Introduction

Sr2RuO4 has attracted much attention recently because it may feature a pairing state hosting Majorana bound states useful for topological quantum computing1,2,3. A large body of experimental data, including that obtained in phase-sensitive measurements4, has shown that the layered perovskite Sr2RuO4 is a spin-triplet, odd-parity superconductor5,6. Assuming that superconductivity in this material is two-dimensional in nature, the fourfold tetragonal crystalline symmetry dictates that the pairing symmetry must be one of the five representations7. Among those, only the two-component, px±ipy state is consistent with the muon spin rotation8 and Kerr rotation9 measurements that suggest the presence of a spontaneous magnetic field in the superconducting state of Sr2RuO4, making it an electronic analogue7 of the superfluid 3He A-phase, and a chiral p-wave superconductor for which a Majorana zero-energy mode bound to the normal core of a half-flux quantum vortex has been predicted1,2. An important question of interest is what is the microscopic mechanism responsible for such a highly exotic superconducting state. In this regard, models based on ferromagnetic10 or antiferromagnetic fluctuations11, spin-orbit coupling12, interaction theory13,14, Hund’s rule coupling15 and interplay of charge and spin fluctuations in the one-dimensional bands16 have been proposed. The debate on these mechanisms is currently ongoing6.

The eutectic phase of Sr2RuO4-Ru featuring crystalline islands of Ru embedded in the bulk crystalline Sr2RuO4, found previously to feature a superconducting transition temperature (Tc) nearly double than that of the bulk Sr2RuO4 (ref. 17), may provide insight into the mechanism issue. The Tc enhancement was attributed to the capillary effect at the Ru/Sr2RuO4 interface18. It was more recently suggested that the enhanced superconductivity occurs on the Sr2RuO4 side away from19 rather than at the Ru/Sr2RuO4 interface as assumed previously18. Meanwhile, dislocations were found to be abundant, which led to the intriguing question as to whether these dislocations are the origin of the enhancement of Tc19. The bulk eutectic phase of Sr2RuO4-Ru possesses both Ru/Sr2RuO4 interfaces and dislocations (Fig. 1a), making it difficult to separate the effect of dislocations from that of the interfaces. Single-crystal flakes of Sr2RuO4 prepared by mechanical exfoliation, in which islands of Ru and dislocations can be identified by scanning electron microscopy (SEM) and/or transmission electron microscopy (TEM; Fig. 1b), provide an opportunity to study separately the respective effect of the Sr2RuO4/Ru interface and dislocations (see below). Interestingly, an edge dislocation in a Sr2RuO4 lattice (Fig. 1c) is expected to possess complicated modifications to the local crystalline structure and electronic states, which may lead to a new way of hosting a Majorana zero mode20. Nevertheless, it has a dominating overall feature that the fourfold rotational symmetry is lowered to one that is basically twofold.

Figure 1: Dislocations and prediction of an enhanced Tc.
figure 1

(a) Transmission electron microscopy (TEM) image of a Sr2RuO4-Ru eutectic crystal showing a Ru island in the middle and large numbers of dislocations. (b) TEM image of a Sr2RuO4 single-crystal flake, showing dislocations and nanodomains of Ru on the edge. The dislocation lines are the superposition of many layers. Scale bars, 500 nm. (c) Schematic of an edge dislocation caused by an extra layer (purple) in a Sr2RuO4 lattice, indicating that the local fourfold rotational symmetry is broken. The edge dislocation scatters the electron beam in a TEM study, manifesting itself as a dark line in the TEM image. (d) Tch plotted as a function of m1/|μ| and m2/|μ| for |μ|/αTc0=0.4. The value of Tch is represented by a colour scale. The highlighted curve represents the contour of Tch=1.5 K.

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Here we report our observation of the enhancement in local Tc near lattice dislocations in Sr2RuO4 up to about twice of the bulk. We develop a phenomenological theory to describe the observation and predict that the enhanced Tc is ubiquitous for symmetry reduction in superconductors with a two-component order parameter.

Results

Phenomenology

A phenomenological theory can be formulated to capture the effect of the symmetry reduction. The general free-energy density of the bulk Sr2RuO4 with a fourfold tetragonal symmetry in zero magnetic field can be written as18,

where ηx and ηy denote the two-component order parameter, a(T)=α(T−Tc0), with α a constant and Tc0=1.5 K the bulk Tc of Sr2RuO4, bi (i=1–3) and Kj (j=1–5) parameters characterizing the bulk superconductor. The effect of the symmetry lowering near a dislocation can be illustrated by first considering a bulk Sr2RuO4 crystal. When an in-plane uniaxial strain is applied to the crystal, the lattice distortion leads naturally to a structural distortion and change in the electronic band structure, which may in turn cause the Tc to change. Without getting into the analysis of the complicated microscopic effect, we will write down a phenomenological theory featuring a set of phenomenological parameters, m1 and m2 used to quantify the effect of structural distortions and μ to measure the hybridization of the two-order parameter components, describing the symmetry-breaking strength.

To obtain only the highest superconducting transition temperature (Tch), it is sufficient to consider the free-energy density up to the quadratic terms. We assume a spatially uniform-order parameter, for which the gradient terms vanish. Therefore, following the idea of degenerate perturbation theory, the free-energy density describing the reduced symmetry (fRS) can be written as21,22

The modified transition temperature Tch, determined by the eigenvalues of the quadratic terms, is given by

where m±=(m1±m2)/2. Equation (3) indicates that the asymmetry-related terms m− and μ always enhance Tch in one superconducting channel, whereas reduce it in the other. In general, the level of crystalline distortion m+ may either enhance or suppress the transition temperature. For certain parameters, the symmetry-lowering effect dominates and Tch>Tc0 can be obtained (Fig. 1d).

The above consideration may be extended to the analysis of a bulk crystal of Sr2RuO4 possessing a single dislocation (Fig. 1c). If the dislocation line features a width d in the x direction, we may obtain the local Tc from Tch (m1, m2 and μ) in the above treatment of a bulk crystal subjecting to a uniaxial stress. The effect of the bulk on the embedded dislocation region in this approach is described by a δ-function in the free-energy density

as done previously to account for the capillary effect at the interface between Ru and Sr2RuO4 (ref. 18). Solving the linearized Ginzburg–Landau equations derived from equation (1) and matching the boundary conditions at x=0 (see Methods), we found that Tc is given by the solution of

which requires that Tch>Tc>Tc0 for self consistency. The spatial dependence of the order parameter was found to be symmetric with respect to the dislocation line at x=0, with the order parameter reaching a maximum value at the dislocation below Tc. As the highest transition temperature found in any Sr2RuO4 including eutectic systems is 3.2 K (ref. 6), we assume that Tch=3.2 K. As a result, the experimentally observed typical onset Tc of 2.5 K corresponds to d=3.5(K2/α Tc0)1/2. This d-value is comparable to the superconducting coherence length given by ξ1,2(0)=(K1,2/α Tc0)1/2 for the anisotropic stripe, suggesting that the use of a δ-function is self-consistent, given that the basic length for order parameter variation in the Ginzburg–Landau theory is ξ1,2(0). It is interesting to note that similar phenomenology can be obtained if two pairing states represented by a single-component order parameter have identical or very close intrinsic Tc. We note that the above theory is not applicable for a point defect for which the perturbation to local crystalline structure is essentially isotropic (rotationally invariant).

The main predictions of our phenomenological theory are threefold. First, an enhanced Tc can be obtained if the symmetry reduction effect dominates over other effects from the presence of a dislocation; second, the resulting local Tc depends strongly on the parameters characterizing the symmetry reduction, suggesting that the local Tc may vary from dislocation to dislocation; third, the magnitudes of the two components of the superconducting order parameter near a dislocation may depend on the temperature differently, leading to a change in the pairing symmetry as the temperature is lowered (see below).

Electrical transport

Single-crystal flakes of Sr2RuO4 were selected under optical microscope and examined by SEM before device fabrication. Some flakes were also examined by TEM. Dislocations and Ru nanodomains can be identified. Ru nanodomains (if observed) appear to always locate at the edge of the crystal (Fig. 1b), perhaps because cleaving tends to occur at the Ru/Sr2RuO4 boundary because of a reduced mechanical strength. For most flakes, however, SEM imaging did not show any Ru nanodomains. These Ru-free flakes of Sr2RuO4 were used to prepare four-point devices for electrical transport measurements. In particular, the temperature dependence of the resistivity for sample A (Fig. 2a) shows an onset Tc of 1.45 K (Fig. 2b), very close to the optimal Tc of Sr2RuO4. For sample B, SEM imaging revealed no Ru nanodomains (Fig. 2d) before the low-temperature transport measurement. However, transport measurements showed an onset Tc of 2.8 K (Fig. 2f). TEM studies of the same device after the transport measurements confirmed the absence of Ru nanodomains and further revealed the presence of dislocations between the two voltage leads (Fig. 2e). Similar results were obtained in sample C (Fig. 2g,h) with onset Tc=1.9 K (Fig. 2i). The magnetic field dependence of the resistivity for field applied along the c axis was also measured. It was found that the upper critical field (Hc2) is 50 mT for sample A (Fig. 2c), comparable to that of the pure phase (75 mT), whereas sample C showed an Hc2 of 0.3T (inset of Fig. 2i), closer to that of the eutectic phase5,6. Therefore, our experimental observations provide a direct confirmation that the presence of dislocations leads to the enhancement of Tc in Sr2RuO4 as predicted by the theory.

Figure 2: Observation of an enhanced Tc with dislocations.
figure 2

(a) False-colour SEM image of sample A. Scale bar, 5 μm. (b) Temperature dependence of the in-plane resistivity ρab for sample A, taken at zero applied magnetic field. (c) Magnetic field dependence of ρab for sample A. (d) False-colour SEM image of sample B. The red square indicates the area examined by TEM. Scale bar, 5 μm. (e) TEM image of the boxed area in d for sample B, showing dislocation lines but no Ru nanodomains. Scale bar, 200 nm. The regions not shown in this image were also checked by TEM and found to possess no Ru nanodomains. (f) Zero-field ρab(T) for sample B. (g) False-colour SEM image of sample C. The red square indicates the area examined by TEM. Scale bar, 10 μm. (h) TEM image of the boxed area in g for sample C. Scale bar, 1 μm. (i) Zero-field ρab(T) for sample C. Inset: ρab(H) for sample C. In the above line plots, the dashed lines indicate a linear fit of the normal-state resistivity. The deviation from such a fit defines the onset Tc and upper critical field.

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The presence of multiple dislocations in our sample, which should be described by different sets of symmetry reduction parameters, should result in multiple phases in the samples with enhanced Tc. This is indeed consistent with the multiple features observed in the ρab(T) curves (Fig. 2f,i). Further, the voltage–current (V–I) characteristics and the dV/dI–I curves were found to show double features suggesting the existence of two different phases at low temperatures in sample C (Fig. 3a), one corresponding to the dislocations and the other the bulk phase. In contrast, in sample A, a single onset Tc (Fig. 2b) and a single feature (Fig. 3b) were found in the ρab(T), and V–I and dV/dI–I curves, respectively.

Figure 3: Observation of multiple superconducting phases.
figure 3

Zero-field V–I curves and corresponding dV/dI–I curves at various temperatures for (a) sample C showing multiple transitions and (b) sample A showing a single transition. The dV/dI–I curves except for those at 0.35 K were shifted for clarity.

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Tunnelling

In addition to an enhanced Tc and inhomogeneous superconducting phase, the phenomenological theory presented above also predicts that the relative magnitude of the two components of the superconducting order parameter varies strongly as the temperature is lowered. In particular, for a system with an onset Tc of 2.5 K, the y component of the order parameter ηy was calculated (see Methods) and found to be much larger than the x component ηx above Tc0 (Fig. 4a, upper panels) at 1.9 K (below Tc). On the other hand, the two components become comparable below Tc0 (Fig. 4a, lower panels). Therefore, the local density of states (DOSs) near a dislocation should vary accordingly as the temperature is lowered.

Figure 4: Variations of the superconducting order parameter as a function of temperature.
figure 4

(a) Spatial dependence of ηx and ηy normalized to the value of ηy at x=0 for T1=1.9 K and T2=1.4 K. Other parameters are Tc0=1.5 K, Tch=3.2 K, Tc=2.5 K and ξ2(T1,2)=[K2/|α(T1,2−Tc0)|]1/2, superconducting coherence lengths at finite temperatures. (b) Quasiparticle tunneling spectra obtained from the junction on sample D, showing a ZBCP below 2.2 K. The ZBCP at 1.9 K shows a triangular shape, whereas that at 0.38 K shows a rectangular shape.

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The experimental detection of the temperature dependence of the order parameter near a dislocation is a significant challenge. Ideally, a local probe such as a scanning tunneling microscope with tunneling spectroscopy capability can be used to probe the local superconducting order parameter. However, as a surface probe, a scanning tunneling microscope cannot locate a dislocation embedded inside the crystal. We performed quasiparticle tunneling measurements on a tunnel junction of Sr2RuO4-TiAl fabricated on a Ru-free flake. Tunneling spectra obtained at an in-plane magnetic field of 40 mT, at which the superconductivity in Al counter electrodes is suppressed, feature zero-bias conductance peaks (ZBCPs). Such ZBCPs were observed in Sr2RuO4 previously23,24, and linked to the p-wave pairing symmetry. For sample D (onset Tc=2.5 K), the ZBCP was visible at 2.2 K, with its shape varying as the temperature is lowered (Fig. 4b). The triangularly shaped ZBCP seen at 1.9 K is particularly akin to those found previously in the break junctions of Sr2RuO4-Ru eutectic crystals above 1.5 K (ref. 23) and is consistent with the expectations of our phenomenological theory that predicts that the ηy component dominates at 1.9 K with a nearly zero ηx component. On the other hand, the ZBCP found at 0.38 K has a rectangular shape, with an onset of superconducting energy gap feature around 0.2 meV, consistent with the weak-coupling value for a Tc of 1.5 K and an order parameter with two equal-magnitude components.

Discussion

The crystalline symmetry lowering characterized by the phenomenological parameters manifests itself microscopically as lattice distortions. An edge dislocation demands the existence of both a compressed region and a stretched region extending along the dislocation line (Fig. 1c), allowing extended regions of distortions to occur. It is relevant to note a recent uniaxial pressure study on pure Sr2RuO4 that revealed an enhancement of Tc up to 3.2 K (onset) and emergence of a continuous distribution of local Tc25. The enhancement of Tc observed in that experiment can be interpreted as a consequence of enhanced interlayer coupling. Indeed, the quantum oscillation measurements carried out under a hydrostatic pressure26 suggest that the pressure, which lowers Tc of Sr2RuO4, makes the Fermi surface more two-dimensional like, implying that weakening the interlayer coupling reduces Tc, consistent with the uniaxial strain result. Meanwhile, the compressed region near a dislocation favours an increased interlayer coupling, suggesting that the observed enhanced Tc in Sr2RuO4 near a dislocation may have its origin also in the strengthening of interlayer coupling.

Electronically, the symmetry lowering could also lead to changes in the DOSs near the Fermi level as well as the average pairing interaction evaluated over the perturbed Fermi surface, both of which may cause the Tc to vary. The change in the DOSs may favour pairing in a specific channel that will result enhancement of Tc for that channel, although the pairing interaction that depends on the exact mechanism is yet to be fully understood. The perturbation on the electronic states by a dislocation should be band dependent, which suggests that our result may have implications on the multiple-band superconductivity picture of Sr2RuO4 (refs 16, 27). The implications of our results on other proposed mechanisms for superconductivity in Sr2RuO4, such as incommensurate fluctuations11 or Coulomb interactions13,14, need to be further clarified.

Our experimental results on Ru-free flakes of Sr2RuO4 are in agreement with the phenomenological theory, providing the first demonstration of symmetry reduction-induced Tc enhancement near a dislocation in a spin-triplet superconductor featuring a two-component order parameter. Our findings have raised an intriguing question on whether it is possible to use symmetry reduction to raise Tc in other exotic superconductors described by a two-component superconducting order parameter. This may also be relevant to multiband superconductors for which the magnitudes of the order parameter on different bands are comparable. More experiments are needed to establish this possibility.

Methods

Phenomenological theory

To solve the linearized Ginzburg–Landau equations on the stripe, we adopt constrains for Kj used in ref. 18, K1/3=K2=K3=K4>>K5. In the geometry we consider, the problem can be simplified into a one-dimensional problem by ignoring the variations along the y and z axis. To examine the onset Tc, we consider the linearized Ginzburg–Landau equations derived from equation (1)

for x≠0. For x=0, we assume a fully transparent boundary at which the solutions have to be continuous. Meanwhile, the solutions satisfy the boundary conditions derived from equation (4)

The solutions for temperatures above Tc0 have the following forms

where ξ1(T)=(K1/|a|)1/2.

where ξ2(T)=(K2/|a|)1/2. Matching the boundary conditions equations (8) and (9), we obtain the following equations

Note that if Tch>Tc0, we always have a solution Tc such that Tch>Tc>Tc0. Meanwhile, because K1/3=K2, equation (15) gives a higher instability temperature, indicating that the y component becomes non-zero first.

To obtain the spatial dependence of the order parameters, we numerically solve the Ginzburg–Landau equations by taking the phenomenological parameters 2b1=3b2=−3b3=0.4α, Tch=3.2 K, K1/αTc0=3K2/αTc0=1, the same as those used in ref. 18. We expand the δ-function in equation (4) into a Gaussian function. The Ginzburg–Landau equations then read

where d=3.5(K2/αTc0)1/2=2, giving rise to Tc=2.5 K. When x goes to infinite, the solutions decay to zero for T>1.5 K and approach a constant of ηx=ηy=(−a/4b1)1/2 for T<1.5 K. The results are plotted in Fig. 4a.

Material

Easily cleavable single crystals of Sr2RuO4 were synthesized by the floating-zone method. Because of the high vapour pressure of RuO2, excess Ru needs to be added to form a crystal with the correct atomic ratio. In the crystal we used, 10% excess Ru (a reduced amount of Ru overcompensation) was added in the starting rod to suppress the formation of Sr2RuO4-Ru and Sr2RuO4-Sr3Ru2O7 eutectic phases, both of which were found previously to possess an enhanced Tc17,28. A bulk crystal from this batch was measured and showed a broad transition with a bulk Tc around 1.35 K, slightly lower than the optimal Tc for Sr2RuO4. Given that the same starting material with proper Ru overcompensation does yield crystals of Sr2RuO4 with optimal Tc, the slightly suppressed bulk Tc value is due to structural imperfection rather than an elevated impurity levels in the crystals. Indeed, devices made on flakes of Sr2RuO4 can show an onset Tc very close to the optimal. The use of easily cleavable Sr2RuO4 crystals is necessary for this study as superconducting films of Sr2RuO4 are not available, despite an early report of initial synthesis success29.

To make a TEM sample of Sr2RuO4 flakes, a bulk Sr2RuO4 crystal was sonicated in methanol. Droplets containing tiny flakes of Sr2RuO4 were dripped onto a TEM grid with carbon membrane. After the methanol dried, Sr2RuO4 flakes remained on the carbon membrane. Thin flakes that are electron-transparent were searched and studied by TEM. Flakes with Ru nanodomains on the edge and a large amount of dislocation lines were observed (Fig. 1b).

Device fabrications

We prepared single-crystal flakes with a lateral dimension of roughly 10–50 μm and a thickness of 300–800 nm by mechanical exfoliation. The flakes were transferred onto a Si/SiO2 substrate with the c axis of the crystal perpendicular to the substrate. A standard four-point probe was prepared by contact photolithography. Electrical leads of 50 nm Ti and 200 nm Au were deposited. After carrying out electric transport measurements at low temperatures, the flakes of sample B and C were further transferred onto a TEM grid using the standard tool of a tungsten tip. TEM study was then carried out on these two samples, revealing dislocations but no Ru nanodomains (Fig. 2e,h).

To fabricate quasiparticle tunneling devices on flakes, a few more steps than those for four-point transport devices were taken. In particular, 200-nm-thick SiO2 was first deposited on top of the flakes as a protection layer. Focused ion beam of 30 kV Ga ions was used to carve ramps on the edge of the flakes. Twenty-minute ion mill of 300 V Ar ions was then used to take off the surface layer damaged by high-energy focused ion beam. After photolithography patterning, 5 nm Ti and 200 nm Al were deposited as a counter electrode. The thin Ti layer was used here to improve the adhesion of Al to Sr2RuO4.

Transport measurements

Low-temperature direct current measurements were performed in a 3He refrigerator with a base temperature of 0.35 K. All leads entering the cryostat are shielded and filtered by low-pass resister-capacitor (RC) filters with a 3-dB cutoff frequency of 600 kHz.

Additional information

How to cite this article: Ying, Y. A. et al. Enhanced spin-triplet superconductivity near dislocations in Sr2RuO4. Nat. Commun. 4:2596 doi: 10.1038/ncomms3596 (2013).