After a brief introduction on nodal superconductors, we review the topological defects in triplet superconductors such as UPt$_{3}$, Sr$_{2}$RuO$_{4}$, etc. This is in part motivated by the surprising discovery of Ana Celia Mota and her colleagues that in some triplet superconductors the flux motion is completely impeded (the ideal pinning). Among topological defects the most prominent is Abrikosov's vortex with quantum flux $ϕ_{0}= \frac{hc}{2e}$. Abrikosov's vortex is universal and ubiquitous and seen in both conventional and unconventional superconductors by the Bitter decoration technique, small angle neutron scattering (SANS), scanning tunneling microscopy (STM), micromagnetometer and more recently by Lorentz electron micrograph. In order to interpret the experiment by Mota et al a variety of textures are proposed. In particular, in analogy to superfluid $^{3}$He-A the $\hat{\ell}$-soliton and $\hat{d}$-soliton play the prominent role. We review these notions and point out possible detection of these domain walls and half-quantum vortices in some triplet superconductors.
A possible mechanism for the horizontal line nodes in triplet superconductor, Sr$_2$RuO$_4$, is proposed. We consider the interlayer Coulomb interaction, as well as the on-site Coulomb repulsion, between electrons in different bands. In the second order perturbation in the interband interaction, the effective interaction becomes dependent on $\cos \frac{q_z}{2}$, resulting in the horizontal line nodes.
We study the Raman spectra of spin-triplet superconductors in Sr$_2$RuO$_4$. The p-wave and f-wave symmetries are considered. We show that there is the clapping mode with frequency of $\sqrt{2} Δ(T)$ and $1.02 Δ(T)$ for p-wave and f-wave superconductors, respectively. This mode is visible as a huge resonance in the B1g and B2g modes of Raman spectra. We discuss the details of the Raman spectra in these superconducting states.
We present direct imaging of magnetic flux structures in the anisotropic, spin-triplet superconductor Sr$_{2}$RuO$_{4}$ using a scanning $μ$SQUID microscope. Individual quantized vortices were seen at low magnetic fields. Coalescing vortices forming flux domains were revealed at intermediate fields. Based on our observations we suggest that a mechanism intrinsic to the material stabilizes the flux domains against the repulsive vortex-vortex interaction. Topological defects like domain walls can provide this, implying proof for unconventional chiral superconductivity.
Despite much effort for over the two decades, the paring symmetry of a Sr$_2$RuO$_4$ superconductor has been still unclear. In this Rapid Communication, motivated by the recent rapid progress in fabrication techniques for Sr$_2$RuO$_4$ thin-films, we propose a promising strategy for identifying the spin-triplet superconductivity in the thin-film geometry by employing an antisymmetric spin-orbit coupling potential and a Zeeman potential due to an external magnetic field. We demonstrate that a spin-triplet superconducting thin-film undergoes a phase transition from a helical state to a chiral st…
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…
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…
We investigate the superconductivity in Sr$_2$RuO$_4$ on the basis of the three-dimensional three-band Hubbard model. We propose a model with Coulomb interactions among the electrons on the nearest-neighbor Ru sites. In our model the intersite Coulomb repulsion and exchange coupling can work as the effective interaction for the spin-triplet paring. This effective interaction is enhanced by the band hybridization, which is mediated by the interlayer transfers. We investigate the possibility of this mechanism in the ground state and find that the orbital dependent spin-triplet superconductivity …
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…
The mechanism of superconductivity in ${\rm Sr}_{2}{\rm RuO}_{4}$ is studied using a degenerate Hubbard model within the weak coupling theory. When the system approaches the orbital instability which is realized due to increasing the on-site Coulomb interaction between the electrons in the different orbitals, it is shown that the triplet superconductivity appears. This superconducting mechanism is only available in orbitally degenerate systems with multiple Fermi surfaces.
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…
Spin-polarized supercurrents can be generated with magnetic inhomogeneity at a ferromagnet/spin-singlet-superconductor interface. In such systems, complex magnetic inhomogeneity makes it difficult to functionalise the spin-polarized supercurrents. However, spin-polarized supercurrents in ferromagnet/spin-triplet-superconductor junctions can be controlled by angle between magnetization and spin of Copper pairs ($d$-vector), that can effectively be utilized in developing of a field of research known as superconducting spintronics. Recently, we found induction of spin-triplet correlation into a f…
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.
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 …
In zero magnetic field the superconductor Sr$_2$RuO$_4$ is believed to have a chiral spin triplet pairing state in which the gap function d-vector is aligned along the crystal c-axis. Using a phenomenological but orbital specific description of the spin dependent electron-electron attraction and a realistic quantitative account of the electronic structure in the normal state we analyze the orientation of the spin triplet Cooper pair d-vector in response to an external c-axis magnetic field. We show that for suitable values of the model parameters a c-axis field of only 20 mT is able to cause a…
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…
We investigate inverse proximity effects in a spin-triplet superconductor (TSC) interfaced with a ferromagnet (FM), assuming different types of magnetic profiles and chiral or helical pairings. The region of the coexistence of spin-triplet superconductivity and magnetism is significantly influenced by the orientation and spatial extension of the magnetization with respect to the spin configuration of the Cooper pairs, resulting into clearcut anisotropy signatures. A characteristic mark of the inverse proximity effect arises in the induced spin-polarization at the TSC interface. This is unexpec…
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 $d_…
We study the spin susceptibility of the orbital-singlet pairings, including the spin-triplet/orbital-singlet/$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 study the spin-orbit coupling in spin-triplet Cooper pairs and clarify multiple superconducting (SC) phases in Sr$_2$RuO$_4$. Based on the analysis of the three-orbital Hubbard model with atomic LS coupling, we show some selection rules of the spin-orbit coupling in Cooper pairs. The spin-orbit coupling is small when the two-dimensional $γ$-band is the main cause of the superconductivity, although the LS coupling is much larger than the SC gap. Considering this case, we investigate multiple SC transitions in the magnetic fields for both H // [001] and H // [100] using the Ginzburg-Landau th…
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…
Supposing the spin-triplet superconducting state of Sr$_2$RuO$_4$, the spin-orbit (SO) coupling associated with relative motion in Cooper pairs is calculated by extending the method for the dipole-dipole coupling given by Leggett in the superfluid $^{3}$He. It is shown that the SO coupling works only in the equal-spin pairing (ESP) state to make the pair angular momentum $\hbar{\vec L}$ and the pair spin angular momentum ${\rm i}{\vec d}\times{\vec d}^{*}$ parallel with each other. The SO coupling gives rise to the internal Josephson effect in a chiral ESP state as in superfluid A-phase of $^3…
The critical current modulation and the spontaneous flux of the vortex states in corner Josephson junctions between Sr$_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.
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…
Tunneling conductance spectra of normal metal/insulator/triplet superconductor junctions are investigated theoretically. As triplet paring states we select several types of symmetries that are promising candidates for the superconducting states in UPt$_{3}$ and in Sr$_{2}$RuO$_{4}$. The calculated conductance spectra are sensitive to the orientation of the junction which reflects the anisotropy of the pairing states. They show either zero-bias conductance peaks or gap-like structures depending on the orientation of the junctions. The existence of a residual density of states, peculiar to nonun…
We investigate the order parameter of Sr$_2$RuO$_4$ in an exchange-only magnetic field. A Ginzburg-Landau symmetry analysis implies three possibilities: a pure $^3$He A phase, a $^3$He A$_1$ or a $^3$He A$_2$ phase. We explore the exchange field dependence of the order parameter and energy gap in a one-band model of Sr$_2$RuO$_4$. The numerical solutions show no A$_1$ phase and that the A$_2$ phase is lower in free energy than the A phase. We explore heat capacity as a function of temperature and field strength and find quantitatively different behaviors for the A and A$_2$ phases.
We investigate the rotation effect of the RuO$_6$ octahedron around the $c$ axis on the topological and transport properties near the surface of the spin-triplet superconductor Sr$_2$RuO$_4$. While the Fermi level of bulk Sr$_2$RuO$_4$ is near the Lifshitz transition, the RuO$_6$ rotation realized near the surface leads to the change of the Fermi surface topology. The edge current resulting from the time-reversal symmetry breaking in the chiral $p$-wave phase with fully opened excitation gap is less affected around Lifshitz transition. The topological property and the edge state are sensitive …
Using a three-band Hubbard Hamiltonian we calculate within the random-phase-approximation the spin susceptibility, $χ({\bf q},ω)$, and NMR spin-lattice relaxation rate, 1/T$_1$, in the normal state of the triplet superconductor Sr$_2$RuO$_4$ and obtain quantitative agreement with experimental data. Most importantly, we find that due to spin-orbit coupling the out-of-plane component of the spin susceptibility $χ^{zz}$ becomes at low temperatures two times larger than the in-plane one. As a consequence strong incommensurate antiferromagnetic fluctuations of the quasi-one-dimensional $xz$- and $y…
We study the tunneling conductance of a ballistic normal metal / ferromagnet / spin-triplet superconductor junction using the extended Blonder-Tinkham-Klapwijk formalism as a model for a $c$-axis oriented Au / SrRuO$_{3}$ / Sr$_{2}$RuO$_{4}$ junction. We compare chiral $p$-wave (CPW) and helical $p$-wave (HPW) pair potentials, combined with ferromagnet magnetization directions parallel and perpendicular to the interface. For fixed $θ_{M}$, where $θ_{M}$ is a direction of magnetization in the ferromagnet measured from the $c$-axis, the tunneling conductance of CPW and HPW clearly show different…
We calculate the tunneling conductance spectra of a normal-metal/insulator/triplet superconductor using the Blonder-Tinkham-Klapwijk (BTK) formulation. Possible states for the superconductor are considered with horizontal lines of nodes, breaking the time reversal symmetry. These results would be useful to discriminate between pairing states in superonductor Sr$_2$RuO$_4$ and also in UPt$_3$.
Sr$_{2}$RuO$_{4}$ is one of the most promising candidates of a topological superconductor with broken time-reversal symmetry, because a number of experiments have revealed evidences for a spin-triplet chiral $p$-wave superconductivity. In order to clarify the time-reversal symmetry of Sr$_{2}$RuO$_{4}$, we introduce a novel test that examines the invariance of the Josephson critical current under the inversion of both the current and magnetic fields, in contrast to the detection of a spontaneous magnetic field employed in past experiments. Analyses of the transport properties of the planar and…
Motivated by a recent angle-resolved thermal conductivity experiment that shows a twofold gap symmetry in the high-field and low-temperature C phase in the heavy-fermion superconductor UPt$_3$, we group-theoretically identify the pairing functions as $E_{1u}$ with the $f$-wave character for all the three phases. The pairing functions are consistent with the observation as well as with a variety of existing measurements. By using a microscopic quasi-classical Eilenberger equation with the identified triplet pairing function under applied fields, we performed detailed studies of the vortex struc…
Half quantum vortices (HQVs) in superfluid $^3He-A$ have been speculated since 1976. Two years ago, Yamashita et al reported a very unusual NMR satellite in the rotating superfluid $^3He-A$ in a parallel plate geometry. Recently this satellite has been interpreted in terms of HQVs. Also, the strong flux pinnings in the triplet superconductors $UPt_3, Sr_2RuO_4$ and $U_{1-x}Th_xBe_{13}$ have been discovered by Mota and her colleagues which are interpreted in terms of HQVs. We shall first review the bound state spectra around a single vortex in unconventional superconductors, which provides the …
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…
Uniaxial strain experiments have become a powerful tool to unveil the character of unconventional phases of electronic matter. Here we propose a combination of the superconducting fitness analysis and density functional theory (DFT) calculations in order to dissect the effects of strain in complex multi-orbital quantum materials from a microscopic perspective. We apply this framework to the superconducting state of Sr$_2$RuO$_4$, and argue that the recently proposed orbitally anti-symmetric spin-triplet (OAST) order parameter candidate has unique signatures under strain which are in agreement …
High-quality single crystals are essentially needed for the investigation of the novel bulk properties of unconventional superconductors. The availability of such crystals grown by the floating-zone method has helped to unveil the unconventional superconductivity of the layered perovskite Sr$_2$RuO$_4$, which is considered as a strong candidate of a topological spin-triplet superconductor. Yet, recent progress of investigations urges further efforts to obtain ultimately high-quality crystalline samples. In this paper, we focus on the method of preparation of feed rods for the floating-zone mel…
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…
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…
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 …
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…
Single crystals of the Sr$_2$RuO$_4$-Ru eutectic system are known to exhibit enhanced superconductivity at 3~K, in addition to the bulk superconductivity of Sr$_2$RuO$_4$ at 1.5~K. The 1.5-K phase is believed to be a spin-triplet, chiral $p$-wave state with the multi-component order parameter, giving rise to chiral domain structure. In contrast, the 3-K phase is attributable to enhanced superconductivity of Sr$_2$RuO$_4$ in the strained interface region between Ru inclusion of a few to tens of micrometers in size and the surrounding Sr$_2$RuO$_4$. We investigate the dynamic behavior of a topol…
We calculate the tunneling conductance spectra of a normal metal / insulator / triplet superconductor from the reflection amplitudes using the Blonder-Tinkham-Klapwijk (BTK) formula. For the triplet superconductor we assume one special p-wave order parameter having line nodes and two two dimensional $f$-wave order parameters with line nodes breaking the time-reversal symmetry. Also we examine nodeless pairing potentials. The tunneling peaks are due to the formation of bound states for each surface orientation at discrete quasiparticles trajectory angles. The tunneling spectra can be used to di…
Magnetic flux structures in single crystals of the layered spin triplet superconductor Sr$\_{2}$RuO$\_{4}$ are studied by scanning micro SQUID Force microscopy. Vortex chains appear as the applied field is tilted along the in-plane direction of the superconductor. The vortex chains align along the direction of the in-plane component of the applied magnetic field. The decoration of in-plane vortices by crossing Abrikosov vortices is observed: two vortex orientations are apparent simultaneously, one along the layers and the other perpendicular to the layers. The crossing vortices appear preferen…
To clarify the superconducting gap structure of the spin-triplet superconductor Sr_2RuO_4, the in-plane thermal conductivity has been measured as a function of relative orientations of the thermal flow, the crystal axes, and a magnetic field rotating within the 2D RuO_2 planes. The in-plane variation of the thermal conductivity is incompatible with any model with line nodes vertical to the 2D planes and indicates the existence of horizontal nodes. These results place strong constraints on models that attempt to explain the mechanism of the triplet superconductivity.
Scanning $μ$SQUID force microscopy is used to study magnetic flux structures in single crystals of the layered spin triplet superconductor Sr$\_{2}$RuO$\_{4}$. Images of the magnetic flux configuration above the $\vec{a}\vec{b}$-face of the cleaved crystal are acquired, mostly after field-cooling the sample. For low applied magnetic fields, individual vortices are observed, each carrying a single quantum of flux. Above 1 gauss, coalescence of vortices is discovered. The coalescing vortices may indicate the presence of domains of a chiral order parameter. When the applied field is tilted from t…
The orbital symmetry of the superconducting order parameter in UPt_3 is identified by evaluating the directionally dependent thermalconductivity and ultrasound attenuation in the clean limit and compared with the existing data for both basal plane and the c-axis of a hexagonal crystal. The resulting two component orbital part expressed by (λ_x(k), λ_y(k)) is combined with the previously determined triplet spin part, leading to clean limit and compared with the existing data for both basal plane and the c-axis of a hexagonal crystal. The resulting two component orbital part expressed by (λ_x(k)…
We calculate the parallel upper critical magnetic field $H_{\parallel}(0)$ for an in-plane isotropic quasi-two-dimensional (Q2D) chiral triplet superconductor at zero temperature, $T=0$. In particular, the ratio $H_{\parallel}(0)/(|dH^{GL}_{\parallel}/dT|_{T=T_c}T_c) = 0.815$ is defined, where $|dH^{GL}_{\parallel}/dT|_{T=T_c}$ is the so-called Ginzburg-Landau slope of the upper critical magnetic field, $T_c$ is a superconducting transition temperature at $H=0$. We show that the theoretically obtained above mentioned value strongly contradicts to the experimentally measured ones in a candidate…
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…
The discovery of noncentrosymmetric superconductors, such as CePt$_3$Si, and chiral superconductors, such as Sr$_2$RuO$_4$, calls for experimental methods to identify the presence of spin-triplet pairing. We here demonstrate a method which accomplishes this in an appealingly simple manner: a spin-sensitive proximity effect in a ferromagnet$\mid$triplet superconductor bilayer. It is shown how the orientation of the field can be used to unambiguously distinguish between different spin-triplet states. Moreover, the proximity effect becomes long-ranged in spite of the presence of an exchange field…
In the last few years the gap symmetries of many new superconductors,including Sr$_2$RuO$_4$, CeCoIn$_5$, $κ$-(ET)$_{2}$Cu(NCS)$_{2}$, YNi$_{2}$B$_{2}$C and PrOs$_{4}$Sb$_{12}$, have been identified via angle-dependent magnetothermal conductivity measurements. However, a controversy still persists as to the nature of the superconductivity in Sr$_2$RuO$_4$. For PrOs$_{4}$Sb$_{12}$, spin-triplet superconductivity has recently been proposed. Here, we also propose g-wave superconductivity for UPd$_2$Al$_3$ (i.e., $Δ({\bf k})=Δ\cos(2χ), χ= ck_{z}$) based on recent thermal conductivity data.
The Josephson effect between triplet and singlet superconductors is studied. Josephson current can flow between triplet and singlet superconductors due to the spin-orbit coupling in the spin-triplet superconductor but it is finite only when triplet superconductor has $L_z=-S_z=\pm 1$, where $L_z$ and $S_z$ are the perpendicular components of orbital angular momentum and spin angular momentum of the triplet Cooper pairs, respectively. The recently observed temperature and orientational dependence of the critical current through a Josephson junction between UPt$_3$ and Nb is investigated by cons…
We calculate the Hall conductivity for a spin-triplet superconductor, using a generalized pairing symmetry dependent on an arbitrary phase, $\vphi$. A promising candidate for such an order parameter is Sr$_{2}$RuO$_{4}$, whose superconducting order parameter symmetry is still subject to investigation. The value of this phase can be determined through Kerr rotation and DC Hall conductivity measurements. Our calculations impose significant constraints on $\vphi$.
A common path to superconducting spintronics, Majorana fermions, and topologically-protected quantum computing relies on spin-triplet superconductivity. While naturally occurring spin-triplet pairing is elusive and even common spin-triplet candidates, such as Sr$_2$RuO$_4$, support alternative explanations, proximity effects in heterostructures can overcome these limitations. It is expected that robust spin-triplet superconductivity in magnetic junctions should rely on highly spin-polarized magnets or complex magnetic multilayers. Instead, we predict that the interplay of interfacial spin-orbi…
We calculate the tunneling conductance spectra of a ferromagnetic metal / insulator / triplet superconductor from the reflection amplitudes using the Blonder-Tinkham-Klapwijk (BTK) formula. For the triplet superconductor, we assume one special $p$-wave order parameter, having line nodes, and two two dimensional $f$-wave order parameters with line nodes, breaking the time reversal symmetry. Also we examine nodeless pairing potentials. The evolution of the spectra with the exchange potential depends solely on the topology of the gap. The weak Andreev reflection within the ferromagnet results in …
We report results of our low-temperature magneto electric transport measurements on micron-sized short cylinders of odd-parity, spin-triplet superconductor Sr$_2$RuO$_4$ with the cylinder axis along the $c$ axis. The in-plane magnetic field and measurement current dependent magnetoresistance oscillations were found to feature an amplitude much larger than that expected from the conventional Little-Parks effect, suggesting that the magnetoresistance oscillations originate from vortex crossing. The free-energy barrier that controls the vortex crossing was modulated by the magnetic flux enclosed …
We investigate the topological aspect of the spin-triplet $f$-wave superconductor UPt$_3$ through microscopic calculations of edge- and vortex-bound states based on the quasiclassical Eilenberger and Bogoliubov-de Gennes theories. It is shown that a gapless and linear dispersion exists at the edge of the $ab$-plane. This forms a Majorana valley, protected by the mirror chiral symmetry. We also demonstrate that, with increasing magnetic field, vortex-bound quasiparticles undergo a topological phase transition from topologically trivial states in the double-core vortex to zero-energy states in t…
We report the field-orientation dependent specific heat of the spin-triplet superconductor Sr2RuO4 under the magnetic field aligned parallel to the RuO2 planes with high accuracy. Below about 0.3 K, striking 4-fold oscillations of the density of states reflecting the superconducting gap structure have been resolved for the first time. We also obtained strong evidence of multi-band superconductivity and concluded that the superconducting gap in the active band, responsible for the superconducting instability, is modulated with a minimum along the [100] direction.
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…
Anomalous features of the proximity effect specific to diffusive normal metal / triplet superconductor (DN/TS) junctions are studied within the quasiclassical Green's function formalism. The pair amplitude $f_{N}$ as a function of energy $ε$ in DN, satisfies an anomalous relation, $% f_{N}(ε)=-f^{*}_{N}(-ε)$, contrary to that in singlet superconductor junctions case where $f_{N}(ε)=f^{*}_{N}(-% ε)$. Such an unusual $ε$ dependence is responsible for a zero energy peak in the local density of states and is a source of an anomalous penetration of the applied magnetic field into DN. % Our results …
