Superconductors without inversion symmetry can exhibit a non-zero magnetization when a supercurrent is present, leading to non-dissipative magnetoelectric effects. Here, we focus on understanding the relation between the spin and orbital properties of these effects in conventional spin-singlet noncentrosymmetric superconductors with orbital Rashba coupling, particularly for multiorbital electronic systems with coupled spin and orbital moments. We investigate how atomic spin-orbit coupling influences the magnitude and direction of the spin and orbital Edelstein effect. Our findings indicate that the correlation between spin and orbital moments induced by supercurrents is generally not determined by the sign of atomic spin-orbit coupling, but rather by the number of bands at the Fermi level and the orbital characteristics of those bands with respect to the mirror parity. We investigate the character of the spin and orbital Edelstein effect by examining its dependence in momentum space and the role of avoiding crossings in determining the induced magnetization. The outcomes demonstrate that the sign change of the orbital Edelstein effect near avoiding crossing bands remains robust despite modifications to the atomic spin-orbit amplitude. Furthermore, we observe that the spin Edelstein effect is typically one order of magnitude smaller than the orbital Edelstein effect, but can be significantly enhanced in certain scenarios with increased atomic spin-orbit coupling. These results indicate that by manipulating the orbital degrees of freedom in noncentrosymmetric materials, the relationship between the spin and orbital Edelstein moments can be effectively controlled.
Unique properties of the optical activity in noncentrosymmetric superconductors: sum rule, missing area, and relation with the superconducting Edelstein effect
Edelstein effect is useful for electric control of magnetic moment. However, Joule heating created by a dissipative current is harmful for its practical applications. In this paper, we investigate two-dimensional noncentrosymmetric superconductors (NCSs) with either $s$-wave or $d$-wave symmetry, and demonstrate that a surface Edelstein effect is significantly enhanced in $d$-wave NCSs. The origin of the enhancement is attributed to surface Majorana states characteristic of gapless spin-singlet superconductors. In the view of superconducting spintronics, this result would give a route to magne…
We report superconducting properties of noncentrosymmetric superconductor LaPtGe which crystallizes in noncentrosymmetric $ α$-ThSi$ _{2} $ structure. The magnetization, resistivity and specific heat measurements confirm that LaPtGe is a type-II bulk superconductor with a transition temperature T$ _{c} $ = 3.05 $ \pm $ 0.05 K. Muon-spin relaxation/rotation measurements confirms that time reversal preserved in the superconducting ground state.
In noncentrosymmetric crystals, nonuniform superconducting states are possible even in the absence of any external magnetic field. The origin of these states can be traced to the Lifshitz invariants in the free energy, which are linear in spatial gradients. We show how various types of the Lifshitz invariants in noncentrosymmetric superconductors can be derived from microscopic theory.
We calculate the spin susceptibility of a superconductor without inversion symmetry, both in the clean and disordered cases. The susceptibility has a large residual value at zero temperature, which is further enhanced in the presence of scalar impurities.
The noncentrosymmetric superconductor TaOs has been characterized using x-ray diffraction, resistivity, magnetization, and specific heat measurements. Magnetization and specific heat measurements show a bulk superconducting transition at 2.07 K. These measurements suggest that TaOs is a weakly coupled type-II superconductor. The electronic specific heat in the superconducting state can be explained by the single-gap BCS model, suggesting s-wave superconductivity in TaOs.
We numerically study the spatially-resolved NMR around a single vortex in a noncentrosymmetric superconductor such as CePt3Si. The nuclear spin-lattice relaxation rate 1/T1 under the influence of the vortex core states is calculated for an s+p-wave Cooper pairing state. The result is compared with that for an s-wave pairing state.
We discuss the Josephson current between two noncentrosymmetric superconductors. The coexistence of superconducting order parameters between spin-singlet $Δ_{\text{S}}$ and helical p-wave spin-triplet $Δ_{\text{T}}$ enriches a variety of low-temperature behavior of Josephson current depending on their relative amplitudes. We will show that characteristic behaviors of the Josephson current for $Δ_{\text{S}} > Δ_{\text{T}}$ are clearly different from those for $Δ_{\text{S}} < Δ_{\text{T}}$. The topologically protected zero-energy surface bound states are responsible for the clear difference. We …
We show how the two-band nature of superconductivity in noncentrosymmetric compounds leads to a variety of novel nonuniform superconducting states induced by a magnetic field. At low fields, a two-band helical state is realized, with a distinctly non-BCS quasiparticle spectrum. At high fields, the superconducting state becomes unstable towards the formation of a lattice of topological phase solitons.
Electronic structure of a novel superconducting noncentrosymmetric compound Mg10Ir19B16 was calculated using the Korringa-Kohn-Rostoker method. Electronic part of the electron-phonon coupling constant, McMillan-Hopfield parameters, were calculated using the rigid-muffin-tin approximation (RMTA). The magnitude of the electron-phonon coupling constant, analysing atomic contributions, is discussed. Our results show, that superconductivity in Mg10Ir19B16 is presumably mediated by electron-phonon interaction.
For a noncentrosymmetric superconductor such as CePt3Si, we consider a Cooper pairing model with a two-component order parameter composed of spin-singlet and spin-triplet pairing components. We demonstrate that such a model on a qualitative level accounts for experimentally observed features of the temperature dependence of the nuclear spin-lattice relaxation rate 1/T1, namely a peak just below Tc and a line-node gap behavior at low temperatures.
We examine intrinsic interfaces separating crystalline twin domains of opposite spin-orbit coupling in a noncentrosymmetric superconductor such as CePt3Si. At these interfaces, low-energy Andreev bound states occur as a consequence of parity-mixed Cooper pairing, and a superconducting phase which violates time reversal symmetry can be realized. This provides an environment allowing flux lines with fractional flux quanta to be formed at the interface. Their presence could have strong implications on the flux creep behavior in such superconductors.
We consider bound states at the vortex core of a non-centrosymmetric superconductor. We show that, despite the mixing of singlet and triplet order parameters, zero energy states survive within certain parameter space as in vortices of some chiral p-wave states.
We show that the two-band nature of noncentrosymmetric superconductors leads naturally to an anomalous Josephson current appearing at zero phase difference in a clean noncentrosymmetric superconductor/ferromagnet/noncentrosymmetric superconductor junction. The two-band nature provides two sets of Andreev bound states which carry two supercurrents with different amplitudes. When the magnetization direction of the ferromagnet is suitably chosen, two supercurrents experience opposite phase shifts from the conventional sinusoidal current-phase relation. Then the total Josephson current results in …
We investigate how the macroscopic spatial structure of broken inversion symmetry manifests in noncentrosymmetric superconductors, by the microscopic broken inversion symmetry of the crystal structure. Based on the time-dependent Ginzburg-Landau theory including the Pauli paramagnetic effect and the Rashba interaction, we demonstrate that the centrosymmetric structures of the internal field and the screening current are broken macroscopically. The flow structure of paramagnetic supercurrent spontaneously induce the flux flow without applying external currents.
The magnetoelectric effect (or Edelstein effect) in noncentrosymmetric superconductors states that a supercurrent can induce spin magnetization. This is an intriguing phenomenon which has potential applications in superconducting spintronic devices. However, the original Edelstein effect only applies to superconductors with polar point group symmetry. In recent years, many new noncentrosymmetric superconductors have been discovered, such as superconductors with chiral lattice symmetry and superconducting transition metal dichalcogenides with various lattice structures. In this Rapid Communicat…
We numerically study the effect of non-magnetic impurities on the vortex bound states in noncentrosymmetric systems. The local density of states (LDOS) around a vortex is calculated by means of the quasiclassical Green's function method. We find that the zero energy peak of the LDOS splits off with increasing the impurity scattering rate.
The superconducting parameters and upper critical field of the noncentrosymmetric superconductor BiPd have proven contentious. This material is of particular interest because it is a singular example of a $4f$-electron-free noncentrosymmetric superconductor of which crystals may be grown and cleaved, enabling surface-sensitive spectroscopies. Here, using bulk probes augmented by tunnelling data on defects, we establish that the lower of the previously reported upper critical fields corresponds to the bulk transition. The material behaves as a nearly-weak-coupled BCS s-wave superconductor, and …
For a noncentrosymmetric superconductor such as CePt3Si, we consider a Cooper pairing model with a two-component order parameter composed of spin-singlet and spin-triplet pairing components. We calculate the superfluid density tensor in the clean limit on the basis of the quasiclassical theory of superconductivity. We demonstrate that such a pairing model accounts for an experimentally observed feature of the temperature dependence of the London penetration depth in CePt3Si, i.e., line-node-gap behavior at low temperatures.
We numerically study the vortex core structure in a noncentrosymmetric superconductor such as CePt3Si without mirror symmetry about the xy plane. A single vortex along the z axis and a mixed singlet-triplet Cooper pairing model are considered. The spatial profiles of the pair potential, local density of states, supercurrent density, and radially-textured magnetic moment density around the vortex are obtained in the clean limit on the basis of the quasiclassical theory of superconductivity.
Noncentrosymmetric superconductors have sparked significant research interests due to their exciting properties, such as the admixture of spin-singlet and spin-triplet Cooper pairs. Here we report $μ$SR and thermodynamic measurements on the noncentrosymmetric superconductor La7Rh3 which indicate a fully established superconducting gap and spontaneous time-reversal-symmetry breaking at the onset of superconductivity. We show that our results pose severe constraints on any microscopic theory of superconductivity in the system. A symmetry analysis identifies ground states compatible with time-rev…
Gap structure in noncentrosymmetric superconductors with spin-orbit band splitting is studied using a microscopic model of pairing mediated by phonons and/or spin fluctuations. The general form of pairing interaction in the band representation is derived, which includes both the intraband and interband pairing terms. In the case of isotropic interaction (in particular, for a BCS-contact interaction), the interband pairing terms vanish identically at any magnitude of the band splitting. The effects of pairing interaction anisotropy are analyzed in detail for a metal of cubic symmetry with stron…
We report measurements of the temperature dependence of the magnetic penetration depth in different quality polycrystalline samples of noncentrosymmetric LaNiC2 down to 0.05 K. This compound has no magnetic phases and breaks time-reversal symmetry. In our highest quality sample we observe a T^2 dependence below 0.4Tc indicative of nodes in the energy gap. We argue that previous results suggesting conventional s-wave behavior may have been affected by magnetic impurities.
We study theoretically the surface of time-reversal-symmetric, noncentrosymmetric superconductor with mixed singlet and triplet order parameters. A pair of counterpropagating subgap quasiparticle surface bound states with opposite spin projections are obtained in the nontrivial Z$_2$ case where the triplet component is larger than the singlet one, contributing to a spin current. In contrast to the pure p-wave cases, these subgap states do not have a fixed spin projections but depend on the momenta along the surface. In the trivial Z$_2$ case where the singlet order parameter is larger, no subg…
We study the stability of topologically protected zero-energy flat bands at the surface of nodal noncentrosymmetric superconductors, accounting for the alteration of the gap near the surface. Within a selfconsistent mean-field theory, we show that the flat bands survive in a broad temperature range below the bulk transition temperature. There is a second transition at a lower temperature, however, below which the system spontaneously breaks time-reversal symmetry. The surface bands are shifted away from zero energy and become weakly dispersive. Simultaneously, a spin polarization and an equili…
Twin boundaries are generic crystalline defects in noncentrosymmetric crystal structures. We study theoretically twin boundaries in time-reversal symmetric noncentrosymmetric superconductors that admit parity-mixed Cooper pairing. Twin boundaries support spin currents as a consequence of this parity mixing. If the singlet and triplet components of the superconducting order parameter are of comparable magnitude, the superconducting state breaks spontaneously the bulk time-reversal symmetry locally near the twin boundary. By self-consistently evaluating the Bogoliubov-de-Gennes equations and the…
Superconductivity in materials whose crystal structure lacks inversion symmetry is a prime candidate for unconventional superconductivity. A new noncentrosymmetric compound Zr$_{3}$Ir crystallizes in a tetragonal $ α$-V$_{3} $S structure. The magnetization, specific heat and muon spin rotation confirm s-wave superconductivity, having a transition temperature T$_{c}$ = 2.3 K. Muon spin relaxation confirms the preservation of time reversal symmetry in the superconducting ground state.
For a noncentrosymmetric superconductor such as Y$_2$C$_3$, we consider a parity-mixing model composed of spin-singlet s-wave and spin-triplet f-wave pairing components. The $d$-vector in f-wave state is chosen to be parallel to the Dresselhaus asymmetric spin-orbit coupling vector. It is found that, the quasiparticle excitation spectrum exhibits distinct nodal structure as a consequence of parity-mixing. Our calculation predict anomalous noninteger power laws for low-temperature nuclear spin-lattice relaxation rate $T_{1}^{-1}$. We demonstrate particularly that such a model can qualitatively …
We propose that in noncentrosymmetric superconductors with weakly asymmetric spin-orbit interaction the field-induced pair correlation between the spin-orbit split different bands ignored in previous studies yields unique effects; i.e. the Pauli depairing effect is anisotropic in the momentum space, and as a result, magnetic fields induce point-node-like anisotropic gap structure of the quasiparticle energy even for isotropic s-wave states, which seriously affects thermodynamic quantities at low temperatures. Also, it is shown that when the magnitude of the spin-orbit interaction is smaller th…
We present a detailed study of the gap symmetry and the quasiparticle wave function topology in two-dimensional superconductors without inversion center. The strong spin-orbit coupling of electrons with the crystal lattice makes it necessary to describe superconductivity in terms of one or more nondegenerate bands characterized by helicity. We develop a topological classification of the superconducting states using the integer-valued Maurer-Cartan invariants and the Bogoliubov Wilson loops, and also calculate the spectrum of fermionic boundary modes.
We investigate the Josephson transport properties in a Josephson junction consisting of a conventional $s$-wave superconductor coupled to a multi-orbital noncentrosymmetric superconductor marked by an orbitally driven inversion asymmetry and isotropic interorbital spin-triplet pairing. Contrary to the canonical single band noncentrosymmetric superconductor, we demonstrate that the local interorbital spin-triplet pairing is tied to the occurrence of sign-changing spin-singlet pair amplitude on different bands with $d$-wave symmetry. Such multi-band $d^{\pm}$-wave state is a unique superconducti…
Noncentrosymmetric superconductors can lead to a variety of exotic properties in the superconducting state such as line nodes, multigap behavior, and time-reversal symmetry breaking. In this paper, we report the properties of the new noncentrosymmetric superconductor TaOs, using muon spin relaxation and rotation measurements. It is shown using the zero-field muon experiment that TaOs preserve the time-reversal symmetry in the superconducting state. From the transverse field muon measurements, we extract the temperature dependence of $λ(T)$ which is proportional to the superfluid density. This …
The unique nonreciprocal responses of superconductors, which stem from the Cooper pairs' quantum condensation, have been attracting attention. Recently, theories of the second-order nonlinear response in noncentrosymmetric superconductors were formulated based on the Bogoliubov-de Gennes theory. In this paper, we study the mechanism and condition for second-order optical responses of time-reversal symmetric superconductors. The numerical results show the characteristic photocurrent and second harmonic generation in the superconducting state. However, the superconductivity-induced nonlinear opt…
The properties of the noncentrosymmetric superconductor ($α$-$\textit{Mn}$ structure) Nb$_{0.5}$Os$_{0.5}$ is investigated using resistivity, magnetization, specific heat, and muon spin relaxation and rotation ($μ$SR) measurements. These measurements suggest that Nb$_{0.5}$Os$_{0.5}$ is a weakly coupled ($λ_{e-ph}$ $\sim$ 0.53) type-II superconductor ($κ_{GL}$ $\approx$ 61) having a bulk superconducting transition temperature $T_c$ = 3.07 K. The specific heat data in the superconductive regime fits well with the single-gap BCS model indicating nodeless s-wave superconductivity in Nb$_{0.5}$Os$…
The spin susceptibility of noncentrosymmetric superconductors is studied when the gap function has line nodes. As examples, d-wave states, where the gap function has an additional odd-parity phase factor, are examined. The curve of the spin susceptibility $χ(T)$ is upward convex when all line nodes are parallel to the magnetic field, while it is downward convex in the other d-wave states and an s-wave state. For polycrystalline powder samples, the temperature dependences of $χ(T)$ are predicted by assuming three explicit conditions of the powder particles. The results are compared with the exp…
We analytically study electronic bound states around a single vortex core in a noncentrosymmetric superconductor such as ${\rm CePt_3Si}$ without mirror symmetry about the $ab$ plane. Considering a mixed spin-singlet-triplet Cooper pairing model, we obtain a formula about the local density of states (LDOS) around a vortex core in any directions of magnetic fields. The LDOS under a magnetic field perpendicular to the c axis are quite different from those of typical s-wave or d-wave superconductors. From the ellipticity of the spatial pattern of the LDOS around a vortex core, one can estimate ex…
Noncentrosymmetric superconductors with broken inversion symmetry $P$ offer rich physical phenomena such as the upper critical magnetic field $H_{c2}$ beyond the Pauli limit and magnetoelectric effect. The relativistic spin-orbit interaction (SOI) plays an essential role in these novel phenomena, which lifts the Kramers degeneracy at each $\mathbf{k}$-point, and leads to the mixing of the spin singlet-even parity and spin triplet odd-parity parings. On the other hand, the time-reversal symmetry $T$ relates the two states at $\mathbf{k}$ and $-\mathbf{k}$ points with the opposite spins, and the…
We report the measurements of de Haas-van Alphen (dHvA) oscillations in the noncentrosymmetric superconductor BiPd. Several pieces of a complex multi-sheet Fermi surface are identified, including a small pocket (frequency 40 T) which is three dimensional and anisotropic. From the temperature dependence of the amplitude of the oscillations, the cyclotron effective mass is ($0.18$ $\pm$ 0.1) $m_e$. Further analysis showed a non-trivial $π$-Berry phase is associated with the 40 T pocket, which strongly supports the presence of topological states in bulk BiPd and may result in topological supercon…
The discovery of new families of unconventional superconductors is important both experimentally and theoretically, especially if it challenges current models and thinking. By using muon spin relaxation in zero-field, time-reversal symmetry breaking has been observed in Re6Hf. Moreover, the temperature dependence of the superfluid density exhibits s-wave superconductivity with an enhanced electron-phonon coupling. This, coupled with the results from isostructural Re6Zr, shows that the Re6X family are indeed a new and important group of unconventional superconductors.
A prime category of superconducting materials in which to look for spin-triplet pairing and topological superconductivity are superconductors without inversion symmetry. It is predicted that the breaking of parity symmetry gives rise to an admixture of spin-singlet / spin-triplet pairing states; a triplet pairing component, being substantial, seems all but guaranteed. However, the experimental confirmation of pair mixing in any particular material remains elusive. In this work, we perform phase-sensitive experiment to examine the pairing state of noncentrosymmetric superconductor $α-$BiPd. The…
Effects of disorder on superconducting properties of noncentrosymmetric compounds are discussed. Elastic impurity scattering, even for scalar impurities, leads to a strongly anisotropic mixing of the electron states in the bands split by the spin-orbit coupling. We focus on the calculation of the critical temperature, the upper critical field, and the spin susceptibility. It is shown that the impurity effects on the critical temperature are similar to those in multi-band centrosymmetric superconductors, in particular, Anderson's theorem holds for isotropic singlet pairing. In contrast, scalar …
We present synthesis and $^{75}$As-nuclear quadrupole resonance (NQR) measurements for the noncentrosymmetric superconductor CaPtAs with a superconducting transition temperature $T_c$ of $\sim 1.5$ K. We discovered two different forms of CaPtAs during synthesis; one is a high-temperature tetragonal form that was previously reported, and the other is a low-temperature form consistent with the orthorhombic structure of CaPtP. According to the $^{75}$As-NQR measurement for superconducting tetragonal CaPtAs, the nuclear spin-lattice relaxation rate $1/T_1$ has an obvious coherence peak below $T_c$…
We study the fermionic boundary modes (Andreev bound states) in a time-reversal invariant one-dimensional superconductor. In the presence of a substrate, spatial inversion symmetry is broken and the electronic properties are strongly affected by an antisymmetric spin-orbit coupling. We assume an arbitrary even number of nondegenerate bands crossing the Fermi level. We show that there is only one possible pairing symmetry in one dimension, an analog of s-wave pairing. The zero-energy Andreev bound states are present if the sign of the gap function in an odd number of bands is different from all…
We have investigated the superconducting state of the noncentrosymmetric superconductor Re6Ti (Tc = 6.0 K) using muon-spin rotation/relaxation (muSR) technique. The zero-field muon experiment shows the presence of spontaneous magnetic fields in the superconducting state, indicating time-reversal symmetry breaking (TRSB). However, the low-temperature transverse field muon measurements suggest nodeless s-wave superconductivity. The time reversal symmetry breaking further confirmed in the stoichiometric composition Re24Ti5. These results indicate that the pairing symmetry is not affected by spin-…
We report hydrostatic pressure study on the Rashba-type noncentrosymmetric superconduc- tors CaMSi3 (M= Ir, Pt). The temperature dependence of the resistivity for both compounds is well described by the conventional Bloch-Grüneisen formalism at each pressure. This fact suggests that electron-phonon scattering is dominant in these compounds. The superconducting critical temperature Tc decreases by pressure as \sim 0.2 K/GPa above 0.41 GPa to 2 GPa for both compounds. This behavior of Tc can be explained with a modest decrease of the density of states based on the conventional BCS theory.
Based on the extended Blonder-Tinkham-Klapwijk formalism, the tunneling conductance characteristics of a planar junction between a ferromagnet and a noncentrosymmetric superconductor are studied. The effects of the Rashba spin-orbit coupling (RSOC), the exchange energy, and the Fermi wave-vector mismatch (FWM) on the conductance are all taken into account. In the absence of the FWM, it is found that far away from the gap edge the conductance is suppressed by the RSOC, while around the gap edge it is almost independent of RSOC. The interplay of the RSOC and the exchange energy causes an enhance…
The nature and symmetry of the superconducting gap function in the noncentrosymmetric superconductor (NCS) Li2Pt3B, even many years after its discovery, appears to be full of contradictions. In this letter based on the existing band structure calculations we find that owing to the considerable nesting near the Fermi surface and the enhanced d-character of the relevant bands that cross the Fermi level,the system gets somewhat strongly correlated. Considering the effect of the onsite Coulomb repulsion on the pairing potential perturbatively, we extract possible superconducting transition. The st…
We present muon spin rotation ($μ$SR) measurements on the noncentrosymmetric superconductor PbTaSe$_2$. From measurements in an applied transverse field between $H_{c1}$ and $H_{c2}$, we extract the superfluid density as a function of temperature in the vortex state. This data can be fit with a fully gapped two-band model, consistent with previous evidence from ARPES, thermal conductivity, and resistivity. Furthermore, zero field measurements show no evidence for a time reversal symmetry breaking field greater than 0.05~G in the superconducting state. This makes exotic fully gapped spin-triple…
One of the key effects which is predicted to arise in superconductors without a centre of inversion is the mixing of singlet and triplet order parameters, which are no longer good quantum numbers on their own due to parity. We have probed the gap structure in the noncentrosymmetric superconductor Ru$_7$B$_3$, through small-angle neutron diffraction from the vortex lattice, in order to search for the proposed mixed order parameter. We find that the measured temperature dependence of the vortex-lattice form factor is well characterised by a model constructed to describe the effects of broken inv…
Precise measurements of the thermodynamic critical field ($B_{\rm c}$) in type-I noncentrosymmetric superconductor BeAu were performed by means of the muon-spin rotation/relaxation technique. The temperature evolution of $B_{\rm c}$ can not be described within the single gap scenario and it requires the presence of at least two different types of the superconducting order parameters. The self-consistent two-gap approach, adapted for analysis of $B_{\rm c}(T)$ behavior, suggests the presence of two superconducing energy gaps with the gap to $T_{\rm c}$ ratios $2Δ/k_{\rm B}T_{\rm c}\simeq4.52$ a…
Specific heat, electrical resistivity, and magnetic susceptibility measurements on a high quality sample of Mg10Ir19B16 provide a self-consistent determination of its superconducting properties. They indicate that Mg10Ir19B16 is a type-II superconductor (Tc=4.45K, k(0)=20), with an electron-phonon coupling constant l_ep=0.66. An analysis of the T-dependent specific heat shows that superconducting properties are dominated by an s-wave gap (D=0.7 meV). Point contact tunneling data provide evidence for multiple superconducting gaps, as expected from strong asymmetric spin-orbit coupling.
