.V.Dubonos,J.G.S.Lok,J.C. Maan, A.E. Filippov and F.M. Peeters, Nature (London), 21πF(Ω2,y)=a(n−φe)2− 4aκ32(n−φe)4+f(a,y,φe) 2P39.G0.,2d5e9G(1e9n9n7e)s,”Superconductivityofmetalsandalloys.” (11) Addison-Wesley (1989) 3V.A. Schweigert and F.M. Peeters, Phys.Rev. B57, 13817 where the function f(a,y,φ ) can explicitely be calcu- (1998) e latedandrepresentsavortexconfiningpotentialinside a 4P. Singha Deo, V.A. Schweigert, F.M. Peeters and A.K. finite superconductorfor κ 1/√2. This correspondsto Geim, Phys.Rev.Lett. 79,4653 (1997) thewell-knownBean-Living≃stonconfiningenergybarrier 5J. Pearl, App.Phys.Lett.5, 65 (1964) 6A.L. Fetter,Phys.Rev.B22, 1200 (1980) in a 3d superconductor which has been obtained in the 3 7E.B. Bogomol’nyi, Sov.J.Nucl.Phys. 24,449 (1977) and D. Saint-James, E.J. Thomasand G.Sarma, ”TypeIISuper- conductivity”,Pergamon Press, (1969) 8For a discussion of topological aspects see E. Akkermans and K. Mallick, Geometrical description of vortices in Ginzburg-Landau billiards, to appear in ‘Topological aspects of low dimensional systems’ Les Houches Lecture Notes (July 1998) cond-mat/9907441 9C. Taubes, Comm. Math.Phys. 72, 277 (1980) 10C.Bolech,G.C.BuscagliaandA.Lopez,Phys.Rev.B 52, R15719 (1995) 11L. Jacobs and C. Rebbi,Phys.Rev. 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