We study the pairing symmetry of Sr$_2$RuO$_4$ through the group-theoretical approach. We emphasize the role of pairing interaction between the quasi-one-dimensional(Q1D) $d_{xz/yz}$ and quasi-two-dimensional(Q2D) $d_{xy}$ orbitals. It is found that two degenerate inter-orbital time-reversal-invariant(TRI) p-wave pairings, one is spin-singlet and the other spin-triplet with out-of-plane $\bm{d}$-vector, could be the most promising candidates. Several important physical quantities are presented, including the near-nodal gap structure, the unchanged out-of-plane Knight shift, and no split transition under strain, which are consistent with the experiments. In addition, these p-wave pairings shed light on resolving the contradiction between the time-reversal breaking and reduced in-plane Knight shift measurements. As the system reaches the Van Hove singularity under applied strain, the pairing symmetry would become a d-wave pairing mainly consisting of inter-orbital components, which could be responsible for the strained 3$K$ phase.
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…
We examine the tunneling spectroscopy of three-dimensional normal-metal/Sr$_2$RuO$_4$ junctions as an experimental means to identify pairing symmetry in Sr$_2$RuO$_4$. In particular, we consider three different possible pairing states in Sr$_2$RuO$_4$: spin-singlet chiral $d$-wave, spin-triplet helical $p$-wave, and spin-nematic $f$-wave ones, all of which are consistent with recent nuclear-magnetic-resonance experiments [A. Pustogow et al., Nature 574, 72 (2019)]. The Blonder-Tinkham-Klapwijk theory is employed to calculate the tunneling conductance, and the cylindrical two-dimensional Fermi …
One key feature of the multi-orbital superconducting Sr$_2$RuO$_4$ is the presence of nodal or near-nodal quasiparticle excitations revealed in a wide variety of experiments. Typically, a nodal gap structure in a two-dimensional model would be inconsistent with the chiral or helical p-wave interpretations. However, we demonstrate that true gap nodes may emerge along certain high-symmetry directions on the quasi-one-dimensional Fermi surfaces, if the multi-orbital chiral or helical p-wave pairings acquire peculiar forms wherein the $d_{xz}$ and $d_{yz}$ orbitals develop $k_y$- and $k_x$-like pa…
Even-parity interorbital spin-triplet pairing emerges as an intriguing candidate in multiorbital superconductors with significant Hund's and spin-orbit coupling. Within such a state, the pairing is dominated by the intraband pseudospin-singlet component via the spin-orbit coupling, distinguishing it from a pure spin triplet and motivating the name, shadowed triplet. With atomic spin-orbit coupling, it was shown that this pairing exhibits an anisotropic $s$-wave character, while higher angular momentum pairing symmetries such as $d$- or $g$-wave have been proposed based on phenomenological anal…
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…
We report a novel phenomenon intimately related to the spin-triplet superconductivity. It is well known that the spin susceptibility decreases below the superconducting transition temperature in almost all superconductors because of spin-singlet pair formation, while it may remain unchanged in a handful of spin-triplet exceptions. Here we report the observation in Sr$_2$RuO$_4$ with nuclear magnetic resonance (NMR) that the spin susceptibility originating from the Ru-4$d$ electron slightly $increases$ by $\sim 2 $\% of total and becomes inhomogeneous in the superconducting state. These are rea…
The unconventional superconductivity in Sr$_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…
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.
The exact nature of unconventional superconductivity in Sr$_2$RuO$_4$ remains a mystery. At the phenomenological level, no superconducting order parameter proposed thus far seems able to coherently account for all essential experimental signatures. Among the latter is the prominent polar Kerr effect, which implies a nonzero ac anomalous Hall conductivity. Assuming the Kerr effect is intrinsic, it can be accounted for by a bulk chiral Cooper pairing with nonzero orbital angular momentum, such as $p+ip$ or $d+id$, which, however, has difficulties in being reconciled with other experimental resul…
We use functional renormalization group method to study a three-orbital model for superconducting Sr$_2$RuO$_4$. Although the pairing symmetry is found to be chiral $p$-wave, the atomic spin-orbit coupling induces near-nodes for quasiparticle excitations. Our theory explains a major experimental puzzle between $d$-wave-like feature observed in thermal experiments and the chiral $p$-wave triplet pairing revealed in nuclear-magnetic-resonance and Kerr effect.
The superconducting pairing in Sr$_2$RuO$_4$ is widely considered to be chiral $p$-wave with $\vec{d}_{\boldsymbol k} \sim (k_x + ik_y)\hat{z}$, which belongs to the $E_u$ representation of the crystalline $D_{4h}$ group. However, this superconducting order appears hard to reconcile with a number of key experiments. In this paper, based on symmetry analysis we discuss the possibility of odd-parity pairing with inherent three-dimensional (3D) character enforced by the inter-orbital interlayer coupling and the sizable spin-orbit coupling (SOC) in the material. We focus on a yet unexplored $E_u$ …
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 quasiparticle interference (QPI) in Sr$_{2}$RuO$_{4}$ is theoretically studied based on two different pairing models in order to propose an experimental method to test them. For a recently proposed two-dimensional model with pairing primarily from the $γ$ band, we found clear QPI peaks evolving with energy and their locations can be determined from the tips of the constant-energy contour (CEC). On the other hand, for a former quasi-one-dimensional model with pairing on the $α$ and $β$ bands, the QPI spectra are almost dispersionless and may involve off-shell contributions to the scattering…
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…
We report first measurements on tunneling into the {\it interior} of a Ru microdomain embedded in bulk chiral p-wave superconductor Sr$_{2}$RuO$_{4}$. The junctions were prepared by pressing a pure In wire onto the ${\it{ab}}$ face of a Ru-containing Sr$_{2}$RuO$_{4}$ single crystal. Below the superconducting transition temperature ($T_{c}$) of Sr$_{2}$RuO$_{4}$ (= 1.5 K), we observed a zero-bias conductance peak (ZBCP) associated with the proximity-induced p-wave superconductivity. However, below 0.44 K, roughly the $T_{c}$ of bulk Ru (= 0.5 K), an unexpected Ru gap and a broad ZBCP superimpo…
Despite of many experimental and theoretical efforts, the pairing symmetry of superconductivity in Sr$_2$RuO$_4$ remains undecided. The accidentally degenerate $d_{x^2-y^2}+ig$ is consistent with most current experiments and seems to be one of the most probable candidates, but we still lack a satisfactory theoretical mechanism for its appearance. Here we construct a phenomenological model combining realistic electronic band structures and all symmetry-allowed multipole fluctuations as potential pairing glues, and make a systematic survey of major pairing states within the Eliashberg framework.…
We study the spin susceptibility of the orbital-singlet pairings, including the spin-triplet/orbital-singlet/$s$-wave $E_g$ representation proposed by Suh \textit{et al}., [H.\ G.\ Suh \textit{et al}., Phys.\ Rev.\ Research \textbf{2}, 032023 (2020)], for a three-orbital model of superconducting Sr$_2$RuO$_4$ in three dimensions. For the pseudospin-singlet states represented in the band basis, the spin susceptibility decreases when reducing the temperature, irrespective of the direction of the applied magnetic fields, even if they are spin-triplet/orbital-singlet pairings in the spin-orbital s…
We investigate pairing states realized at the (001) interface of a spin-triplet superconductor Sr$_2$RuO$_4$ on the basis of microscopic calculations. Because of a Rashba-type spin-orbit interaction induced at the interface, strong parity-mixing of Cooper pairs between a spin-singlet state and a spin-triplet state occurs in this system. There are also strong inter-band pair correlations between the spin-orbit split bands, in spite of the considerably large spin-orbit splitting. This is due to frustration between the spin-orbit interaction and pairing interactions. In this pairing state, time-r…
The possibility of non-s-wave superconductivity induced by phonons is investigated using a simple model that is inspired by Sr$_2$RuO$_4$. The model assumes a two-dimensional electronic structure, a two-dimensional spin-fluctuation spectrum, and three-dimensional electron-phonon coupling. Taken separately, each interaction favors formation of spin-singlet pairs (of s symmetry for the phonon interaction and d$_{x^2-y^2}$ symmetry for the spin interaction), but in combination, a variety of more unusual singlet and triplet states are found, depending on the interaction parameters. This may have i…
The microscopic origin of triplet superconductivity in Sr$_2$RuO$_4$ is discussed, paying attention to the role of Coulomb interaction, $U_{pp}$, at the O site. It is shown on the $d$-$p$ model that $U_{pp}$ induces a ferromagnetic exchange interaction between "d-electrons" (molecular orbital with $d_{xy}$-symmetry) at adjacent Ru sites, leading to short-range ferromagnetic correlations and promoting Cooper pairing with ($\sin p_x\pm\ii\sin p_y$)-symmetry on the $γ$-band. The reason why such ferromagnetic correlations work effectively may be traced back to the fact that the level of 4d-electro…
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…
In order to probe the effect of charge fluctuations on triplet pairing, we study the pairing symmetry in the one-band Hubbard model having the off-site Coulomb repulsion ($V$) on top of the on-site repulsion as a model for the $γ$ band of Sr$_2$RuO$_4$, a strong candidate for triplet pairing superconductor. The result, obtained with the dynamical cluster approximation combined with the quantum Monte-Carlo method, and confirmed from the fluctuation exchange approximation, shows that while $d_{x^2-y^2}$-pairing dominates over $p$ in the absence of $V$, introduction of $V$ makes $p_{x+y}$ and $d_…
The critical current modulation and the spontaneous flux of the vortex states in corner Josephson junctions between Sr$_2$RuO$_4$ and a conventional s-wave superconductor are calculated as a function of the crystal orientation, and the magnetic field. For Sr$_2$RuO$_4$ we assume two nodeless p-wave pairing states. Also we use the nodal $f$-wave states $B_{1g}\times E_u$ and $B_{2g} \times E_u$, and one special p-wave state having line nodes. It is seen that the critical current depends solely on the topology of the gap.
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 establish the existence of bulk odd-frequency superconductivity in Sr$_2$RuO$_4$ and show that an intrinsic Kerr effect is a direct evidence of this state. We use both general two- and three-orbital models, as well as a realistic tight-binding description of Sr$_2$RuO$_4$ to demonstrate that odd-frequency pairing arises due to finite hybridization between different orbitals in the normal state, and is further enhanced by finite inter-orbital pairing.
The superconducting state of Sr$_2$RuO$_4$ was once thought to be a leading candidate for $p$-wave superconductivity. A constant Knight shift below the transition temperature provided evidence for spin-triplet pairing, and a $π$ phase shift observed in Josephson junction tunneling experiments suggested odd-parity pairing, both of which are described by $p$-wave states. However, with recent experiments observing a significant decrease in the Knight shift below the transition temperature, signifying a spin-singlet state, the odd-parity results are left to be reconciled. In this work, we show tha…
Using a two-dimensional Hubbard Hamiltonian for the three electronic bands crossing the Fermi level in Sr$_2$RuO$_4$ we calculate the band structure and spin susceptibility $χ({\bf q}, ω)$ in quantitative agreement with nuclear magnetic resonance (NMR) and inelastic neutron scattering (INS) experiments. The susceptibility has two peaks at {\bf Q}$_i = (2π/3, 2π/3)$ due to the nesting Fermi surface properties and at {\bf q}$_i = (0.6π, 0)$ due to the tendency towards ferromagnetism. Applying spin-fluctuation exchange theory as in layered cuprates we determine from $χ({\bf q}, ω)$, electronic di…
Due to the presence of a nontrivial three-dimensional spin-orbital entanglement, Sr$_2$RuO$_4$ may be a time-reversal invariant nematic $p$-wave superconductor with coexisting in-plane and out-of-plane pairings. Here we discuss various signatures of such a state if the out-of-plane pairing component is dominant. First, the enhancement of the superconducting $T_c$ under in-plane uniaxial strains is nearly a quadratic function of the strain, because the out-of-plane pairing lacks a linear-order coupling to the strain. Second, when the strain applies along a certain in-plane direction, the nemati…
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…
Until recently it has been believed that the superconductivity in Sr$_2$RuO$_4$ is described by p-wave pairing. However, both the recent specific heat and the magnetic penetration depth measurements on the purest single crystals of Sr$_2$RuO$_4$ appear to be explained more consistently in terms of f-wave superconductivity. In order to further this hypothesis, we study theoretically the thermodynamics and thermal conductivity of f-wave superconductors in a planar magnetic field. We find the simple expressions for these quantities when $H \ll H_{c2} $ and $T \ll T_{c}$, which should be readily a…
The polar Kerr effect in superconducting Sr$_2$RuO$_4$ implies finite ac anomalous Hall conductivity. Since intrinsic anomalous Hall effect (AHE) is not expected for a chiral superconducting pairing developed on the single Ru $d_{xy}$ orbital, multiorbital chiral pairing actively involving the Ru $d_{xz}$ and $d_{yz}$ orbitals has been proposed as a potential mechanism. Here we propose that AHE could still arise even if the chiral superconductivity is predominantly driven by the $d_{xy}$ orbital. This is demonstrated through two separate models which take into account subdominant orbitals in t…
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…
We study the (001) surface state of a recently proposed $E_g$ symmetry inter-orbital-odd spin-triplet s-wave superconducting (SC) state in Sr$_2$RuO$_4$ (SRO). We confirm that this pair potential is transformed into a chiral $d$-wave pair potential and a pseudo-Zeeman field in the band basis for a low-energy range. Due to the chiral $d$-wave pair potential, the surface states appear near zero energy in the momentum range enclosed by the nodal lines of the chiral d-wave pair potential for each band at the (001) surface. Nevertheless, the pseudo-Zeeman field gives band splitting of the surface s…
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…
Directional Andreev-Reflection Signatures of Inter-Orbital Pairing in Sr$_2$RuO$_4$
CNR-SPIN, c/o Università di Salerno（意大利萨莱诺大学CNR-SPIN研究所）；Faculty of Environmental Life, Natural Science and Technology, Okayama University（冈山大学环境生命理工学部）；Department of Applied Physics, Nagoya University（名古屋大学应用物理系）；Max-Planck Institute for Solid State Research（斯图加特马克斯·普朗克固体研究所）；University of St Andrews（圣安德鲁斯大学）；Department of Applied Science and Technology, Politecnico di Torino（都灵理工大学应用科学与技术系）
We have measured the diffracted neutron scattering intensities from the square magnetic flux lattice in the perovskite superconductor Sr$_2$RuO$_4$, which is thought to exhibit p-wave pairing with a two-component order parameter. The relative intensities of different Bragg reflections over a wide range of field and temperature have been shown to be inconsistent with a single component Ginzburg-Landau theory but qualitatively agree with a two component p-wave Ginzburg-Landau theory.
Exotic superconducting properties of Sr$_{2}$RuO$_{4}$ have provided strong support for an unconventional pairing symmetry. However, the extensive efforts over the past decade have not yet unambiguously resolved the controversy about the pairing symmetry in this material. While recent phase-sensitive experiments using flux modulation in Josephson junctions consisting of Sr$_{2}$RuO$_{4}$ and a conventional superconductor have been interpreted as conclusive evidence for a chiral spin-triplet pairing, we propose here an alternative interpretation. We show that an overlooked chiral spin-singlet p…
A major challenge to the chiral $p$-wave hypothesis for the pairing symmetry of the unconventional superconductor Sr$_2$RuO$_4$ 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 $p$-wave superconductor near a normal metal / superconductor interface using the lattice Bogoliubov-de Gennes equations and Ginzburg-Landau theor…
A detailed theoretical interpretation of the Josephson interference experiment between Sr$_2$RuO$_4$ and Pb reported by Kidwingira \textit{et al} is given. Assuming chiral p-wave pairing symmetry a Ginzburg-Landau theory is derived in order to investigate the structure of domain walls between chiral domains. It turns out that anisotropy effects of the Fermi surface and the orientation of the domain walls are essential for their internal structure. Introducing a simple model for a Josephson junction the effect of domain walls intersecting the interface between Sr$_2$RuO$_4$ and Pb is discussed.…
We study Josephson current between s-wave/spin-triplet superconductor junctions by taking into account details of band structures in Sr$_{2}$RuO$_{4}$ such as three conduction bands, spin-orbit interaction in the bulk and that at the interface. We assume five superconducting order parameters in Sr$_{2}$RuO$_{4}$: a chiral p-wave symmetry and four helical p-wave symmetries. We calculate current-phase relationship $I(φ)$ in these junctions, where $φ$ is the macroscopic phase difference between two superconductors. The results for a chiral p-wave pairing symmetry show that $\cos(φ)$ term appears …
Using an asymptotically exact weak coupling analysis of a multi-orbital Hubbard model of the electronic structure of \SRO, we show that the interplay between spin and charge fluctuations leads unequivocally to triplet pairing which originates in the quasi-one dimensional bands. The resulting superconducting state spontaneously breaks time-reversal symmetry and is of the form $Δ\sim p_x + i p_y \hat{z}$ with sharp gap minima and a d-vector that is only {\it weakly} pinned. The supercondutor is topologically {\it trivial} and hence lacks robust chiral Majorana fermion modes along the boundary. T…
We propose a general scheme to probe the compatibility of arbitrary pairing states with a given normal state Hamiltonian by the introduction of a concept called superconducting fitness. This new quantity gives a direct measure of the suppression of the superconducting critical temperature in presence of key symmetry breaking fields. A merit of the superconducting fitness is that it can be used as a tool to identify non-trivial mechanisms to suppress superconductivity under various external influences, in particular, magnetic fields or distortions, even in complex multi-orbital systems. In the …
We propose to realize second-order topological superconductivity in bilayer spin-polarized Fermi gas superfluids. We focus on systems with intralayer chiral $p$-wave pairing and with tunable interlayer hopping and interlayer interactions. Under appropriate circumstances, an interlayer even-parity $s$- or $d$-wave pairing may coexist with the intralayer $p$-wave. Our model supports localized Majorana zero modes not only at the corners of the system geometry, but also at the terminations of certain one-dimensional defects, such as lattice line defects and superfluid domain walls. We show how suc…
We study theoretically the mixed state properties of a strong uniaxially-anisotropic type II superconductor with the Pauli paramagnetic effect, focusing on their behaviors when the magnetic field orientation is tilted from the conduction layer ab plane. On the basis of Eilenberger theory, we quantitatively estimate significant contributions of the Pauli paramagnetic effects on a variety of physical observables, including transverse and longitudinal components of the flux line lattice form factors, magnetization curves, Sommerfeld coefficient, field distributions and magnetic torques. We apply …
Conditions for orbital selective altermagnetism in Sr$_2$RuO$_4$: tight binding model, similarities with cuprates and implications on superconductivity
Applying in-plane uniaxial pressure to strongly correlated low-dimensional systems has been shown to tune the electronic structure dramatically. For example, the unconventional superconductor Sr$_2$RuO$_4$ can be tuned through a single Van Hove singularity which results in a strong enhancement of both $T_\text{c}$ and $H_\text{c2}$. Out-of-plane ($c$ axis) uniaxial pressure is expected to tune the quasi-two-dimensional structure even more strongly, causing it to approach two Van Hove singularities simultaneously. Here we achieve a record value of $3.2$ GPa compression along the $c$ axis of Sr$…
Soon after the discovery of superconductivity in Sr2RuO4 (SRO) in 1994, it was conjectured that its order parameter (OP) has a form similar to that realized in the superfluid He-3, namely, odd parity and spin triplet. The chiral p-wave pairing realized in He-3 A-phase was favored by several early experiments, in particular, the muon spin rotation and the Knight shift measurements published in 1998. However, the original Knight shift result was called into question in early 2019, raising the question as to whether the chiral p-wave, or even the spin-triplet pairing itself, is indeed realized in…
Sr$_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…
One of the main developments in unconventional superconductivity in the past two decades has been the discovery that most unconventional superconductors form phase diagrams that also contain other strongly correlated states. Many systems of interest are therefore close to more than one instability, and tuning between the resultant ordered phases is the subject of intense research1. In recent years, uniaxial pressure applied using piezoelectric-based devices has been shown to be a particularly versatile new method of tuning, leading to experiments that have advanced our understanding of the fas…
To establish the mechanism of unconventional superconductivity in Sr$_2$RUO$_4$, a prerequisite is direct information concerning the momentum-space structure of the energy gaps $Δ_i(k)$, and in particular whether the pairing strength is stronger ("dominant") on the quasi-1D ($α$ and $β$) or on the quasi-2D ($γ$) Fermi surfaces. We present scanning tunneling microscopy (STM) measurements of the density-of-states spectra in the superconducting state of Sr$_2$RuO$_4$ for $0.1 T_C<T<T_C$, and analyze them, along with published thermodynamic data, using a simple phenomenological model. We show that…
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 $T_c$. Th…
The possible superconducting states of strontium ruthenate (Sr$_2$RuO$_4$) are organized into irreducible representations of the point group $D_{4h}$, with a special emphasis on nodes occurring within the superconducting gap. Our analysis covers the cases with and without spin-orbit coupling and takes into account the possibility of inter-orbital pairing within a three-band, tight-binding description of Sr$_2$RuO$_4$. No dynamical treatment if performed: we are confining ourselves to a group-theoretical analysis. The case of uniaxial deformations, under which the point group symmetry is reduce…
The superconducting state formed due to direct intra-orbital pairing in tetragonal multi-band superconductor Sr$_2$RuO$_4$ has different properties than the state formed due to intra-band pairing. In particular, the theory operating with direct intra-orbital pairing successfully explains the Kerr rotation of reflected light polarization observed several years ago. Here we apply intra-orbital approach to the problem of Ginzburg-Landau description of basal plane upper critical field in this material. It is shown that typical for two component superconducting state additional phase transition in …
