Equal-spin Pairing State Research Articles

Kramers Fulde-Ferrell state and superconducting spin diode effect

We study a novel Fulde-Ferrell equal-spin pairing state with opposite center of mass Cooper pair momentum for each spin polarization. This state respects time-reversal symmetry and is dubbed a Kramers Fulde-Ferrell (KFF) state. It can be realized in a one-dimensional system with spin-orbit coupling and nearest neighbor attraction. We find that the KFF state supports nonreciprocal spin transport for both bulk superconductor and Josephson junctions when inversion symmetry is broken. In addition to the spin Josephson diode effect, the charge transport is controlled by intriguing dynamics of bound states whose transitions can be manipulated by the length of the KFF superconductor. We discuss the relevance of the effective theory to the novel physics observed in monolayer Fe-based superconductor along line defects and domain walls, and the potential for using spins to make dissipationless superconducting spintronics.

Anomalous Thermal Hall Effect in Chiral Phases of $$^3$$He-Aerogel

We report theoretical results for heat transport by quasiparticle excitations in superfluid $^3$He infused into silica aerogel engineered with uniaxial anisotropy. For this system two distinct equal spin pairing (ESP) superfluid phases have been reported based on NMR spectroscopy. Theoretical analysis predicts the first ESP state to be the chiral A phase with chiral axis aligned along the strain axis, and the lower temperature phase to be a polar-distorted chiral phase with random transverse fluctuations in the orientation of the chiral axis. We report calculations of heat transport for the high temperature chiral phase, including an anomalous (zero field) thermal Hall current originating from branch conversion scattering of Bogoliubov quasiparticles by the chiral order parameter induced by potential scattering by the silica aerogel. Observation of an anomalous thermal Hall current would provide a direct signature of the underlying chirality and topology of the superfluid phase of $^3$He in "stretched" silica aerogels.

Controllable spin pairing states in silicene-based superconducting hybrid structures with noncollinear magnetizations

We have theoretically investigated the modulation between the opposite- and equal-spin pairings and the related transport properties in silicene-based ferromagnet/ferromagnet/superconductor hybrid structures with noncollinear magnetizations. Due to the exotic electronic properties of silicene, the exclusive fully spin-polarized equal-spin pairing state appears without any contamination from the opposite-spin pairing state in a perpendicular magnetic configuration. Furthermore, the switch effect between fully spin-polarized opposite- and equal-spin pairings can be realized by tuning the Fermi level. In addition, the fully spin-polarized equal-spin pairing correlation can be enhanced by modulation of the magnitude and orientation of the exchange field in the central region. It is also a significant finding that for the formation of the fully spin-polarized equal-spin pairing state, the length of the central layer can be taken to be large, but not for opposite-spin pairing. Our findings provide an ideal platform to explore fully spin-polarized opposite- and equal-spin pairing states separately.

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.

Spin-Triplet Superconductivity Induced by Longitudinal Ferromagnetic Fluctuations in UCoGe Probed by 59Co NMR Measurement

In this review, we present a summary of our experimental results suggesting that the superconducting ferromagnet UCoGe is a spin-triplet superconductor, along with some of the basic physical properties of UCoGe in the normal state. We show experimental evidence that superconductivity in UCoGe is mediated by longitudinal ferromagnetic fluctuations predominant above the superconducting-transition temperature TSC. This is obtained from the tuning of ferromagnetic fluctuations and superconductivity by changing the external field along the magnetic easy axis, the c-axis. In addition, we show that the spin susceptibility perpendicular to the spontaneous magnetic moment, which is probed by Knight-shift measurement, is almost constant in the superconducting state. The behavior of the susceptibility and the invariance of the spontaneous moment below TSC can be consistently explained by the equal-spin pairing state with a large exchange field along the c-axis.

Dissipation signatures of the normal and superfluid phases in torsion pendulum experiments withHe3in aerogel

We present data for energy dissipation factor (Q^{-1}) over a broad temperature range at various pressures of a torsion pendulum setup used to study 3He confined in a 98% open silica aerogel. Values for Q^{-1} above T_c are temperature independent and have a weak pressure dependence. Below T_c, a deliberate axial compression of the aerogel by 10% widens the range of metastability for a superfluid Equal Spin Pairing (ESP) state; we observe this ESP phase on cooling and the B phase on warming over an extended temperature region. While the dissipation for the B phase tends to zero as T goes to 0, Q^{-1} exhibits a peak value greater than that at T_c at intermediate temperatures. Values for Q^{-1} in the ESP phase are consistently higher than in the B phase and are proportional to \rho_s/\rho until the ESP to B phase transition is attained. We apply a viscoelastic collision-drag model, which couples the motion of the helium and the aerogel through a frictional relaxation time \tau_f. Our dissipation data is not sensitive to the damping due to the presumed small but non-zero value of \tau_f. The result is that an additional mechanism to dissipate energy not captured in the collision-drag model and related to the emergence of the superfluid order must exist. The extra dissipation below T_c is possibly associated with mutual friction between the superfluid phases and the clamped normal fluid. The pressure dependence of the measured dissipation in both superfluid phases is likely related to the pressure dependence of the gap structure of the "dirty" superfluid. The large dissipation in the ESP state is consistent with the phase being the A or the Polar with the order parameter nodes oriented in the plane of the cell and perpendicular to the aerogel anisotropy axis.

Chiral phases of superfluid3He in an anisotropic medium

I report theoretical analysis and predictions for the equilibrium phases of superfluid 3He infused into a low-density, homogeneous uniaxial aerogel. Ginzburg-Landau (GL) theory for a class of equal-spin-pairing (ESP) states in a medium with uniaxial anisotropy is developed and used to analyze recent experiments on uniaxially strained aerogels. For 3He in an axially "stretched" aerogel GL theory predicts a transition from normal liquid into a chiral ABM phase at T_c1 in which the chirality axis, l, is aligned along the strain axis. This orbitally aligned state, is protected from random fluctuations in the anisotropy direction, has a positive NMR frequency shift, a sharp NMR resonance line and is identified with the high-temperature ESP-1 phase of superfluid 3He in axially stretched aerogel. A second transition into a biaxial phase is predicted to onset at a slightly lower temperature, T_c2 < T_c1. This phase is an ESP state, breaks time-reversal symmetry, and is defined by an orbital order parameter that spontaneously breaks axial rotation symmetry. This transition is driven by the coupling of an axially aligned 1D "polar" order parameter to the two time-reversed 2D axial ABM states. The biaxial phase has a continuous degeneracy associated with the projection of its chiral axis in the plane normal to the anisotropy axis. Theoretical predictions for the NMR frequency shifts of the biaxial phase provide an identification of the ESP-2 as the biaxial state, partially disordered by random anisotropy (Larkin-Imry-Ma effect). The "width" of the jump in the NMR frequency shift at T_c2 provides an estimate of the orbital domain size, xi_LIM ~ 5 mu-m at 18 bar.

Dc Josephson effect in a junction between two ferromagnetic superconductors

By using the Furusaki–Tsukada formula, we calculate Josephson current in a junction between two ferromagnetic superconductors (FSs) as a function of temperature, exchange field and insulator barrier strength of interface between FSs. We consider FSs in three different Cooper pairing states: spin singlet s-wave pairing (SWP) state, spin triplet opposite spin pairing (OSP) state and spin triplet equal spin pairing (ESP) state. It is found that the FS exchange field suppresses dc Josephson current for SWP and OSP states while it does not show any sensitivity to exchange field in the case of ESP state. Also, at low temperature and high barrier strength, dc Josephson current shows clearly different curves for SWP, OSP and ESP states. Therefore, our results can help to discriminate the kind of Cooper pair symmetry in ferromagnetic superconductors.

Orientation of the Angular Momentum in Superfluid 3He-A in a Stretched Aerogel

Superfluid 3He-A in a fully characterized stretched aerogel, used in previous work by Pollanen et al., has been studied for parallel and perpendicular orientations of the magnetic field relative to the anisotropy axis of the aerogel. Consistently, we find that an equal spin pairing state (ESP) is stabilized down to the lowest temperature. From our pulsed NMR frequency shifts as a function of temperature and tip angle, the orientation of the orbital angular momentum $\hat{l}$ has been determined. The aerogel anisotropy introduced by uniaxial stretching tends to align $\hat{l}$ in the axial state parallel to the strain axis, consistent with the theory proposed by Sauls and contrary to Volovik’s prediction based on an impurity calculation of Rainer and Vuorio.

Spin susceptibility in the superconducting state of the ferromagnetic superconductor UCoGe

In order to determine the superconducting paring state in the ferromagnetic superconductor UCoGe, the ${}^{59}$Co NMR Knight shift, which is directly related to the microscopic spin susceptibility, was measured in the superconducting state under magnetic fields perpendicular to the spontaneous magnetization axis ${}^{59}{K}^{a,b}$. ${}^{59}{K}^{a,b}$ shows to be almost constant, but does not decrease below a superconducting transition. These behaviors as well as the invariance of the internal field at the Co site in the superconducting state exclude the spin-singlet pairing, and can be interpreted with the equal-spin pairing state with a large exchange field along the $c$ axis, which was studied by Mineev [Phys. Rev. B 81, 180504 (2010)].

Topologically protected surface Majorana arcs and bulk Weyl fermions in ferromagnetic superconductors

A number of ferromagnetic superconductors have been recently discovered which are believed to be in the so-called ``equal spin pairing'' state. In the equal spin pairing state, the Cooper pairs condense forming order parameters ${\ensuremath{\Delta}}_{\ensuremath{\uparrow}\ensuremath{\uparrow}},\phantom{\rule{4pt}{0ex}}{\ensuremath{\Delta}}_{\ensuremath{\downarrow}\ensuremath{\downarrow}}$ which are decoupled in the spin sector. We show that these three-dimensional systems should generically support topologically protected surface Majorana arcs and bulk Weyl fermions as gapless excitations. Similar protected low-energy exotic quasiparticles should also appear in the recently discovered noncentrosymmetric superconductors in the presence of Zeeman splitting. The protected surface arcs can be probed by angle-resolved photoemission as well as Fourier transform scanning tunneling spectroscopy experiments.

Interference effect on pairing correlations in a metal-superconductor-metal layer with magnetic interfaces

We study the proximity effect in a multilayer composed of a normal metal and a superconductor with spin-active interfaces. The symmetry of the induced pair amplitude is analyzed while varying the interface distance and the direction of the interface spins. We apply the general Green’s function formalism without making the quasiclassical approximation in order to study the interference effect from the surrounding interfaces of the superconducting layer. By utilizing the formalism developed by A. Millis et al., we compute the Keldysh Green’s function while incorporating the general boundary condition for magnetic interfaces of a superconducting heterostructure. The interference can have a profound effect on the proximity phenomena in the ballistic limit when the interface distance becomes less than the superconducting coherence length. We also discuss the effect of the direction of the interface spins on the induction of the odd-frequency equal-spin pairing state, which is important for applications of ferromagnetic quantum devices.

Effect of Spin–Orbit Interaction in Spin-Triplet Superconductor: Structure of d-Vector and Anomalous 17O-NQR Relaxation in Sr2RuO4

Supposing the spin-triplet superconducting state of Sr$_2$RuO$_4$, the spin-orbit (SO) coupling associated with relative motion in Cooper pairs is calculated by extending the method for the dipole-dipole coupling given by Leggett in the superfluid $^{3}$He. It is shown that the SO coupling works only in the equal-spin pairing (ESP) state to make the pair angular momentum $\hbar{\vec L}$ and the pair spin angular momentum ${\rm i}{\vec d}\times{\vec d}^{*}$ parallel with each other. The SO coupling gives rise to the internal Josephson effect in a chiral ESP state as in superfluid A-phase of $^3$He with a help of an additional anisotropy arising from SO coupling of atomic origin which works to direct the {\bf d}-vector into $ab$-plane. This resolves the problem of the anomalous relaxation of $^{17}$O-NQR and the structure of {\bf d}-vector in Sr$_2$RuO$_4$.

Unusual Upper Critical Field of the Ferromagnetic Superconductor UCoGe

We report upper critical field B(c2)(T) measurements on a single-crystalline sample of the ferromagnetic superconductor UCoGe. B(c2)(0) obtained for fields applied along the orthorhombic axes exceeds the Pauli limit for B parallela,b and shows a strong anisotropy B(c2)(a) approximately B(c2)(b)>>B(c2)(c). This provides evidence for an equal-spin pairing state and a superconducting gap function of axial symmetry with point nodes along the c axis, which is also the direction of the uniaxial ferromagnetic moment m(0)=0.07micro(B). An unusual curvature or kink is observed in the temperature variation of B(c2) which possibly indicates UCoGe is a two-band ferromagnetic superconductor.

Equal-spin pairing state of superfluidHe3in aerogel

The equal-spin pairing state, the so-called A-like phase, of superfluid $^{3}\mathrm{He}$ in aerogels is studied theoretically in the Ginzburg-Landau region by examining thermodynamics, and the resulting equilibrium phase diagram is mapped out. We find that the ABM pairing state with presumably quasi-long-ranged superfluid order has a lower free energy than the planar and ``robust'' states and is the best candidate of the A-like phase with a strange lowering of the polycritical point observed experimentally.

Superfluid transition to ABM state of 3He in aerogel

Stability of the ABM pairing state in aerogel is studied in order to identify the equal-spin pairing (ESP) state of superfluid 3He there close to the transition temperature T c( P). It is shown by examining T c( P) beyond the mean-field approximation that the quenched disorder makes the normal to ABM transition point higher compared with that of the normal to BW transition over a wider pressure range than in the bulk case. If the ESP state in 3He in aerogel is the ABM state, the present result naturally explains why the ESP state in aerogel at least upon cooling is realized even at lower pressures than the so-called PCP in the bulk case.

NMR studies of superfluid [formula omitted] in low density silica aerogels

Continuous wave NMR studies of superfluid 3 He in 99.4%, 98.7% and 98.3% porosity silica aerogels show similar phase diagrams, with an equal spin pairing (ESP) A-like state stable near T c,a, and a B-like phase stable at lower temperatures. On cooling through T c,a, the sample magnetization and frequency shift show substantial supercooling of the A-like phase, with discontinuous changes to B-like behavior seen at about 0.64 T c,b (bulk) for all samples. On warming, both the frequency shift and magnetization change continuously as the sample enters an ESP state. This suggests that this phase transition is broadened by pinning of the A–B interface on density fluctuations in the aerogel, even very close to T c,a.

Magnetic field suppression of3He superfluidity in aerogel

We report nuclear magnetic resonance (NMR) measurements of superfluid3He confined in 98% porosity aerogel*.3He superfluidity in aerogel has been identified as an equal-spin pairing superfluid with a transition temperature,T c, suppressed from that of the bulk. In magnetic fields, 0.38≤H≤2.2 kOe,T c is further suppressed by 61 ±1μK/kOe2 at 18.7 bar. Such a quadratic magnetic field dependence of the superfluid transition is unexpected for an equal-spin pairing state. The effects of small additions of4He on the NMR frequency shifts and on the phase diagram are described.