Inversion and time reversal are essential symmetries for the structure of Cooper pairs in superconductors. The loss of one or both leads to modifications to this structure and can change the properties of the superconducting phases in profound ways. Lacking inversion, superconductivity in noncentrosymmetric materials has become an important topic, in particular, in the context of topological superconductivity as well as unusual magnetic and magneto-electric properties. Recently, crystal structures with local, but not global inversion-symmetry breaking have attracted attention, as superconductivity can exhibit phenomena not naively expected in centrosymmetric materials. After introducing the concept of locally noncentrosymmetric crystals and different material realizations, we discuss consequences of such local symmetry breaking on the classification, the expected and, in parts, already observed phenomenology of unconventional superconductivity, and possible topological superconducting phases.
Unlike other pnictides, SrPtAs has a hexagonal structure, containing layers with As-Pt atoms that form a honeycomb lattice. These layers lack inversion symmetry which allows for a spin-orbit coupling that we show has a dramatic effect on superconductivity in this material. In particular, for conventional s-wave superconductivity in SrPtAs, both the spin susceptibility and the paramagnetic limiting field are enhanced significantly with respect to that usually expected for s-wave superconductors. SrPtAs provides a prime example of a superconductor with locally broken inversion symmetry.
We argue the emergence of nodal excitations due to the coupling with static antiferromagnetic order in fully-gapped superconducting states in both cases with and without inversion symmetry. This line node structure is not accompanied with the sign change of the superconducting gap, in contrast to usual unconventional Cooper pairs with higher angular momenta. In the case without inversion symmetry, the stability of the nodal excitations crucially depends on the direction of the antiferromagnetic staggered magnetic moment. A possible realization of this phenomenon in CePt$_3$Si is discussed.
We study how an inversion-breaking quantum critical point affects the ground state of a one-dimensional electronic liquid with repulsive interaction and spin-orbit coupling. We find that regardless of the interaction strength, the critical fluctuations always lead to a gap in the electronic spin-sector. The origin of the gap is a two-particle backscattering process, which becomes relevant due to renormalization of the Luttinger parameter near the critical point. The resulting spin-gapped state is a one-dimensional version of spin triplet superconductivity. Interestingly, in the case of a ferro…
Vortex phase in a singlet superconductor in the absence of impurities is shown to be absolutely unstable with respect to the appearance of a triplet component which breaks both inversion and time-reversal symmetries of Cooper pairs. Symmetry breaking paramagnetic effects are demonstrated to be of the order of unity if the orbital upper critical field, Hc2(0), is of the order of Clogston paramagnetic limiting field, Hp. We suggest a generic phase diagram of such type-IV superconductor, which is singlet one at H=0 and characterized by mixed singlet-triplet order parameter with broken time-revers…
This Chapter gives a brief introduction to some basic aspects metals and superconductors in crystal without inversion symmetry. In a first part we analyze some normal state properties which arise through antisymmetric spin-orbit coupling existing in non-centrosymmetric materials and show its influence on the de Haas-van Alphen effect. For the superconducting phase we introduce a multi-band formulation which naturally arises due the spin splitting of the bands by spin-orbit coupling. It will then be shown how the states can be symmetry classified and their relation to the original classificatio…
We investigated the chemical pressure effects on structural and electronic properties of SnTe-based material using partial substitution of Sn by Ag0.5Bi0.5, which results in lattice shrinkage. For Sn1-2x(AgBi)xTe, single-phase polycrystalline samples were obtained with a wide range of x. On the basis of band calculations, we confirmed that the Sn1-2x(AgBi)xTe system is basically possessing band inversion and topologically preserved electronic states. To explore new superconducting phases related to the topological electronic states, we investigated the In-doping effects on structural and super…
The symmetry properties and the general overview of the superconductivity theory in the itinerant ferromagnets and in materials without space parity are presented. The basic notions of unconventional superconductivity are introduced in broad context of multiband superconductivity which is inherent property of ferromagnetic metals or metals without centre of inversion.
The traditional symmetries that protect superconductivity are time-reversal and inversion. Here, we examine the minimal symmetries protecting superconductivity in two dimensions and find that time-reversal symmetry and inversion symmetry are not required, and having a combination of either symmetry with a mirror operation on the basal plane is sufficient. We classify superconducting states stabilized by these two symmetries, when time-reversal and inversion symmetries are not present, and provide realistic minimal models as examples. Interestingly, several experimentally realized systems, such…
This work presents the emergence of superconductivity in Ir - doped Weyl semimetal T$_d$ - MoTe$_{2}$ with broken inversion symmetry. Chiral anomaly induced planar Hall effect and anisotropic magneto-resistance confirm the topological semimetallic nature of Mo$_{1-x}$Ir$_{x}$Te$_{2}$. Observation of weak anisotropic, moderately coupled type-II superconductivity in T$_d$ -Mo$_{1-x}$Ir$_{x}$Te$_{2}$ makes it a promising candidate for topological superconductor.
We study superconductivity in spin-orbit-coupled systems in the vicinity of inversion symmetry breaking. We find that due to the presence of spin-orbit coupling, fluctuations of the incipient parity-breaking order generate an attractive pairing interaction in an odd-parity pairing channel, which competes with the s-wave pairing. We show that applying a Zeeman field suppresses the s-wave pairing and promotes the odd-parity superconducting state. Our work provides a new mechanism for odd-parity pairing and opens a route to novel topological superconductivity.
While multi-layer systems can possess global inversion centers, they can have regions with locally broken inversion symmetry. This can modify the superconducting properties of such a system. Here we analyze two dimensional multi-layer systems yielding spatially modulated antisymmetric spin-orbit coupling (ASOC) and discuss superconductivity with mixed parity order parameters. In particular, the influence of ASOC on the spin susceptibility is investigated at zero temperature. For weak inter-layer coupling we find an enhanced spin susceptibility induced by ASOC, which hints the potential importa…
We analyze a mean-field model of electrons on a square lattice with two types of interaction: forward scattering favoring a d-wave Pomeranchuk instability and a BCS pairing interaction driving d-wave superconductivity. Tuning the interaction parameters a rich variety of phase diagrams is obtained. If the BCS interaction is not too strong, Fermi surface symmetry breaking is stabilized around van Hove filling, and coexists with superconductivity at low temperatures. For pure forward scattering Fermi surface symmetry breaking occurs typically via a first order transition at low temperatures. The …
Superconductivity with $T_{\rm c}$ $\simeq$ 2 K was discovered in the intermetallic binary compounds Rh$_2$Ga$_9$ and Ir$_2$Ga$_9$. This is the first observation of superconductivity in the Rh-Ga and Ir-Ga binary systems. Both compounds crystallize in a distorted Co$_2$Al$_9$-type structure (monoclinic, space group: $Pc$), which lacks spatial inversion symmetry. Specific heat measurements revealed that both compounds are weak-coupling BCS superconductors having an isotropic superconducting gap. Measurements in magnetic fields indicated type-I superconductivity with a critical field $H_{\rm c}(…
It is pointed out that erroneous Bardeen-Cooper-Schrieffer model equations have been used by Haranath Ghosh in his recent treatment of time-reversal symmetry-breaking superconductivity. Consequently, his numerical results are misleading, and his conclusions are not to the point.
We report muon spin relaxation measurements on the superconductor Sr2RuO4 that reveal the spontaneous appearance of an internal magnetic field below the transition temperature: the appearance of such a field indicates that the superconducting state in this material is characterized by the breaking of time-reversal symmetry. These results, combined with other symmetry considerations, suggest that superconductivity in Sr2RuO4 is of p-wave (odd-parity) type, analogous to superfluid 3He.
The theory of paramagnetic limit of superconductivity in metals without inversion center is developed. There is in general the paramagnetic suppression of superconducting state. The effect is strongly dependent on field orientation in respect to crystal axes. The reason for this is that the degeneracy of electronic states with opposite momenta forming of Cooper pairs is lifted by magnetic field but for some field directions this lifting can be small or even absent.
We report superconductivity in the ternary half-Heusler compound LuPtBi, with Tc = 1.0 K and Hc2 = 1.6 T. The crystal structure of LuPtBi lacks inversion symmetry, hence the material is a noncentrosymmetric superconductor. Magnetotransport data show semimetallic behavior in the normal state, which is evidence for the importance of spin-orbit interaction. Theoretical calculations indicate that the strong spin-orbit interaction in LuPtBi should cause strong band inversion, making this material a promising candidate for 3D topological superconductivity.
The Weyl semimetal MoTe$_2$ offers a rare opportunity to study the interplay between Weyl physics and superconductivity. Recent studies have found that Se substitution can boost the superconductivity up to 1.5K, but suppress the Td structure phase that is essential for the emergence of Weyl state. A microscopic understanding of possible coexistence of enhanced superconductivity and the Td phase has not been established so far. Here, we use scanning tunneling microscopy (STM) to study a optimally doped new superconductor MoTe$_{1.85}$Se$_{0.15}$ with bulk Tc ~ 1.5K. By means of quasiparticle in…
We report on muon spin relaxation measurements of the 4f^2-based heavy-fermion superconductor filled-skutterudite PrOs4Sb12. The results reveal the spontaneous appearance of static internal magnetic fields below the superconducting transition temperature, providing unambiguous evidence for the breaking of time-reversal symmetry in the superconducting state. A discussion is made on which of the spin or orbital component of Cooper pairs carries a nonzero momentum.
We study time reversal symmetry breaking superconductivity with $Δ_k = Δ_{x^2-y^2} (k) +e^{iθ} Δ_α$ ($α= s$ or $d_{xy}$) symmetries. It is shown that the behavior of such superconductors could be {\em qualitatively} different depending on the minor components ($α$) and its phase at lower temperatures. It is argued that such {\em qualitatively different} behaviors in thermal as well as in angular dependencies could be a {\em source} of consequences in transport and Josephson physics. Orthorhombicity is found to be a strong mechanism for mixed phase (in case of $α= s$). We show that due to elect…
We study nonlinear optical responses in superconducting systems with inversion ($\mathcal{I}$) symmetry-breaking order parameters. We first show that any superconducting system with $\mathcal{I}$ and time-reversal ($\mathcal{T}$) symmetries requires an $\mathcal{I}$-breaking order parameter to support optical transitions between particle-hole pair bands. We then use a 1D toy model of an $\mathcal{I}$-breaking superconductor to numerically calculate linear and nonlinear conductivities, including shift current and second harmonic generations (SHG) responses. We find that the magnitude of the sig…
We study electric, thermal, and thermoelectric conductivities in the vicinity of a z=2 superconductor-diffusive metal transition in two dimensions, both in the high and low frequency limits. We find violation of the Wiedemann-Franz law and a dc thermoelectric conductivity $α$ that does not vanish at low temperatures, in contrast to Fermi liquids. We introduce a Langevin equation formalism to study critical dynamics over a broad region surrounding the quantum critical point.
We study the superconducting state of multi-orbital spin-orbit coupled systems in the presence of an orbitally driven inversion asymmetry assuming that the inter-orbital attraction is the dominant pairing channel. Although the inversion symmetry is absent, we show that superconducting states that avoid mixing of spin-triplet and spin-singlet configurations are allowed, and remarkably, spin-triplet states that are topologically nontrivial can be stabilized in a large portion of the phase diagram. The orbital-dependent spin-triplet pairing generally leads to topological superconductivity with po…
We investigate the macroscopic effects of charge density waves (CDW) and superconductivity in layered superconducting systems with broken lattice inversion symmetry (allowing for piezoelectricity) such as two dimensional (2D) transition metal dichalcogenides (TMD). We work with the low temperature time dependent Ginzburg-Landau theory and study the coupling of lattice distortions and low energy CDW collective modes to the superconducting order parameter in the presence of electromagnetic fields. We show that superconductivity and piezoelectricity can coexist in these singular metals. Furthermo…
Since the discovery of superconductivity in the alkali-doped fullerenes \cite{hebard}, the electron-phonon interaction has been the primary suspect for superconductivity in this class of compounds. In this paper we first provide a pedagogical review of how the question of mechanism has traditionally been settled, and then some well-known properties of the optical reflectance (and the derived conductance) are summarized. Finally we demonstrate how a recently derived inversion procedure can use the optical properties of a metal to infer the magnitude of the electron-phonon interaction. We conclu…
Time-reversal symmetry breaking (TRSB) with a Bardeen-Cooper-Schrieffer (BCS) -like superconductivity occurs in a small, but growing number of noncentrosymmetric (NCS) materials, although the mechanism is poorly understood. We present heat capacity, magnetization, resistivity, and muon spin resonance/relaxation ($μ$SR) measurements on polycrystalline samples of NCS La$_{7}$Pd$_{3}$. Transverse-field $μ$SR and heat capacity data show La$_{7}$Pd$_{3}$ is a type-II superconductor with a BCS-like gap structure, while zero-field $μ$SR results provide evidence of TRSB. We discuss the implications of…
We argue that superconductivity in the coexistence region with spin-density-wave (SDW) order in weakly doped Fe-pnictides differs qualitatively from the ordinary $s^{+-}$ state outside the coexistence region, as it develops an additional gap component which is a mixture of intra-pocket singlet ($s^{++}$) and inter-pocket spin-triplet pairings (the $t-$state). The coupling constant for the $t-$channel is proportional to the SDW order and involves interactions that do not contribute to superconductivity outside of the SDW region. We argue that the $s^{+-}$ and $t-$type superconducting orders coe…
Monolayer WTe$_2$, a centrosymmetric transition metal dichacogenide, has recently been established as a quantum spin Hall insulator and found superconducting upon gating. Here we study the pairing symmetry and topological nature of superconducting WTe$_2$ with a microscopic model at mean-field level. Surprisingly, we find that the spin-triplet phases in our phase diagram all host Majorana modes localized on two opposite corners. Even when the conventional pairing is favored, we find that an intermediate in-plane magnetic field exceeding the Pauli limit stabilizes an unconventional equal-spin p…
Although multilayer systems possess global inversion symmetry, some of the layers lack local inversion symmetry because no global inversion centers are present on such layers. Such locally non-centrosymmetric systems exhibit spatially modulated Rashba spin-orbit coupling. In this study, the superconductivity in multilayer models exhibiting inhomogeneous Rashba spin-orbit coupling is investigated. We study the electronic structure, superconducting gap, and spin susceptibility in the superconducting state with mixed parity order parameters. We show the enhancement of the spin susceptibility by R…
Recently, in certain flat band lattice systems at commensurate fillings, fractional quantum Hall states have been found -- which have anyonic excitations. We study such systems away from commensuration, i.e. the ground state of an anyon gas in such a system. The presence of the underlying lattice allows access to an entirely new regime where the anyon kinetic energy can be larger than their interaction energy. Within the flux-attachment approach, using mean-field then adding fluctuations, we find several possible superfluid states. Two have intrinsic topological order, i.e. fractionalized quas…
We study theoretically the superconductivity in doped Weyl semimetals with an inversion symmetry based on the Bogoliubov-de Gennes equations. In principle, the two superconducting states, i.e., the zero momentum BCS-like pairing and the finite momentum Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) pairing are competing in this kind of systems. Our self-consistent calculation indicates that the BCS-type state may be the ground state. The competition between these two pairing states is studied in detail through normal state Fermi surface and the finite energy spectral functions. Generally, the Fermi s…
Mineral calaverite AuTe2 is a layered compound with an incommensurately modulated structure. The modulation is characterized by the formation of molecular-like Te2 dimers. We have found that the breaking of Te2 dimers that occurs in Au1-xPtxTe2 results in the emergence of superconductivity at Tc = 4.0 K.
We propose a route to achieve odd-parity spin-triplet superconductivity in metallic collinear antiferromagnets with inversion symmetry. Owing to the existence of hidden antiunitary symmetry, which we call the effective time-reversal symmetry (eTRS), the Fermi surfaces of ordinary antiferromagnetic metals are generally spin-degenerate, and spin-singlet pairing is favored. However, by introducing a local inversion symmetry breaking perturbation that also breaks the eTRS, we can lift the degeneracy to obtain spin-polarized Fermi surfaces. In the weak-coupling limit, the spin-polarized Fermi surfa…
FeSe is a unique member of the family of iron-based superconductors, not only because of the high values of $T_{c}$ in FeSe monolayer, but also because in bulk FeSe superconductivity emerges inside a nematic phase without competing with long-range magnetic order. Near $T_{c}$, superconducting order necessarily has $s+d$ symmetry, because nematic order couples linearly the $s$-wave and $d$-wave harmonics of the superconducting order parameter. Here we argue that the near-degeneracy between $s$-wave and $d$-wave pairing instabilities in FeSe, combined with the sign-change of the nematic order pa…
This article will give an overview on both theoretical and experimental developments concerning states with lattice symmetry breaking in the cuprate high-temperature superconductors. Recent experiments have provided evidence for states with broken rotation as well as translation symmetry, and will be discussed in terms of nematic and stripe physics. Of particular importance here are results obtained using the techniques of neutron and x-ray scattering and scanning tunneling spectroscopy. Ideas on the origin of lattice-symmetry-broken states will be reviewed, and effective models accounting for…
Roles of paramagnetic and diamagnetic pair-breaking effects in superconductivity in electric-field-induced surface metallic state are studied by Bogoliubov-de Gennes equation, when magnetic fields are applied parallel to the surface. The multi-gap states of sub-bands are related to the depth dependence and the magnetic field dependence of superconductivity. In the Fermi-energy density of states and the spin density, sub-band contributions successively appear from higher-level sub-bands with increasing magnetic fields. The characteristic magnetic field dependence may be a key feature to identif…
UTe$_2$ has recently been found to be a likely spin-triplet superconductor that exhibits evidence for chiral Majorana edge states. A characteristic structural feature of UTe$_2$ is inversion-symmetry related pairs of U atoms, forming rungs of ladders. Here we show how each rung's two sublattice degrees of freedom play a key role in understanding the electronic structure and the origin of superconductivity. In particular, we show that DFT+$U$ calculations generically reveal a topological band that originates from a band inversion associated with $5f$ electrons residing on these rung sublattice …
IrTe2, a layered compound with a triangular iridium lattice, exhibits a structural phase transition at approximately 250 K. This transition is characterized by the formation of Ir-Ir bonds along the b-axis. We found that the breaking of Ir-Ir bonds that occurs in Ir1-xPtxTe2 results in the appearance of a structural critical point in the T = 0 limit at xc = 0.035. Although both IrTe2 and PtTe2 are paramagnetic metals, superconductivity at Tc = 3.1 K is induced by the bond breaking in a narrow range of x > xc in Ir1-xPtxTe2. This result indicates that structural fluctuations can be involved in …
The noncentrosymmetric superconductor Re$_{24}$Ti$_{5}$, a time-reversal symmetry (TRS) breaking candidate with $T_c = 6$\,K, was studied by means of muon-spin rotation/relaxation ($μ$SR) and tunnel-diode oscillator (TDO) techniques. At a macroscopic level, its bulk superconductivity was investigated via electrical resistivity, magnetic susceptibility, and heat capacity measurements. The low-temperature penetration depth, superfluid density and electronic heat capacity all evidence an $s$-wave coupling with an enhanced superconducting gap. The spontaneous magnetic fields revealed by zero-field…
Superconductivity in materials without spatial inversion symmetry is studied. We show that in contrast to common believe, spin-triplet pairing is not entirely excluded in such systems. Moreover, paramagnetic limiting is analyzed for both spin-singlet and triplet pairing. The lack of inversion symmetry reduces the effect of the paramagnetic limiting for spin-singlet pairing. These results are applied to MnSi and CePt_3Si.
Electron energy-loss spectroscopy (EELS) was combined with heat capacity measurements to follow the change of superconductivity with systematic Al doping of MgB$_2$. By using x-ray diffraction and Vegard's law to assess the actual Al content in the samples, changes in behavior were found to be much more in agreement with theoretical predictions than in earlier studies. EELS data show that $σ$-band hole states disappear above 33% Al, approximately the composition at which the $σ$ band Fermi surface is predicted to lose its cylindrical shape in reciprocal space and break apart into ellipsoidal p…
Topological materials which possess topologically protected surface states have attracted much attention in recent years. In this work, we study the critical current of superconductor/inversion symmetry breaking topological material/superconductor junctions. We found surprisingly that, in topological materials with broken inversion symmetry, the magnitude of the critical Josephson currents $|I^{+}_c(B)|$ at fixed magnetic field $B$ is not the same for critical currents $|I^{-}_c(B)|$ flowing in the opposite direction. Moreover, the critical currents violate the $| I_{c}^{\pm}(B)| = |I_{c}^{\pm…
Strong-coupling approach to temperature dependence of competing orders of superconductivity: Possible time-reversal symmetry breaking and nontrivial topology
We report a globally reversible effect of electronic tuning on the magnetic phase diagram in CeCoIn_{5} driven by electron (Pt and Sn) and hole (Cd, Hg) doping. Consequently, we are able to extract the superconducting pair breaking component for hole and electron dopants with pressure and co-doping studies, respectively. We find that these nominally non-magnetic dopants have a remarkably weak pair breaking effect for a d-wave superconductor. The pair breaking is weaker for hole dopants, which induce magnetic moments, than for electron dopants. Furthermore, both Pt and Sn doping have a similar …
$^{75}$As and $^{139}$La NMR and nuclear quadrupole resonance (NQR) studies on Zn-substituted LaFeAsO$_{0.85}$ have been performed to investigate the Zn-impurity effects microscopically. Although superconductivity in LaFeAsO$_{0.85}$ disappears by 3% Zn substitution, we found that NMR/NQR spectra and NMR physical quantities in the normal state are hardly changed, indicating that the crystal structure and electronic states are not modified by Zn substitution. Our results suggest that the suppression of superconductivity by Zn substitution is not due to the change of the normal-state properties,…
By employing a series of experimental techniques, we provide clear evidence that CaPtAs represents a rare example of a noncentrosymmetric superconductor which simultaneously exhibits nodes in the superconducting gap and broken time-reversal symmetry (TRS) in its superconducting state (below $T_c$ $\approx$ 1.5 K). Unlike in fully-gapped superconductors, the magnetic penetration depth $λ(T)$ does not saturate at low temperatures, but instead it shows a $T^2$-dependence, characteristic of gap nodes. Both the superfluid density and the electronic specific heat are best described by a two-gap mode…
We discuss superconductivity in a model on a cubic lattice for a $Γ_3$ non-Kramers system. In previous studies, it is revealed that $d$-wave superconductivity with $E_g$ symmetry occurs in a wide parameter range in a $Γ_3$ system. Such anisotropic superconductivity can break the cubic symmetry of the lattice. In a $Γ_3$ system, the quadrupole degrees of freedom are active and the effect of the cubic symmetry breaking should be important. Here, we investigate the coexisting states of the $d$-wave superconductivity and quadrupole order by a mean-field theory. In particular, we discuss possible c…
We consider mixed symmetry superconducting phases in Dirac materials in the odd parity channel, where pseudoscalar and vector order parameters can coexist due to their similar critical temperatures when attractive interactions are of finite range. We show that the coupling of these order parameters to unordered magnetic dopants favors the condensation of novel time-reversal symmetry breaking (TRSB) phases, characterized by a condensate magnetization, rotation symmetry breaking, and simultaneous ordering of the dopant moments. We find a rich phase diagram of mixed TRSB phases characterized by p…
The occurrence of high-T_c superconductivity in systems including the cuprates and the iron-based superconductors, is known to coincide with the existence of anomalous normal-state properties which have been associated with quantum criticality. We argue here that this observation results from the fact that quantum criticality can allow the occurrence of very-strong-coupling superconductivity by preventing its suppression due to competing symmetry-breaking instabilities. Treating the electrons through a large-U ansatz yields their separation into boson quasiparticles which are directly involved…
We study theoretically a moiré superlattice geometry consisting of mirror-symmetric twisted trilayer graphene surrounded by identical transition metal dichalcogenide layers. We show that this setup allows to switch on/off and control the spin-orbit splitting of the Fermi surfaces via application of a perpendicular displacement field $D_0$, and explore two manifestations of this control: first, we compute the evolution of superconducting pairing with $D_0$; this features a complex admixture of singlet and triplet pairing and, depending on the pairing state in the parent trilayer system, phase t…
