Suppression of spontaneous currents in SrRuO by surface disorder
A major challenge to the chiral -wave hypothesis for the pairing symmetry of the unconventional superconductor SrRuO is the null result of sensitive scanning magnetometry experiments designed to detect the expected spontaneous charge currents. Motivated by junction tunneling conductance measurements which indicate the quenching of superconductivity at the surfaces of even high-purity samples, we examine the spontaneous currents in a chiral -wave superconductor near a normal metal / superconductor interface using the lattice Bogoliubov-de Gennes equations and Ginzburg-Landau theory, and find that the edge current is suppressed by more than an order of magnitude compared to previous estimates. These calculations demonstrate that interface details can have a quantitatively meaningful effect on the expectations for magnetometry experiments.
Strontium Ruthenate, SrRuO, is an unconventional superconductor ()Mackenzie2003 for which there exists substantial evidence for odd-parity pairingIshida1998; Nelson2004; Kidwingira2006; Jang2011 as well as for the spontaneous breaking of time reversal symmetry below Luke1998; Luke2000; Xia2006. These observations lead naturally to the conclusion that the pairing symmetry is chiral -wave ( ), a two dimensional analog of the A-phase of superfluid He. Though this is the leading phenomenological hypothesis, it is seemingly contradicted by several experiments. Prominent among these are high resolution scanning magnetometry measurementsKirtley2007; Hicks2010, which image magnetic fields across several of sample (including the sample edge) and see no sign of the expected spontaneous currents.
The presence of spontaneous, persistent charge currents at edges and domain walls is a robust consequence of time-reversal symmetry breaking superconductivity. However, the magnitude of these currents is determined by microscopic details – they are neither quantized nor universal. The reason that the null result of the scanning magnetometry experiments poses such a challenge to the chiral -wave hypothesis is quantitative – spontaneous currents of size comparable to theoretical estimatesMatsumoto1999; Furusaki2001; Stone2004; Imai2012; Sauls2011 would give a magnetic signal more than two orders of magnitude greater than the experimental resolution. Magnetometry measurements on mesoscopic samplesJang2011 also see no signs of these currents.
In this paper we calculate the spontaneous surface currents for a family of models consistent with the phenomenology of superconductivity in SrRuO. Motivated by -axis tunneling experimentsKashiwaya2011, we employ a different interface condition than previous studies, modeling the surface region as a normal metal layer adjoining the superconducting bulk. We find that, compared to previous estimates, the expected magnetic signal from edge currents is reduced by over an order of magnitude. These calculations demonstrate that interface details can have a quantitatively meaningful effect on the expectations for magnetometry experiments.
Ii Surface imperfection
The assumption of specular surface scattering as employed inMatsumoto1999; Furusaki2001; Stone2004; Imai2012; Sauls2011 requires an atomically smooth surface. faces of SrRuO can be cleaved, but and faces are typically polished to a smoothness of several Kirtley2007, on the order of ten lattice constants. In -axis junction tunneling conductance measurements, signatures of superconductivity at the surface are present only at the sub- level on top of a substantial smooth backgroundKashiwaya2011, as shown in Fig. 2 of that reference. Accordingly, the best indication from experiment is that the edge region is metallicKashiwaya2014, with a superconducting gap developing only further into the sample.
Such a scenario is plausible given the fragility of unconventional superconductivity to elastic scattering (i.e. the inapplicability of Anderson’s Theorem to a sign-changing order parameter), which has been explicitly verified for this materialMackenzie1998. Rough or pair-breaking surface effects have been shownAshby2009; Nagato1998 to sharply reduce the superconducting order parameter at the surface, although not to meaningfully alter the surface density of states. Accordingly, the observation of metallic behavior suggests that there is a higher density of defects near the surface (presumably introduced during crystal growth or preparation procedures), leading to a reduced mean free path and the quenching of superconductivity near the surface.
To facilitate calculations, we do not directly treat a rough surface or defects in the surface region, but rather adopt a model consisting of a clean interface between vacuum and a metallic region, which in turn has a clean interface with the superconducting bulk. The metallic region is arranged by setting appropriate coupling constants to zero in lattice Bogoliubov de-Gennes Hamiltonians. This introduces artifacts which will be discussed in section VII.
Iii Model Hamiltonians
We consider spinless fermions on a 2D square lattice corresponding to the RuO plane, and work in a cylinder geometry: periodic boundary conditions are taken in the direction, and open boundary conditions in . We will consider two different Bogoliubov-de-Gennes Hamiltonians:
is a minimal Hamiltonian for chiral -wave superconductivity on the band of SrRuO, which arises principally from Ru orbitals (represented by the index on fermion operators), for which we include the tight binding matrix elements , , . corresponds to the quasi-one-dimensional and bands, which arise principally from the and orbitals (fermion indices and respectively), with tight binding matrix elements , , . For this model there is also an important next-nearest-neighbor orbital hybridization matrix element , whose presence is crucial for establishing a chiral superconducting gap. We take values which are consistent with the Fermi surface measured in ARPESDamascelli2000 and the quasiparticle effective masses measured in quantum oscillationsBergemann2003.
Nearest-neighbor pairing for the orbital and next-nearest neighbor pairing for the and orbitals represent the lowest lattice harmonics consistent with a weak coupling analysisRaghu2010, which predicts a fully gapped orbital and “accidental” nodes on and which are lifted to parametrically deep gap minima in the presence of orbital mixing . Calculations are performed with the self-consistency conditions , , , with attractive interactions and which are allowed to vary along the direction. We model the metallic edge region adjoining the superconducting bulk by setting and to zero in a region of width sites, and nonzero and uniform in a region of width sites, with value chosen to yield the desired bulk values of and . In this model, superconductivity arises independently on the quasi-two-dimensional band and on the quasi-one-dimensional and bands (i.e. there is no inter-band proximity effect) and our estimate for the SrRuO edge current will be the sum of contributions from and . The consequences of this artificial assumption will be considered in section VII.
The current operator for the link from site to site can be derived from the lattice version of the equation of continuity and the Heisenberg equation of motion. It has an intra-orbital part
where is the orbital index. For the model of the and bands there is also an inter-orbital part for the current between next-nearest neighbors
We neglect the effect of screening, whose effects have been explored elsewhere Matsumoto1999; Furusaki2001; Ashby2009. Accordingly, our figure of merit for edge currents will be the total amount of current flowing through the metal region and half of the superconducting bulk, i.e.
where the two terms in the sum are for nearest neighbor and next-nearest neighbor links, including intra- and inter-orbital contributions as appropriate, and the angle brackets represent a thermal average. Note that only net currents in the direction are allowed by continuity in the cylinder geometry.
Iv Ginzburg Landau Theory
Ginzburg-Landau theory represents an approximate solution to the BdG equations that becomes exact in the limit , but provides valuable intuition even at low temperatures. The expression for the free energy can be found in the literatureSigrist1991:
For our purposes, we need not treat quartic terms or those with more than two derivatives. The equations for the order parameter fields must be supplemented by appropriate conditions for a boundary at fixed :
The conditions for an insulating boundary follow from the fact that specular scattering is fully pair-breaking for (which is by construction odd under )Ambegaokar1974. The conditions for a metallic boundary involve phenomenological parameters which capture the fact that a metal interface is partially pair-breaking for both components DeGennes1991.
We continue to ignore screening, and focus on the spontaneous current (i.e. the current which exists in the absence of phase gradients imposed by an external field):
In these expressions we have implemented translation symmetry in the direction and assumed a uniform relative phase factor of between and (i.e. positive chirality). Here the coefficients and determine the coherence lengths of the two order parameter components, the inter-component gradient coupling sets the scale of the currents, and is the usual parameter which tunes through the critical point. The coefficients can be treated as phenomenological parameters or computed directly from the microscopic Hamiltonians given above.
V BdG Results
As previously mentioned, our estimate for the edge current in SrRuO is the sum of contributions due to the quasi-2D band and the quasi-1D bands; we initially plot and discuss these contributions separately. Values of net current are given in units of , which is the net current due to the band with an insulating interface () at , in the weak coupling limit . is approximately equal to the value of the total current per spin in a quasi-classical approximation (such as the Matsumoto-Sigrist predictionMatsumoto1999 used in Kirtley2007; Hicks2010) when screening is neglected. If our model predicts a current and screening alters our predictions in the same way as it does the quasi-classical results of Matsumoto-Sigrist, then our prediction of a magnetic signal (such as the peak flux) is equal to the Matsumoto-Sigrist prediction times .
Plots of the current and both components of the order parameter as a function of distance from the edge are shown in Figures 1. Figures 2 show the two current contributions versus temperature for several choices of . Data points near are not included due to computational cost. Figures 3 show the current contributions as a function of the bulk order parameter ( and respectively, with fixed values of and . Before considering the effect of the normal-metal region, we note basic results for a clean insulator (or vacuum) / superconductor (IS) interface (). In that case, compared to the contribution from , the net current from is reduced by a factor of approximately three at zero temperature and six at the experimental temperature of .
Turning to the results for a normal metal / superconductor (NS) interface (i.e. ), one feature of the curves for different values of is that they all coincide at zero temperature and at sufficiently high temperature, differing only in an intermediate crossover region. This follows from the proximity effect: while the superconducting gap is zero in the metal (where ), pair correlations do penetrate. The length scale for this penetration is set by , (where is the Fermi velocity), and thus diverges at zero temperature, so that the width of the metallic region is effectively zero. By contrast, at temperatures such that , pairing correlations decay to zero before the edge is encountered, so that the metallic region is effectively infinite. In both cases, an increase in should have a negligible effect on the currents, consistent with the calculation.
For there is a pronounced suppression of the current in both the one and quasi-1D cases compared with the current without a metallic region (). The amount of this suppression depends on the size of the pairing gap. For SrRuO, the pairing gap is on the order of , so that extrapolation to the weak coupling limit is necessary for a quantitative estimate. For a model including all three bands in this weak coupling limit, we find a suppression of approximately twenty compared to the initial Matsumoto-Sigrist predictions.
Vi Qualitative explanation from Ginzburg-Landau theory
The results of the previous section can be summarized as follows: 1) the contribution from the bands is a several times smaller than that of the band for the IS geometry. 2) both contributions are substantially suppressed in the NS geometry. 3) the suppression due to the NS geometry is considerably larger for the band than for the bands. Ginzburg-Landau theory, though it is not quantitatively valid at low temperatures, can nonetheless qualitatively explain each of these results.
1) With a conventional insulating interface, the scale of spontaneous currents is set by the coefficient . 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 . Since , it follows that is substantially smaller than and similarly for the currents. A microscopic calculation gives .
2) The suppression in current in the NS geometry can be viewed as a consequence of the different boundary conditions on the order parameter. The boundary values of of and are respectively increased and decreased compared to the insulating case. At a fixed distance from the edge and are larger while and are smaller than their corresponding values for the insulating boundary. Eq. (IV) for the current shows that this yields a numerical (though not parametric) reduction in the current for any choice of G-L coefficients.
3) The tremendous suppression of the current in the quasi-2D NS model is a lattice effect. For the fine-tuned case , one can show that and the two components of the order parameter heal away from the metal in precisely the same way, leading to a vanishing current in lowest-order G-L theoryAshby2009. For a quadratic dispersion and an order parameter , as is often used to describe the bandMatsumoto1999; Sauls2011; Furusaki2001; Stone2004, the coefficients satisfy . However, for a lattice-compatible order parameter as treated here and for an appropriate tight-binding band structure for the band, . The large suppression of the band current due to the NS geometry can be roughly identified with the proximity of this result to the fine-tuned case .
Superconductivity on the quasi-1D bands was previously conjecturedRaghu2010 to lead to dramatically reduced edge currents compared to a quasi-2D scenario due to trivial topology (i.e. the Chern numbers of the two bands add to zero, yielding no net chiral edge modes). The results shown above for the IS interface show a substantial reduction (by a factor between three and six), but nonetheless of order one, falsifying the initial conjecture and illustrating the tenuous connection between topology and edge currents in chiral -wave superconductors (this topic will be treated in depth in a forthcoming paper).
Even if the quasi-1D bands had vastly reduced currents in the IS case, the contribution from the band would generically be large, even if it were not the “dominant” band. The neglect of the current contribution from the subdominant band(s) is only justified if the experimental temperature exceeds the subdominant gap scale. However, thermodynamic evidence shows that the gaps on all bands are at least comparable to Firmo2013. At low temperatures, the edge currents should then correspond to the sum of contributions from the quasi-1D and quasi-2D bands, with the weak coupling limit taken for both and . At low temperatures and with a clean interface, the generic scale of edge currents is “of order one” regardless of microscopic mechanism details such as the identity of the dominant band(s).
Though there does not seem to be any physical reason for a parametric suppression of edge currents, we find a meaningful quantitative reduction of over an order of magnitude compared to previous estimates by considering the effect of surface imperfection. Within a model of a clean metal of width abutting a clean superconductor, with , the total current from all three bands is suppressed by a factor of more than twenty in the weak coupling limit compared to the result for the band and an IS interface. Within our model, there is essentially no suppression in the limit of sufficiently low temperatures and/or narrow metallic regions, where superconducting correlations induced by the proximity effect extend all the way to the edge. This is an artifact of our model, however, which does not treat surface roughness or disorder directly. For example, pair-breaking and diffuse scattering effects are known to reduce the zero-temperature current Ashby2009; Nagato1998.
The calculations presented here are not expected to be quantitatively correct for the actual superconducting gap structure and surface physics of SrRuO. Our model of spinless fermions entirely neglects spin-orbit coupling (SOC), which has been proposed to qualitatively affect pairingVeenstra2014. However, as far as the edge current is concerned, the primary effect of SOC is to modestly renormalize the band structure; hence, its explicit inclusion would not change any of our results substantially. A more serious unphysical assumption is the neglect of the inter-band proximity effect, without which superconductivity would generically arise at very different temperatures on the and bands. While inter-band proximity coupling would not change the additivity of the current contributions from the different bands, it would alter the length scale over which the various order parameter components heal away from an interface. The resulting currents could be reduced or increased compared to our results, depending on microscopic details.
These defects notwithstanding, the model treated above illustrates that substantial reductions in magnetic signal can arise from interface effects. We now consider the consequences of a twenty-fold reduction for the interpretation of magnetometry experiments. Even with this reduction the magnetic signal at the edge would still be estimated to be several times the resolution of scanning magnetometry experiments, and should therefore be observable. However, if multiple domains of sufficiently small size are present in the sample and intersect the edge, the magnetic fields from spontaneous currents would be unobservable. Kirtley et alKirtley2007 find that, to be consistent with the Matsumoto-Sigrist predictionsMatsumoto1999, -plane domains below about in size are necessary. To be consistent with a prediction twenty times smaller, the domains could be as large as perhaps . However, the presence of multiple -plane domains within the sample would lead to spontaneous currents at the domain walls, which have not been treated here. Unless domain walls are pinned by crystal defects that, like a rough edge, lead to quenched superconductivity (an unlikely proposition), the suppression indicated in the foregoing calculations would not apply to the domain wall currents.
One scenario for the lack of an edge signal which would not imply a signal at interior domain boundaries is the c-axis stacking of planar domains of macroscopic horizontal extent and alternating chirality. The energetic cost of the domain boundaries would be small, due to the very weak dispersion of the electronic band structure along the c direction, and symmetry requires that no spontaneous current would flow at these boundaries. The measurements of Hicks et al Hicks2010 place an upper bound of on the height of such domains (depending on microscopic domain details, and again assuming Matsumoto-Sigrist predictions for edge currentsMatsumoto1999). Here, a twenty-fold reduction of expected edge currents for a single domain would revise upward the experimental bound on domain size, possibly reconciling the null result of scanning magnetometry experiments with the spontaneous time reversal symmetry breaking seen in Kerr effect measurements with mesoscopic spot size () and skin depth (Kapitulnik2014).
We have shown that spontaneous currents in a chiral -wave superconductor are highly sensitive to interface details, in particular that surface disorder leading to a -thickness metallic surface region can cause a suppression of more than an order of magnitude compared to naive estimates. We propose that a scenario of c-axis domain stacking, along with surface disorder, might resolve the seeming disagreement between scanning magnetometry and Kerr probes, and further suggest that the edge of a crystal fractured in vacuum might host a much lower defect density, and potentially lead to observable edge currents.
Acknowledgements: SL thanks Aharon Kapitulnik, Steven Kivelson, Kathryn Moler, and Boris Spivak for helpful discussions. This work is supported by NSERC and CIFAR at McMaster and by the Canada Research Chair and Canada Council Killam programs (CK). At Stanford, this work is supported in part by the DOE Office of Basic Energy Sciences, contract DE-AC02-76SF00515 (SL and SR), an ABB fellowship (SL), and the Alfred P. Sloan Foundation (SR).