Triplet Order Parameter Research Articles

Light-induced switching between singlet and triplet superconducting states

While the search for topological triplet-pairing superconductivity has remained a challenge, recent developments in optically stabilizing metastable superconducting states suggest a new route to realizing this elusive phase. Here, we devise a testable theory of competing superconducting orders that permits ultrafast switching to an opposite-parity superconducting phase in centrosymmetric crystals with strong spin-orbit coupling. Using both microscopic and phenomenological models, we show that dynamical inversion symmetry breaking with a tailored light pulse can induce odd-parity (spin triplet) order parameter oscillations in a conventional even-parity (spin singlet) superconductor, which when driven strongly can send the system to a competing minimum in its free energy landscape. Our results provide new guiding principles for engineering unconventional electronic phases using light, suggesting a fundamentally non-equilibrium route toward realizing topological superconductivity.

Spectral properties of a mixed singlet-triplet Ising superconductor

Conventional two-dimensional superconductivity is destroyed when the critical in-plane magnetic field exceeds the so-called Pauli limit. Some monolayer transition-metal dichalcogenides lack inversion symmetry and the strong spin-orbit coupling leads to a valley-dependent Zeeman-like spin splitting. The resulting spin-valley locking lifts the valley degeneracy and results in a strong enhancement of the in-plane critical magnetic field. In these systems, it was predicted that the density of states in an in-plane field exhibits distinct mirage gaps at finite energies of about the spin-orbit coupling strength, which arise from a coupling of the electron and hole bands at energy larger than the superconducting gap. In this study, we investigate the impact of a triplet pairing channel on the spectral properties, primarily the mirage gap and the superconducting gap, in the clean limit. Notably, in the presence of the triplet pairing channel, the mirage-gap width is reduced for the low magnetic fields. Furthermore, when the temperature is lower than the triplet critical temperature, the mirage gaps survive even in the strong-field limit due to the finite singlet and triplet order parameters. Our work provides insights into controlling and understanding the properties of spin-triplet Cooper pairs.

Classification of Yu-Shiba-Rusinov states of magnetic impurities on superconducting surfaces: A fully relativistic first-principles study

Magnetic adatoms at surfaces of superconductors may induce localized bound states within the superconducting energy gap referred to as Yu-Shiba-Rusinov (YSR) states. We solve the Kohn-Sham-Dirac Bogoliubov--de Gennes equations within the fully relativistic multiple scattering Green's function method to study the nature of the YSR states for $3d$ magnetic impurities (Cr, Mn, Fe, and Co) placed on a pristine Nb(110) surface, as well as capped by a monolayer of Bi, Re, or Ir. First, we perform self-consistent relativistic calculations in the normal state and determine the direction of the easy axis and the magnetic anisotropy energy. Then we study the relation between the electronic structure in the normal state and the YSR states. We show that the YSR states can be categorized according to the energy-resolved singlet and triplet order parameters as normal, superconducting singlet, or induced triplet states. Finally, we demonstrate that the rotation of the adatom magnetic moment shifts the energy of the YSR states as a consequence of the spin orbit coupling. During this rotation, some branches of the YSR states might cross the Fermi energy, implying that at certain paths in the configuration space a peak can be found at zero energy.

Stabilizing Fluctuating Spin-Triplet Superconductivity in Graphene via Induced Spin-Orbit Coupling.

A recent experiment showed that a proximity-induced Ising spin-orbit coupling enhances the spin-triplet superconductivity in Bernal bilayer graphene. Here, we show that, due to the nearly perfect spin rotation symmetry of graphene, the fluctuations of the spin orientation of the triplet order parameter suppress the superconducting transition to nearly zero temperature. Our analysis shows that both an Ising spin-orbit coupling and an in-plane magnetic field can eliminate these low-lying fluctuations and can greatly enhance the transition temperature, consistent with the recent experiment. Our model also suggests the possible existence of a phase at small anisotropy and magnetic field which exhibits quasilong-range ordered spin-singlet charge 4e superconductivity, even while the triplet 2e superconducting order only exhibits short-ranged correlations. Finally, we discuss relevant experimental signatures.

Surface Cooper-Pair Spin Waves in Triplet Superconductors.

We study the electrodynamics of spin triplet superconductors including dipolar interactions, which give rise to an interplay between the collective spin dynamics of the condensate and orbital Meissner screening currents. Within this theory, we identify a class of spin waves that originate from the coupled dynamics of the spin-symmetry breaking triplet order parameter and the electromagnetic field. In particular, we study magnetostatic spin wave modes that are localized to the sample surface. We show that these surface modes can be excited and detected using experimental techniques such as microwave spin wave resonance spectroscopy or nitrogen-vacancy magnetometry, and propose that the detection of these modes offers a means for the identification of spin triplet superconductivity.

The effect of nonmagnetic disorder in the superconducting energy gap of strontium ruthenate

We have performed for the strontium ruthenate compound, a disordered tight-binding numerical study in the unitary and intermedia scattering regimes analyzing the imaginary part of the self-consistent elastic scattering cross-section at zero temperature as a function of the superconducting gap parameter using the anisotropic limit of the γ sheet. We start this research, mentioning a set of different zero energy superconducting spectrum values inferred from experimental data. Furthermore, it is reproduced the Miyake–Narikiyo tiny triplet gap by varying the zero energy superconducting microscopic parameter. Additionally, using the finite temperature Matsubara Green functions formalism, it is investigated the energy gap as a function temperature and disorder. We conclude that the evolution of the reduced phase space using the elastic scattering cross-section for a triplet order parameter is a useful numerical technique and predicts the two elastic scattering regimes for strontium ruthenate.

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.

Singlet-triplet mixing in the order parameter of the noncentrosymmetric superconductor Ru7B3

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 inversion symmetry on the superconducting state, indicating the presence of a mixed singlet-triplet gap and confirming the theoretical predictions.

Interplay between singlet and triplet pairings in multiband two-dimensional oxide superconductors

We theoretically study the superconducting properties of multi-band two-dimensional transition metal oxide superconductors by analyzing not only the role played by conventional singlet pairings, but also by the triplet order parameters, favored by the spin-orbit couplings present in these ma- terials. In particular, we focus on the two-dimensional electron gas at the (001) interface between LaAlO3 and SrTiO3 band insulators where the low electron densities and the sizeable spin-orbit couplings affect the superconducting features. Our theoretical study is based on an extended su- perconducting mean-field analysis of the typical multi-band tight-binding Hamiltonian, as well as on a parallel analysis of the effective electronic bands in the low-momentum limit, including static on-site and inter-site intra-band attractive potentials under applied magnetic fields. The presence of triplet pairings is able to strongly reduce the singlet order parameters which, as a result, are no longer a monotonic function of the charge density. The interplay between the singlet and the triplet pairings affects the dispersion of quasi-particle excitations in the Brillouin zone and also induces anisotropy in the superconducting behavior under the action of an in-plane and of an out- of-plane magnetic fields. Finally, non-trivial topological superconducting states become stable as a function of the charge density, as well as of the magnitude and of the orientation of the magnetic field. In addition to the chiral, time-reversal breaking, topological superconducting phase, favored by the linear Rashba couplings and by the on-site attractive potentials in the presence of an out- of-plane magnetic field, we find that a time-reversal invariant topological helical superconducting phase is promoted by not-linear spin-orbit couplings and by the inter-site attractive interactions in the absence of magnetic field.

Signatures of triplet correlations in density of states of Ising superconductors

The few-layer transition metal dichalcogenides (TMDs) have been recently suggested as a platform for controlled unconventional superconductivity. We study the manifestations of unconventional triplet pairing in the density of states of a disordered TMD based monolayer. The conventional singlet pairing attraction is assumed to be the dominant pairing interaction. We map the phase diagrams of disordered Ising superconductors in the plane of temperature and the in-plane magnetic field. The latter suppresses singlet and promote triplet correlations. The triplet order parameters of a trivial (non-trivial) symmetry compete (cooperate) with the singlet order parameter which gives rise to a rich phase diagram. We locate the model-dependent phase boundaries and compute the order parameters in each of the distinct phases. With this information, we obtain the density of states by solving the Gorkov equation. The triplet components of the order parameters may change an apparent width of the density of states by significantly increasing the critical field. The triplet components of the order parameters lead to the density of states broadening significantly exceeding the broadening induced by magnetic field and disorder in the singlet superconductor.

Magnetic impurities in thin films and 2D Ising superconductors

In the theory of dilute magnetic impurities in superconductors, the effect of all impurity spin-components is expressed via a single magnetic scattering rate Γm. In a more realistic setting, magnetic impurities are anisotropic. In this case, the spatial randomness of three spin-components of impurities gives rise to generally different scattering rates Γi (i=1,2,3). We explore the effects of anisotropic magnetic impurities on the in-plane critical field in 2D superconductors. We discuss singlet, triplet, and parity-mixed order parameters allowed in systems without the inversion center. Also, the addition of a small amount of magnetic impurities may cause singlet to triplet crossovers. In all cases, different components of impurity spin affect the magnetic field—temperature phase diagram differently. We show that anisotropy of the magnetic impurities can serve as a probe of unconventional triplet or parity-mixed superconductivity.

Time-reversal symmetry-breaking in noncentrosymmetric superconductors

In this paper we examine the appearance of time-reversal symmetry-breaking (TRSB) states in the bulk and at the surface of a noncentrosymmetric superconductor. We utilize a Ginzburg–Landau theory, with coefficients derived from a microscopic model of the superconductor. We show that suppression of the triplet order parameter at the surface stabilizes a TRSB state by locally tuning the system into the bulk TRSB phase. We compare these results with those from centrosymmetric systems, and examine the appearance of a magnetization at the surface.

Excitonic magnet in external field: Complex order parameter and spin currents

We investigate spin-triplet exciton condensation in the two-orbital Hubbard model close to half filling by means of dynamical mean-field theory. Employing an impurity solver that handles complex off-diagonal hybridization functions, we study the behavior of excitonic condensate in stoichiometric and doped systems subject to external magnetic field. We find a general tendency of the triplet order parameter to lay perpendicular with the applied field and identify exceptions from this rule. For solutions exhibiting k-odd spin textures, we discuss the Bloch theorem which, in the absence of spin-orbit coupling, forbids the appearance of spontaneous net spin current. We demonstrate that the Bloch theorem is not obeyed by the dynamical mean-field theory.

Current-Phase Relation of a Ferromagnetic Josephson Junction Between Triplet Superconductors

We study the Josephson effect of a $\rm{T_1 F T_2}$ junction, consisting of spin-triplet superconductors (T), a weak ferromagnetic metal (F), and ferromagnetic insulating interfaces. Two types of the triplet order parameters are considered; $(k_x +ik_y)\hat{z}$ and $k_x \hat{x}+k_y\hat{y}$. We compute the current density in the ballistic limit by using the generalized quasiclassical formalism developed to take into account the interference effect of the multilayered ferromagnetic junction. We discuss in detail how the current-phase relation is affected by orientations of the d-vectors of superconductor and the magnetizations of the ferromagnetic tunneling barrier. General condition for the anomalous Josephson effect is also derived.

Induced spin-triplet pairing in the coexistence state of antiferromagnetism and singlet superconductivity: Collective modes and microscopic properties

The close interplay between superconductivity and antiferromagnetism in several quantum materials can lead to the appearance of an unusual thermodynamic state in which both orders coexist microscopically, despite their competing nature. A hallmark of this coexistence state is the emergence of a spin-triplet superconducting gap component, called $\pi$-triplet, which is spatially modulated by the antiferromagnetic wave-vector, reminiscent of a pair-density wave. In this paper, we investigate the impact of these $\pi$-triplet degrees of freedom on the phase diagram of a system with competing antiferromagnetic and superconducting orders. Although we focus on a microscopic two-band model that has been widely employed in studies of iron pnictides, most of our results follow from a Ginzburg-Landau analysis, and as such should be applicable to other systems of interest, such as cuprates and heavy fermions. The Ginzburg-Landau functional reveals not only that the $\pi$-triplet gap amplitude couples tri-linearly with the singlet gap amplitude and the staggered magnetization magnitude, but also that the $\pi$-triplet $d$-vector couples linearly with the magnetization direction. While in the mean field level this coupling forces the $d$-vector to align parallel or anti-parallel to the magnetization, in the fluctuation regime it promotes two additional collective modes - a Goldstone mode related to the precession of the $d$-vector around the magnetization and a massive mode, related to the relative angle between the two vectors, which is nearly degenerate with a Leggett-like mode associated with the phase difference between the singlet and triplet gaps. We also investigate the impact of magnetic fluctuations on the superconducting-antiferromagnetic phase diagram, showing that due to their coupling with the $\pi$-triplet order parameter, the coexistence region is enhanced.

Theory of surface spectroscopy for noncentrosymmetric superconductors

We study noncentrosymmetric superconductors with the tetrahedral $T_d$, tetragonal $C_{4v}$, and cubic point group $O$. The order parameter is computed self-consistently in the bulk and near a surface for several different singlet to triplet order parameter ratios. It is shown that a second phase transition below $T_c$ is possible for certain parameter values. In order to determine the surface orientation's effect on the order parameter suppression, the latter is calculated for a range of different surface orientations. For selected self-consistent order parameter profiles the surface density of states is calculated showing intricate structure of the Andreev bound states (ABS) as well as spin polarization. The topology's effect on the surface states and the tunnel conductance is thoroughly investigated, and a topological phase diagram is constructed for open and closed Fermi surfaces showing a sharp transition between the two for the cubic point group $O$.

Abrikosov vortices in SF bilayers

We study the spatial distribution of supercurrent circulated around an Abrikosov vortex in an SF bilayer in perpendicular magnetic field. Within the dirty limit regime and circular cell approximation for the vortex lattice, we derive the conditions when the Usadel equations the F layer can be solved analytically. Using the obtained solutions, we demonstrate the possibility of reversal of direction of proximity induced supercurrents around the vortex in the F layer compared to that in the S-layer. The direction of currents can be controlled either by varying transparency of the SF interface or by changing an exchange field in a ferromagnet. We argue that the origin of this effect is due the phase shift between singlet and triplet order parameter components induced in the F-layer. Possible ways of experimental detection of the predicted effect are discussed.

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.

Gapped tripletp-wave superconductivity in strong spin-orbit-coupled semiconductor quantum wells in proximity tos-wave superconductor

We show that the {\it gapped} triplet superconductivity, i.e., a triplet superconductor with triplet order parameter, can be realized in strong spin-orbit-coupled quantum wells in proximity to $s$-wave superconductor. It is revealed that with the singlet order parameter induced from the superconducting proximity effect, in quantum wells, not only can the triplet pairings arise due to the spin-orbit coupling, but also the triplet order parameter can be induced due to the repulsive effective electron-electron interaction, including the electron-electron Coulomb and electron-phonon interactions. This is a natural extension of the work of de Gennes, in which the repulsive-interaction-induced singlet order parameter arises in the normal metal in proximity to $s$-wave superconductor [Rev. Mod. Phys. {\bf 36}, 225 (1964)]. Specifically, we derive the effective Bogoliubov-de Gennes equation, in which the self-energies due to the effective electron-electron interactions contribute to the singlet and triplet order parameters. It is further shown that for the singlet order parameter, it is efficiently suppressed due to this self-energy renormalization; whereas for the triplet order parameter, it is the $p$-wave ($p_x\pm ip_y$) one with the ${\bf d}$-vector parallel to the effective magnetic field due to the spin-orbit coupling. Finally, we perform the numerical calculation in InSb (100) quantum wells. Specifically, we reveal that the Coulomb interaction is much more important than the electron-phonon interaction at low temperature. Moreover, it shows that with proper electron density, the minimum of the renormalized singlet and the maximum of the induced triplet order parameters are comparable, and hence can be experimentally distinguished.

Control of magnetism in singlet-triplet superconducting heterostructures

We analyze the magnetization at the interface between singlet and triplet superconductors and show that its direction and dependence on the phase difference across the junction are strongly tied to the structure of the triplet order parameter as well as to the pairing interactions. We consider equal spin helical, opposite spin chiral, and mixed symmetry pairing on the triplet side and show that the magnetization vanishes at $\phi=0$ only in the first case, follows approximately a $\cos\phi$ behavior for the second, and shows higher harmonics for the last configuration. We trace the origin of the magnetization to the magnetic structure of the Andreev bound states near the interface, and provide a symmetry-based explanation of the results. Our findings can be used to control the magnetization in superconducting heterostructures and to test symmetries of spin-triplet superconductors.