Spin-triplet Pairing States Research Articles

A high-throughput screening and discovery of lanthanum based ternary noncentrosymmetric superconductors with ZrNiAl-structure from ab initio calculations

Noncentrosymmetric superconductors act as promising candidates for exploring unconventional properties like spin-triplet pairing states and intrinsic topological superconductivity. In this work, we provide an illustration of a comprehensive materials discovery process for the development of lanthanum-based superconductors without inversion center similar to ZrNiAl. Our approach begins with a comprehensive screening based on the first principles calculations by substituting different element species and materials database. Throughout a series of strict criteria, we expand the range of materials in this family to 35 compounds with superior stabilities which indicate that they are highly likely to be synthesized. Among these compounds we have identified 15 promising candidates with large energy splitting at the Fermi level. Moreover, the conventional superconducting transition temperatures have also been explored, the results show that 30 compounds are superconducting phases. This work may help with the design and the expansion of the viable phase space of La-based ZrNiAl-like materials for investigating novel quantum phenomena in noncentrosymmetric superconductors.

μSR Study of Unconventional Pairing Symmetry in the Quasi-1D Na2Cr3As3 Superconductor

We report the finding of a novel pairing state in a newly discovered superconductor Na2Cr3As3. This material has a non-centrosymmetric quasi-one-dimensional crystal structure and is superconducting at TC∼ 8.0 K. We find that the magnetic penetration depth data suggests the presence of a nodal line pz-wave pairing state with zero magnetic moment using transverse-field muon-spin rotation (TF-μSR) measurements. The nodal gap observed in Na2Cr3As3 compound is consistent with that observed in isostructural (K,Cs)2Cr3As3 compounds using TF-μSR measurements. The observed pairing state is consistent with a three-band model spin-fluctuation calculation, which reveals the Sz=0 spin-triplet pairing state with the sinkz pairing symmetry. The long-sought search for chiral superconductivity with topological applications could be aided by such a novel triplet Sz=0p-wave pairing state.

Anomalous electromagnetic response in the spin-triplet superconductor UTe2

The recently discovered heavy-fermion superconductor ${\mathrm{UTe}}_{2}$, as a promising candidate for an odd-parity superconductivity, exhibits anomalous magnetic-field reinforced superconductivity with a huge upper critical field of 35 T. However, the thermodynamic properties of this unconventional superconductivity still remain unclear. Herein we present the results of dc magnetization and heat capacity measurements in ${\mathrm{UTe}}_{2}$ to probe the magnetic response at low temperatures. The obtained isothermal magnetization in the superconducting state of ${\mathrm{UTe}}_{2}$ clearly demonstrates the anomalous magnetic flux pinning effect, which is most likely related to the peculiar electromagnetic response of the spin-triplet pairing state in the hard-magnetization axis (orthorhombic $b$ axis). The ac susceptibility measurements support the anomalous vortex states in ${\mathrm{UTe}}_{2}$. We also found evidence for the Lifshitz transition just above ${H}_{\mathrm{c}2}$ and the anomalous flux pinning anomalies for the easy-magnetization axis ($a$ axis).

Superconductivity from repulsive interactions on the kagome lattice

The discovery of superconductivity in layered vanadium-based kagome metals $A$V$_3$Sb$_5$ ($A$: K, Rb, Cs) has added a new family of materials to the growing class of possible unconventional superconductors. However, the nature of the superconducting pairing in these materials remains elusive. We present a microscopic theoretical study of the leading superconducting instabilities on the kagome lattice based on spin- and charge-fluctuation mediated Cooper pairing. The applied methodology includes effects of both on-site and nearest-neighbor repulsive Coulomb interactions. Near the upper van Hove filling -- relevant for the $A$V$_3$Sb$_5$ materials -- we find a rich phase diagram with several pairing symmetries being nearly degenerate. In particular, while a substantial fraction of the phase diagram is occupied by a spin-singlet order parameter transforming as a two-dimensional irreducible representation of the point group, several nodal spin-triplet pairing states remain competitive. We compute the band and interaction parameter-dependence of the hierarchy of the leading superconducting instabilities, and determine the detailed momentum dependence of the resulting preferred gap structures. Crucially, for moderate values of the interaction parameters, the individual pairing states depend strongly on momentum and exhibit multiple nodes on the Fermi surface. We discuss the properties of these superconducting gap structures in light of recent experimental developments of the $A$V$_3$Sb$_5$ materials.

Observation of robust zero-energy state and enhanced superconducting gap in a trilayer heterostructure of MnTe/Bi2Te3/Fe(Te, Se).

The interface between magnetic material and superconductors has long been predicted to host unconventional superconductivity, such as spin-triplet pairing and topological nontrivial pairing state, particularly when spin-orbital coupling (SOC) is incorporated. To identify these unconventional pairing states, fabricating homogenous heterostructures that contain such various properties are preferred but often challenging. Here, we synthesized a trilayer-type van der Waals heterostructure of MnTe/Bi2Te3/Fe(Te, Se), which combined s-wave superconductivity, thickness-dependent magnetism, and strong SOC. Via low-temperature scanning tunneling microscopy, we observed robust zero-energy states with notably nontrivial properties and an enhanced superconducting gap size on single unit cell (UC) MnTe surface. In contrast, no zero-energy state was observed on 2-UC MnTe. First-principle calculations further suggest that the 1-UC MnTe has large interfacial Dzyaloshinskii-Moriya interaction and a frustrated AFM state, which could promote noncolinear spin textures. It thus provides a promising platform for exploring topological nontrivial superconductivity.

Anomalous normal fluid response in a chiral superconductor UTe2

Chiral superconductors have been proposed as one pathway to realize Majorana normal fluid at its boundary. However, the long-sought 2D and 3D chiral superconductors with edge and surface Majorana normal fluid are yet to be conclusively found. Here, we report evidence for a chiral spin-triplet pairing state of UTe2 with surface normal fluid response. The microwave surface impedance of the UTe2 crystal was measured and converted to complex conductivity, which is sensitive to both normal and superfluid responses. The anomalous residual normal fluid conductivity supports the presence of a significant normal fluid response. The superfluid conductivity follows the temperature behavior predicted for an axial spin-triplet state, which is further narrowed down to a chiral spin-triplet state with evidence of broken time-reversal symmetry. Further analysis excludes trivial origins for the observed normal fluid response. Our findings suggest that UTe2 can be a new platform to study exotic topological excitations in higher dimension.

Shear viscosity in the strong interaction regime of a p-wave superfluid Fermi gas

The p-wave superfluid state is a promising spin-triplet and non s-wave pairing state in an ultracold Fermi gas. In this work we study the low-temperature shear viscosity of a one-component p-wave superfluid Fermi gas, by means of Kubo formalism. Our study is done in the strong-coupling limit where Fermi superfluid reduces into a system of composite bosons. Taking into account px-wave Cooper channel in the self-energy, the viscous relaxation rates are determined. The relaxation rates related to these interactions are calculated as a function of temperature. Their temperature dependence is different from the s-wave superfluid Fermi gas, and this is due to the anisotropic pairing interaction in the p-wave superfluid. Our results contribute to understand how this anisotropy affects transport properties of this unconventional superfluid Fermi gas in low temperature limit.

Renormalized quasiparticles, topological monopoles, and superconducting line nodes in heavy-fermion CeTX3 compounds

Non-centrosymmetric superconductors have recently attracted much attention, since the lack of inversion symmetry mixes spin singlet and triplet pairing states, which may allow the realization of topological superconductivity. In this work, we study the electronic properties of the family of inversion-broken CeTX$_3$ heavy-fermion superconductors, finding topological nodal lines as well as Dirac and Weyl points, which are renormalized closer to the Fermi energy by correlations. We find that the Weyl nodal lines have a substantial effect on the Fermi surface spin structure of the normal state, and lead to line nodes in the superconducting phase.

Spin-Triplet Pairing State Evidenced by Half-Quantum Flux in a Noncentrosymmetric Superconductor.

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 broken parity symmetry gives rise to an admixture of spin-triplet and spin-singlet pairing states. However, experimental confirmation of pairing mixing in any material remains elusive. In this work, we perform a phase-sensitive experiment to examine the pairing state of noncentrosymmetric superconductor α-BiPd. The Little-Parks effect observed in mesoscopic polycrystalline α-BiPd rings reveals the presence of half-integer magnetic flux quantization, which provides a decisive evidence for the spin-triplet pairing state. We find both half-quantum fluxes and integer-quantum fluxes of different proportions, consistent with the scenario of an admixture of singlet-triplet pairing.

Time-reversal symmetry breaking in topological superconductor Sr0.1Bi2Se3

The single helical Fermi surface on the surface state of three-dimensional topological insulator ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ is constrained by the time-reversal invariant bulk topology to possess a spin-singlet superconducting pairing symmetry. In fact, the Cu-doped and pressure-tuned superconducting ${\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$ show no evidence of the time-reversal symmetry (TRS) breaking. We report on the detection of the TRS breaking in the topological superconductor ${\mathrm{Sr}}_{0.1}{\mathrm{Bi}}_{2}{\mathrm{Se}}_{3}$, probed by zero-field $\ensuremath{\mu}\mathrm{SR}$ measurements. The TRS breaking provides strong evidence for the existence of a spin-triplet pairing state. The existence of TRS breaking is also verified by longitudinal-field $\ensuremath{\mu}\mathrm{SR}$ measurements, which negates the possibility of magnetic impurities as the source of TRS breaking. The temperature-dependent superfluid density deduced from transverse-field $\ensuremath{\mu}\mathrm{SR}$ measurements yields nodeless superconductivity with low superconducting carrier density and penetration depth $\ensuremath{\lambda}=1622(134)\phantom{\rule{0.16em}{0ex}}\mathrm{nm}$. From the microscopic theory of unconventional pairing, we find that such a fully gapped spin-triplet pairing channel is promoted by the complex interplay between the structural hexagonal warping and higher order Dresselhaus spin-orbit-coupling terms. Based on Ginzburg-Landau analysis, we delineate the mixing of singlet- to triplet-pairing symmetry as the chemical potential is tuned far above from the Dirac cone. Our observation of such spontaneous TRS breaking chiral superconductivity on a helical surface state, protected by the TRS invariant bulk topology, can open avenues for interesting research and applications.

Little-Parks oscillations with half-quantum fluxoid features in Sr2RuO4 microrings

In a micro ring of a superconductor with a spin-triplet equal-spin pairing state, a fluxoid, a combined object of magnetic flux and circulating supercurrent, can penetrate as half-integer multiples of the flux quantum. A candidate material to investigate such half-quantum fluxoids is Sr$_\mathsf{2}$RuO$_\mathsf{4}$. We fabricated Sr$_\mathsf{2}$RuO$_\mathsf{4}$ micro rings using single crystals and measured their resistance behavior under magnetic fields controlled with a three-axes vector magnet. Proper Little-Parks oscillations in the magnetovoltage as a function of an axially applied field, associated with fluxoid quantization are clearly observed, for the first time using bulk single crystalline superconductors. We then performed magnetovoltage measurements with additional in-plane magnetic fields. By carefully analyzing both the voltages $V_+$ ($V_-$) measured at positive (negative) current, we find that, above an in-plane threshold field of about 10 mT, the magnetovoltage maxima convert to minima. We interpret this behavior as the peak splitting expected for the half-quantum fluxoid states.

Thermal transport in topological-insulator-based superconducting hybrid structures with mixed singlet and triplet pairing states

In the framework of the Bogoliubov–de Gennes equation, we investigate the thermal transport properties in topological-insulator-based superconducting hybrid structures with mixed spin-singlet and spin-triplet pairing states, and emphasize the different manifestations of the spin-singlet and spin-triplet pairing states in the thermal transport signatures. It is revealed that the temperature-dependent differential thermal conductance strongly depends on the components of the pairing state, and the negative differential thermal conductance only occurs in the spin-singlet pairing state dominated regime. It is also found that the thermal conductance is profoundly sensitive to the components of the pairing state. In the spin-singlet pairing state controlled regime, the thermal conductance obviously oscillates with the phase difference and junction length. With increasing the proportion of the spin-triplet pairing state, the oscillating characteristic of the thermal conductance fades out distinctly. These results suggest an alternative route for distinguishing the components of pairing states in topological-insulator-based superconducting hybrid structures.

Cooperon condensation and intravalley pairing states in honeycomb Dirac systems

Motivated by recent developments in the experimental study of superconducting graphene and transition metal dichalcogenides, we investigate superconductivity of the Kane-Mele (KM) model with short-range attractive interactions on the two-dimensional honeycomb lattice. We show that intra-valley spin-triplet pairing arises from nearest-neighbor (NN) attractive interaction and the intrinsic spin-orbit coupling. We demonstrate this in two independent approaches: We study superconducting instability driven by condensation of Cooperons, which are in-gap bound states of two conduction electrons, within the $T$-matrix approximation and also study the superconducting ground state within the mean-field theory. We find that Cooperons with antiparallel spins condense at the $K$ and $K'$ points. This leads to the emergence of an intra-valley spin-triplet pairing state belonging to the irreducible representation A$_1$ of the point group $C_{6v}$. The fact that this pairing state has opposite chirality for $K$ and $K'$ identifies this state as a "helical" valley-triplet state, the valley-analog to the $^3$He-B phase in two dimension. Because of the finite center of mass momentum of Cooper pairs, the pair amplitude in NN bonds exhibits spatial modulation on the length scale of lattice constant, such that this pairing state may be viewed as a pair-density wave state. We find that the pair amplitude spontaneously breaks the translational symmetry and exhibits a $p$-Kekul\'e pattern. We also discuss the selection rule for pairing states focusing the characteristic band structure of the KM model and the Berry phase effects to the emergence of the intra-valley pairing state.

Characterizingp-wave superconductivity using the spin structure of Shiba states

Cooper pairs in two-dimensional unconventional superconductors with broken inversion symmetry are in a mixture of an even-parity spin-singlet pairing state with an odd-parity spin-triplet pairing state. We study the magnetic properties of the impurity bound states in such superconductors and find striking signatures in their spin polarization which allow to unambiguously discriminate a non-topological superconducting phase from a topological one. Moreover, we show how these properties, which could be measured using spin-polarized scanning tunneling microscopy (STM), also enable to determine the direction of the spin-triplet pairing vector of the host material and thus to distinguish between different types of unconventional pairing.

Evidence for spin-triplet odd-parity superconductivity close to type-II van Hove singularities

Searching for unconventional Cooper pairing states has been at the heart of superconductivity research since the discovery of BCS superconductors. In particular, spin-triplet odd-parity pairing states were recently revisited due to the possibility of tuning towards topological superconductors. In this context, it is interesting to note a recent proposal that such a spin-triplet pairing instability occurs when the band filling is near van Hove singularities (vHS) associated with momenta away from time-reversal invariant momenta named type-II vHS. However, this result was obtained within a weak coupling renormalization group with Fermi surface patch approximation. To explore superconducting instabilities beyond this weak coupling Fermi surface patch approximation, we perform systematic study on Hubbard model in a two-dimensional square lattice using three different methods: random phase approximation, large-scale dynamical mean field theory simulations with continuous time quantum Monte Carlo (CTQMC) impurity solver, and large-scale dynamical cluster simulations with CTQMC cluster solver. We find, in a wide doping range centered around the type-II van Hove filling, a two-fold degenerate, spin-triplet, odd-parity $p$-wave pairing state emerges due to repulsive interaction, when the Fermi surface is not sufficiently nested. Possible relevance of our findings to the recently discovered superconductors LaO$_{1-x}$F$_{x}$BiS$_{2}$, Ir$_{1-x}$Pt$_{x}$Te$_{2}$ and proposed doped BC$_{3}$ are also discussed.

Anisotropic superconductivity in noncentrosymmetric BiPd

We report measurements of London penetration depth $\lambda(T)$ for the noncentrosymmetric superconductor BiPd by using a tunnel diode oscillator. Pronounced anisotropic behavior is observed in the low-temperature penetration depth; the in-plane penetration depth $\lambda_{ac}(T)$ follows an exponential decrease, but the interplane penetration depth $\lambda_b(T)$ shows power-law-type behavior. The superfluid density $\rho_s(T)$, converted from the penetration depth $\lambda(T)$, is best fitted by an anisotropic two-band BCS model. We argue that such a complex order parameter is attributed to the admixture of spin-singlet and spin-triplet pairing states as a result of antisymmetric spin-orbit coupling in BiPd.

Spin-Triplet Pairing State of Sr2RuO4in thec-Axis Magnetic Field

We investigate the spin-triplet superconducting state of Sr2RuO4 in the magnetic field along the c-axis on the basis of the four-component Ginzburg-Landau (GL) model with a weak spin-orbit coupling. We consider superconducting states described by the d-vector parallel to the ab-plane, and find that three spin-triplet pairing states are stabilized in the magnetic field-temperature phase diagram. Although a helical state is stable at low magnetic fields, a chiral II state is stabilized at high magnetic fields. A non-unitary spin-triplet pairing state appears near the transition temperature owing to the coupling of magnetic field and chirality. We elucidate synergistic and/or competing roles of the magnetic field, chirality, and spin-orbit coupling. It is shown that a fractional vortex lattice is stabilized in the chiral II phase owing to the spin-orbit coupling.

Odd-frequency spin-triplet pairing states in half-metal/d-wave superconductor junctions

Based on the extended Blonder–Tinkham–Klapwijk formalism, differential conductance spectrum of d-wave superconductor/half-metal junctions is investigated as a function of the interfacial spin–orbit coupling (SOC). It shows that the SOC modifies the interfacial spin-flip scattering, enabling spin-singlet and spin-triplet pairing conversion with the aid of equal-spin-band Andreev reflection. A zero-bias conductance peak has been found, which is closely related to the zero energy Andreev bound state of equal-spin pairing correlation induced by the broken electron-hole symmetry of model Hamiltonian and gives strong evidences for the odd-frequency spin-triplet pairing states.

Twofold spin-triplet pairing states and tunneling conductance in ferromagnet/ferromagnet/iron pnictide superconductor heterojunctions

By applying an extended eight-component Bogoliubov–de Gennes equation, we study theoretically the tunneling conductance in clean ferromagnet/ferromagnet/iron pnictide superconductor (FM/FM/iron-based SC) heterojunctions. Under the condition of noncollinear magnetizations, twofold novel Andreev reflections exist due to the existence of two bands in the SC, in which the incident electron and the two Andreev-reflected holes, belonging to the same spin subband, form twofold spin-triplet pairing states near the FM/iron-based SC interface. It is shown that the conversions of the conductance not only between the zero-bias peak and valley at zero energy but also between the peaks and dips at two gap energies are strongly dependent on both the interband coupling strength in the SC and the spin polarization in the FM. The qualitative differences from tunneling into a conventional s-wave SC are also presented, which may help with experimentally probing and identifying the antiphase s-wave pairing symmetry in the iron-based SC.

Dynamical spin structure factors of quantum spin nematic states

Dynamical spin structure factors of quantum spin nematic states are calculated in a spin-$\frac{1}{2}$ square-lattice ${J}_{1}$-${J}_{2}$ model with ferromagnetic ${J}_{1}$ and competing antiferromagnetic ${J}_{2}$ interactions. To this end, we use a fermion representation, generalizing it to $N$ flavors. We begin with a spin-triplet pairing state of fermion fields, called ${Z}_{2}$ planar state, which is a stable saddle-point solution in the large-$N$ limit in a finite parameter range where the couplings ${J}_{1}$ and ${J}_{2}$ compete strongly [R. Shindou and T. Momoi, Phys. Rev. B 80, 064410 (2009)]. Using a large-$N$ expansion, we take into account fluctuations around this saddle point up to corrections of order $1/N$. The dynamical spin structure factors thus obtained signify the existence of gapless $q$-linear director-wave (spin-wave) modes at $\mathbit{q}=(0,0)$ and gapped ``gauge-field''-like collective modes at $\mathbit{q}=(\ensuremath{\pi},\ensuremath{\pi})$, whose spectral weight vanishes as a linear and quadratic function of the momentum, respectively. The low-energy collective modes contain fluctuations of nematic-director, spin, and gauge degrees of freedom. Associated with the gapless $q$-linear modes, we evaluate the temperature dependence of the nuclear spin relaxation rate $1/{T}_{1}$ in the low-temperature regime as $1/{T}_{1}\ensuremath{\propto}{T}^{2d\ensuremath{-}1}$, where $d$ is the effective spatial dimension.