Suppression of spontaneous currents in Sr 2 RuO 4 by surface disorder
Samuel Lederer, Wen Huang, Edward Taylor, Srinivas Raghu, Catherine Kallin
aa r X i v : . [ c ond - m a t . s up r- c on ] A p r Suppression of spontaneous currents in Sr RuO by surface disorder Samuel Lederer , Wen Huang , Edward Taylor , Srinivas Raghu , , and Catherine Kallin , Department of Physics, Stanford University, Stanford, California, 94305, USA Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, L8S 4M1, Canada SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025 and Canadian Institute for Advanced Research, Toronto, Ontario M5G 1Z8, Canada
A major challenge to the chiral p -wave hypothesis for the pairing symmetry of the unconventional super-conductor Sr RuO is the null result of sensitive scanning magnetometry experiments designed to detect theexpected spontaneous charge currents. Motivated by junction tunneling conductance measurements which in-dicate the quenching of superconductivity at the surfaces of even high-purity samples, we examine the sponta-neous currents in a chiral p -wave superconductor near a normal metal / superconductor interface using the latticeBogoliubov-de Gennes equations and Ginzburg-Landau theory, and find that the edge current is suppressed bymore than an order of magnitude compared to previous estimates. These calculations demonstrate that interfacedetails can have a quantitatively meaningful effect on the expectations for magnetometry experiments. I. INTRODUCTION
Strontium Ruthenate, Sr RuO , is an unconventional super-conductor ( T c = 1 . K ) for which there exists substantial ev-idence for odd-parity pairing as well as for the spontaneousbreaking of time reversal symmetry below T c . These obser-vations lead naturally to the conclusion that the pairing sym-metry is chiral p -wave ( p x ± ip y ), a two dimensional analogof the A-phase of superfluid He. Though this is the leadingphenomenological hypothesis, it is seemingly contradicted byseveral experiments. Prominent among these are high reso-lution scanning magnetometry measurements , which im-age magnetic fields across several µm of sample (includingthe sample edge) and see no sign of the expected spontaneouscurrents.The presence of spontaneous, persistent charge currentsat edges and domain walls is a robust consequence of time-reversal symmetry breaking superconductivity. However, themagnitude of these currents is determined by microscopic de-tails – they are neither quantized nor universal. The reasonthat the null result of the scanning magnetometry experimentsposes such a challenge to the chiral p -wave hypothesis isquantitative – spontaneous currents of size comparable to the-oretical estimates would give a magnetic signal more thantwo orders of magnitude greater than the experimental resolu-tion. Magnetometry measurements on mesoscopic samples also see no signs of these currents.In this paper we calculate the spontaneous surface currentsfor a family of models consistent with the phenomenology ofsuperconductivity in Sr RuO . Motivated by a -axis tunnel-ing experiments , we employ a different interface conditionthan previous studies, modeling the surface region as a nor-mal metal layer adjoining the superconducting bulk. We findthat, compared to previous estimates, the expected magneticsignal from edge currents is reduced by over an order of mag-nitude. These calculations demonstrate that interface detailscan have a quantitatively meaningful effect on the expecta-tions for magnetometry experiments. II. SURFACE IMPERFECTION
The assumption of specular surface scattering as employedin requires an atomically smooth surface. ab faces ofSr RuO can be cleaved, but ac and bc faces are typicallypolished to a smoothness of several nm , on the order of tenlattice constants. In a -axis junction tunneling conductancemeasurements, signatures of superconductivity at the surfaceare present only at the sub- level on top of a substantialsmooth background , as shown in Fig. 2 of that reference.Accordingly, the best indication from experiment is that theedge region is metallic , with a superconducting gap devel-oping only further into the sample.Such a scenario is plausible given the fragility of unconven-tional superconductivity to elastic scattering (i.e. the inappli-cability of Anderson’s Theorem to a sign-changing order pa-rameter), which has been explicitly verified for this material .Rough or pair-breaking surface effects have been shown to sharply reduce the superconducting order parameter at thesurface, although not to meaningfully alter the surface densityof states. Accordingly, the observation of metallic behaviorsuggests that there is a higher density of defects near the sur-face (presumably introduced during crystal growth or prepa-ration procedures), leading to a reduced mean free path andthe quenching of superconductivity near the surface.To facilitate calculations, we do not directly treat a roughsurface or defects in the surface region, but rather adopt amodel consisting of a clean interface between vacuum and ametallic region, which in turn has a clean interface with the su-perconducting bulk. The metallic region is arranged by settingappropriate coupling constants to zero in lattice Bogoliubovde-Gennes Hamiltonians. This introduces artifacts which willbe discussed in section VII. III. MODEL HAMILTONIANS
We consider spinless fermions on a 2D square lattice corre-sponding to the RuO plane, and work in a cylinder geometry:periodic boundary conditions are taken in the y direction, andopen boundary conditions in x . We will consider two differentBogoliubov-de-Gennes Hamiltonians: H γ = − X i,j T zij c † z,i c z,j + X i h ∆ γx ( i ) c † z,i c † z,i +ˆ x + ∆ γy ( i ) c † z,i c † z,i +ˆ y + h.c. i (1) H αβ = − X i,j X η = x,y T ηij c † η,i c η,j − t ′ X i X s = ± s h c † x,i c y,i +ˆ x + s ˆ y + h.c. i + X i X s = ± h ∆ αβx ( i ) c † x,i c † x,i +ˆ x + s ˆ y + s ∆ αβy ( i ) c † y,i c † y,i +ˆ x + s ˆ y + h.c. i (2) H γ is a minimal Hamiltonian for chiral p -wave supercon-ductivity on the γ band of Sr RuO , which arises princi-pally from Ru d d xy orbitals (represented by the index z on fermion operators), for which we include the tight bind-ing matrix elements t z ≡ T zi,i ± ˆ x = T zi,i ± ˆ y , t ′ z ≡ T zi,i ± ˆ x ± ˆ y , µ z ≡ T zi,i . H αβ corresponds to the quasi-one-dimensional α and β bands, which arise principally from the d xz and d yz or-bitals (fermion indices x and y respectively), with tight bind-ing matrix elements t ≡ T xi,i ± ˆ x = T yi,i ± ˆ y , t ⊥ ≡ T xi,i ± ˆ y = T yi,i ± ˆ x , µ ≡ T xi,i = T yi,i . For this model there is also an im-portant next-nearest-neighbor orbital hybridization matrix el-ement t ′ , whose presence is crucial for establishing a chiral su-perconducting gap. We take values { t, t ⊥ , t ′ , µ, t z , t ′ z , µ z } = { , . , . , , . , . , . } which are consistent with theFermi surface measured in ARPES and the quasiparticle ef-fective masses measured in quantum oscillations .Nearest-neighbor pairing for the d xy orbital and next-nearest neighbor pairing for the d xz and d yz orbitals rep-resent the lowest lattice harmonics consistent with a weakcoupling analysis , which predicts a fully gapped d xy or-bital and "accidental" nodes on d xz and d yz which are liftedto parametrically deep gap minima in the presence of or-bital mixing t ′ . Calculations are performed with the self-consistency conditions ∆ γx ( i ) = − g γ ( i ) h c z,i +ˆ x c z,i i , ∆ γy ( i ) = − g γ ( i ) h c z,i +ˆ y c z,i i , ∆ αβx ( i ) = − g αβ ( i ) h c x,i +ˆ x +ˆ y c x,i i , ∆ αβy ( i ) = − g αβ ( i ) h c y,i +ˆ x +ˆ y c y,i i with attractive interactions g αβ ( i ) and g γ ( i ) which are allowed to vary along the x direc-tion. We model the metallic edge region adjoining the super-conducting bulk by setting g αβ and g γ to zero in a region ofwidth N m sites, and nonzero and uniform in a region of width N s sites, with value chosen to yield the desired bulk valuesof ∆ αβ and ∆ γ . In this model, superconductivity arises in-dependently on the quasi-two-dimensional γ band and on thequasi-one-dimensional α and β bands (i.e. there is no inter-band proximity effect) and our estimate for the Sr RuO edgecurrent will be the sum of contributions from H γ and H αβ .The consequences of this artificial assumption will be consid-ered in section VII.The current operator for the link from site i to site j can bederived from the lattice version of the equation of continuity and the Heisenberg equation of motion. It has an intra-orbitalpart ˆ J ηi,j = iT ηi,j h c † η,i, c η,j − h.c. i (3)where η = x, y, z is the orbital index. For the model of the α and β bands there is also an inter-orbital part for the currentbetween next-nearest neighbors ˆ J xyi,i + s ˆ x + s ˆ y = it ′ s s h c † x,i, c y,i + s ˆ x + s ˆ y + c † y,i, c x,i + s ˆ x + s ˆ y − h.c. i (4)where s , s = ± .We neglect the effect of screening, whose effects have beenexplored elsewhere . Accordingly, our figure of meritfor edge currents will be the total amount of current I flowingthrough the metal region and half of the superconducting bulk,i.e. I = N m + N s / X n =1 h ˆ J n ˆ x,n ˆ x +ˆ y + ˆ J n ˆ x,n ˆ x +ˆ x +ˆ y i (5)where the two terms in the sum are for nearest neighbor andnext-nearest neighbor links, including intra- and inter-orbitalcontributions as appropriate, and the angle brackets representa thermal average. Note that only net currents in the ˆ y direc-tion are allowed by continuity in the cylinder geometry. IV. GINZBURG LANDAU THEORY
Ginzburg-Landau theory represents an approximate solu-tion to the BdG equations that becomes exact in the limit T − T c → − , but provides valuable intuition even at lowtemperatures. The expression for the free energy can be foundin the literature : F = r (cid:0) | ψ x | + | ψ y | (cid:1) + K (cid:0) | ∂ x ψ x | + | ∂ y ψ y | (cid:1) + K (cid:0) | ∂ y ψ x | + | ∂ x ψ y | (cid:1) + K ([ ∂ x ψ x ] ∗ [ ∂ y ψ y ] + [ ∂ y ψ x ] ∗ [ ∂ x ψ y ] + c.c. )+ higher order terms . (6)For our purposes, we need not treat quartic terms or those withmore than two derivatives. The equations for the order param-eter fields must be supplemented by appropriate conditions fora boundary at fixed x : ψ x = 0 , ∂ x ψ y = 0 , insulating boundary (7) ∂ x ψ x = ψ x b x , ∂ x ψ y = ψ y b y , metallic boundary (8)The conditions for an insulating boundary follow from the factthat specular scattering is fully pair-breaking for ψ x (which isby construction odd under x → − x ) . The conditions for ametallic boundary involve phenomenological parameters b x,y which capture the fact that a metal interface is partially pair-breaking for both components .We continue to ignore screening, and focus on the sponta-neous current (i.e. the current which exists in the absence ofphase gradients imposed by an external field): J spont ∝ − iK ( ψ y [ ∂ x ψ x ] ∗ + ψ x [ ∂ x ψ y ] ∗ − c.c. ) ∝ K ( | ψ y | ∂ x | ψ x | − | ψ x | ∂ x | ψ y | ) (9)In these expressions we have implemented translation sym-metry in the y direction and assumed a uniform relative phasefactor of i between ψ x and ψ y (i.e. positive chirality). Herethe coefficients K and K determine the coherence lengthsof the two order parameter components, the inter-componentgradient coupling K sets the scale of the currents, and r ∝ T − T c is the usual parameter which tunes through the criti-cal point. The coefficients can be treated as phenomenologicalparameters or computed directly from the microscopic Hamil-tonians given above. V. BDG RESULTS
As previously mentioned, our estimate for the edge currentin Sr RuO is the sum of contributions due to the quasi-2D γ band and the quasi-1D α, β bands; we initially plot and dis-cuss these contributions separately. Values of net current aregiven in units of I ≡ . et/ ~ , which is the net currentdue to the γ band with an insulating interface ( N m = 0 ) at T = 0 . T c , in the weak coupling limit ∆ γ → + . I is ap-proximately equal to the value of the total current per spinin a quasi-classical approximation (such as the Matsumoto-Sigrist prediction used in ) when screening is neglected.If our model predicts a current I and screening alters our pre-dictions in the same way as it does the quasi-classical resultsof Matsumoto-Sigrist, then our prediction of a magnetic sig-nal (such as the peak flux) is equal to the Matsumoto-Sigristprediction times I/I .Plots of the current and both components of the order pa-rameter as a function of distance from the edge are shown inFigures 1. Figures 2 show the two current contributions ver-sus temperature for several choices of N m . Data points near T c are not included due to computational cost. Figures 3 showthe current contributions as a function of the bulk order param-eter ( ∆ γ and ∆ αβ respectively, with fixed values of T /T c and N m /ξ . Before considering the effect of the normal-metal re-gion, we note basic results for a clean insulator (or vacuum)/ superconductor (IS) interface ( N m = 0 ). In that case, com-pared to the contribution from H γ , the net current from H αβ is reduced by a factor of approximately three at zero tempera-ture and six at the experimental temperature of . T c .Turning to the results for a normal metal / superconductor(NS) interface (i.e. N m = 0 ), one feature of the I − T curvesfor different values of N m is that they all coincide at zero tem-perature and at sufficiently high temperature, differing only inan intermediate crossover region. This follows from the prox-imity effect: while the superconducting gap ∆( i ) ≡ − g h cc i is zero in the metal (where g = 0 ), pair correlations h cc i do penetrate. The length scale for this penetration is set by v F /T , (where v F is the Fermi velocity), and thus diverges Distance from edgeArb.units ∆ γ x ∆ γ y J γ (a) Distance from edgeArb. units ∆ αβ y ∆ αβ x J αβ (b) Figure 1: Current and the two components of the orderparameter as a function of position for (a) H γ and (b) H αβ .The first 40 sites are the metallic region, in which the gapvanishes, and clean interfaces with vacuum are present atpositions and . Pair correlations in the metallic regionare shown in dashed lines. The bulk order parameter valuesare ∆ αβ = ∆ γ = 0 . t , T = 0 . T c . at zero temperature, so that the width of the metallic regionis effectively zero. By contrast, at temperatures such that v F /T < N m , pairing correlations decay to zero before theedge is encountered, so that the metallic region is effectivelyinfinite. In both cases, an increase in N m should have a negli-gible effect on the currents, consistent with the calculation.For v F /T < N m there is a pronounced suppression of thecurrent in both the one and quasi-1D cases compared with thecurrent without a metallic region ( N m = 0 ). The amount ofthis suppression depends on the size of the pairing gap. ForSr RuO , the pairing gap is on the order of − t , so thatextrapolation to the weak coupling limit ∆ → + is nec-essary for a quantitative estimate. For a model including allthree bands in this weak coupling limit, we find a suppressionof approximately twenty compared to the initial Matsumoto- T/T c I γ /I N m =0 (IS)N m =10N m =20N m =40 (a) T/T c I αβ /I N m =0 (IS)N m =10N m =20N m =40 (b) Figure 2: Contributions to the current near a metallic edgeregion from (a) the γ band and (b) the α, β bands vs.temperature for several values of N m , the thickness of thismetallic edge region abutting the superconducting bulk. Thesuperconducting bulk is of width N s = 100 sites, andcurrents are quoted in units of I , which is essentially theMatsumoto-Sigrist result in the absence of screening. Thebulk order parameter values are ∆ αβ = ∆ γ = 0 . t . For thecurrent from α, β there are finite size effects associated withnear-nodal quasiparticles which render the results at very lowtemperature less well behaved. We have verified that thezero-temperature current values in the thermodynamic limitare within 15 % of those shown here.Sigrist predictions. VI. QUALITATIVE EXPLANATION FROMGINZBURG-LANDAU THEORY
The results of the previous section can be summarized asfollows: 1) the contribution from the α, β bands is a severaltimes smaller than that of the γ band for the IS geometry. 2)both contributions are substantially suppressed in the NS ge- ∆ γ /t I/I I γ NS I γ IS NS ExtrapolationIS Extrapolation (a) ∆ αβ /t I/I I αβ NS I αβ IS NS ExtrapolationIS Extrapolation (b)
Figure 3: Extrapolation of current contributions of (a) the γ band and (b) the α, β bands to the weak coupling limit ∆ αβ , ∆ γ → + for the both insulator/superconductor (IS)and normal metal/superconductor (NS) interfaces. Currentsare quoted in units of I which is essentially theMatsumoto-Sigrist result in the absence of screening. Asthe bulk gap is reduced, the temperature is reduced and thelength scales N m , N s are increased in order to fix the values T /T c = 0 . , N m /ξ ≈ , N s /ξ ≈ . The metallic boundaryleads to suppression of over an order of magnitude in boththe quasi-1D and quasi-2D cases.ometry. 3) the suppression due to the NS geometry is consid-erably larger for the γ band than for the α, β bands. Ginzburg-Landau theory, though it is not quantitatively valid at low tem-peratures, can nonetheless qualitatively explain each of theseresults.1) With a conventional insulating interface, the scale ofspontaneous currents is set by the coefficient K . In the quasi-2D model, this is a number of order one, whereas in the quasi-1D model, it vanishes in the limit of zero inter-orbital mixing t ′ . Since t ′ = 0 . t , it follows that K αβ is substantially smallerthan K γ and similarly for the currents. A microscopic calcu-lation gives K αβ ≈ . K γ .2) The suppression in current in the NS geometry can beviewed as a consequence of the different boundary conditionson the order parameter. The boundary values of of | ψ x | and | ψ y | are respectively increased and decreased compared to theinsulating case. At a fixed distance from the edge | ψ x | and ∂ x | ψ y | are larger while | ψ y | and ∂ x | ψ x | are smaller than theircorresponding values for the insulating boundary. Eq. (9)for the current shows that this yields a numerical (though notparametric) reduction in the current for any choice of G-L co-efficients.3) The tremendous suppression of the current in the quasi-2D NS model is a lattice effect. For the fine-tuned case K = K , one can show that b x = b y and the two componentsof the order parameter heal away from the metal in preciselythe same way, leading to a vanishing current in lowest-orderG-L theory . For a quadratic dispersion and an order param-eter k x + ik y , as is often used to describe the γ band ,the coefficients satisfy K = 3 K . However, for a lattice-compatible order parameter sin k x + i sin k y as treated hereand for an appropriate tight-binding band structure for the γ band, K = 0 . K . The large suppression of the γ bandcurrent due to the NS geometry can be roughly identified withthe proximity of this result to the fine-tuned case K = K . VII. DISCUSSION
Superconductivity on the quasi-1D bands was previouslyconjectured to lead to dramatically reduced edge currentscompared to a quasi-2D scenario due to trivial topology (i.e.the Chern numbers of the two bands add to zero, yielding nonet chiral edge modes). The results shown above for the ISinterface show a substantial reduction (by a factor betweenthree and six), but nonetheless of order one, falsifying theinitial conjecture and illustrating the tenuous connection be-tween topology and edge currents in chiral p -wave supercon-ductors (this topic will be treated in depth in a forthcomingpaper).Even if the quasi-1D bands had vastly reduced currents inthe IS case, the contribution from the γ band would generi-cally be large, even if it were not the "dominant" band. The ne-glect of the current contribution from the subdominant band(s)is only justified if the experimental temperature exceeds thesubdominant gap scale. However, thermodynamic evidenceshows that the gaps on all bands are at least comparable to T c = 1 . K ≈ . meV . At low temperatures, the edgecurrents should then correspond to the sum of contributionsfrom the quasi-1D and quasi-2D bands, with the weak cou-pling limit taken for both ∆ αβ and ∆ γ . At low temperaturesand with a clean interface, the generic scale of edge currentsis "of order one" regardless of microscopic mechanism detailssuch as the identity of the dominant band(s).Though there does not seem to be any physical reason for aparametric suppression of edge currents, we find a meaning-ful quantitative reduction of over an order of magnitude com-pared to previous estimates by considering the effect of sur-face imperfection. Within a model of a clean metal of width ∼ ξ abutting a clean superconductor, with T = 0 . T c , the total current from all three bands is suppressed by a factor ofmore than twenty in the weak coupling limit compared to theresult for the γ band and an IS interface. Within our model,there is essentially no suppression in the limit of sufficientlylow temperatures and/or narrow metallic regions, where su-perconducting correlations induced by the proximity effectextend all the way to the edge. This is an artifact of our model,however, which does not treat surface roughness or disorderdirectly. For example, pair-breaking and diffuse scattering ef-fects are known to reduce the zero-temperature current .The calculations presented here are not expected to bequantitatively correct for the actual superconducting gapstructure and surface physics of Sr RuO . Our model ofspinless fermions entirely neglects spin-orbit coupling (SOC),which has been proposed to qualitatively affect pairing .However, as far as the edge current is concerned, the pri-mary effect of SOC is to modestly renormalize the band struc-ture; hence, its explicit inclusion would not change any ofour results substantially. A more serious unphysical assump-tion is the neglect of the inter-band proximity effect, with-out which superconductivity would generically arise at verydifferent temperatures on the γ and α/β bands. While inter-band proximity coupling would not change the additivity ofthe current contributions from the different bands, it wouldalter the length scale over which the various order parametercomponents heal away from an interface. The resulting cur-rents could be reduced or increased compared to our results,depending on microscopic details.These defects notwithstanding, the model treated above il-lustrates that substantial reductions in magnetic signal canarise from interface effects. We now consider the conse-quences of a twenty-fold reduction for the interpretation ofmagnetometry experiments. Even with this reduction themagnetic signal at the edge would still be estimated to be sev-eral times the resolution of scanning magnetometry experi-ments, and should therefore be observable. However, if multi-ple domains of sufficiently small size are present in the sampleand intersect the edge, the magnetic fields from spontaneouscurrents would be unobservable. Kirtley et al find that, to beconsistent with the Matsumoto-Sigrist predictions , ab -planedomains below about . µm in size are necessary. To be con-sistent with a prediction twenty times smaller, the domainscould be as large as perhaps µm . However, the presenceof multiple ab -plane domains within the sample would leadto spontaneous currents at the domain walls, which have notbeen treated here. Unless domain walls are pinned by crystaldefects that, like a rough edge, lead to quenched supercon-ductivity (an unlikely proposition), the suppression indicatedin the foregoing calculations would not apply to the domainwall currents.One scenario for the lack of an edge signal which wouldnot imply a signal at interior domain boundaries is the c-axisstacking of planar domains of macroscopic horizontal extentand alternating chirality. The energetic cost of the domainboundaries would be small, due to the very weak dispersionof the electronic band structure along the c direction, and sym-metry requires that no spontaneous current would flow at theseboundaries. The measurements of Hicks et al place an up-per bound of − nm on the height of such domains(depending on microscopic domain details, and again assum-ing Matsumoto-Sigrist predictions for edge currents ). Here,a twenty-fold reduction of expected edge currents for a sin-gle domain would revise upward the experimental bound ondomain size, possibly reconciling the null result of scanningmagnetometry experiments with the spontaneous time rever-sal symmetry breaking seen in Kerr effect measurements withmesoscopic spot size ( ∼ µm ) and skin depth ( ∼ nm ).We have shown that spontaneous currents in a chiral p -wavesuperconductor are highly sensitive to interface details, in par-ticular that surface disorder leading to a µm -thickness metal-lic surface region can cause a suppression of more than anorder of magnitude compared to naive estimates. 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