Spin-triplet Order Parameter Research Articles

Anomalous Josephson coupling and high-harmonics in non-centrosymmetric superconductors with S-wave spin-triplet pairing

We study the Josephson effects arising in junctions made of non-centrosymmetric superconductors with spin-triplet pairing having s-wave orbital-singlet symmetry. We demonstrate that the orbital dependent character of the spin-triplet order parameter determines its non-trivial texture in the momentum space due to the inversion symmetry breaking and spin-orbit interactions. The emergence of this pattern is responsible for the occurrence of an anomalous Josephson coupling and a dominance of high-harmonics in the current phase relation. Remarkably, due to the spin-orbital couplings, variations in the electronic structure across the heterostructure can generally turn the ground state of the junction from 0- to a generic value of the Josephson phase, thus realizing the so-called φ-junction. Hallmarks of the resulting Josephson behavior, apart from non-standard current-phase relation, are provided by an unconventional temperature and magnetic field dependence of the critical current. These findings indicate the path for the design of superconducting orbitronics devices and account for several observed anomalies of the supercurrent in oxide interface superconductors.

Effects of strain in multiorbital superconductors: The case of Sr2RuO4

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 calculations to dissect the effects of strain in complex multiorbital quantum materials from a microscopic perspective. We apply this framework to the superconducting state of ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ and argue that the recently proposed orbitally antisymmetric spin triplet order parameter candidate has unique signatures under strain which are in agreement with recent observations. In particular, we can account for the asymmetric splitting of the critical temperatures for compressive strain along the $\ensuremath{\langle}100\ensuremath{\rangle}$ direction and the reduction of the critical temperature for compressive strain along the $\ensuremath{\langle}001\ensuremath{\rangle}$ and $\ensuremath{\langle}110\ensuremath{\rangle}$ directions with a single free parameter.

Isospin magnetism and spin-polarized superconductivity in Bernal bilayer graphene.

In conventional superconductors, Cooper pairing occurs between electrons of opposite spin. We observe spin-polarized superconductivity in Bernal bilayer graphene when doped to a saddle-point van Hove singularity generated by large applied perpendicular electric field. We observe a cascade of electrostatic gate-tuned transitions between electronic phases distinguished by their polarization within the isospin space defined by the combination of the spin and momentum-space valley degrees of freedom. Although all of these phases are metallic at zero magnetic field, we observe a transition to a superconducting state at finite B‖ ≈ 150mT applied parallel to the two-dimensional sheet. Superconductivity occurs near a symmetry breaking transition, and exists exclusively above the B‖-limit expected of a paramagnetic superconductor with the observed transition temperature TC ≈ 30mK, consistent with a spin-triplet order parameter.

Effects of hybridization and spin–orbit coupling to induce odd-frequency pairing in two-band superconductors

The effect of spin-independent hybridization potential and spin–orbit coupling on two-band superconductor with equal-time s-wave interband pairing order parameter is investigated theoretically. To study symmetry classes in two-band superconductors, the Gor’kov equations are solved analytically. By defining spin-singlet and spin-triplet s-wave order parameter due to two-band degree of freedom, the symmetry classes of Cooper pair are studied. For spin singlet case, it is shown that spin-independent hybridization generates Cooper pair that belongs to even-frequency spin singlet even-momentum even-band parity (ESEE) symmetry class for both intraband and interband pairing correlations. For spin-triplet order parameter, intraband pairing correlation generates odd-frequency spin triplet even-momentum even-band parity (OTEE) symmetry class, whereas interband pairing correlation generates even-frequency spin-triplet even-momentum odd-band parity (ETEO) class. For the spin singlet, spin–orbit coupling generates pairing correlation that belongs to odd-frequency spin-singlet odd-momentum even-band parity (OSOE) symmetry class and even-frequency spin-singlet even-momentum even-band parity (ESEE) for intraband and interband pairing correlation, respectively. In the spin-triplet case for intraband and interband correlation, spin–orbit coupling generates even-frequency spin-triplet odd-momentum even-band parity (ETOE) and even-frequency spin-triplet even-momentum odd-band parity (ETEO), respectively. Our results are in agreement with previous experimental and theoretical studies.

In situ transport characterization of magnetic states in Nb/Co superconductor/ferromagnet heterostructures.

Employment of the non-trivial proximity effect in superconductor/ferromagnet (S/F) heterostructures for the creation of novel superconducting devices requires accurate control of magnetic states in complex thin-film multilayers. In this work, we study experimentally in-plane transport properties of microstructured Nb/Co multilayers. We apply various transport characterization techniques, including magnetoresistance, Hall effect, and the first-order-reversal-curves (FORC) analysis. We demonstrate how FORC can be used for detailed in situ characterization of magnetic states. It reveals that upon reduction of the external field, the magnetization in ferromagnetic layers first rotates in a coherent scissor-like manner, then switches abruptly into the antiparallel state and after that splits into the polydomain state, which gradually turns into the opposite parallel state. The polydomain state is manifested by a profound enhancement of resistance caused by a flux-flow phenomenon, triggered by domain stray fields. The scissor state represents the noncollinear magnetic state in which the unconventional odd-frequency spin-triplet order parameter should appear. The non-hysteretic nature of this state allows for reversible tuning of the magnetic orientation. Thus, we identify the range of parameters and the procedure for in situ control of devices based on S/F heterostructures.

Superconductivity in Cu-doped Bi2Se3 with potential disorder

We study the effects of random nonmagnetic impurities on superconducting transition temperature ${T}_{c}$ in a Cu-doped ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, for which four types of pair potentials have been proposed. Although all the candidates belong to $s$-wave symmetry, two orbital degrees of freedom in electronic structures enrich the symmetry variety of a Cooper pair such as even-orbital-parity and odd-orbital-parity. We consider realistic electronic structures of Cu-doped ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ by using a tight-binding Hamiltonian on a hexagonal lattice and consider effects of impurity scatterings through the self-energy of the Green's function within the Born approximation. We find that even-orbital-parity spin-singlet superconductivity is basically robust even in the presence of impurities. The degree of the robustness depends on the electronic structures in the normal state and on the pairing symmetry in orbital space. On the other hand, two odd-orbital-parity spin-triplet order parameters are always fragile in the presence of potential disorder.

Cooper-Pair Spin Current in a Strontium Ruthenate Heterostructure.

It has been recognized that the condensation of spin-triplet Cooper pairs requires not only broken gauge symmetry but also spin ordering as well. One consequence of this is the possibility of a Cooper-pair spin current analogous to the magnon spin current in magnetic insulators, the analogy also extending to the existence of the Gilbert damping of the collective spin-triplet dynamics. The recently fabricated heterostructure of the thin film of the itinerant ferromagnet SrRuO_{3} on bulk Sr_{2}RuO_{4}, the best-known candidate material for a spin-triplet superconductor, offers a promising platform for generating such spin current. We show how such heterostructure allows us to not only realize the long-range spin valve but also electrically drive the collective spin mode of the spin-triplet order parameter. Our proposal represents both a novel experimental realization of superfluid spin transport and a transport signature of the spin-triplet superconductivity therein.

Full-gap superconductivity in spin-polarised surface states of topological semimetal \u03b2-PdBi2

A bulk superconductor possessing a topological surface state at the Fermi level is a promising system to realise long-sought topological superconductivity. Although several candidate materials have been proposed, experimental demonstrations concurrently exploring spin textures and superconductivity at the surface have remained elusive. Here we perform spectroscopic-imaging scanning tunnelling microscopy on the centrosymmetric superconductor β-PdBi2 that hosts a topological surface state. By combining first-principles electronic-structure calculations and quasiparticle interference experiments, we determine the spin textures at the surface, and show not only the topological surface state but also all other surface bands exhibit spin polarisations parallel to the surface. We find that the superconducting gap fully opens in all the spin-polarised surface states. This behaviour is consistent with a possible spin-triplet order parameter expected for such in-plane spin textures, but the observed superconducting gap amplitude is comparable to that of the bulk, suggesting that the spin-singlet component is predominant in β-PdBi2.

Applications of the conformal transformation method in studies of composed superconducting systems

A framework for analytical studies of superconducting systems is presented and illustrated. The formalism, based on the conformal transformation of momentum space, allows one to study the effects of both the dispersion relation and the structure of the pairing interaction in two-dimensional anisotropic high-Tc superconductors. In this method, the number of employed degrees of freedom coincides with the dimension of the momentum space, which is different compared to that in the standard Van Hove scenario with a single degree of freedom. A new function, the kernel of the density of states, is defined and its relation to the standard density of states is explained. The versatility of the method is illustrated by analyzing coexistence and competition between spin-singlet and spin-triplet order parameters in superconducting systems with a tight-binding-type dispersion relation and an anisotropic pairing potential. Phase diagrams of stable superconducting states in the coordinates I· (the ratio of hopping parameters) and n (the carrier concentration) are presented and discussed. Moreover, the role of attractive and repulsive on-site interactions for the stability of the s-wave order parameter is explained.

Anomalous superconducting proximity effect and coherent charge transport in semiconducting thin film with spin-orbit interaction

We present a microscopic theory of the superconducting proximity effect in a semiconducting thin film with a spin-orbit interaction in an external magnetic field. We demonstrate that an effective 1D Hamiltonian which describes induced superconductivity in NSO in contact with a usual s-wave superconductor possesses not only a spin-singlet induced superconducting order parameter term, as commonly adopted, but also a spin triplet order parameter term. Using this new effective Hamiltonian we confirm previous results for a normal current across contacts of NSO with a normal metal and for a Josephson current with the same NSO with induced superconductivity, obtained previously in the framework of the phenomenological Hamiltonian without spin-triplet terms. However, a calculated current-phase relation across the transparent contact between NSO with induced superconductivity in magnetic field and a usual s-wave superconductor differs significantly from previous results. We suggest the experiment which can confirm our theoretical predictions.

Unconventional superconducting states of interlayer pairing in bilayer and trilayer graphene

We develop a theory for interlayer pairing of chiral electrons in graphene materials which results in an unconventional superconducting state with an $s$-wave spin-triplet order parameter. In a pure bilayer graphene, this superconductivity exhibits a gapless property with an exotic effect of temperature-induced condensation causing an increase of the pairing amplitude with increasing temperature. We find that a finite doping opens a gap in the excitation spectrum and weakens this anomalous temperature dependence. We further explore the possibility of realizing a variety of pairing patterns with different topologies of the Fermi surface, by tuning the difference in the doping of the two layers. In trilayer graphene, the interlayer superconductivity is characterized by a two-component order parameter which can be used to define two distinct phases in which only one of the components is nonvanishing. For ABA stacking the stable state is determined by a competition between these two phases. On variation of the relative amplitude of the corresponding coupling strength, a first-order phase transition can occur between these two phases. For ABC stacking, we find that the two phases coexist with the possibility of a similar phase transition, which turns out to be second order.

Interplay between spin-singlet and spin-triplet order parameters in a model of an anisotropic superconductor with cuprate planes

Model of a two-dimensional anisotropic superconductor with a relatively wide, partially filled conduction band is considered. Attractive nearest neighbor interaction with the amplitude V1, and on-site interaction (with the amplitude V0) taken either as repulsive or attractive are included in the model, along with the tight-binding dispersion relation implying singularities in the density of states. The analytic study is based on the conformal transformation method, which can be regarded as an extension of the Van Hove Scenario, plausible for anisotropic systems such as high-Tc cuprates. Employing the Fourier harmonics as a complete and orthogonal basis for irreducible representations of the cuprate plane symmetry group (C4v), the symmetry of the superconducting order parameter is identified for various values of the model parameters. A wide range of parameters η = 2t1/t0 (the ratio of the second and the first nearest neighbor hopping integrals) and n (the carrier concentration) is studied. The obtained diagrams allow one to identify and mark the areas of stability for the superconducting spin-singlet s- and d-wave and the spin-triplet p-wave order parameters. In particular the problem of coexistence of the s-, p- and d-wave order parameters is addressed and solved for selected values of the ratio V0/V1. A possible island of stability of the d-wave order parameter in the s-wave order parameter environment for a relatively strong on-site interaction is revealed. The triple points, around which the s-, d-, and p-wave order parameters coexist, are identified on the diagrams. It is shown that results of the calculations performed for the two-dimensional tight-binding band model are dissimilar with some obtained within the standard BCS-type approximation. The influence of the on-site interaction on the stability of the s-wave order parameter is explained in detail. The developed and widely illustrated formalism can be easily applied to some more composed models.

Critical current of the spin-triplet superconducting phase inSr2RuO4

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 $d \ll \xi$ where $\xi$ 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 identification of the correct order parameter.

μ +-Knight shift in the superconducting state of U 1− xTh xBe 13, x=0 and 0.035 single crystals

We report μ +-Knight shift measurements as a function of temperature in U 1− x Th x Be 13, x=0 and 0.035, single crystals. An anomalous change in the hyperfine coupling at low temperatures is observed which is similar to that reported in nuclear magnetic resonance studies of U- and Ce-based heavy fermion compounds. In the superconducting phase the static spin susceptibility, χ spin, determined from the μ +-Knight shift decreases continuously with decreasing temperature in pure UBe 13, but saturates below the second transition temperature in U 0.965Th 0.035Be 13. We briefly discuss these behaviors in terms of a two-component spin-triplet order parameter and propose μSR experiments that can test our hypothesis.

The Josephson effect in contacts between the spin-triplet superconductor Sr 2RuO 4 and conventional superconductors

The Josephson effect between the spin-triplet superconductor Sr 2RuO 4 and conventional superconductors has been investigated for Sr 2RuO 4/Nb and Sr 2RuO 4/Sn junctions, which are fabricated on the cleaved (0 0 1) plane of single-crystalline Sr 2RuO 4. Contrary to the proposed candidates of the spin-triplet order parameter in Sr 2RuO 4, which are described in terms of the d -vector aligned to the c-axis, the Josephson effect has been observed for both junctions. This may be explained by the assumption that the presence of the steps and/or the Ru lamellas causes the contribution of the current the direction of which is other than the c-axis direction.

Mean-field study of the interplay between antiferromagnetism andd-wave superconductivity

The interplay between antiferromagnetism and d-wave superconductivity is studied in a mean-field approximation for a generic microscopic Hamiltonian with short-range repulsion and near-neighbor attraction. In the presence of competing microscopic interactions, the phase boundaries of antiferromagnetic and superconducting states are significantly modified in some region of the doping-temperature plane. The transition between superconductivity and antiferromagnetism occurs through a phase where both order parameters coexist with a third, dynamically generated, spin-triplet amplitude. This dynamical generation of a new order parameter is not restricted to a system with antiferromagnetism and d-wave superconductivity, but is a generic feature for fermionic systems. The dynamically generated spin-triplet order parameter is found to be robust to variations in the mean-field Hamiltonian.

Spin fluctuations in the ruthenium oxidesRuO2,SrRuO3,CaRuO3,andSr2RuO4probed by Ru NMR

By means of Ru NMR, magnetic properties in three-dimensional (3D) perovskites ${\mathrm{SrRuO}}_{3}$ (ferromagnet) and ${\mathrm{CaRuO}}_{3}$ (exchange enhanced paramagnet) were investigated and compared with those in the two-dimensional (2D) layered perovskite ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ (superconductor), ${\mathrm{RuO}}_{2},$ and Ru metal (Pauli paramagnet). We found that $4d$-spin contributions in the Knight shift and ${}^{101}{(1/T}_{1})$ of Ru are predominant in ${\mathrm{SrRuO}}_{3},$ ${\mathrm{CaRuO}}_{3},$ and ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4},$ but not in nonmagnetic ${\mathrm{RuO}}_{2}$ and Ru metal. The experimental results that a Stoner factor for ${\mathrm{CaRuO}}_{3}$ is close to 1 and that a correlation factor estimated from modified-Korringa behavior of $\mathrm{Ru}\ensuremath{-}{T}_{1}$ is much less than 1 indicate that ${\mathrm{CaRuO}}_{3}$ is a nearly ferromagnetic (FM) metal dominated by low-frequency and long-wavelength components of the spin fluctuations (SF). Even though the Stoner factor for ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ also indicates the closeness to the ferromagnetism, the SF in ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ are different from those in ${\mathrm{CaRuO}}_{3}.$ The results combined with the previous ${}^{17}\mathrm{O}$ NMR study in ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}$ indicate that the in-plane low-frequency components of SF are exchange enhanced without any significant q dependence due to 2D electronic character (2D nearly FM), while the out-of-plane low-frequency component of SF shows the existence of antiferromagnetic (AFM) SF between layers at low temperature. We propose that this evolution from 3D to 2D nearly FM SF is relevant to the onset of spin-triplet superconductivity in ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}.$ The 2D nearly FM SF in the in-plane ${4d}_{\mathrm{xy}}\ensuremath{-}{p}_{\mathrm{\ensuremath{\pi}}\mathrm{}}$ band may play a significant role for the stabilization of parallel spin pairing state within the basal plane among various representations of spin-triplet order parameter in ${\mathrm{Sr}}_{2}{\mathrm{RuO}}_{4}.$

17O Knight shift study in the superconducting state of Sr2RuO4

17O Knight shift measurements in Sr2RuO4 were performed over the wide range of magnetic field 3.2-11.4kOe parallel to the basal RuO2 planes. The spin susceptibility is totally unchanged through its Tc, evidencing that the spin-triplet superconducting state is realized in Sr2RuO4. The result indicates that the Cooper pairs consist of the parallel spin pairs | ↑↑> and | ↓↓> with their quantization axis perpendicular to the c-axis direction. The in-plane 2D nearly ferromagnetic spin fluctuations may play a role for the stabilization of this state among various representations of spin-triplet order parameter.