Triplet Order Parameter Research Articles

Domain walls in superconductors: Andreev bound states and tunneling features

Domain walls can be formed in superconductors with a discrete degeneracy of the ground state, for instance, due to the breaking of time reversal symmetry. We study all cases where the formation of domain walls is possible in a tetragonal superconductor with the point group ${D}_{4h}$. We discuss both triplet and mixed singlet order parameters. It is found that in all cases the domain walls support subgap Andreev bound states, whose energies strongly depend on the direction of semiclassical propagation. The bound state contribution to the density of quasiparticle states exhibits peculiar features, which can be observed in tunneling experiments.

Spin Josephson current in a ferromagnet/noncentrosymmetric superconductor junction

We investigate the equilibrium spin transport in a ferromagnet/noncentrosymmetric superconductor (FM/NCS) junction where the NCS has a dominant triplet order parameter and helical edge state. Based on the symmetry analysis and numerical calculation, we demonstrate that there is a nonzero spin supercurrent flowing in the junction, which stems from the exchange coupling between the FM magnetization and triplet Cooper-pair spin. It is also found that a transverse spin current other than the helical edge spin current is flowing along the interface of the junction, and its polarization is related to the longitudinal spin supercurrent. Besides, an equilibrium Hall current is also shown to flow along the junction’s interface due to the broken time-reversal symmetry from the FM.

Collective modes of the order parameter in a triplet superfluid neutron liquid

The complete spectrum of collective modes of the triplet order parameter in the superfluid neutron matter is examined in the BCS approximation below the pair-breaking threshold. The dispersion equations both for the unitary and nonunitary excitations are derived and solved in the limit of $q\rightarrow0$ by taking into account the anisotropy of the energy gap for the case of $P$-wave pairing. By our analysis, there is only one Goldstone mode which is associated with the broken gauge symmetry. We found no additional Goldstone modes associated with the broken rotational symmetry but found that the oscillations of the total angular momentum are qualitatively similar to the "normal-flapping" mode in the A-phase of superfluid Helium. There are also two collective modes associated with internal vibrations of the structure of the order parameter oscillating with $\omega(T=0) =1.20\Delta_{0}$ and $\omega(T=0) =0.61\Delta_{0}$.

Properties of Magnetic-Superconducting Proximity Systems

We discuss properties of proximity systems made of superconducting, ferromagnetic, and graphite nanoscale thin films. In superconducting-graphite proximity systems, we show how size quantization results in oscillations of the density of states with increasing graphite film thickness. In superconducting-ferromagnetic multilayers, we discuss the critical temperature variations of the proximity system as a function of ferromagnetic layer thickness and magnetic inhomogeneity. We also review the Josephson effect through a superconductor–ferromagnet–superconductor multilayer. In this context, we discuss the presence of a long-range odd-frequency triplet order parameter.

Interfacial spin Hall current in a Josephson junction with Rashba spin—orbit coupling

We theoretically investigate the spin transport properties of the Cooper pairs in a conventional Josephson junction with Rashba spin—orbit coupling considered in one of the superconducting leads. It is found that an angle-resolved spin supercurrent flows through the junction and a nonzero interfacial spin Hall current driven by the superconducting phase difference also appears at the interface. The physical origin of this is that the Rashba spin—orbit coupling can induce a triplet order parameter in the s-wave superconductor. The interfacial spin Hall current dependences on the system parameters are also discussed.

D-Wave-like nodal superconductivity in the organic conductor (TMTSF)2ClO4

We suggest theoretical explanation of the high upper critical magnetic field, perpendicular to conducting chains, Hc2b′, experimentally observed in the superconductor (TMTSF)2ClO4, in terms of singlet superconducting pairing. In particular, we compare the results of d-wave-like nodal, d-wave-like node-less, and s-wave scenarios of superconductivity. We show that, in d-wave-like nodal scenario, superconductivity can naturally exceed both the orbital upper critical magnetic field and Clogston–Shandrasekhar paramagnetic limit as well as reach experimental value, Hc2b′≃6T, in contrast to d-wave-like node-less and s-wave scenarios. In our opinion, the obtained results are strongly in favor of d-wave-like nodal superconductivity in (TMTSF)2ClO4, whereas, in a sister compound, (TMTSF)2PF6, we expect either the existence of triplet order parameter or the coexistence of triplet and singlet order parameters.

Josephson Hall current in a noncentrosymmetric superconductor/ferromagnet/superconductor junction

We theoretically study the transverse spin and charge transport in a 2D noncentrosymmetric superconductor (NCS)/ferromagnet (FM)/superconductor Josephson junction, in which the NCS has a dominant triplet order parameter and helical edge states. It is predicted that both equilibrium spin and charge Hall currents can flow transversely in the junction and are driven by the superconducting phase gradient. The physical origin of the Josephson Hall current is the broken time-reversal symmetry due to the FM and the mixing of the helical edge states. The Hall currents are sensitive to the FM magnetization direction, and the planar magnetization component is vitally important because it can lead to a spin flip effect that makes the Cooper pairs tunnel between unequal chiral edge states of the two NCSs.

Singlet versus triplet particle-hole condensates in quantum oscillations in cuprates

Quantum oscillations in a tilted magnetic field offer the possibility of distinguishing singlet versus triplet order parameters in the particle hole channel provided the measurements reflect a putative ``normal'' state of a density wave obtained by applying by high magnetic field at low temperatures. A theoretical analysis is given that compares spin density wave, a singlet $d$-density wave, and a triplet $d$-density wave. While the existence of a spin zero in the oscillation amplitude is a necessary consequence of a singlet order parameter, a triplet order parameter may or may not exhibit a spin zero, making it a quantitative issue that depends on the actual extremal orbits on the Fermi surface. Nonetheless, a theoretical analysis can shed light on the striking recent measurements in $\mathrm{YBa_{2}Cu_{3}O_{6+\delta}}$.

Spin current in topologically trivial and nontrivial noncentrosymmetric superconductors

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 subgap surface bound states show up. In both cases, there is also a finite contribution to the spin current from the continuum states with energies between the two gaps. The method for obtaining the quasiclassical Green's functions associated with the noncentrosymmetric superconductors is also presented.

Long-range odd triplet order parameter with equal spin pairing in diffusive Co/In contacts

We have measured the conductance of single, double, and multiple junction superconductor (S)/ferromagnet (F) devices made of Co and In. The contacts are created by pinholes in the CoO which prevent the local destruction of superconductivity. The conductance of the single junction has a V-shape cusp at zero bias typical of a nodal order parameter such as $p$ wave. In the double S/F device the nonlocal conductance at low energies is positive and larger in the parallel state than in the antiparallel state by 30%. The same effect was found in multiple S/F junctions in parallel. The results can be explained by an odd triplet order parameter inside S, up to a distance of 150 nm from the S/F interface.

Zero-energy vortex bound states in noncentrosymmetric 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.

F-wave versusP-wave superconductivity in organic conductors

Current experimental results suggest that some organic quasi-one-dimensional superconductors exhibit triplet pairing symmetry. Thus, we discuss several potential triplet order parameters for the superconducting state of these systems within the functional integral formulation. We compare weak spin-orbit coupling $f_{xyz}$, $p_x$, $p_y$ and $p_z$ symmetries via several thermodynamic quantities. For each symmetry, we analyse the temperature dependences of the order parameter, condensation energy, specific heat, and superfluid density tensor.

Domain wall superconductivity in ferromagnetic superconductors

On the basis of a phenomenological Ginzburg-Landau approach we investigate the problem of triplet order parameter nucleation in a ferromagnetic superconductor with a domain structure in an applied external magnetic field. The critical temperature of the superconductivity nucleation is shown to increase near the domain boundaries and to have a peculiar field dependence. Our results are also applied to a description of the superconductivity localized near the domain wall in $S/F$ heterostructures.

TRIPLET SUPERCONDUCTIVITY ORDER PARAMETER IN AN ORGANIC SUPERCONDUCTOR (TMTSF)2PF6

We argue that an organic superconductor (TMTSF)2PF6 under pressure P = 6 kbar is a triplet superconductor1,2 with a spin part of the triplet order parameter being d(k) = (dx (k) ≠ 0, dy (k) = 0, dz (k) = 0). We obtain these results by means of an analysis of the anisotropic experimental critical fields H c2 (see I. J. Lee et al., Phys. Rev. Lett.78, 3555 (1997)). By using symmetry arguments, we also discuss the most probably orbital part of the triplet order parameter in (TMTSF)2PF6. Our analysis is in accordance with the very recent Knight shift data obtained on (TMTSF)2PF6.

MAGNETIC TESTS TO REVEAL TRIPLET SUPERCONDUCTIVITY IN (TMTSF)2PF6 AND A POSSIBLE BREAKING OF A TIME REVERSAL SYMMETRY IN Sr2RuO4, LBCO, and YBCO

We propose methods to reveal the existence of a two-component triplet order parameter (which is recently suggested to break a time reversal symmetry in Sr 2 RuO 4) and to test a possible two-component singlet d-wave order parameter in high- T c superconductors (which may also break the time reversal symmetry). We have also determined a spin part (i.e., d(k) = [dx(k) ≠ 0, dy(k) = 0, dz(k) = 0)] of a triplet order parameter1,2 in (TMTSF) 2 PF 6 under pressure by means of an analysis of the experimental H c2 measurements (I. J. Lee et al., Phys. Rev. Lett. 78, 3555 (1997)).

Possible triplet order parameter in a Q1D organic superconductor (TMTSF) 2PF 6

We point out that (TMTSF) 2PF 6 compound at P ≅ 6 kbar is possibly a triplet superconductor with the vector order parameter d= ( d x ≠ 0, d y , d z = 0). It corresponds to an anisotropic spin susceptibility in a superconducting state: χ b′ ( z) = χ 0, χ a ( x) ≠ χ 0, where χ 0 is its value in a metallic phase. (Here, the spin quantization axis, z, is parallel to the easy axis, b′, of an antiferromagnetic SDW phase existing at P = 0). It is shown that the suggested order parameter can explain why the upper critical field along b′-axis (I.J. Lee, M.J. Naughton, G. Danner, and P.M. Chaikin, Phys. Rev. Lett. 78 3555 (1997)) exceeds all paramagnetic limiting fields (including that for a non-uniform superconductivity) whereas the upper critical field along the a-axis is restricted by the Pauli paramagnetic effects. The triplet order parameter is consistent with the earlier results on a destruction of superconductivity by nonmagnetic impurities and the absence of the Hebel-Slichter peak in the NMR data as well as with the recent Knight shift measurements by S. Brown et al and by I. J. Lee et al.

Ferromagnetic-semiconductor–singlet-(or triplet) superconductor–ferromagnetic-semiconductor systems as possible logic circuits and switches

We consider thin superconducting (S) films of thickness d $\ll \xi_{0}$, sandwiched between two ferromagnetic semiconducting insulators (FI) with differently orientated magnetizations - the FI-S-FI system. We calculate the dependence of the superconducting critical temperature on the orientation of the magnetization in the insulators and on the thickness of the superconducting film. The calculations are done for singlet as well as triplet superconductors. In the singlet case T_{c} depends on the relative orientation of the left and right magnetization only, while in the triplet case T_{c} depends on the absolute orientation of magnetization. The latter property can serve as a kind of spin-spectroscopy of triplet and unconventional superconductors, for instance in resolving the structure of the triplet order parameter in the recently discovered layered superconductor Sr_{2}RuO_{4}. The possibility of logic circuits and switches, which are based on the FI-S-FI systems with arbitrary orientation of magnetizations in FI films, is analyzed too.

Excitonic order at strong coupling: Pseudospin, doping, and ferromagnetism

A tight binding model is introduced to describe the strong interaction limit of excitonic ordering. At stoichiometry, the model reduces in the strong coupling limit to a pseudo-spin model with approximate U(4) symmetry. Excitonic order appears in the pseudo-spin model as in-plane pseudo-magnetism. The U(4) symmetry unifies all possible singlet and triplet order parameters describing such states. Super-exchange, Hunds-rule coupling, and other perturbations act as anisotropies splitting the U(4) manifold, ultimately stabilizing a paramagnetic triplet state. The tendency to ferromagnetism with doping (observed experimentally in the hexaborides) is explained as a spin-flop transition to a different orientation of the U(4) order parameter. The physical mechanism favoring such a reorientation is the enhanced coherence (and hence lower kinetic energy) of the doped electrons in a ferromagnetic background relative to the paramagnet. A discussion of the physical meaning of various excitonic states and their experimental consequences is also provided.

Triplet electron pairing and anisotropic spin susceptibility in organic superconductors(TMTSF)2X

We argue that (TMTSF)_2 PF_6 compound under pressure is likely a triplet superconductor with a vector order parameter d(k) \equiv (d_a(k) \neq 0, d_c(k) = ?, d_{b'}(k) = 0); |d_a(k)| > |d_c(k)|. It corresponds to an anisotropic spin susceptibility at T=0: \chi_{b'} = \chi_0, \chi_a \ll \chi_0, where \chi_0 is its value in a metallic phase. [The spin quantization axis, z, is parallel to a so-called b'-axis]. We show that the suggested order parameter explains why the upper critical field along the b'-axis exceeds all paramagnetic limiting fields, including that for a nonuniform superconducting state, whereas the upper critical field along the a-axis (a \perp b') is limited by the Pauli paramagnetic effects [I. J. Lee, M. J. Naughton, G. M. Danner and P. M. Chaikin, Phys. Rev. Lett. 78, 3555 (1997)]. The triplet order parameter is in agreement with the recent Knight shift measurements by I. J. Lee et al. as well as with the early results on a destruction of superconductivity by nonmagnetic impurities and on the absence of the Hebel-Slichter peak in the NMR relaxation rate.

Relativistic framework for microscopic theories of superconductivity. I. The Dirac equation for superconductors

We present a unified treatment of relativistic effects in superconductors. The relativistically correct (Dirac-type) single-particle Hamiltonian describing the quasiparticle spectrum of superconductors is deduced from symmetry considerations and the requirement of the correct nonrelativistic limit. We provide a complete list of all order parameters consistent with the requirement of Lorentz covariance. This list contains the relativistic generalizations of the BCS and the triplet order parameters, among others. Furthermore, we present a symmetry classification of the order parameters according to their behavior under the Lorentz group, generalizing previous treatments that were based on the Galilei group. The considerations in this paper are based only on the concepts of pairing and Lorentz covariance. They can therefore be applied to all situations in which pairing takes place. This includes BCS-type superconductors, as well as the heavy-fermion compounds, high-temperature superconductors, pairing of neutrons and protons in neutron stars, and superfluid helium 3.