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Stability of topological defects in chiral superconductors: London theory. Victor Vakaryuk∗ Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA Thispaperexaminesthethermodynamicstabilityofchiraldomainwallsandvortices–topological defects which can exist in chiral superconductors. Using London theory it is demonstrated that at sufficiently small applied and chiral fields the existence of domain walls and vortices in the sample is not favored and the sample’s configuration is a single domain. The particular chirality of the single-domain configuration is neither favored nor disfavored by the applied field. Increasing the fieldleadstoanentryofadomain-walllooporavortexintothesample. Theformationofastraight domain wall is never preferred in equilibrium. Values of the entry (critical) fields for both types of 2 defects, as well as the equilibrium size of the domain-wall loop, are calculated. We also consider a 1 mesoscopic chiral sample and calculate its zero-field magnetization, susceptibility, and a change in 0 the magnetic moment due to a vortex or a domain wall entry. We show that in the case of a soft 2 domain wall whose energetics is dominated by the chiral current (and not by the surface tension) n its behavior in mesoscopic samples is substantially different from that in the bulk case and can be a used for a controllable transfer of edge excitations. The applicability of these results to Sr RuO – 2 4 J a tentative chiral superconductor – is discussed. 2 1 I. INTRODUCTION respondstoasingle-domainsampleandthatthespecific ] chirality of this domain is neither favored nor disfavored n by the applied field. Upon increasing the applied field o Chiral superconductors belong to an exotic class of c physical systems whose many-body ground state carries either a domain wall loop or a vortex enters the sample; - theconfigurationinwhichadomainwallformsastraight r nonzero current and hence breaks time-reversal symme- p try. Because the current can assume two time-reversal line which terminates at the edges of the sample is never u favoredinequilibrium. Valuesoftheentry(critical)fields connected directions, the ground state of a chiral super- s as well as of the domain wall loop size are given in terms . conductorisdoublydegenerate(chiral). Thisdegeneracy t of model parameters such as the magnitude of the chiral a opensapossibilityfortheexistenceofextendedtopolog- m ical defects – domain walls – which connect regions of current and a domain wall surface tension. opposite chirality and exist along with conventional de- - d fects such as vortex lines. n Itwasrecentlysuggestedthatgrapheneatspecificdop- o Motivated by recent cantilever magnetometry mea- ingcansupportchiralsuperconductivity.1,2 Anotherten- c surementsinSr2RuO4 (Ref.14),wealsoconsiderameso- [ tative candidate for a chiral superconductor is Sr2RuO4 scopic chiral sample with a hole for which we calculate below 1.5K (see Refs. 3,4 for review) which is corrob- zero-field magnetic moment, susceptibility, and a mag- 3 orated by µSR,5,6 the Kerr effect,7 and phase-sensitive v netic moment change due to a vortex or domain wall measurements.8,9 The candidacy of Sr RuO is, how- 5 2 4 entry. We demonstrate that in the mesoscopic limit the ever, undermined by the fact that several attempts to 2 size of a loop formed by a very soft domain wall can be 0 detect a surface magnetic field generated by the chiral controlled with the applied field and speculate that such 6 currents10–13 or the magnetic moment associated with an effect can give rise to a controllable transfer of edge . them14 have not yielded a positive result. 9 excitations such as Majorana modes. 0 OneofthepossibleexplanationsaimedtocutthisGor- 1 dian knot of seemingly contradicting observations is to 1 assume the presence, on a mesoscopic scale, of an alter- : nating chiral domain structure which leads to the sub- Thepaperisorganizedinthefollowingway. InSection v i stantial field cancellation. Previous studies reported in II we give a phenomenological description of a chiral do- X the literature have focused either on the calculation of main wall used throughout the paper. In Section III we r the domain wall properties such as surface tension and derive an expression for the Gibbs energy of a chiral su- a accompanying chiral current (see, e.g., Refs. 15,16) or perconductor with topological defects in the strong type on finding the distribution of magnetic fields assuming a II limit. In Section IV we apply results of the previous particular domain structure without attempts to justify section to several domain configurations in macroscopic the latter.10–12,17,18 samples. This is the core section of the paper. In Sec- InthisworkweuseLondontheorytoaddresstheques- tion V we focus on a mesoscopic geometry. Section VI is tion of the thermodynamic stability of several domain devoted to overall conclusions. Appendix A contains the configurations for a simple sample’s geometry (such as solutionoftheLondonequationforatwo-domaincircular cylindrical), where demagnetizing effects can be easily cylinder of arbitrary dimensions with a hole. Appendix taken into account. We show in particular that in small B contains some useful results involving modified Bessel applied fields the equilibrium domain configuration cor- functions. 2 II. DOMAIN WALL DESCRIPTION configurationsisthatwhereasacirculardomainwallcre- ates a nonzero net flux ((cid:96) (cid:54)=0) the net flux created by dw Even in the simplest case of an isotropic chiral super- astraightdomainwallvanishes((cid:96)dw =0). Itshouldalso conductor the domain wall structure can be quite com- be pointed out that both σ and i will in general be dif- plicatedandisingeneraldeterminedbytheinterplaybe- ferentforthetwoconfigurations;wedonotindicatesuch tweenthematerial–Ginzburg-Landau–parametersand a difference explicitly, unless otherwise stated. the domain wall geometry. To describe chiral domain Wenoteinpassingthatthedomainstructureinchiral walls we use a simplified model in which it is modeled as superconductorsneednotbesimilartothatinferromag- a sheet-like object with surface tension σ which carries netic materials because the latter do not exhibit field chiral current with linear density 2i and is characterized screening. by a winding number (cid:96) . dw The presence of the current along a domain wall is ne- cessitated by the chiral nature of the state in which the III. SURFACE REPRESENTATION OF THE internal orbital motion of Cooper pairs, while compen- GIBBS POTENTIAL FOR A CHIRAL sated in the bulk, produces a nonzero charge current i SUPERCONDUCTOR on a boundary with vacuum and 2i on a boundary with another domain. The presence of such a current leads Let us start by considering a superconducting sample to a discontinuity of magnetic field (or, rather, magnetic placed in a uniform external magnetic field. The distri- induction) across the domain wall. bution of currents and fields in the sample is a function Ageneraldescriptionofadomainwallrequiresseveral of the applied field and, in thermal equilibrium, can be windingnumbers,whichreflectsthemulticomponentna- found through minimization of the corresponding Gibbs tureoftheunderlyingchiralstate.19Thewindingnumber potential, defined as30 relevant to our model, (cid:96) , has a physical meaning of a dw net flux (in units of flux quantum, cf. below) generated 1 (cid:90) G=F + d3r (B2−2B·H), (1) by the domain wall’s chiral and screening currents in an s 8π infinite superconducting medium.29 Defined in this way the winding number (cid:96) depends on the geometry of the where H and B are magnetic field and induction respec- dw domain wall and, in general, need not be an integer. tively; the volume integration extends over the space oc- The domain wall surface tension σ complements our cupied by the superconductor and over any cavities con- treatmentoftopologicaldefectsbyspecifyingitsintrinsic tained in it. The free energy F of the sample, which by s energyperunitarea. Althoughexistingcalculationsseem our definition excludes the field energy given by the B2 to indicate that for a chiral p+ip superconductor σ >0, term in eqn. (1), may contain terms describing kinetic the author is not aware of a general proof that would energies of charge and spin currents,20 spin-orbit inter- exclude the opposite. action energy,21 effects of kinematic spin polarization,22 Weconsideramodelinwhichboththesampleandthe etc. domainwallaretranslationallyinvariantalongthedirec- In London theory the free energy of a superconductor tion of the applied field and focus only on two domain is approximated by the kinetic energy of supercurrents wall configurations – a straight line and a circle – which, described by the superfluid velocity v . The sum of the s due to their high symmetry, admit straightforward ana- kinetic energy of supercurrents and the magnetic field lytical treatment. The main difference between the two energy can be written in the following form:23 (cid:90) d3r(cid:18)1ρ v2(r)+ 1 B2(cid:19)=−|Φ0|(cid:73) d2s·(B×∇θ)− 1 (cid:73) d2s·(cid:0)B×A(cid:1), (2) 2 s s 8π 16π2 8π sc where Φ ≡ hc/2e (< 0), A is the vector potential, ρ rem. The convenience of such a representation relates to 0 s is the superfluid density and θ is the phase of the order the fact that for relevant geometries the surface integra- parameter. The volume integration extends over the re- tion is usually more straightforward to perform than the gion of space occupied by the superconductor and (cid:72) d2s volume one. Moreover, using eqn. (2) one can avoid di- denotes the integration over its surface. The above re- rect calculation of the magnetic field contribution which sult, derived under the main assumption of the London is usually quite cumbersome. approximation – uniform superfluid density31 – is valid Eqn. (2) can also be used in the presence of topologi- for a superconductor of an arbitrary geometry and relies cal defects if they are treated in the following way: The onlyontheuseofMaxwell’sequationsandGauss’stheo- volume integrals should exclude regions of nonuniform 3 superfluid density associated with the defects, while the surface integrals should be complemented by an integra- tion over the surface that encloses the excluded volume. - 3 2 The “missing” contribution to the free energy can be ac- 1 counted for by introducing defects’ surface energy Fσ, + + - which can be computed from a more general description (e.g., Ginzburg-Landau theory). Such an approach leads to the following representation of the free energy32 (cid:90) 1 F =F + d3r ρ v2(r), (3) s σ 2 s s sc−d FIG.1: Onionliketwo-domainconfigurationofachiralsample wheretheintegralnowexcludesregionswithnonuniform used in the calculation of the Gibbs potential (6). Arrows superfluiddensityassociatedwiththedefects. Intheex- indicate the direction of the chiral currents. treme type II limit λ/ξ (cid:29) 1, this approach should give a quantitatively good approximation for the Ginzburg- Landau energy of a superconductor with topological de- and 3 – the outer surface of the sample. The surfaces fects,whereasinthemarginalcaseλ(cid:38)ξ onemighthope are characterized by their respective radii Rj. For the to get a qualitatively reasonable description. domain chirality shown in Fig. 1 surfaces 1 and 3 carry counterclockwise chiral current i while the domain wall 2 carries a clockwise current 2i. A. Gibbs potential of a two-domain cylinder Owing to the presence of the chiral current, the mag- netic field across each surface is discontinuous. Let us Forageneralsample’sgeometry,evaluationofthesur- denote the field values on the inner (−) and outer (+) face integrals in representation (2) is complicated by the sides of the surface j as Hj±. In this notation H3+ and spatial dependence of the magnetic field. One exception H1−areequivalenttotheappliedfieldHaandtothefield is a geometry that has a translational symmetry along in the hole Hh, respectively, and the field discontinuities the direction of the applied field (i.e., a cylinder with an are given by arbitrary cross section). In this geometry the value of H −H =˜i, H −H =2˜i, H −H =˜i, (5) the field on the sample’s surface is constant, which can 3− a 2+ 2− h 1+ be seen from the application of Amp`ere’s law to a rect- where we introduced the field jump ˜i ≡ 4πi/c which is angular contour with a side parallel to the field. analogous to a domain wall magnetization used, for ex- Motivated by recent cantilever magnetometry mea- ample, in Refs. 16,17. surementsonmesoscopicannularSr RuO samples,14we 2 4 Let Φ denote the total flux through the area limited consider a circular hollow cylinder with an onionlike do- j by the surface j and Φ ≡ Φ −Φ . Then, using the main structure shown in Fig. 1. The hole which is char- ij i j definition of the Gibbs potential (4) and the surface rep- acterized by an integer winding number (cid:96) provides, for s resentation of the free energy (3) given by eqn. (2), the small applied fields, the only place where vortices can Gibbs potential of this configuration is given by reside,33 and the two-domain configuration is the sim- plest one in which a reduction of the total magnetic mo- 8πg = 8πf −|Φ |(cid:96) (H −H ) (6) ment can be achieved (as observed in Ref. 14). σ 0 s h a − |Φ |(cid:96) (H −H +˜i)−˜i(Φ −Φ )−H Φ , Results of this section can also be used for cylindrical 0 dw 2− a 21 32 a 3 samples with an arbitrarily shaped cross section, pro- where g and f indicate that energies are taken per unit videdthedistancebetweenthedefectsandtheboundary length in the direction of the applied field. As discussed is much larger than λ. In this limit, as shown in Sec- in Section II, (cid:96) is a measure of the flux carried by a tion IV, a circular domain wall configuration is favored dw domain wall in an infinitely large sample and f is the energetically over a configuration in which a straight do- σ surface energy of the domain wall per unit length. main wall runs across the sample and terminates on the Intheabsenceofchiralcurrentsanddomainwalls,i.e., sample’s boundaries. when both˜i and (cid:96) are set to zero, expression (6) coin- For a cylindrical geometry with the axis parallel to dw cides,uptoanadditiveconstant34,withthatobtainedin the applied field H we have B = H and expression (1) a Ref. 24 for a hollow nonchiral cylinder. Notice that one simplifies to should not expect G to be of a simple form G∝M ·H a 1 (cid:90) where M is the magnetic moment of the cylinder since, G=F + d3r (H2−2H·H ), (4) s 8π a in general, in the absence of the applied field M (cid:54)=0. Theapplicationofeqn.(6)requiresknowledgeoffields To make use of eqn. (4) we notice that the onionlike ge- andfluxesinthesystemintermsoftheappliedfieldand ometryshowninFig.1consistsofthreesurfaces: 1–the parameters (cid:96) , (cid:96) , ˜i, and R . Such knowledge can be s dw j inner surface of the sample, 2 – the domain wall surface, obtained from a solution of the London equation. The 4 general solution of the London equation for an onionlike cylinderandcanbeshowntoholdevenwhenthescreen- geometryisgiveninAppendixA.Wenowproceedtothe ing is geometrically limited as in the mesoscopic settings analysisofthe“macroscopic”limitofthissolution,where considered in Section V. our results have simple analytical form. In Section V While it is natural to expect the invariance under the we relax the “macroscopic” constraint and consider this full time-reversal operation, which in the case of the geometry for a sample with arbitrary dimensions. Meissner state involves the reversal of both chiral cur- rent˜i and the applied field H , the invariance under the a reversal of either ˜i or H alone (partial time-reversal op- a IV. STABILITY OF TOPOLOGICAL DEFECTS eration) as demonstrated by (9) might be considered a IN MACROSCOPIC LIMIT surprising feature.36 An intriguing question is whether this feature is just a peculiarity of the cylindrical geom- In this section we calculate the Gibbs potential in the etry which possesses translational invariance along the macroscopiclimitwhenallrelevantdistances,suchasthe direction of the applied field or has a broader validity. size of the sample, the distance between defects, and the While the author does not have a proof of the latter, a sample’s boundary are larger than λ. In this limit the plausibilityargumentcanbegiventhatsuggeststhatthe precise shape of the boundary and the defect’s location invariance under the partial time-reversal operation can relative to it are irrelevant. We consider the four defect be expected if the sample has a mirror symmetry in the configurationsshowninFig.2(a): theMeissnerstate(no plane perpendicular to the field. defects), the state with a vortex, and the states with Wealsonotethat,asevidentfromeqn.(9),chiralcur- a domain wall shaped as a loop or as a straight line. rents give a positive contribution to the electromagnetic Thefirstthreeconfigurationscanbeobtainedaslimiting energyofthesystem. Thisstatementshouldalsoholdfor cases of the onionlike geometry of Fig. 1 and hence are samples with dimensions of the order of or smaller than described by eqn. (6). In the macroscopic, limit Gibbs λ. For such mesoscopic samples positive chiral contribu- potentialsofmorecomplicateddefectconfigurationssuch tiontotheelectromagneticenergymaybecomecompara- ascombinationsofthosementionedabovecanbewritten ble with the negative condensation energy whose scale is in a similar way. setbythethermodynamiccriticalfield. Thismechanism may hinder the formation of the chiral superconducting state and has to be borne in mind when considering the A. Meissner state possibility of a chiral pairing in very small samples.25 The Meissner state of a cylinder corresponds to a B. Vortex state single-domain configuration with no trapped defects, Fig. 2(a). Its Gibbs potential is obtained from eqn. (6) bysettingf ,(cid:96) ,(cid:96) ,R ,andR tozeroandisgivenby Avortexstatecorrespondstoasingle-domainconfigu- σ s dw 2 1 the expression rationwithaholewithnonzerophasewinding(cid:96)s around it. Suchaconfigurationisobtainedfromanonionlikege- 8πg =(˜i−H )Φ , (7) ometry by setting (cid:96) =0, R =ξ (cid:28)λ, and then taking M a M dw 1 the limit R → R . Using eqn. (6) the Gibbs potential 2 1 whereΦ ≡Φ isthenetfluxthroughthesamplegener- of the vortex state relative to that of the Meissner state M 3 atedbybothchiralandscreeningcurrents. Inthemacro- is given by the expression scopiclimitforasampleofcircumferenceP,theMeissner 8π(g −g )=−|Φ |(cid:96) (H −2H ), (10) flux Φ is given by a plausible expression: v M 0 s v a M where H is the value of the magnetic field on the outer Φ =λP(H +˜i), (8) v M a side of the surface which defines the normal vortex core; in the notation used in eqn. (6), H corresponds to H which is a direct consequence of the fact that both the v 2+ after taking the limit R →R . In deriving eqn. (10) we applied field and the field created by the chiral current 2 1 haveneglectedthevortexcoreenergyandthefluxcarried arescreenedoveraregionofthicknessλaroundtheouter by it – a step which is well justified in the extreme type edge of the sample.35 Given expression (8), the Gibbs II limit used here. potential of a solid macroscopic cylinder in the Meissner For large applied fields, the energy difference (10) is state takes the following form: negative, which means that the Meissner state is ther- 8πg =λP(˜i2−H2). (9) modynamicallyunstable. Thecriticalfieldforthevortex M a entry is determined by the following equation: This at first sight counterintuitive result implies that a H =H /2. (11) c1,v v chirality of a single-domain Meissner state specified by the sign of˜i is neither favored nor disfavored by an ex- The field H can be found by either solving the London v ternal field. Although proven in the macroscopic limit, equation in the macroscopic limit or by taking appropri- this statement is in fact independent of the size of the ate limits in the results for the onionlike geometry given 5 + - + + + Meissner Vortex + - + - - + DW as a loop Straight DW (a) (b) FIG. 2: (a) Configurations of the defects considered in Section IV. The positive direction of the applied field is out of the page toward the reader. (b) Values of winding numbers and chiral currents for vortex, loop and straight domain wall (DW) configurations. The size of the vortex core is exaggerated for visual purposes. in the Appendix A. In doing so one obtains C. State with a domain wall loop We now proceed to the configuration in which a do- |Φ |(cid:96) main wall forms a circular loop, i.e., terminates in the H =− 0 s ln(2λ/ξ). (12) c1,v 4πλ2 sample forming an “island” of opposite chirality. The Gibbs potential g of such a configuration is obtained dw◦ from eqn. (6) by setting (cid:96) = 0 and taking the limit s R →0. This leads to the expression Thus, the value of the first critical field for a vortex en- 1 try in a chiral superconductor is independent of both the magnitude and the sign on the chiral current ˜i 8πgdw◦ = 8πfσ−|Φ0|(cid:96)dw(H2−−Ha+˜i) and, within logarithmic precision, coincides with H for − 2˜iΦ +Φ (˜i−H ), (13) c1 2 3 a nonchiral superconductors37 (see, e.g., Ref. 27). Alter- natively, for a fixed chirality, eqn. (12) demonstrates the wheref isthesurfaceenergydefinedaftereqn.(6). H σ 2− absence of the field-reversal splitting of Hc1,v. It should isthefieldontheinnersideofthedomain’sboundary,Φ2 be noted, however, that inclusion of the vortex core en- is the flux through the area limited by it, and Φ is the 3 ergyin(10),whichtendstobechirality-dependent,26will totalfluxthroughthesample,whichincludesthescreen- result in nonzero field-reversal splitting of Hc1,v. ing contribution and the flux created by the domain. In the macroscopic limit these quantities can be found by One might wonder what happens if the magnitude taking the appropriate limit in the general solution for of the chiral current ˜i is such that the magnetic field the field distribution, which is given in Appendix A. In created in its immediate neighborhood is larger than this way we obtain H . Whilethisquestioncannotbeansweredwithinthe c1,v macroscopicapproximationusedinthissection,eqn.(12) |Φ |(cid:96) suggests that, for applied fields smaller than Hc1,v, vor- H2− =− 4π0Rdλw −˜i, ticesgeneratedbythe chiralcurrent’smagneticfield will 1 (14) tend to stay away from the bulk “decorating” edges of Φ =− |Φ |(cid:96) −˜i2πRλ, 2 2 0 dw the sample and domain boundaries (if present). Φ =Φ −(cid:96) |Φ |, 3 M dw 0 Another conclusion that can be drawn from the re- sultsofthissectionisthat, inthermalequilibriuminthe where R is the radius of the domain island [≡R in the 2 absence of the applied field, the total flux trapped by a notationofeqn.(6)]. ThefluxΦ consistsofthefluxΦ 3 M chiralcavity,locatedatdistancemuchlargerthanλaway generated by the boundary of the sample [cf. eqn. (8)] from the sample’s boundary, is zero. This follows from and of the flux carried by a domain wall loop, (cid:96) Φ . dw 0 settingH =0inexpression(10)andthenminimizingit Plugging these results into eqn. (13) yields the following a with respect to (cid:96) . expressionfortheGibbspotentialofthecirculardomain s 6 wall configuration: To estimate the actual value of H for a given ma- c1,dw terial a knowledge of chiral current ˜i and domain wall 8π(g −g ) = |Φ0|2(cid:96)2dw +(˜i2+4πσ/λ)4πRλ surface tension σ is required. These can be calculated dw◦ M 4πRλ using Ginzburg-Landau theory and turn out to depend + 2|Φ |(cid:96) H , (15) on various material and geometrical parameters such as 0 dw a Ginzburg-Landau expansion coefficients and the orien- where gM is the Gibbs potential of the Meissner state tation of the domain wall relative to the crystal axes and σ is the surface tension of the domain wall, such (see, e.g., Refs. 15,16). Ignoring for simplicity material thatfσ =2πRσ. Wefirstminimizetheaboveexpression anisotropy, one can conclude that withrespecttoRwhichyieldstheequilibriumsizeofthe domain island: 4πσ/λ=(cid:15) |Φ0|2 , ˜i2 =(cid:15) |Φ0|2 , (19) 14π2λ3ξ 24π2λ4 |Φ ||(cid:96) | R = 0 dw . (16) ◦ (cid:113) where(cid:15) aredimensionlessparameterswhichinaweak- 4πλ ˜i2+4πσ/λ 1,2 coupling BCS limit are of the order of 1.15,16 We now define the following parameter: Evaluated at R the difference (15) is positive for small ◦ applied fields; upon increasing the field the difference κ ≡4πσ/(˜i2λ)=(cid:15) λ/((cid:15) ξ). (20) d 1 2 (15)becomesnegativeatsomevalueofH whichdefines a the critical field for the creation of a domain wall loop. As can be seen from eqns. (16) and (17) this parame- In other words, circular domains such as that shown in terdetermineswhethertheenergeticsofthedomainwall Fig. 2(a) will be thermodynamically stable only if the is dominated by the chiral current (“soft” domain wall, applied field Ha exceeds a critical value Hc1,dw, defined κd (cid:28) 1) or by the surface tension (“hard” domain wall, by κd (cid:29) 1). In the weak-coupling limit when (cid:15)1,2 ∼ 1, one generally expects that κ ≈ λ/ξ. Provided the weak- d (cid:113) H = ˜i2+4πσ/λ. (17) coupling limit is applicable for Sr2RuO4 (λ/ξ ∼ 1 and c1,dw hence κ ∼ 1), one would expect that H ∼ H d c1,v c1,dw and the domain size R ∼λ. However, given the uncon- Notice that unlike H for a vortex entry, eqn. (12), ◦ c1,v ventional nature of superconductivity in Sr RuO , the the critical field for the domain wall loop entry depends 2 4 applicability of the weak-coupling results to this mate- on the chiral current ˜i but is independent of the flux rial remains an open question. carriedbythedefect. Thereasonforthelatteristhe(cid:96) dw dependence of the equilibrium domain size as specified by eqn. (16). D. State with a straight domain wall that Let+−+denotethechiralityarrangementofadomain terminates at the edges wall loop state shown in Fig. 2(a). Unlike the Meissner state discussed in Section IVA, the energy of this state, A straight domain wall configuration that terminates eqn. (15), is not invariant under reversal of the applied at the edges of the sample is qualitatively different from field. Equivalently, for a fixed applied field the energies theclosedconfigurationdiscussedearlier. Unlikethelat- of the +−+ arrangement and of its time-reversal coun- ter, the total flux carried by a straight domain wall is terpart−+−(obtainedbychangingthedirectionofthe zero (see, e.g., Ref. 19), which substantially changes its chiral currents and the sign of (cid:96) ) are different. In par- dw energetics. Let Φ denote the total flux carried by ± ticular,forapositiveappliedfieldtheenergyofthe−+− ± domainsshowninFig.2(a). TheGibbspotentialofsuch arrangement is larger than that of +−+. Although the a state can be found along the lines that led to eqn. (6), energy of the former state can be lowered by adding vor- and is given by tices,itwillstillbelargerthaneitherthe+−+arrange- mentorapurevortexstateandhencecannotcorrespond 8πg =8πf +˜i(Φ −Φ )−H (Φ +Φ ). (21) dw| σ + − a + − to a true equilibrium. Putting together eqns. (12, 16, and 17) leads to the In the macroscopic limit, fluxes Φ can be easily com- ± following expression for the size of the domain island: puted, which leads to the following expression for g : dw| R◦/λ∝Hc1,v/Hc1,dw, (18) 8π(gdw|−gM)=2Rλ(˜i2+4πσ/λ), (22) i.e., R scales as the ratio of critical fields of the vortex whereg istheGibbspotentialoftheMeissnerstateand ◦ M and domain entries. Strictly speaking, the formulation R is the length of the domain wall segment. Provided thatleadstothisscalingisvalidonlyifR /λ(cid:29)1. How- that the surface energy σ > 0, the field-independent ◦ ever, because of the exponential falloff of the screening expression (22) is always positive, which implies that a currents,onemightexpectthatitisqualitativelycorrect straightdomainwallconfigurationisthermodynamically eveninthelimitingcaseofrelativelysmalldomainswhen unstable relative either to the Meissner state or to the R /λ(cid:38)1. state with a domain wall loop. ◦ 7 (a) (b) (c) 6 1.6 0.9 1.4 4 0.7 1.2 2 0.5 1 0 0.8 0.3 0.6 -2 0.1 0.4 0 0.5 1 1.5 2 2.5 3 3.5 4 0 2 4 6 8 10 12 0 2 4 6 8 10 12 FIG. 3: (a) The spatial profile of the Gibbs potential of a mesoscopic chiral cylinder with a domain wall loop as a function of its size for several applied fields. The potential is given relative to that of the Meissner state. Two metastable equilibria are clearly visible at zero applied field. (b,c) Critical field H and the equilibrium size Req of the domain wall loop at c1,dw 2 H =H as functions of the parameter κ , defined by eqn. (20). It is assumed that the variation of κ is entirely due to a c1,dw d d thevariationofeither(cid:15) or(cid:15) . Surprisingly,evenforamesoscopicsample,valuesofH andReq shownin(b)and(c)agree 1 2 c1,dw 2 quantitatively well with analytical results (17) and (16) obtained for a macroscopic sample. E. Summary of Section IV V. MAGNETIC RESPONSE OF A MESOSCOPIC CHIRAL SAMPLE Tosummarize,inthissectionweobtainedcriticalfields Inthissectionweconsiderthethermodynamicstability for a domain wall H , eqn. (17), and a vortex H , ofdomainwallsandvorticesinmesoscopicchiralsamples c1,dw c1,v eqn. (12), entries into a chiral superconductor in the whose relevant dimensions are comparable to λ.38 This macroscopic limit (all relevant dimensions much larger problemismotivatedbyrecentcantilevermagnetometry than λ). It was shown that the preferred domain wall measurements done on small Sr2RuO4 particles.14 Al- configuration is that of a loop whose equilibrium size is though the main aim of Ref. 14 was to probe the ex- given by eqn. (16). These results imply, in particular, istence of half-quantum vortices, it is interesting to ex- that, at fields above H , a cross section of the do- amine whether the observations reported there shed any c1,dw main structure of a macroscopic sample in the direction light on the question of the chiral nature of Sr2RuO4. perpendicular to the field is that of a plum pudding – a As in Section IV we will make use of expression (6) single domain populated by domain islands of opposite to evaluate the Gibbs potential for the onionlike geome- chirality. try shown in Fig. 1. It is assumed that vortices present in the system reside only in the cylinder’s hole, which guarantees a contour-independent definition of the vor- Itwasalsopointedoutthatastatement,oftenencoun- tex winding number (cid:96) . This assumption excludes the teredintheliterature,thatcoolinginthefieldshouldre- s possibility of wall vortices and limits our consideration ducedomainsbybiasingthesystemtoonechirality(field torelativelysmallappliedfieldsandmoderatechiralcur- training) does not refer to the thermodynamic equilib- rents (cf. discussion at the end of Section IVB). Notice, rium. This can be seen from the eqn. (9) for the Gibbs however, that, as demonstrated in Ref. 14, in a confined potentialofasingle-domainsamplethatdoesnotcontain geometry with geometrically reduced screening the field termslinearinthechiralcurrentandhencecannotdiffer- required for the wall vortex entry can be substantially entiate between domains of opposite chirality. Upon in- larger than the bulk H [given by eqn. (12)]. creasing the field a topological defect which corresponds c1,v It is convenient to introducing the notation to the minimal of the two fields H and H enters c1,v c1,dw the sample and for large fields both vortices and domain a =K (j)I (k)−I (j)K (k), jk 0 0 0 0 wall loops will be present. (23) b ≡K (j)I (k)−I (j)K (k), jk 2 0 2 0 However, even in thermal equilibrium, the direction where I and K are modified Bessel functions of n-th n n of the applied field can affect relative chiralities; for ex- orderandK (j)≡K (R /λ),etc.;R ,R ,andR stand 0 0 j 1 3 2 ample, for a positive field the +−+ domain wall loop for the radii of the inner and outer surfaces and for the configurationofFig.2(a)isfavoredoveritstimereversal radius of the circular domain wall loop, respectively (see −+−. Fig. 1). 8 The magnetic moment of the cylinder is given by the are given in Fig. 3(b) and (c), where they are plotted as following expression: a function of the parameter κ which characterizes the d interplaybetweenchiralcurrentsandthesurfacetension, M =L(Φ −πR2H )/4π, (24) 3 3 a eqn.(20). Ithasalsobeencheckedthatthedependencies showninnFig.3(b)and(c)alsodescribeacylinderwith where L is the height of the cylinder and Φ is the total 3 a hole, provided that Req is constrained to lie between flux through the area limited by outer boundary 3. Un- 2 R and R and H is constrained by the values of κ like the macroscopic limit, the values of fields and fluxes 1 3 c1,dw d which correspond to Req =R and Req =R . required to evaluate quantities of interest are no longer 2 1 2 3 It is important to emphasize that both H and given by simple analytical expressions and are relegated c1,v H discussed above are computed for a defect-free toAppendixA,wherethegeneralsolutionoftheLondon c1,dw sample. Only one of these fields has a physical meaning; equation for this geometry is presented. for example, if it turns out that H <H then the Inthezeroappliedfieldthegroundstateofthesystem c1,v c1,dw valueofthelatterneedstoberecalculatedinthepresence is obtained through minimization of (6) and corresponds of a vortex. to(cid:96) =0,(cid:96) =0i.e.toasingle-domaindefect-freestate. dw s The entry of a defect into the sample leads to a jump The zero-field magnetic moment M of a single-domain 0 in the magnetic moment. Such a jump can be evaluated chiral cylinder is then equal to using the results of Appendix A and is given by M =(˜iL/4π)(cid:0)πR2−2πλ2/b (cid:1)+˜iχ , (25) 0 3 13 m |Φ |L (cid:18) 2λ2 (cid:19) ∆M = 0 ∆(cid:96) 1− (28) whereχm isthemagneticsusceptibility,χm =∂M/∂Ha. v 4π s R12b13 The magnetic susceptibility is determined by the sys- tem’s dimensions R and R and does not depend on for a hole vortex entry and 1 3 the chiral current˜i: (cid:18) (cid:19) |Φ |L b 4πL−1χm = πaR32 (cid:18)b31− R42λR42 b1 (cid:19). (26) ∆Mdw = 40π ∆(cid:96)dw 1− b1123 (29) 13 1 3 13 for a circular domain wall entry. While ∆M is inde- v Itisinterestingtonotethattheresult(26)alsoholdsfor pendentofchiralcurrent, ∆M dependsonthedomain dw nonzero (cid:96) and (cid:96) , i.e., in the presence of either (hole) s dw wall size R , which is determined through the energy 2 vortices or a domain wall loop as long as the radius of minimization and hence implicitly depends on˜i. In the thelatterisfieldindependent. Inmesoscopicsettingsthe limitR →R wehave∆M /∆M =∆(cid:96) /∆(cid:96) andif 2 1 v dw s dw independence of equilibrium R on H can be expected 2 a R →R then ∆M →0. 2 3 dw for a “hard” domain wall whose energetics is dominated We now come back to the case of an extremely soft bythesurfacetensionσ. Intheoppositelimitofa“soft” domain wall mentioned earlier. Fig. 4 shows the radius domain wall whose behavior is dominated by the chiral of a circular domain wall with σ → 0 as a function of current˜iandnotbythesurfacetensionσ onecanexpect the applied field. As the applied field is increased the significant variations of R with H , as demonstrated 2 a domain wall moves continuously from the outer to the below. Suchvariationsleadtoadeviationoftheresponse innersurfaceofthecylinder.39 Giventhepossibilitythat from the simple linear form described by (26). chiral boundaries can carry topologically nontrivial exci- Upon increasing the applied field the system under- tations such as Majorana modes (see, e.g., Ref. 28) one goes a transition into a state in which either (cid:96) or (cid:96) is s dw mayspeculatethatsuchaprocesscanbeusedtoperform nonzero. Theorderingoftheseeventscanbedetermined a controllable transfer of excitations between the edges fromthecomparisonofthecriticalfieldsrequiredforthe of the sample. entry of the defects. The critical field for a hole vortex entry is given by A. Application to Sr2RuO4 |Φ | a H = 0 13 . (27) c1,v 2πR2 b −2λ2/R2 1 13 1 Wenowturntothequestionoftheinterpretationofthe Setting R → ξ and R → ∞ in the expression above, results of Jang et al.14 in terms of possible chiral super- 1 3 one recovers the bulk limit given by eqn. (12). conductivity. Janget al.reportedcantilevermagnetome- To find the critical field for a circular domain wall en- trymeasurementsofamesoscopicSr RuO particlewith 2 4 try one first needs to know its equilibrium size, which approximatedimensionsR =390nm,R =850nm,and 1 3 can be found through the minimization of the Gibbs po- L=350nm; the magnetic moment sensitivity was of the tential (6) with respect to R . Unlike the macroscopic order of 10−15e.m.u.40 The range of fields used in the 2 limit (Section IVC), the spatial profile of the Gibbs po- measurementswassuchastocoverthefirstexpecteden- tential in geometries with constrained screening can be try for a hole vortex, eqn. (27). quite complicated and may include several metastable The quantities χ , ∆M and H computed earlier m v c1,v equilibria [see Fig. 3a], which obstructs transparent ana- are independent of chiral current and the only quantity lytical treatment. Numerical results for H and Req that can be used to estimate˜i independently of σ is M . c1,dw 2 0 9 It was shown that a preferred configuration of the do- 1.6 main wall is that of a loop; a straight domain wall is never favored in thermodynamic equilibrium. Domain wall loops can exist in the superconducting bulk only if 1.4 the applied field is larger than H , which depends on c1,dw both the magnitude of the chiral current and the surface tensionofthedomainwall. Thecriticalfieldrequiredfor 1.2 a bulk vortex entry is not affected by the presence of the chiral currents. 1 We have also considered magnetic response and de- fect stability in mesoscopic chiral samples. It was shown 0.8 thatforaverysoftdomainwallitssizecanbecontrolled by the applied field. This phenomenon can potentially provideamechanismforacontrolledtransferofedgeex- 0.6 citations such as Majorana modes. There are several possible extensions to this work 0.4 0 1 2 3 4 5 which can be treated in the general framework outlined in SectionIII.An obvious one isto generalize theresults presented here for sample geometries that are not trans- lationallyinvariantalongtheappliedfield. Inparticular, FIG. 4: Radius of a soft domain wall loop as a function of it is interesting to inquire whether the conclusion that theappliedfield. ThedemonstrateddependenceofR onH 2 a the chirality of a single-domain sample is not favored by might be useful for performing controllable transfer of edge theappliedfieldholdsforothertypesofgeometries. One excitations. geometrywhichseemstobeanalyticallytractableisthat ofthePearllimit,inwhichthethicknessofthesampleis Within the noise resolution, measurements reported in smaller than the penetration depth. This might be par- Ref. 14 did not observe zero-field moment, which sets ticular relevant in connections with recent speculations the limit M < 10−15e.m.u. Using expressions (25) and about chiral superconductivity in graphene1,2. 0 (26) and the dimensions of the sample quoted above, we One can also consider domain structures that are not obtain the following upper bound for the magnitude of translationally invariant along the direction of the field. the chiral current: Such a possibility which was suggested in Ref. 12 is very attractivebecauseithasthepotentialtoreconciletheab- i<10−3×i (30) wc sence of the chiral field in Sr RuO as observed by scan- 2 4 ningmeasurementswithnonzeroKerrandµSRsignals.41 where the weak-coupling value i for Sr RuO is ob- wc 2 4 tained from eqns. (19) by setting (cid:15)2 = 1, λ = 200nm Another possible extension would deal with the inter- and is approximately equal to 1.9 × 1011 statamp/cm. actionbetweendomainwallloopsanddomainwallloops Limit(30)isconsistentwithscanningSQUIDmicroscopy and vortices. The interaction between such defects may measurements12 where, assuming the domain size of the have measurable signatures in the magnetization curves order of 1µ, measured i was estimated to be less than of macroscopic chiral samples. 0.1% of the weak-coupling value. VI. CONCLUSIONS Letusreviewthequantitativeresultsofthispaper. We VII. ACKNOWLEDGMENTS have considered thermodynamic stability of two types of topological defects – vortices and domain walls – which can exist in a chiral superconductor. Using the London ItisapleasuretothankDavidFerguson,TonyLeggett theoryitwasshownthatinthezeroappliedfieldamacro- and Catherine Kallin for useful discussions and Valentin scopic chiral sample is either defect-free or has defects Stanev for critical reading of the manuscript. I am also which are expelled toward the edges. The first situation very grateful to Alex Levchenko and Mike Norman for is realized if the chiral currents are small and the second continuous encouragement and support. The financial requiresthemtobesufficientlylargesuchthatthemagni- supportwasprovidedbytheCenterforEmergentSuper- tude of the field created in the immediate neighborhood conductivity,anEnergyFrontierResearchCenterfunded of the edge chiral current is larger than a critical field bytheU.S.DOE,OfficeofScience,underAwardNo.DE- required for a defect entry. AC0298CH1088. 10 Appendix A: Solution of the London equation for an The equations above determine the field distribution onion-like geometry through yet-unknown values of the fields on the bound- aries. To find the latter, one can use additional con- InthisappendixweconsiderthesolutionoftheLondon straints such as those provided by the Feynman-Onsager equationfortheonionlikecirculartwo-domainconfigura- (FO) quantization condition, which is obtained from the tion shown in Fig. 1; a single-domain configuration can London form of the Ginzburg-Landau equation for the beobtainedasalimitingcasebymovingthedomainwall current: to the inner or outer boundary of the sample. As mentioned earlier, in the extreme type II limit, the j =− c|Φ0| (cid:0)∇θ+ 2π A(cid:1), (A6) s 8π2λ2 |Φ | magnetic properties of a chiral domain wall can be mod- 0 eled by replacing it with a sheet current i if the domain where θ is a phase of the superconducting order parame- wall is on the surface and 2i if it is in the bulk. The cur- terandAisavectorpotentialofthetotalmagneticfield. rent carried by e.g. a surface domain wall is iL, where Using the symmetry of the problem and integrating the L is the length of the domain wall across the direction expression above along a circular contour R, one obtains of the current. In a cylindrical geometry with an exter- nisaleqfiueilvdalpeanrtaltloelatobtohuendcyarliyndceorn’sdiatxioins,faorshteheetmcuargrnenettici −4cπ ∂∂Hr (cid:12)(cid:12)(cid:12)(cid:12) =−8πc|2ΦR0λ|2 (cid:0)(cid:96)+1/|Φ0| (cid:73) A·dl(cid:1), (A7) field, R R wherethecurrentdensityhasbeenexpressedintermsof 4π H −H = i≡˜i, (A1) thefieldderivativewiththehelpoftheMaxwellequation. ·↑ ↑· c The constant (cid:96) characterizes the order parameter phase where the arrow in the subscripts indicates the direction winding around the integration contour. of the current and a dot indicates the side at which the In applying the FO relation (A7), one needs to bear field is taken. In the London limit the calculation of the in mind that the domain wall, being a phase defect, may magneticresponseofasamplewithagivendomainstruc- possess a nonzero vorticity and, hence, along with vor- ture is thus reduced to solving the London equation in tices, contributes to the winding number (cid:96) for appropri- each domain and then matching solutions using appro- ate integration contours. For a circular contour the do- priate boundary conditions. Let R2 denote the radius of main wall vorticity is denoted as (cid:96)dw. For a nonzero (cid:96)dw the domain wall and R and R be the inner and outer notonlythemagneticfieldbutalsothescreeningcurrent 1 3 radiiofthecylinder. Usingeqn.(A1),theboundarycon- experience a jump across the domain wall. This follows ditions for the field can be written in terms of the fields from writing down (A7) for inner and outer boundaries on the domain boundaries: of the domain wall. Taking into account that the flux is a continuous function of the integration contour, one H =H +˜i, H −H =2˜i, H =H −˜i, (A2) obtains 3− a 2+ 2− 1+ h where H3− ≡ H(R3 − 0), H1+ ≡ H(R1 + 0), H2± ≡ (∂H/∂r)| −(∂H/∂r)| = |Φ0| (cid:96) , (A8) H(R ±0) and H and H are the applied field and the 2+ 2− 2πR λ2 dw 2 a h 2 field in the hole respectively. In cylindrical coordinates, where R is the radius of the circular domain wall. the solution of the London equation can be written in 2 Recalling that due to the presence of the chiral cur- terms of Bessel functions I and K : 0 0 rent the magnetic field itself experiences a jump across r ∈(R ,R ): H(r)=c I (r/λ)+c(cid:48) K (r/λ), the domain wall and using (A8) and (A3) we obtain an 1 2 12 0 12 0 r ∈(R ,R ): H(r)=c I (r/λ)+c(cid:48) K (r/λ), expressionthatrelatesfieldvaluesonboundaries1,2and 2 3 23 0 23 0 3: (A3) where the constants c and c(cid:48) are determined by fields H2−a13 = H1+a23+H3−a12−˜iR˜22a12(b23−a23) jk jk on the domain boundaries, + |Φ0| (cid:96) a a , (A9) 2πλ2 dw 12 23 c =a−1(H K (1)−H K (2)), 12 12 2− 0 1+ 0 where R˜ ≡R/λ and b is defined by42 c(cid:48) =a−1(H I (2)−H I (1)), 23 12 12 1+ 0 2− 0 (A4) c =a−1(H K (2)−H K (3)), b ≡K (j)I (k)−I (j)K (k). (A10) 23 23 3− 0 2+ 0 jk 2 0 2 0 c(cid:48) =a−1(H I (3)−H I (2)), 23 23 2+ 0 3− 0 As a useful check of various identities one can consider limitingcasesofdomainwall2movingtoeithertheinner with a defined as jk or the outer surface of the cylinder: 2→1 or 2→3. For ajk =K0(j)I0(k)−I0(j)K0(k), (A5) exampleinthelimit2→1relation(A9)becomesH2− = H which implies that in a very thin domain the field 1+ where K (j)≡K (R ), etc. is uniform. In the limit 2→3 we have H =H −2˜i 0 0 j 2− 3−

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Stability of topological defects in chiral superconductors: London theory PDF

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Preview Stability of topological defects in chiral superconductors: London theory

Stability of topological defects in chiral superconductors: London theory. Victor Vakaryuk∗ Materials Science Division, Argonne National Laboratory, Argonne, Illinois 60439, USA Thispaperexaminesthethermodynamicstabilityofchiraldomainwallsandvortices–topological defects which can exist in chiral superconductors. Using London theory it is demonstrated that at sufficiently small applied and chiral fields the existence of domain walls and vortices in the sample is not favored and the sample’s configuration is a single domain. The particular chirality of the single-domain configuration is neither favored nor disfavored by the applied field. Increasing the fieldleadstoanentryofadomain-walllooporavortexintothesample. Theformationofastraight domain wall is never preferred in equilibrium. Values of the entry (critical) fields for both types of 2 defects, as well as the equilibrium size of the domain-wall loop, are calculated. We also consider a 1 mesoscopic chiral sample and calculate its zero-field magnetization, susceptibility, and a change in 0 the magnetic moment due to a vortex or a domain wall entry. We show that in the case of a soft 2 domain wall whose energetics is dominated by the chiral current (and not by the surface tension) n its behavior in mesoscopic samples is substantially different from that in the bulk case and can be a used for a controllable transfer of edge excitations. The applicability of these results to Sr RuO – 2 4 J a tentative chiral superconductor – is discussed. 2 1 I. INTRODUCTION respondstoasingle-domainsampleandthatthespecific ] chirality of this domain is neither favored nor disfavored n by the applied field. Upon increasing the applied field o Chiral superconductors belong to an exotic class of c physical systems whose many-body ground state carries either a domain wall loop or a vortex enters the sample; - theconfigurationinwhichadomainwallformsastraight r nonzero current and hence breaks time-reversal symme- p try. Because the current can assume two time-reversal line which terminates at the edges of the sample is never u favoredinequilibrium. Valuesoftheentry(critical)fields connected directions, the ground state of a chiral super- s as well as of the domain wall loop size are given in terms . conductorisdoublydegenerate(chiral). Thisdegeneracy t of model parameters such as the magnitude of the chiral a opensapossibilityfortheexistenceofextendedtopolog- m ical defects – domain walls – which connect regions of current and a domain wall surface tension. opposite chirality and exist along with conventional de- - d fects such as vortex lines. n Itwasrecentlysuggestedthatgrapheneatspecificdop- o Motivated by recent cantilever magnetometry mea- ingcansupportchiralsuperconductivity.1,2 Anotherten- c surementsinSr2RuO4 (Ref.14),wealsoconsiderameso- [ tative candidate for a chiral superconductor is Sr2RuO4 scopic chiral sample with a hole for which we calculate below 1.5K (see Refs. 3,4 for review) which is corrob- zero-field magnetic moment, susceptibility, and a mag- 3 orated by µSR,5,6 the Kerr effect,7 and phase-sensitive v netic moment change due to a vortex or domain wall measurements.8,9 The candidacy of Sr RuO is, how- 5 2 4 entry. We demonstrate that in the mesoscopic limit the ever, undermined by the fact that several attempts to 2 size of a loop formed by a very soft domain wall can be 0 detect a surface magnetic field generated by the chiral controlled with the applied field and speculate that such 6 currents10–13 or the magnetic moment associated with an effect can give rise to a controllable transfer of edge . them14 have not yielded a positive result. 9 excitations such as Majorana modes. 0 OneofthepossibleexplanationsaimedtocutthisGor- 1 dian knot of seemingly contradicting observations is to 1 assume the presence, on a mesoscopic scale, of an alter- : nating chiral domain structure which leads to the sub- Thepaperisorganizedinthefollowingway. InSection v i stantial field cancellation. Previous studies reported in II we give a phenomenological description of a chiral do- X the literature have focused either on the calculation of main wall used throughout the paper. In Section III we r the domain wall properties such as surface tension and derive an expression for the Gibbs energy of a chiral su- a accompanying chiral current (see, e.g., Refs. 15,16) or perconductor with topological defects in the strong type on finding the distribution of magnetic fields assuming a II limit. In Section IV we apply results of the previous particular domain structure without attempts to justify section to several domain configurations in macroscopic the latter.10–12,17,18 samples. This is the core section of the paper. In Sec- InthisworkweuseLondontheorytoaddresstheques- tion V we focus on a mesoscopic geometry. Section VI is tion of the thermodynamic stability of several domain devoted to overall conclusions. Appendix A contains the configurations for a simple sample’s geometry (such as solutionoftheLondonequationforatwo-domaincircular cylindrical), where demagnetizing effects can be easily cylinder of arbitrary dimensions with a hole. Appendix taken into account. We show in particular that in small B contains some useful results involving modified Bessel applied fields the equilibrium domain configuration cor- functions. 2 II. DOMAIN WALL DESCRIPTION configurationsisthatwhereasacirculardomainwallcre- ates a nonzero net flux ((cid:96) (cid:54)=0) the net flux created by dw Even in the simplest case of an isotropic chiral super- astraightdomainwallvanishes((cid:96)dw =0). Itshouldalso conductor the domain wall structure can be quite com- be pointed out that both σ and i will in general be dif- plicatedandisingeneraldeterminedbytheinterplaybe- ferentforthetwoconfigurations;wedonotindicatesuch tweenthematerial–Ginzburg-Landau–parametersand a difference explicitly, unless otherwise stated. the domain wall geometry. To describe chiral domain Wenoteinpassingthatthedomainstructureinchiral walls we use a simplified model in which it is modeled as superconductorsneednotbesimilartothatinferromag- a sheet-like object with surface tension σ which carries netic materials because the latter do not exhibit field chiral current with linear density 2i and is characterized screening. by a winding number (cid:96) . dw The presence of the current along a domain wall is ne- cessitated by the chiral nature of the state in which the III. SURFACE REPRESENTATION OF THE internal orbital motion of Cooper pairs, while compen- GIBBS POTENTIAL FOR A CHIRAL sated in the bulk, produces a nonzero charge current i SUPERCONDUCTOR on a boundary with vacuum and 2i on a boundary with another domain. The presence of such a current leads Let us start by considering a superconducting sample to a discontinuity of magnetic field (or, rather, magnetic placed in a uniform external magnetic field. The distri- induction) across the domain wall. bution of currents and fields in the sample is a function Ageneraldescriptionofadomainwallrequiresseveral of the applied field and, in thermal equilibrium, can be windingnumbers,whichreflectsthemulticomponentna- found through minimization of the corresponding Gibbs tureoftheunderlyingchiralstate.19Thewindingnumber potential, defined as30 relevant to our model, (cid:96) , has a physical meaning of a dw net flux (in units of flux quantum, cf. below) generated 1 (cid:90) G=F + d3r (B2−2B·H), (1) by the domain wall’s chiral and screening currents in an s 8π infinite superconducting medium.29 Defined in this way the winding number (cid:96) depends on the geometry of the where H and B are magnetic field and induction respec- dw domain wall and, in general, need not be an integer. tively; the volume integration extends over the space oc- The domain wall surface tension σ complements our cupied by the superconductor and over any cavities con- treatmentoftopologicaldefectsbyspecifyingitsintrinsic tained in it. The free energy F of the sample, which by s energyperunitarea. Althoughexistingcalculationsseem our definition excludes the field energy given by the B2 to indicate that for a chiral p+ip superconductor σ >0, term in eqn. (1), may contain terms describing kinetic the author is not aware of a general proof that would energies of charge and spin currents,20 spin-orbit inter- exclude the opposite. action energy,21 effects of kinematic spin polarization,22 Weconsideramodelinwhichboththesampleandthe etc. domainwallaretranslationallyinvariantalongthedirec- In London theory the free energy of a superconductor tion of the applied field and focus only on two domain is approximated by the kinetic energy of supercurrents wall configurations – a straight line and a circle – which, described by the superfluid velocity v . The sum of the s due to their high symmetry, admit straightforward ana- kinetic energy of supercurrents and the magnetic field lytical treatment. The main difference between the two energy can be written in the following form:23 (cid:90) d3r(cid:18)1ρ v2(r)+ 1 B2(cid:19)=−|Φ0|(cid:73) d2s·(B×∇θ)− 1 (cid:73) d2s·(cid:0)B×A(cid:1), (2) 2 s s 8π 16π2 8π sc where Φ ≡ hc/2e (< 0), A is the vector potential, ρ rem. The convenience of such a representation relates to 0 s is the superfluid density and θ is the phase of the order the fact that for relevant geometries the surface integra- parameter. The volume integration extends over the re- tion is usually more straightforward to perform than the gion of space occupied by the superconductor and (cid:72) d2s volume one. Moreover, using eqn. (2) one can avoid di- denotes the integration over its surface. The above re- rect calculation of the magnetic field contribution which sult, derived under the main assumption of the London is usually quite cumbersome. approximation – uniform superfluid density31 – is valid Eqn. (2) can also be used in the presence of topologi- for a superconductor of an arbitrary geometry and relies cal defects if they are treated in the following way: The onlyontheuseofMaxwell’sequationsandGauss’stheo- volume integrals should exclude regions of nonuniform 3 superfluid density associated with the defects, while the surface integrals should be complemented by an integra- tion over the surface that encloses the excluded volume. - 3 2 The “missing” contribution to the free energy can be ac- 1 counted for by introducing defects’ surface energy Fσ, + + - which can be computed from a more general description (e.g., Ginzburg-Landau theory). Such an approach leads to the following representation of the free energy32 (cid:90) 1 F =F + d3r ρ v2(r), (3) s σ 2 s s sc−d FIG.1: Onionliketwo-domainconfigurationofachiralsample wheretheintegralnowexcludesregionswithnonuniform used in the calculation of the Gibbs potential (6). Arrows superfluiddensityassociatedwiththedefects. Intheex- indicate the direction of the chiral currents. treme type II limit λ/ξ (cid:29) 1, this approach should give a quantitatively good approximation for the Ginzburg- Landau energy of a superconductor with topological de- and 3 – the outer surface of the sample. The surfaces fects,whereasinthemarginalcaseλ(cid:38)ξ onemighthope are characterized by their respective radii Rj. For the to get a qualitatively reasonable description. domain chirality shown in Fig. 1 surfaces 1 and 3 carry counterclockwise chiral current i while the domain wall 2 carries a clockwise current 2i. A. Gibbs potential of a two-domain cylinder Owing to the presence of the chiral current, the mag- netic field across each surface is discontinuous. Let us Forageneralsample’sgeometry,evaluationofthesur- denote the field values on the inner (−) and outer (+) face integrals in representation (2) is complicated by the sides of the surface j as Hj±. In this notation H3+ and spatial dependence of the magnetic field. One exception H1−areequivalenttotheappliedfieldHaandtothefield is a geometry that has a translational symmetry along in the hole Hh, respectively, and the field discontinuities the direction of the applied field (i.e., a cylinder with an are given by arbitrary cross section). In this geometry the value of H −H =˜i, H −H =2˜i, H −H =˜i, (5) the field on the sample’s surface is constant, which can 3− a 2+ 2− h 1+ be seen from the application of Amp`ere’s law to a rect- where we introduced the field jump ˜i ≡ 4πi/c which is angular contour with a side parallel to the field. analogous to a domain wall magnetization used, for ex- Motivated by recent cantilever magnetometry mea- ample, in Refs. 16,17. surementsonmesoscopicannularSr RuO samples,14we 2 4 Let Φ denote the total flux through the area limited consider a circular hollow cylinder with an onionlike do- j by the surface j and Φ ≡ Φ −Φ . Then, using the main structure shown in Fig. 1. The hole which is char- ij i j definition of the Gibbs potential (4) and the surface rep- acterized by an integer winding number (cid:96) provides, for s resentation of the free energy (3) given by eqn. (2), the small applied fields, the only place where vortices can Gibbs potential of this configuration is given by reside,33 and the two-domain configuration is the sim- plest one in which a reduction of the total magnetic mo- 8πg = 8πf −|Φ |(cid:96) (H −H ) (6) ment can be achieved (as observed in Ref. 14). σ 0 s h a − |Φ |(cid:96) (H −H +˜i)−˜i(Φ −Φ )−H Φ , Results of this section can also be used for cylindrical 0 dw 2− a 21 32 a 3 samples with an arbitrarily shaped cross section, pro- where g and f indicate that energies are taken per unit videdthedistancebetweenthedefectsandtheboundary length in the direction of the applied field. As discussed is much larger than λ. In this limit, as shown in Sec- in Section II, (cid:96) is a measure of the flux carried by a tion IV, a circular domain wall configuration is favored dw domain wall in an infinitely large sample and f is the energetically over a configuration in which a straight do- σ surface energy of the domain wall per unit length. main wall runs across the sample and terminates on the Intheabsenceofchiralcurrentsanddomainwalls,i.e., sample’s boundaries. when both˜i and (cid:96) are set to zero, expression (6) coin- For a cylindrical geometry with the axis parallel to dw cides,uptoanadditiveconstant34,withthatobtainedin the applied field H we have B = H and expression (1) a Ref. 24 for a hollow nonchiral cylinder. Notice that one simplifies to should not expect G to be of a simple form G∝M ·H a 1 (cid:90) where M is the magnetic moment of the cylinder since, G=F + d3r (H2−2H·H ), (4) s 8π a in general, in the absence of the applied field M (cid:54)=0. Theapplicationofeqn.(6)requiresknowledgeoffields To make use of eqn. (4) we notice that the onionlike ge- andfluxesinthesystemintermsoftheappliedfieldand ometryshowninFig.1consistsofthreesurfaces: 1–the parameters (cid:96) , (cid:96) , ˜i, and R . Such knowledge can be s dw j inner surface of the sample, 2 – the domain wall surface, obtained from a solution of the London equation. The 4 general solution of the London equation for an onionlike cylinderandcanbeshowntoholdevenwhenthescreen- geometryisgiveninAppendixA.Wenowproceedtothe ing is geometrically limited as in the mesoscopic settings analysisofthe“macroscopic”limitofthissolution,where considered in Section V. our results have simple analytical form. In Section V While it is natural to expect the invariance under the we relax the “macroscopic” constraint and consider this full time-reversal operation, which in the case of the geometry for a sample with arbitrary dimensions. Meissner state involves the reversal of both chiral cur- rent˜i and the applied field H , the invariance under the a reversal of either ˜i or H alone (partial time-reversal op- a IV. STABILITY OF TOPOLOGICAL DEFECTS eration) as demonstrated by (9) might be considered a IN MACROSCOPIC LIMIT surprising feature.36 An intriguing question is whether this feature is just a peculiarity of the cylindrical geom- In this section we calculate the Gibbs potential in the etry which possesses translational invariance along the macroscopiclimitwhenallrelevantdistances,suchasthe direction of the applied field or has a broader validity. size of the sample, the distance between defects, and the While the author does not have a proof of the latter, a sample’s boundary are larger than λ. In this limit the plausibilityargumentcanbegiventhatsuggeststhatthe precise shape of the boundary and the defect’s location invariance under the partial time-reversal operation can relative to it are irrelevant. We consider the four defect be expected if the sample has a mirror symmetry in the configurationsshowninFig.2(a): theMeissnerstate(no plane perpendicular to the field. defects), the state with a vortex, and the states with Wealsonotethat,asevidentfromeqn.(9),chiralcur- a domain wall shaped as a loop or as a straight line. rents give a positive contribution to the electromagnetic Thefirstthreeconfigurationscanbeobtainedaslimiting energyofthesystem. Thisstatementshouldalsoholdfor cases of the onionlike geometry of Fig. 1 and hence are samples with dimensions of the order of or smaller than described by eqn. (6). In the macroscopic, limit Gibbs λ. For such mesoscopic samples positive chiral contribu- potentialsofmorecomplicateddefectconfigurationssuch tiontotheelectromagneticenergymaybecomecompara- ascombinationsofthosementionedabovecanbewritten ble with the negative condensation energy whose scale is in a similar way. setbythethermodynamiccriticalfield. Thismechanism may hinder the formation of the chiral superconducting state and has to be borne in mind when considering the A. Meissner state possibility of a chiral pairing in very small samples.25 The Meissner state of a cylinder corresponds to a B. Vortex state single-domain configuration with no trapped defects, Fig. 2(a). Its Gibbs potential is obtained from eqn. (6) bysettingf ,(cid:96) ,(cid:96) ,R ,andR tozeroandisgivenby Avortexstatecorrespondstoasingle-domainconfigu- σ s dw 2 1 the expression rationwithaholewithnonzerophasewinding(cid:96)s around it. Suchaconfigurationisobtainedfromanonionlikege- 8πg =(˜i−H )Φ , (7) ometry by setting (cid:96) =0, R =ξ (cid:28)λ, and then taking M a M dw 1 the limit R → R . Using eqn. (6) the Gibbs potential 2 1 whereΦ ≡Φ isthenetfluxthroughthesamplegener- of the vortex state relative to that of the Meissner state M 3 atedbybothchiralandscreeningcurrents. Inthemacro- is given by the expression scopiclimitforasampleofcircumferenceP,theMeissner 8π(g −g )=−|Φ |(cid:96) (H −2H ), (10) flux Φ is given by a plausible expression: v M 0 s v a M where H is the value of the magnetic field on the outer Φ =λP(H +˜i), (8) v M a side of the surface which defines the normal vortex core; in the notation used in eqn. (6), H corresponds to H which is a direct consequence of the fact that both the v 2+ after taking the limit R →R . In deriving eqn. (10) we applied field and the field created by the chiral current 2 1 haveneglectedthevortexcoreenergyandthefluxcarried arescreenedoveraregionofthicknessλaroundtheouter by it – a step which is well justified in the extreme type edge of the sample.35 Given expression (8), the Gibbs II limit used here. potential of a solid macroscopic cylinder in the Meissner For large applied fields, the energy difference (10) is state takes the following form: negative, which means that the Meissner state is ther- 8πg =λP(˜i2−H2). (9) modynamicallyunstable. Thecriticalfieldforthevortex M a entry is determined by the following equation: This at first sight counterintuitive result implies that a H =H /2. (11) c1,v v chirality of a single-domain Meissner state specified by the sign of˜i is neither favored nor disfavored by an ex- The field H can be found by either solving the London v ternal field. Although proven in the macroscopic limit, equation in the macroscopic limit or by taking appropri- this statement is in fact independent of the size of the ate limits in the results for the onionlike geometry given 5 + - + + + Meissner Vortex + - + - - + DW as a loop Straight DW (a) (b) FIG. 2: (a) Configurations of the defects considered in Section IV. The positive direction of the applied field is out of the page toward the reader. (b) Values of winding numbers and chiral currents for vortex, loop and straight domain wall (DW) configurations. The size of the vortex core is exaggerated for visual purposes. in the Appendix A. In doing so one obtains C. State with a domain wall loop We now proceed to the configuration in which a do- |Φ |(cid:96) main wall forms a circular loop, i.e., terminates in the H =− 0 s ln(2λ/ξ). (12) c1,v 4πλ2 sample forming an “island” of opposite chirality. The Gibbs potential g of such a configuration is obtained dw◦ from eqn. (6) by setting (cid:96) = 0 and taking the limit s R →0. This leads to the expression Thus, the value of the first critical field for a vortex en- 1 try in a chiral superconductor is independent of both the magnitude and the sign on the chiral current ˜i 8πgdw◦ = 8πfσ−|Φ0|(cid:96)dw(H2−−Ha+˜i) and, within logarithmic precision, coincides with H for − 2˜iΦ +Φ (˜i−H ), (13) c1 2 3 a nonchiral superconductors37 (see, e.g., Ref. 27). Alter- natively, for a fixed chirality, eqn. (12) demonstrates the wheref isthesurfaceenergydefinedaftereqn.(6). H σ 2− absence of the field-reversal splitting of Hc1,v. It should isthefieldontheinnersideofthedomain’sboundary,Φ2 be noted, however, that inclusion of the vortex core en- is the flux through the area limited by it, and Φ is the 3 ergyin(10),whichtendstobechirality-dependent,26will totalfluxthroughthesample,whichincludesthescreen- result in nonzero field-reversal splitting of Hc1,v. ing contribution and the flux created by the domain. In the macroscopic limit these quantities can be found by One might wonder what happens if the magnitude taking the appropriate limit in the general solution for of the chiral current ˜i is such that the magnetic field the field distribution, which is given in Appendix A. In created in its immediate neighborhood is larger than this way we obtain H . Whilethisquestioncannotbeansweredwithinthe c1,v macroscopicapproximationusedinthissection,eqn.(12) |Φ |(cid:96) suggests that, for applied fields smaller than Hc1,v, vor- H2− =− 4π0Rdλw −˜i, ticesgeneratedbythe chiralcurrent’smagneticfield will 1 (14) tend to stay away from the bulk “decorating” edges of Φ =− |Φ |(cid:96) −˜i2πRλ, 2 2 0 dw the sample and domain boundaries (if present). Φ =Φ −(cid:96) |Φ |, 3 M dw 0 Another conclusion that can be drawn from the re- sultsofthissectionisthat, inthermalequilibriuminthe where R is the radius of the domain island [≡R in the 2 absence of the applied field, the total flux trapped by a notationofeqn.(6)]. ThefluxΦ consistsofthefluxΦ 3 M chiralcavity,locatedatdistancemuchlargerthanλaway generated by the boundary of the sample [cf. eqn. (8)] from the sample’s boundary, is zero. This follows from and of the flux carried by a domain wall loop, (cid:96) Φ . dw 0 settingH =0inexpression(10)andthenminimizingit Plugging these results into eqn. (13) yields the following a with respect to (cid:96) . expressionfortheGibbspotentialofthecirculardomain s 6 wall configuration: To estimate the actual value of H for a given ma- c1,dw terial a knowledge of chiral current ˜i and domain wall 8π(g −g ) = |Φ0|2(cid:96)2dw +(˜i2+4πσ/λ)4πRλ surface tension σ is required. These can be calculated dw◦ M 4πRλ using Ginzburg-Landau theory and turn out to depend + 2|Φ |(cid:96) H , (15) on various material and geometrical parameters such as 0 dw a Ginzburg-Landau expansion coefficients and the orien- where gM is the Gibbs potential of the Meissner state tation of the domain wall relative to the crystal axes and σ is the surface tension of the domain wall, such (see, e.g., Refs. 15,16). Ignoring for simplicity material thatfσ =2πRσ. Wefirstminimizetheaboveexpression anisotropy, one can conclude that withrespecttoRwhichyieldstheequilibriumsizeofthe domain island: 4πσ/λ=(cid:15) |Φ0|2 , ˜i2 =(cid:15) |Φ0|2 , (19) 14π2λ3ξ 24π2λ4 |Φ ||(cid:96) | R = 0 dw . (16) ◦ (cid:113) where(cid:15) aredimensionlessparameterswhichinaweak- 4πλ ˜i2+4πσ/λ 1,2 coupling BCS limit are of the order of 1.15,16 We now define the following parameter: Evaluated at R the difference (15) is positive for small ◦ applied fields; upon increasing the field the difference κ ≡4πσ/(˜i2λ)=(cid:15) λ/((cid:15) ξ). (20) d 1 2 (15)becomesnegativeatsomevalueofH whichdefines a the critical field for the creation of a domain wall loop. As can be seen from eqns. (16) and (17) this parame- In other words, circular domains such as that shown in terdetermineswhethertheenergeticsofthedomainwall Fig. 2(a) will be thermodynamically stable only if the is dominated by the chiral current (“soft” domain wall, applied field Ha exceeds a critical value Hc1,dw, defined κd (cid:28) 1) or by the surface tension (“hard” domain wall, by κd (cid:29) 1). In the weak-coupling limit when (cid:15)1,2 ∼ 1, one generally expects that κ ≈ λ/ξ. Provided the weak- d (cid:113) H = ˜i2+4πσ/λ. (17) coupling limit is applicable for Sr2RuO4 (λ/ξ ∼ 1 and c1,dw hence κ ∼ 1), one would expect that H ∼ H d c1,v c1,dw and the domain size R ∼λ. However, given the uncon- Notice that unlike H for a vortex entry, eqn. (12), ◦ c1,v ventional nature of superconductivity in Sr RuO , the the critical field for the domain wall loop entry depends 2 4 applicability of the weak-coupling results to this mate- on the chiral current ˜i but is independent of the flux rial remains an open question. carriedbythedefect. Thereasonforthelatteristhe(cid:96) dw dependence of the equilibrium domain size as specified by eqn. (16). D. State with a straight domain wall that Let+−+denotethechiralityarrangementofadomain terminates at the edges wall loop state shown in Fig. 2(a). Unlike the Meissner state discussed in Section IVA, the energy of this state, A straight domain wall configuration that terminates eqn. (15), is not invariant under reversal of the applied at the edges of the sample is qualitatively different from field. Equivalently, for a fixed applied field the energies theclosedconfigurationdiscussedearlier. Unlikethelat- of the +−+ arrangement and of its time-reversal coun- ter, the total flux carried by a straight domain wall is terpart−+−(obtainedbychangingthedirectionofthe zero (see, e.g., Ref. 19), which substantially changes its chiral currents and the sign of (cid:96) ) are different. In par- dw energetics. Let Φ denote the total flux carried by ± ticular,forapositiveappliedfieldtheenergyofthe−+− ± domainsshowninFig.2(a). TheGibbspotentialofsuch arrangement is larger than that of +−+. Although the a state can be found along the lines that led to eqn. (6), energy of the former state can be lowered by adding vor- and is given by tices,itwillstillbelargerthaneitherthe+−+arrange- mentorapurevortexstateandhencecannotcorrespond 8πg =8πf +˜i(Φ −Φ )−H (Φ +Φ ). (21) dw| σ + − a + − to a true equilibrium. Putting together eqns. (12, 16, and 17) leads to the In the macroscopic limit, fluxes Φ can be easily com- ± following expression for the size of the domain island: puted, which leads to the following expression for g : dw| R◦/λ∝Hc1,v/Hc1,dw, (18) 8π(gdw|−gM)=2Rλ(˜i2+4πσ/λ), (22) i.e., R scales as the ratio of critical fields of the vortex whereg istheGibbspotentialoftheMeissnerstateand ◦ M and domain entries. Strictly speaking, the formulation R is the length of the domain wall segment. Provided thatleadstothisscalingisvalidonlyifR /λ(cid:29)1. How- that the surface energy σ > 0, the field-independent ◦ ever, because of the exponential falloff of the screening expression (22) is always positive, which implies that a currents,onemightexpectthatitisqualitativelycorrect straightdomainwallconfigurationisthermodynamically eveninthelimitingcaseofrelativelysmalldomainswhen unstable relative either to the Meissner state or to the R /λ(cid:38)1. state with a domain wall loop. ◦ 7 (a) (b) (c) 6 1.6 0.9 1.4 4 0.7 1.2 2 0.5 1 0 0.8 0.3 0.6 -2 0.1 0.4 0 0.5 1 1.5 2 2.5 3 3.5 4 0 2 4 6 8 10 12 0 2 4 6 8 10 12 FIG. 3: (a) The spatial profile of the Gibbs potential of a mesoscopic chiral cylinder with a domain wall loop as a function of its size for several applied fields. The potential is given relative to that of the Meissner state. Two metastable equilibria are clearly visible at zero applied field. (b,c) Critical field H and the equilibrium size Req of the domain wall loop at c1,dw 2 H =H as functions of the parameter κ , defined by eqn. (20). It is assumed that the variation of κ is entirely due to a c1,dw d d thevariationofeither(cid:15) or(cid:15) . Surprisingly,evenforamesoscopicsample,valuesofH andReq shownin(b)and(c)agree 1 2 c1,dw 2 quantitatively well with analytical results (17) and (16) obtained for a macroscopic sample. E. Summary of Section IV V. MAGNETIC RESPONSE OF A MESOSCOPIC CHIRAL SAMPLE Tosummarize,inthissectionweobtainedcriticalfields Inthissectionweconsiderthethermodynamicstability for a domain wall H , eqn. (17), and a vortex H , ofdomainwallsandvorticesinmesoscopicchiralsamples c1,dw c1,v eqn. (12), entries into a chiral superconductor in the whose relevant dimensions are comparable to λ.38 This macroscopic limit (all relevant dimensions much larger problemismotivatedbyrecentcantilevermagnetometry than λ). It was shown that the preferred domain wall measurements done on small Sr2RuO4 particles.14 Al- configuration is that of a loop whose equilibrium size is though the main aim of Ref. 14 was to probe the ex- given by eqn. (16). These results imply, in particular, istence of half-quantum vortices, it is interesting to ex- that, at fields above H , a cross section of the do- amine whether the observations reported there shed any c1,dw main structure of a macroscopic sample in the direction light on the question of the chiral nature of Sr2RuO4. perpendicular to the field is that of a plum pudding – a As in Section IV we will make use of expression (6) single domain populated by domain islands of opposite to evaluate the Gibbs potential for the onionlike geome- chirality. try shown in Fig. 1. It is assumed that vortices present in the system reside only in the cylinder’s hole, which guarantees a contour-independent definition of the vor- Itwasalsopointedoutthatastatement,oftenencoun- tex winding number (cid:96) . This assumption excludes the teredintheliterature,thatcoolinginthefieldshouldre- s possibility of wall vortices and limits our consideration ducedomainsbybiasingthesystemtoonechirality(field torelativelysmallappliedfieldsandmoderatechiralcur- training) does not refer to the thermodynamic equilib- rents (cf. discussion at the end of Section IVB). Notice, rium. This can be seen from the eqn. (9) for the Gibbs however, that, as demonstrated in Ref. 14, in a confined potentialofasingle-domainsamplethatdoesnotcontain geometry with geometrically reduced screening the field termslinearinthechiralcurrentandhencecannotdiffer- required for the wall vortex entry can be substantially entiate between domains of opposite chirality. Upon in- larger than the bulk H [given by eqn. (12)]. creasing the field a topological defect which corresponds c1,v It is convenient to introducing the notation to the minimal of the two fields H and H enters c1,v c1,dw the sample and for large fields both vortices and domain a =K (j)I (k)−I (j)K (k), jk 0 0 0 0 wall loops will be present. (23) b ≡K (j)I (k)−I (j)K (k), jk 2 0 2 0 However, even in thermal equilibrium, the direction where I and K are modified Bessel functions of n-th n n of the applied field can affect relative chiralities; for ex- orderandK (j)≡K (R /λ),etc.;R ,R ,andR stand 0 0 j 1 3 2 ample, for a positive field the +−+ domain wall loop for the radii of the inner and outer surfaces and for the configurationofFig.2(a)isfavoredoveritstimereversal radius of the circular domain wall loop, respectively (see −+−. Fig. 1). 8 The magnetic moment of the cylinder is given by the are given in Fig. 3(b) and (c), where they are plotted as following expression: a function of the parameter κ which characterizes the d interplaybetweenchiralcurrentsandthesurfacetension, M =L(Φ −πR2H )/4π, (24) 3 3 a eqn.(20). Ithasalsobeencheckedthatthedependencies showninnFig.3(b)and(c)alsodescribeacylinderwith where L is the height of the cylinder and Φ is the total 3 a hole, provided that Req is constrained to lie between flux through the area limited by outer boundary 3. Un- 2 R and R and H is constrained by the values of κ like the macroscopic limit, the values of fields and fluxes 1 3 c1,dw d which correspond to Req =R and Req =R . required to evaluate quantities of interest are no longer 2 1 2 3 It is important to emphasize that both H and given by simple analytical expressions and are relegated c1,v H discussed above are computed for a defect-free toAppendixA,wherethegeneralsolutionoftheLondon c1,dw sample. Only one of these fields has a physical meaning; equation for this geometry is presented. for example, if it turns out that H <H then the Inthezeroappliedfieldthegroundstateofthesystem c1,v c1,dw valueofthelatterneedstoberecalculatedinthepresence is obtained through minimization of (6) and corresponds of a vortex. to(cid:96) =0,(cid:96) =0i.e.toasingle-domaindefect-freestate. dw s The entry of a defect into the sample leads to a jump The zero-field magnetic moment M of a single-domain 0 in the magnetic moment. Such a jump can be evaluated chiral cylinder is then equal to using the results of Appendix A and is given by M =(˜iL/4π)(cid:0)πR2−2πλ2/b (cid:1)+˜iχ , (25) 0 3 13 m |Φ |L (cid:18) 2λ2 (cid:19) ∆M = 0 ∆(cid:96) 1− (28) whereχm isthemagneticsusceptibility,χm =∂M/∂Ha. v 4π s R12b13 The magnetic susceptibility is determined by the sys- tem’s dimensions R and R and does not depend on for a hole vortex entry and 1 3 the chiral current˜i: (cid:18) (cid:19) |Φ |L b 4πL−1χm = πaR32 (cid:18)b31− R42λR42 b1 (cid:19). (26) ∆Mdw = 40π ∆(cid:96)dw 1− b1123 (29) 13 1 3 13 for a circular domain wall entry. While ∆M is inde- v Itisinterestingtonotethattheresult(26)alsoholdsfor pendentofchiralcurrent, ∆M dependsonthedomain dw nonzero (cid:96) and (cid:96) , i.e., in the presence of either (hole) s dw wall size R , which is determined through the energy 2 vortices or a domain wall loop as long as the radius of minimization and hence implicitly depends on˜i. In the thelatterisfieldindependent. Inmesoscopicsettingsthe limitR →R wehave∆M /∆M =∆(cid:96) /∆(cid:96) andif 2 1 v dw s dw independence of equilibrium R on H can be expected 2 a R →R then ∆M →0. 2 3 dw for a “hard” domain wall whose energetics is dominated We now come back to the case of an extremely soft bythesurfacetensionσ. Intheoppositelimitofa“soft” domain wall mentioned earlier. Fig. 4 shows the radius domain wall whose behavior is dominated by the chiral of a circular domain wall with σ → 0 as a function of current˜iandnotbythesurfacetensionσ onecanexpect the applied field. As the applied field is increased the significant variations of R with H , as demonstrated 2 a domain wall moves continuously from the outer to the below. Suchvariationsleadtoadeviationoftheresponse innersurfaceofthecylinder.39 Giventhepossibilitythat from the simple linear form described by (26). chiral boundaries can carry topologically nontrivial exci- Upon increasing the applied field the system under- tations such as Majorana modes (see, e.g., Ref. 28) one goes a transition into a state in which either (cid:96) or (cid:96) is s dw mayspeculatethatsuchaprocesscanbeusedtoperform nonzero. Theorderingoftheseeventscanbedetermined a controllable transfer of excitations between the edges fromthecomparisonofthecriticalfieldsrequiredforthe of the sample. entry of the defects. The critical field for a hole vortex entry is given by A. Application to Sr2RuO4 |Φ | a H = 0 13 . (27) c1,v 2πR2 b −2λ2/R2 1 13 1 Wenowturntothequestionoftheinterpretationofthe Setting R → ξ and R → ∞ in the expression above, results of Jang et al.14 in terms of possible chiral super- 1 3 one recovers the bulk limit given by eqn. (12). conductivity. Janget al.reportedcantilevermagnetome- To find the critical field for a circular domain wall en- trymeasurementsofamesoscopicSr RuO particlewith 2 4 try one first needs to know its equilibrium size, which approximatedimensionsR =390nm,R =850nm,and 1 3 can be found through the minimization of the Gibbs po- L=350nm; the magnetic moment sensitivity was of the tential (6) with respect to R . Unlike the macroscopic order of 10−15e.m.u.40 The range of fields used in the 2 limit (Section IVC), the spatial profile of the Gibbs po- measurementswassuchastocoverthefirstexpecteden- tential in geometries with constrained screening can be try for a hole vortex, eqn. (27). quite complicated and may include several metastable The quantities χ , ∆M and H computed earlier m v c1,v equilibria [see Fig. 3a], which obstructs transparent ana- are independent of chiral current and the only quantity lytical treatment. Numerical results for H and Req that can be used to estimate˜i independently of σ is M . c1,dw 2 0 9 It was shown that a preferred configuration of the do- 1.6 main wall is that of a loop; a straight domain wall is never favored in thermodynamic equilibrium. Domain wall loops can exist in the superconducting bulk only if 1.4 the applied field is larger than H , which depends on c1,dw both the magnitude of the chiral current and the surface tensionofthedomainwall. Thecriticalfieldrequiredfor 1.2 a bulk vortex entry is not affected by the presence of the chiral currents. 1 We have also considered magnetic response and de- fect stability in mesoscopic chiral samples. It was shown 0.8 thatforaverysoftdomainwallitssizecanbecontrolled by the applied field. This phenomenon can potentially provideamechanismforacontrolledtransferofedgeex- 0.6 citations such as Majorana modes. There are several possible extensions to this work 0.4 0 1 2 3 4 5 which can be treated in the general framework outlined in SectionIII.An obvious one isto generalize theresults presented here for sample geometries that are not trans- lationallyinvariantalongtheappliedfield. Inparticular, FIG. 4: Radius of a soft domain wall loop as a function of it is interesting to inquire whether the conclusion that theappliedfield. ThedemonstrateddependenceofR onH 2 a the chirality of a single-domain sample is not favored by might be useful for performing controllable transfer of edge theappliedfieldholdsforothertypesofgeometries. One excitations. geometrywhichseemstobeanalyticallytractableisthat ofthePearllimit,inwhichthethicknessofthesampleis Within the noise resolution, measurements reported in smaller than the penetration depth. This might be par- Ref. 14 did not observe zero-field moment, which sets ticular relevant in connections with recent speculations the limit M < 10−15e.m.u. Using expressions (25) and about chiral superconductivity in graphene1,2. 0 (26) and the dimensions of the sample quoted above, we One can also consider domain structures that are not obtain the following upper bound for the magnitude of translationally invariant along the direction of the field. the chiral current: Such a possibility which was suggested in Ref. 12 is very attractivebecauseithasthepotentialtoreconciletheab- i<10−3×i (30) wc sence of the chiral field in Sr RuO as observed by scan- 2 4 ningmeasurementswithnonzeroKerrandµSRsignals.41 where the weak-coupling value i for Sr RuO is ob- wc 2 4 tained from eqns. (19) by setting (cid:15)2 = 1, λ = 200nm Another possible extension would deal with the inter- and is approximately equal to 1.9 × 1011 statamp/cm. actionbetweendomainwallloopsanddomainwallloops Limit(30)isconsistentwithscanningSQUIDmicroscopy and vortices. The interaction between such defects may measurements12 where, assuming the domain size of the have measurable signatures in the magnetization curves order of 1µ, measured i was estimated to be less than of macroscopic chiral samples. 0.1% of the weak-coupling value. VI. CONCLUSIONS Letusreviewthequantitativeresultsofthispaper. We VII. ACKNOWLEDGMENTS have considered thermodynamic stability of two types of topological defects – vortices and domain walls – which can exist in a chiral superconductor. Using the London ItisapleasuretothankDavidFerguson,TonyLeggett theoryitwasshownthatinthezeroappliedfieldamacro- and Catherine Kallin for useful discussions and Valentin scopic chiral sample is either defect-free or has defects Stanev for critical reading of the manuscript. I am also which are expelled toward the edges. The first situation very grateful to Alex Levchenko and Mike Norman for is realized if the chiral currents are small and the second continuous encouragement and support. The financial requiresthemtobesufficientlylargesuchthatthemagni- supportwasprovidedbytheCenterforEmergentSuper- tude of the field created in the immediate neighborhood conductivity,anEnergyFrontierResearchCenterfunded of the edge chiral current is larger than a critical field bytheU.S.DOE,OfficeofScience,underAwardNo.DE- required for a defect entry. AC0298CH1088. 10 Appendix A: Solution of the London equation for an The equations above determine the field distribution onion-like geometry through yet-unknown values of the fields on the bound- aries. To find the latter, one can use additional con- InthisappendixweconsiderthesolutionoftheLondon straints such as those provided by the Feynman-Onsager equationfortheonionlikecirculartwo-domainconfigura- (FO) quantization condition, which is obtained from the tion shown in Fig. 1; a single-domain configuration can London form of the Ginzburg-Landau equation for the beobtainedasalimitingcasebymovingthedomainwall current: to the inner or outer boundary of the sample. As mentioned earlier, in the extreme type II limit, the j =− c|Φ0| (cid:0)∇θ+ 2π A(cid:1), (A6) s 8π2λ2 |Φ | magnetic properties of a chiral domain wall can be mod- 0 eled by replacing it with a sheet current i if the domain where θ is a phase of the superconducting order parame- wall is on the surface and 2i if it is in the bulk. The cur- terandAisavectorpotentialofthetotalmagneticfield. rent carried by e.g. a surface domain wall is iL, where Using the symmetry of the problem and integrating the L is the length of the domain wall across the direction expression above along a circular contour R, one obtains of the current. In a cylindrical geometry with an exter- nisaleqfiueilvdalpeanrtaltloelatobtohuendcyarliyndceorn’sdiatxioins,faorshteheetmcuargrnenettici −4cπ ∂∂Hr (cid:12)(cid:12)(cid:12)(cid:12) =−8πc|2ΦR0λ|2 (cid:0)(cid:96)+1/|Φ0| (cid:73) A·dl(cid:1), (A7) field, R R wherethecurrentdensityhasbeenexpressedintermsof 4π H −H = i≡˜i, (A1) thefieldderivativewiththehelpoftheMaxwellequation. ·↑ ↑· c The constant (cid:96) characterizes the order parameter phase where the arrow in the subscripts indicates the direction winding around the integration contour. of the current and a dot indicates the side at which the In applying the FO relation (A7), one needs to bear field is taken. In the London limit the calculation of the in mind that the domain wall, being a phase defect, may magneticresponseofasamplewithagivendomainstruc- possess a nonzero vorticity and, hence, along with vor- ture is thus reduced to solving the London equation in tices, contributes to the winding number (cid:96) for appropri- each domain and then matching solutions using appro- ate integration contours. For a circular contour the do- priate boundary conditions. Let R2 denote the radius of main wall vorticity is denoted as (cid:96)dw. For a nonzero (cid:96)dw the domain wall and R and R be the inner and outer notonlythemagneticfieldbutalsothescreeningcurrent 1 3 radiiofthecylinder. Usingeqn.(A1),theboundarycon- experience a jump across the domain wall. This follows ditions for the field can be written in terms of the fields from writing down (A7) for inner and outer boundaries on the domain boundaries: of the domain wall. Taking into account that the flux is a continuous function of the integration contour, one H =H +˜i, H −H =2˜i, H =H −˜i, (A2) obtains 3− a 2+ 2− 1+ h where H3− ≡ H(R3 − 0), H1+ ≡ H(R1 + 0), H2± ≡ (∂H/∂r)| −(∂H/∂r)| = |Φ0| (cid:96) , (A8) H(R ±0) and H and H are the applied field and the 2+ 2− 2πR λ2 dw 2 a h 2 field in the hole respectively. In cylindrical coordinates, where R is the radius of the circular domain wall. the solution of the London equation can be written in 2 Recalling that due to the presence of the chiral cur- terms of Bessel functions I and K : 0 0 rent the magnetic field itself experiences a jump across r ∈(R ,R ): H(r)=c I (r/λ)+c(cid:48) K (r/λ), the domain wall and using (A8) and (A3) we obtain an 1 2 12 0 12 0 r ∈(R ,R ): H(r)=c I (r/λ)+c(cid:48) K (r/λ), expressionthatrelatesfieldvaluesonboundaries1,2and 2 3 23 0 23 0 3: (A3) where the constants c and c(cid:48) are determined by fields H2−a13 = H1+a23+H3−a12−˜iR˜22a12(b23−a23) jk jk on the domain boundaries, + |Φ0| (cid:96) a a , (A9) 2πλ2 dw 12 23 c =a−1(H K (1)−H K (2)), 12 12 2− 0 1+ 0 where R˜ ≡R/λ and b is defined by42 c(cid:48) =a−1(H I (2)−H I (1)), 23 12 12 1+ 0 2− 0 (A4) c =a−1(H K (2)−H K (3)), b ≡K (j)I (k)−I (j)K (k). (A10) 23 23 3− 0 2+ 0 jk 2 0 2 0 c(cid:48) =a−1(H I (3)−H I (2)), 23 23 2+ 0 3− 0 As a useful check of various identities one can consider limitingcasesofdomainwall2movingtoeithertheinner with a defined as jk or the outer surface of the cylinder: 2→1 or 2→3. For ajk =K0(j)I0(k)−I0(j)K0(k), (A5) exampleinthelimit2→1relation(A9)becomesH2− = H which implies that in a very thin domain the field 1+ where K (j)≡K (R ), etc. is uniform. In the limit 2→3 we have H =H −2˜i 0 0 j 2− 3−

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