The quasi-2D metal Sr$_2$RuO$_4$ is one of the best characterized unconventional superconductors, yet the nature of its superconducting order parameter is still highly debated. This information is crucial to determine the pairing mechanism of Cooper pairs. Here we use ultrasound velocity to probe the superconducting state of Sr$_2$RuO$_4$. This thermodynamic probe is symmetry-sensitive and can help to identify the superconducting order symmetry. Indeed, we observe a sharp jump in the shear elastic constant $c_{66}$ as the temperature is raised across the superconducting transition at $T_c$. This directly implies that the superconducting order parameter is of a two-component nature. Based on symmetry argument and given the other known properties of Sr$_2$RuO$_4$, we discuss what states are compatible with this requirement and propose that the two-component order parameter, namely $\lbrace d_{xz}; d_{yz} \rbrace$, is the most likely candidate.
Recent experiments suggest that the superconducting order parameter of Sr$_2$RuO$_4$ has two components. A two-component order parameter has multiple degrees of freedom in the superconducting state that can result in low-energy collective modes or the formation of domain walls -- a possibility that would explain a number of experimental observations including the smallness of the time reversal symmetry breaking signal at T$_\mathrm{c}$ and telegraph noise in critical current experiments. We perform ultrasound attenuation measurements across the superconducting transition of Sr$_2$RuO$_4$ using…
We present a study of the electronic ultrasound attenuation $α$ in the unconventional superconductor Sr$_2$RuO$_4$ . The power law behavior of $α$ at temperatures down to $T_c/30$ clearly indicates the presence of nodes in the gap. In the normal state, we find an enormous anisotropy of $α$ in the basal plane of the tetragonal structure. In the superconducting state, the temperature dependence of $α$ also exhibits significant anisotropy. We discuss these results in relation to possible gap functions.
The unconventional superconductivity in Sr$_2$RuO$_4$ continues to attract considerable interest. While many measurements can be interpreted on the basis of chiral $p$-wave pairing with intriguing topological character, a number of exceptions hinder an unambiguous verification of such pairing. The pairing mechanism also remains under debate. In this paper, with effects of the sizable spin-orbit coupling accounted for, we reexamine the superconducting instabilities in Sr$_2$RuO$_4$ through systematic microscopic analysis within random phase approximation. Our calculations show that the odd-pari…
There have been two different proposals for the spin-triplet order parameter of the superconducting phase in Sr$_2$RuO$_4$; an $f$-wave order parameter and the multigap model where some of the bands have the line node. In an effort to propose an experiment that can distinguish two cases, we study the behavior of the supercurrent and compute the critical current for these order parameters when the sample is a thin film with the thickness $d \ll ξ$ where $ξ$ is the coherence length. It is found that the supercurrent behaves very differently in two models. This will serve as a sharp test for the …
We compute and compare even- and odd-parity superconducting order parameters of strontium ruthenate ($\mathrm{Sr}_2\mathrm{RuO}_4$) in the limit of weak interactions, resulting from a fully microscopic three-dimensional model including spin-orbit coupling. We find that odd-parity helical and even-parity $d$-wave order are favored for smaller and larger values of the Hund's coupling parameter $J$, respectively. Both orders are found compatible with specific heat data and the recently-reported nuclear magnetic resonance (NMR) Knight shift drop [A. Pustogow et al. Nature 574, 72 (2019)]. The chir…
We present results from point-contact measurements on Sr$_2$RuO$_4$ that show a linear dependence of the excess current as a function of temperature and applied magnetic fields over a surprisingly wide range of the phase diagram. We propose an explanation of this finding in terms of a p-wave triplet-pairing state with coupling to a low-energy fluctuation mode. Within this model we obtain a quantitative description of the temperature dependence of the excess current. The impact of surface effects on order parameter and excess current is addressed.
Shortly after the discovery of superconductivity in Sr$_2$RuO$_4$, spin-triplet pairing was proposed and further corroborated by a constant Knight shift ($K$) across the transition temperature (T$_c$). However, a recent experiment observed a drop in $K$ at T$_c$ which becomes larger under uniaxial strain, ruling out several spin-triplet scenarios. Here we show that inter-orbital spin-triplet pairing features a d-vector that rotates when uniaxial strain is applied, leading to a larger drop in the spin polarization perpendicular to the strain direction, distinct from spin-singlet pairing. We pro…
We study the spin dynamics in a p-wave superconductor at the nesting vector associated with $α$ and $β$ bands in Sr$_2$RuO$_4$. We find a collective mode at the nesting vector in the superconducting phase identified as the odd-parity pairing state which breaks time reversal symmetry. This mode in the spin channel only exists in the p-wave superconductor, not in s- or d-wave superconductors. We propose that probing this mode would clarify the pairing symmetry in Sr$_2$RuO$_4$. The possibility of second superconducting phase transition is also discussed.
After more than 25 years of research, three even-parity superconducting states -- the $d+id$-wave, $d+ig$-wave, and $s+id$-wave states -- have emerged as leading candidates for the superconducting states of Sr$_2$RuO$_4$. In the present work, we propose a tunneling spectroscopy experiment for distinguishing among these three superconducting states. The key component of our proposal is that we examine the conductance spectra of normal-metal/Sr$_2$RuO$_4$ junctions with various angles between the junction interface and the crystal axis of the Sr$_2$RuO$_4$. The angle dependence of the conductanc…
Quite recently, a pronounced drop of $^{17}$O NMR Knight shift in the superconducting (SC) state of an unstrained Sr$_2$RuO$_4$ was reported by Pustogow and Luo ${\it et al.}$ They revealed such behavior from the free-induction decay (FID) signal after a weak RF pulse. We examined this behavior with our single-crystalline Sr$_2$RuO$_4$, and reproduced their result: the peaks of the $^{17}$O-NMR spectra shift in the SC state as long as the RF-pulse power is smaller than a threshold. Furthermore, we measured the temperature variation of the Knight shift by a standard spin-echo method with small-…
We discovered a fractional Chern structure in chiral superconducting Sr$_2$RuO$_4$ nanofilms by employing electric transport. By using Sr$_2$RuO$_4$ single crystals with nanoscale thickness, a fractional Hall conductance was observed without an external magnetic field. The Sr$_2$RuO$_4$ nanofilms enhanced the superconducting transition temperature to about 3 K. We found an anomalous induced voltage with temperature and thickness dependence, and the switching behavior of the induced voltage appeared when we applied a magnetic field. We suggest that there was fractional magnetic-field-induced el…
Recent experiments make it appear more likely that the order parameter of the unconventional superconductor Sr$_2$RuO$_4$ has a spin-triplet $f$-wave symmetry. We study ultrasonic absorption and thermal conductivity of superconducting Sr$_2$RuO$_4$ and fit to the recent data for various $f$-wave candidates. It is shown that only $f_{x^2-y^2}$-wave symmetry can account qualitatively for the transport data.
It is shown that the properties of the superconductivity observed in Sr$_2$RuO$_4$ can be accounted for by properly taking into account the internal degrees of freedom of the system. The electrons are strongly correlated in the RuO$_2$ planes while the superconducting phase transition is driven by ``umklapp scattering'' between the planes. The in-plane magnetic correlations are determined via a non-perturbative approach. The orbital degeneracy and the magnetic moments are internal degrees of freedom of the Cooper pairs that account for the thermodynamic properties in the superconducting phase.
We review electronic transport in superconducting junctions with Sr$_2$RuO$_4$. Transport measurements provide evidence for chiral domain walls and, therefore, chiral superconductivity in superconducting Sr$_2$RuO$_4$, but so far, the symmetry of the underlying superconducting state remains inconclusive. Further studies involving density of states measurements and spin-polarised transport in local/non--local Sr$_2$RuO$_4$ junctions with magnetic materials could lead to fundamental discoveries and a better understanding of the superconducting state.
The superconducting gap function of Sr$_2$RuO$_4$ was investigated by means of quasiparticle reflection and transmission at the normal conductor-superconductor interface of Sr$_2$RuO$_4$-Pt point contacts. We found two distinguishly different types of $dV/dI$ vs $V$ spectra either with a double-minimum structure or with a zero-bias conductance anomaly. Both types of spectra are expected in the limit of high and low transparency, respectively, of the interface barrier between a normal metal and a spin-triplet superconductor. Together with the temperature dependence of the spectra this result st…
A comprehensive theory of superconductivity in Sr$_2$RuO$_4$ must explain experiments that suggest even parity superconducting order and others that suggest broken time reversal symmetry. Completeness further requires that the theory applies to Ca$_2$RuO$_4$, a Mott-Hubbard semiconductor that exhibits an unprecedented insulator-to-metal transition driven by very small electric field, and also by doping with very small concentration of electrons, leading to a metallic state proximate to ferromagnetism. A valence transition model, previously proposed for superconducting cuprates [Phys. Rev. B {\…
Among unconventional superconductors, Sr$_2$RuO$_4$ has become a benchmark for experimentation and theoretical analysis because its normal-state electronic structure is known with exceptional precision, and because of experimental evidence that its superconductivity has, very unusually, a spontaneous angular momentum, i.e. a chiral state. This hypothesis of chirality is however difficult to reconcile with recent evidence on the spin part of the order parameter. Measurements under uniaxial stress offer an ideal way to test for chirality, because under uniaxial stress the superconducting and chi…
The measurements of intensity of ultrasonic resonances below the transition to the superconducting state in a tetragonal metal cannot distinguish between the magnetic and nonmagnetic superconducting states with two-component order parameters.
We review the normal and superconducting state properties of the unconventional triplet superconductor Sr$_2$RuO$_4$ with an emphasis on the analysis of the magnetic susceptibility and the role played by strong electronic correlations. In particular, we show that the magnetic activity arises from the itinerant electrons in the Ru $d$-orbitals and a strong magnetic anisotropy occurs ($χ^{+-} < χ^{zz}$) due to spin-orbit coupling. The latter results mainly from different values of the $g$-factor for the transverse and longitudinal components of the spin susceptibility (i.e. the matrix elements d…
We study the dynamical spin susceptibility, $χ({\bf q}, ω)$, in the normal and superconducting state of Sr$_2$RuO$_4$. In the normal state, we find a peak in the vicinity of ${\bf Q}_i\simeq (0.72π,0.72π)$ in agreement with recent inelastic neutron scattering (INS) experiments. We predict that for spin triplet pairing in the superconducting state a {\it resonance peak} appears in the out-of-plane component of $χ$, but is absent in the in-plane component. In contrast, no resonance peak is expected for spin singlet pairing.
The character of the superconducting phase of Sr$_2$RuO$_4$, is topic of a longstanding discussion. The classification of the symmetry allowed order parameters has relied on the tetragonal symmetry of the lattice and on cylindrical Fermi surfaces, usually taken to be featureless, not including the non-trivial symmetry aspects related to their orbital content. Here we show how the careful account of the orbital degree of freedom in Sr$_2$RuO$_4$, leads to a much richer classification of order parameters. We analyse the stability and degeneracy of these new order parameters from the perspective …
This paper gives a detailed numerical study of the superconducting electronic heat transport in the unconventional multiband superconductor Strontium Ruthenate Sr$_2$RuO$_4$. The study demostrates that a model with different nodal structures on different sheets of the Fermi surface is able to describe quantitatively experimental heat transport data. The contribution of the density of states DOS is given for each sheet of the Fermi surface and the total contribution is also calculated. Finally, a discussion of the universal character of the electronic heat transport in unconventional supercondu…
Clarifying the chiral domains structure of superconducting Sr$_{2}$RuO$_{4}$ has been a long-standing issue in identifying its peculiar topological superconducting state. We evaluated the critical current $I_{c}$ versus the magnetic field $H$ of Nb/Sr$_{2}$RuO$_{4}$ Josephson junctions, changing the junction dimension in expectation of that the number of domains in the junction is controlled. $I_{c}(H)$ exhibits a recovery from inversion symmetry breaking to invariance when the dimension is reduced to several microns. This inversion invariant behavior indicates the disappearance of domain wall…
The unusual superconducting state in Sr$_2$RuO$_4$ has long been viewed as being analogous to a superfluid state in liquid $^3$He. Nevertheless, calculations based on this odd-parity state are presently unable to completely reconcile the properties of Sr$_2$RuO$_4$. Using a self-consistent quantum many-body scheme that employs realistic parameters, we are able to model several signature properties of the normal and superconducting states of Sr$_2$RuO$_4$. We find that the dominant component of the model superconducting state is of even parity and closely related to superconducting state for th…
Muon Knight shift as a precise probe of the superconducting symmetry of Sr$_2$RuO$_4$
Toyota Riken–Kyoto University Research Center (TRiKUC), Kyoto University（丰田理研-京都大学研究中心）；Institute for Chemical Research, Kyoto University（京都大学化学研究所）；PSI Center for Neutron and Muon Sciences CNM（保罗谢尔研究所中子与缪子科学中心）；Institute for Solid State Physics, the University of Tokyo（东京大学固体物理研究所）；Department of Physics, University of British Columbia（不列颠哥伦比亚大学物理系）；Department of Electronic Science and Engineering, Graduate School of Engineering, Kyoto University（京都大学大学院工学研究科电子科学与工程系）
Comprehensive measurements of dc and ac susceptibility, dc resistance, magnetoresistance, Hall resistivity, and microwave absorption and dispersion in fields up to 8 T have been carried out on RuSr_2GdCu_2O_8 with the aim to establish the properties of RuO_2 and CuO_2 planes. At ~130 K, where the magnetic order develops in the RuO_2 planes, one observes a change in the slope of dc resistance, change in the sign of magnetoresistance, and the appearance of an extraordinary Hall effect. These features indicate that the RuO_2 planes are conducting. A detailed analysis of the ac susceptibility and …
The unconventional superconductivity in Sr$_2$RuO$_4$ continues to defy a unified interpretation. In this paper, we focus on some novel aspects of its superconducting pairing by exploiting the orbital degree of freedom in this material. The multi-orbital nature, combined with the symmetry of the orbitals involved, leads to a plethora of exotic Cooper pairings not accessible in single-orbital systems. Essential physics is illustrated first using a two-orbital model with $d_{xz}$- and $d_{yz}$-orbitals. We classify the gap functions according to the underlying lattice symmetries, analyze the eff…
Although correlated electronic-structure calculations explain very well the normal state of Sr$_2$RuO$_4$, its superconducting symmetry is still unknown. Here we construct the spin and charge fluctuation pairing interactions based on its correlated normal state. Correlations significantly reduce ferromagnetic in favor of antiferromagnetic fluctuations and increase inter-orbital pairing. From the normal-state Eliashberg equations, we find spin-singlet $d$-wave pairing close to magnetic instabilities. Away from these instabilities, where charge fluctuations increase, we find two time-reversal sy…
Sr$_2$RuO$_4$ is the best candidate for spin-triplet superconductivity, an unusual and elusive superconducting state of fundamental importance. In the last three decades Sr$_2$RuO$_4$ has been very carefully studied and despite its apparent simplicity when compared with strongly correlated high-$T_{c}$ cuprates, for which the pairing symmetry is understood, there is no scenario that can explain all the major experimental observations, a conundrum that has generated tremendous interest. Here we present a density-functional based analysis of magnetic interactions in Sr$_{2}$RuO$_{4}$ and discuss…
It has been shown [1] that many seemingly contradictory experimental findings concerning the superconducting state in Sr$_2$RuO$_4$ can be accounted for as resulting from the existence of an assumed tetra-critical point at near ambient pressure at which $d_{x^2-y^2}$ and $g_{xy(x^2-y^2)}$ superconducting states are degenerate. We perform both a Landau-Ginzburg and a microscopic mean-field analysis of the effect of spatially varying strain on such a state. In the presence of finite $xy$ shear strain, the superconducting state consists of two possible symmetry-related time-reversal symmetry (TRS…
In the unconventional superconductor Sr$\mathsf{_2}$RuO$\mathsf{_4}$, unusual first-order superconducting transition has been observed in the low-temperature and high-field region, accompanied by a four-fold anisotropy of the in-plane upper critical magnetic field $H_{c2}$. The origin of such unusual $H_{c2}$ behavior should be closely linked to the debated superconducting symmetry of this oxide. Here, toward clarification of the unusual $H_{c2}$ behavior, we performed the resistivity measurements capable of switching in-plane current directions as well as precisely controlling the field direc…
The nature of the Cooper pairing in the paradigmatic unconventional superconductor Sr$_2$RuO$_4$ is an outstanding puzzle in condensed matter physics. Despite the tremendous efforts made in the past twenty-seven years, neither the pairing symmetry nor the underlying pairing mechanism in this material has been understood with clear consensus. This is largely due to the lack of a superconducting order that is capable of interpreting in a coherent manner the numerous essential experimental observations. At this stage, it may be desirable to reexamine the existing theoretical descriptions of super…
The superconducting symmetry of Sr$_2$RuO$_4$ remains a puzzle. Time-reversal symmetry breaking $d_{x^2-y^2} + ig_{xy(x^2-y^2)}$ pairing has been proposed for reconciling multiple key experiments. However, its stability remains unclear. In this work, we theoretically study the superconducting instabilities in Sr$_2$RuO$_4$, including the effects of spin-orbit coupling (SOC), in the presence of both local and longer-range interactions within a random phase approximation. We show that the inclusion of second nearest neighbor repulsions, together with non-local SOC in the $B_{2g}$ channel or orbi…
The symmetry of the superconducting condensate in Sr$_2$RuO$_4$ remains controversial after nuclear magnetic resonance (NMR) experiments recently overturned the dominant chiral $p$-wave paradigm. Several theoretical proposals have been put forward to account for existing experiments, including a $d+ig$-wave admixture, conjectured to be stabilized by longer-range Coulomb interactions. We perform a material-specific microscopic theoretical study of pairing by spin- and charge-fluctuations in Sr$_2$RuO$_4$, including the effects of spin-orbit coupling, and both local and longer-range Coulomb repu…
We observe an unconventional superconducting minigap induced into a ferromagnet SrRuO$_3$ from a spin-triplet superconductor Sr$_2$RuO$_4$ using a Au/SrTiO$_3$/SrRuO$_3$/Sr$_2$RuO$_4$ tunnel junction. Voltage bias differential conductance of the tunnel junctions exhibits V-shaped gap features around zero bias, corresponding to a decrease in the density-of-states with an opening of a superconducting minigap in SrRuO$_3$. Observation of a minigap at a surface of a 15~nm thick SrRuO$_3$ layers confirms the spin-triplet nature of induced superconductivity. The shape and temperature dependence of t…
The superconducting state in the quasi-two-dimensional and strongly correlated Sr$_2$RuO$_4$ is uniquely held up as a solid state analog to superfluid $^3$He-$A$, with an odd-parity order parameter that also breaks time reversal symmetry, and for which the vector order parameter has the same direction in spin space for all electron momenta. The recent discovery that uniaxial pressure causes a steep rise and maximum in transition temperature ($T_c$) in strained samples motivated the study of $^{17}$O nuclear magnetic resonance (NMR) that we describe in this article. A reduction of Knight shifts…
Strontium ruthenate (Sr$_2$RuO$_4$) has long been thought to host a spin-triplet chiral $p$-wave superconducting state. However, the singletlike response observed in recent spin-susceptibility measurements casts serious doubts on this pairing state. Together with the evidence for broken time-reversal symmetry and a jump in the shear modulus $c_{66}$ at the superconducting transition temperature, the available experiments point towards an even-parity chiral superconductor with $k_z(k_x\pm ik_y)$-like $E_g$ symmetry, which has consistently been dismissed based on the quasi-two-dimensional electr…
Ultrasonic Observation of Slowing Down of Multipole Fluctuations in Sr$_2$RuO$_4$
Tokyo University of Science（东京科学大学）；Niigata University（新潟大学）；Toyama Prefectural University（富山県立大学）；National Institute of Advanced Industrial Science and Technology (AIST)（产业技术综合研究所）
The long-standing quest to determine the superconducting order of Sr$_2$RuO$_4$ (SRO) has received renewed attention after recent nuclear magnetic resonance (NMR) Knight shift experiments have cast doubt on the possibility of spin-triplet pairing in the superconducting state. As a putative solution, encompassing a body of experiments conducted over the years, a $d+ig$-wave order parameter caused by an accidental near-degeneracy has been suggested [S. A. Kivelson et al., npj Quantum Materials $\bf{5}$, 43 (2020)]. Here we develop a general Ginzburg--Landau theory for multiband superconductors. …
The vortex lattice structures of Sr$_2$RuO$_4$ for the odd parity representations of the superconducting state are examined for the magnetic field along the crystallographic directions. Particular emphasis is placed upon the two dimensional representation which is believed to be relevant to this material. It is shown that when the zero-field state breaks time reversal symmetry, there must exist two superconducting transitions when there is a finite field along a high symmetry direction in the basal plane. Also it is shown that a square vortex lattice is expected when the field is along the $c$…
In this work, a numerical study of the superconducting specific heat of the unconventional multiband superconductor Strontium Ruthenate is performed. Two band gaps models are employed, and the results rendered for each of them are compared. One of the models, previously proposed by one of the authors to explain the experimental temperature behavior of the ultrasound attenuation, considers two gaps with point nodes of different magnitude on different gap surface sheets, while the other one is an isotropic and line node model, reported in the literature for describing quantitatively experimental…
Sr$_2$RuO$_4$ has long been the focus of intense research interest because of conjectures that it is a correlated topological superconductor. It is the momentum space (k-space) structure of the superconducting energy gap $Δ_i(\mathbf{k})$ on each band $i$ that encodes its unknown superconducting order-parameter. But, because the energy scales are so low, it has never been possible to directly measure the $Δ_i(\mathbf{k})$ of Sr$_2$RuO$_4$. Here we implement Bogoliubov quasiparticle interference (BQPI) imaging, a technique capable of high-precision measurement of multiband $Δ_i(\mathbf{k})$. At…
We present an analysis of the Hall conductivity $σ_{xy}(ω, T)$ in time reversal symmetry breaking states of exotic superconductors. We find that the dichroic signal is non-zero in systems with inter-band order parameters. This new intrinsic mechanism may explain the Kerr effect observed in strontium ruthenate and possibly other superconductors. We predict coherence factor effects in the temperature dependence of the imaginary part of the ac Hall conductivity $ Imσ_{xy}(ω, T)$, which can be tested experimentally.
In this short review, some bulk threshold phenomena of the unconventional superconductivity in the strontium ruthenate compound (Sr2RuO4) are mentioned. Particularly, in Sr2RuO4, the Ehrenfest relations, the velocity propagation of the elastic field modes, and the ultrasound attenuation find a common point to study elastic anisotropic lattice field effects at the transition temperature (Tc) between the normal and the superconducting state. Ehrenfest relations are suitable for establishing a phase diagram of the Tc point according to symmetry consideration of the electrons pairs in unconvention…
The binary ruthenate, RuO$_2$, has been the subject of intense interest due to its itinerant antiferromagnetism and strain-induced superconductivity. The strain mechanism and its effect on the microscopic electronic states leading to the normal and superconducting state, however, remain undisclosed. Here, we investigate highly-strained epitaxial (110) RuO$_2$ films using polarization-dependent oxygen K-edge X-ray absorption spectroscopy (XAS). Through the detection of pre-edge peaks, arising from O:$2p$ - Ru:$4d$ hybridization, we uncover the effects of epitaxial strain on the orbital/electron…
Muon spin rotation/relaxation ($μ$SR) and polar Kerr effect measurements provide evidence for a time-reversal symmetry breaking (TRSB) superconducting state in Sr$_2$RuO$_4$. However, the absence of a cusp in the superconducting transition temperature ($T_{\rm c}$) vs. stress and the absence of a resolvable specific heat anomaly at TRSB transition temperature ($T_{\rm TRSB}$) under uniaxial stress challenge a hypothesis of TRSB superconductivity. Recent $μ$SR studies under pressure and with disorder indicate that the splitting between $T_{\rm c}$ and $T_{\rm TRSB}$ occurs only when the structu…
There is considerable evidence that the superconducting state of Sr$_2$RuO$_4$ breaks time reversal symmetry. In the experiments showing time reversal symmetry breaking its onset temperature, $T_\text{TRSB}$, is generally found to match the critical temperature, $T_\text{c}$, within resolution. In combination with evidence for even parity, this result has led to consideration of a $d_{xz} \pm id_{yz}$ order parameter. The degeneracy of the two components of this order parameter is protected by symmetry, yielding $T_\text{TRSB} = T_\text{c}$, but it has a hard-to-explain horizontal line node at…
