It has been commonly accepted that magnetic field suppresses superconductivity by inducing the ordered motion of Cooper pairs. We demonstrate that magnetic field can instead provide a generation of superconducting correlations by inducing the motion of superconducting condensate. This effect arises in superconductor/ferromagnet heterostructures in the presence of Rashba spin-orbital coupling. We predict the odd-frequency spin-triplet superconducting correlations called the Berezinskii order to be switched on at large distances from the superconductor/ferromagnet interface by the application of a magnetic field. This is shown to result in the unusual behaviour of Josephson effect and local density of states in superconductor/ferromagnet structures.
We study proximity induced triplet superconductivity in a spin-orbit-coupled system, and show that the \textbf{d} vector of the induced triplet superconductivity undergoes precession that can be controlled by varying the relative strengths of Rashba and Dresselhaus spin-orbit couplings. In particular, a long-range spin-triplet helix is predicted when these two spin-orbit couplings have equal strengths. We also study the Josephson junction geometry and show that a transition between 0 and $π$ junctions can be induced by controlling the spin-orbit coupling with a gate voltage. An experimental se…
We present an overview of the contributions made by Konstantin B. Efetov and the authors to the field of triplet odd-frequency superconductivity in hybrid ferromagnet-superconductor structures and the repercussions they had on experimental and theoretical research. Konstantin B. Efetov passed away in August 2021. We hope the present short review gives a faithful testimony of an important part of his scientific legacy.
We study graphene ferromagnet/superconductor/ferromagnet (F/S/F) nanostructures via a microscopic self-consistent Dirac Bogoliubov-de Gennes formalism. We show that as a result of proximity effects, experimentally accessible spin switching phenomena can occur as one tunes the Fermi level $μ_F$ of the F regions or varies the angle $θ$ between exchange field orientations. Superconductivity can then be switched on and off by varying either $θ$ or $μ_F$ (a spin-controlled superconducting graphene switch). The induced equal-spin triplet correlations in S can be controlled by tuning $μ_F$, effective…
We study a proximity junction between a conventional s-wave superconductor and a conductor with Rashba spin-orbit coupling, with a specific focus on the spin structure of the induced pairing amplitude. We find that spin-triplet pairing correlations are induced by spin-orbit coupling in both one- and two-dimensional systems due to the lifted spin degeneracy. Additionally, this induced triplet pairing has a component with an odd frequency dependence that is robust to disorder. Our predictions are based on the solutions of the exact Gor'kov equations and are beyond the quasiclassical approximatio…
We have studied the dependence of the superconducting (SC) transition temperature on the mutual orientation of magnetizations of Fe1 and Fe2 layers in the spin valve system CoO_x/Fe1/Cu/Fe2/Pb. We find that this dependence is nonmonotonic when passing from the parallel to the antiparallel case and reveals a distinct minimum near the orthogonal configuration. The analysis of the data in the framework of the SC triplet spin valve theory gives direct evidence for the long-range triplet superconductivity arising due to noncollinearity of the two magnetizations.
Unconventional superconductivity and in particular triplet superconductivity have been front and center of topological materials and quantum technology research. Here we report our observation of triplet superconductivity in nonmagnetic CoSi$_2$/TiSi$_2$ heterostructures on silicon. CoSi$_2$ undergoes a sharp superconducting transition at a critical temperature $T_c \approx$ 1.5 K, while TiSi$_2$ is a normal metal. We investigate conductance spectra of both two-terminal CoSi$_2$/TiSi$_2$ tunnel junctions and three-terminal T-shaped CoSi$_2$/TiSi$_2$ superconducting proximity structures. We rep…
The superconducting state in uranium compounds UGe2, URhGe and UCoGe is formed at temperatures far below the Curie temperature pointing on nonconventional nature of superconductivity in these materials - namely the superconductivity with triplet pairing. The emergence of superconductivity is accompanied by the slight magnetization expulsion typical for the type-II superconductors. Following classic Abrikosov paper I develop the theory of type-II superconductivity in application to two-band ferromagnetic metal with equal spin triplet pairing.
Spin-triplet superconductivity is known to be a rare quantum phenomenon. Here we show that nonsymmorphic crystalline symmetries can dramatically assist spin-triplet superconductivity in the presence of spin-orbit coupling. Even with a weak spin-orbit coupling, the spin-triplet pairing can be the leading pairing instability in a lattice with a nonsymmorphic symmetry. The underlining mechanism is the spin-sublattice-momentum lock on electronic bands that are protected by the nonsymmorphic symmetry. We use the nonsymmorphic space group P4/nmm to demonstrate these results and discuss related exper…
We argue that odd-frequency triplet superconductivity can be conveniently realized in hybrid superconductor-ferromagnet (SF) structures with a ferromagnetic vortex. We demonstrate that due to proximity-induced long-range triplet pairing such SFS junctions can sustain appreciable supercurrent which can be directly measured in experiments.
We consider a multidomain superconductor/ferromagnet (SF) structure with an in-plane magnetization, assuming that the neighboring domains are separated by the Neel domain walls. We show that an odd triplet long-range component arises in the domain walls and spreads into domains over a long distance of the order ξ_T = \sqrt{D/2πT} (in the dirty limit). The density of states variation in the domains due to this component changes over distances of the order ξ_T and turns to zero in the middle of domains if the magnetization rotates in the same direction in all domain walls.
The coexistence of a homogeneous (Meissner-like) phase of spin-triplet superconductivity and ferromagnetism is investigated within the framework of a phenomenological model of spin-triplet ferromagnetic superconductors. The results are discussed in view of application to metallic ferromagnets as UGe$_2$, ZrZn$_2$, URhGe, and Fe.
We calculate the conductance of a superconductor/ferromagnet (S/F) mesoscopic structure in the dirty limit. First we assume that the ferromagnet exhibits a homogeneous magnetization and consider the case that the penetration of the condensate into the F wire is negligible and the case in which the proximity effect is taken into account. It is shown that if the exchange field is large enough, the conductance below the critical temperature $T_C$, is always smaller than the conductance in the normal state. At last, we calculate the conductance for a F/S structure with a local inhomogeneity of the…
We study proximity effect in diffusive ferromagnet/normal-metal/superconductor junction with precessing magnetization of the ferromagnet. We find that the odd-frequency pairing induced in the normal metal is modified by spin pumping from the ferromagnet and hence can be tuned by changing the precessional frequency. At the frequency corresponding to twice the superconducting gap, the odd-frequency pairing is strongly enhanced. We find a crossover from the even- to the odd-frequency superconductivity in the normal metal by tuning the precessional frequency. This gives a clearcut signature of the…
A generalization of the Cooper pairing mechanism is proposed which allows for a triplet state of lower energy. This is achieved by incorporating spin into the canonical commutation relations and by modifying the $δ$ potential contact interaction. The gap equation contain as solutions both singlet and triplet states. It is shown that the triplet state is lower in energy than the singlet state which may explain the spin-triplet superconductivity observed in heavy fermion compound UPt3 and in Sr$_2$RuO$_4$.
We consider a system with electron-phonon interaction, antiferromagnetic fluctuations and disconnected open Fermi surfaces. The existence of odd-parity superconductivity in this circumstance is shown for the first time. If it is applied to the quasi-one-dimensional systems like the organic conductors (TMTSF)_2X we obtain spin-triplet superconductivity with nodeless gap. Our result is also valid in higher dimensions(2d and 3d).
Quasiperiodicity has long been known to be a potential platform to explore exotic phenomena, realizing an intricate middle point between ordered solids and disordered matter. In particular, quasiperiodic structures are promising playgrounds to engineer critical wavefunctions, a powerful starting point to engineer exotic correlated states. Here we show that systems hosting a quasiperiodic modulation of antiferromagnetism and spin-singlet superconductivity, as realized by atomic chains in twisted van der Waals materials, host a localization-delocalization transition as a function of the coupling…
Non-centrosymmetric superconductors exhibit the magnetoelectric effect which manifests itself in the appearance of the magnetic spin polarization in response to a dissipationless electric current (supercurrent). While much attention has been dedicated to the thermodynamic version of this phenomenon (Edelstein effect), non-equilibrium transport magnetoelectric effects have not been explored yet. We propose the magnetoelectric Andreev effect (MAE) which consists in the generation of spin-polarized triplet Andreev conductance by an electric supercurrent. The MAE stems from the spin polarization o…
We report observation of spin triplet superconductivity in epitaxial Bi/Ni bilayers with TC up to 4 K and 2Delta/kBTC = 12. Andreev reflection spectroscopy (ARS) with ballistic injection of unpolarized and spin-polarized electrons conclusively reveals spin triplet p-wave superconductivity. The gap structure measured by ARS in multiple crystal directions shows the ABM (Anderson-Brinkman-Morel) state, the same as that in superfluid 3He.
We have measured a long-range supercurrent in Josephson junctions containing Co (a strong ferromagnetic material) when we insert thin layers of either PdNi or CuNi weakly-ferromagnetic alloys between the Co and the two superconducting Nb electrodes. The critical current in such junctions hardly decays for Co thicknesses in the range of 12-28 nm, whereas it decays very steeply in similar junctions without the alloy layers. The long-range supercurrent is controllable by the thickness of the alloy layer, reaching a maximum for a thickness of a few nm. These experimental observations provide stron…
The mechanism of superconductivity in ${\rm Sr}_{2}{\rm RuO}_{4}$ is studied using a degenerate Hubbard model within the weak coupling theory. When the system approaches the orbital instability which is realized due to increasing the on-site Coulomb interaction between the electrons in the different orbitals, it is shown that the triplet superconductivity appears. This superconducting mechanism is only available in orbitally degenerate systems with multiple Fermi surfaces.
A mechanism leading to the spin-triplet superconductivity is proposed based on the antiferromagnetic spin fluctuation. The effects of anisotropy in spin fluctuation on the Cooper pairing and on the direction of d vector are examined in the one-band Hubbard model with RPA approximation. The gap equations for the anisotropic case are derived and applied to Sr_2RuO_4. It is found that a nesting property of the Fermi surface together with the anisotropy leads to the triplet superconductivity with the d=z(sin{k_x}\pm isin{k_y}), which is consistent with experiments.
The authors discuss the possibility of coexistence of antiferromagnetism and triplet superconductivity as a particular example of a broad class of systems where the interplay of magnetism and superconductivity is important. This paper focuses on the case of quasi-one-dimensional metals, where it is known experimentally that antiferromagnetism is in close proximity to triplet superconductivity in the temperature versus pressure phase diagram. Over a narrow range of pressures, the authors propose an intermediate non-uniform phase consisting of alternating insulating antiferromagnetic and triplet…
In a Kondo lattice system, suppression of effective Kondo coupling leads to the breakdown of the heavy-electron metal and a change in the electronic structure. Spin triplet superconductivity in the Kondo lattice UTe2 appears to be associated with spin fluctuations originating from incipient ferromagnetic order. Here we show clear evidence of twofold enhancement of spin-triplet superconductivity near the pressure-driven suppression of the Kondo coherence, implying that superconductivity is strengthened by the affiliated growth of both spin and charge fluctuations. The coherent Kondo state disco…
We investigate the spin-triplet superconductivity mediated by phonons in quasi-one-dimensional (Q1D) systems with open Fermi surfaces. We obtain the ground state phase diagrams. It is found that spin-triplet superconductivity occurs for weak screening and strong on-site Coulomb interaction, even in the absence of any additional nonphonon pairing interactions. We find that the nodeless spin-triplet state is more favorable than the spin-triplet state with line nodes, for the parameter values of the Q1D superconductors (TMTSF)_2X. We also find that Q1D open Fermi surface, which is the specific fe…
We reveal that in a magnetically polarized medium, a specific commensurate triplet pair density wave (TPDW) superconducting (SC) state, the staggered d-wave $Π$-triplet state, may coexist with homogeneous triplet SC states and even dominate eliminating them under generic conditions. When only this TPDW SC state is present, we have the remarkable phenomenon of gapless superconductivity. This may explain part of the difficulties in the realization of the engineered localized Majorana fermion modes for topological quantum computation. We point out qualitative characteristics of the tunneling dens…
We investigate the possibility of superconductivity on the graphene lattice within the repulsive Hubbard model using the variational cluster approximation (VCA). We find that singlet superconductivity is impossible; instead, triplet superconductivity is favored, with four solutions that are close to each other in energy and differ by their symmetry; one is f-wave, the other three p-wave, including a p+ip solution that breaks time-reversal invariance.
We revisit the exciton mechanism of superconductivity in the framework of microcavity physics, replacing virtual excitons as a binding agent of Cooper pairs by excitations of an exciton-polariton Bose-Einstein condensate. We consider a model microcavity where a quantum well with a two dimensional electron gas is sandwiched between two undoped quantum wells, where a polariton condensate is formed. We show that the critical temperature for superconductivity dramatically increases with the condensate population, opening a new route towards high temperature superconductivity.
We theoretically consider the upper critical magnetic field, perpendicular to a conducting axis in a triplet quasi-one-dimensional superconductor. In particular, we demonstrate that, at high magnetic fields, the orbital effects against superconductivity in a magnetic field are reversible and, therefore, superconductivity can restore. It is important that the above mentioned quantum limit can be achieved in presumably triplet quasi-one-dimensional superconductor Li$_{0.9}$Mo$_6$O$_{17}$ [J.-F. Mercure et al., Phys. Rev. Lett. {\bf 108}, 187003 (2012)] at laboratory available pulsed magnetic fie…
We discuss the possibility of coexistence of spin density wave (antiferromagnetism) and triplet superconductivity as a particular example of a broad class of systems where the interplay of magnetism and superconductivity is important. We focus on the case of quasi-one-dimensional metals, where it is known experimentally that antiferromagnetism is in close proximity to triplet superconductivity in the temperature versus pressure phase diagram. Over a narrow range of pressures, we propose an intermediate non-uniform phase consisting of alternating antiferromagnetic and triplet superconducting st…
There is mounting evidence for triplet superconductivity in the recently discovered skutterudite compound PrOs_4Sb_12. In this work, we propose nodal order parameters for the A- and B-phases of this superconductor which are consistent with angle dependent magnetothermal conductivity measurements and with low-temperature thermal conductivity data in the range T > 150 mK. The quasiparticle density of states and the thermal conductivity kappa_zz are derived within the quasiclassical approximation.
In PrOs4Sb12, the lowest-lying singlet and triplet states in a Pr 4f^2 configuration hybridize with conduction electrons having local a_u and t_u point-group symmetries. It is shown that for an attractive triplet pairing interaction, the orbital degrees of freedom of the t_u component are important. In addition, the Th point-group symmetry characteristic of skutterudites plays an important role in stabilizing triplet superconductivity.
We characterize the coexistence of itinerant ferromagnetism and spin-triplet superconductivity within a single mechanism involving local (Hund's rule) exchange among $d$ electrons. The ratio of transition temperatures and the spin anisotropy of the superconducting gap is estimated for $ZrZn_2$. The $A$ phase is stable in very low applied and molecular fields, whereas the $A1$ phases persists in higher fields. A small residual magnetic moment is present below the Stoner threshold in the superconducting phase.
We consider a SFS Josephson junction made of two superconductors S and a multidomain ferromagnet F with an in-plane magnetization. We assume that the neighboring domains of the ferromagnet are separated by Neel domain walls. An odd-frequency triplet long-range component of superconducting correlations arises in the domain walls and spreads into the domains over a long distance of the order $ξ_T = \sqrt{D / 2 πT}$, where $D$ is the diffusion coefficient (dirty limit is implied). We calculate the contribution of this component to the Josephson current in the situation when conventional short-ran…
To investigate relations between long-range antiferromagnetic (AF) order, superconductivity and two particle triplet collective excitations we consider a modified two dimensional t-J model at doping close to half filling. The model includes additional hopping t'' and nearest sites Coulomb repulsion V. The additional parameters allow us to control closeness of the system to the AF instability. We demonstrate the possibility of co-existence of long-range AF order and d-g-wave superconductivity. In the phase with long-range AF order we find, analytically, superconducting gaps and spin wave renorm…
We discuss the possibility of coexistence of spin density wave (antiferromagnetism) and triplet superconductivity as a particular example of a broad class of systems where the interplay of magnetism and superconductivity is important. We focus on the case of quasi-one-dimensional metals, where it is kown that antiferromagnetism is in close proximity to triplet superconductivity in the pressure versus temperature phase diagram. Over a range of pressures, we propose an intermediate non-uniform phase consisting of antiferromagnetic and triplet superconducting orders. In the coexistence region, we…
We show how in principle a coherent coupling between two superconductors of opposite parity can be realized in a three-layer oxide heterostructure. Due to strong intraionic spin-orbit coupling in the middle layer, singlet Cooper pairs are converted into triplet ones and vice versa. This results in a large enhancement of the triplet superconductivity, persisting well above the native triplet critical temperature.
We show that the nonuniform state (Fulde-Ferrel-Larkin-Ovchinnikov (FFLO) state) of the spin triplet superconductivity in noncentrosymmetric systems is stabilized by antisymmetric spin-orbit coupling even if the magnetic field is absent. The transition temperature of the spin triplet superconductivity is reduced by the antisymmetric spin-orbit coupling in general. This pair breaking effect is shown to be similar to the Pauli pair breaking effect due to magnetic field for the spin singlet superconductivity, in which FFLO state is stabilized near the Pauli limit (or Chandrasekhar-Clogston limit)…
Have you been lying awake wondering what symmetries determine whether a superconductor is spin singlet, triplet, or both? We show that if BCS theory is supplied with additional degrees of freedom, spin singlet can coexist with spin triplet superconductivity. In the first part, we didactically guide the reader to the most general superconducting state using symmetry arguments. If both singlet and triplet pairing channels are present, a magnetic field can convert between spin singlet and triplet states. In the second part, we discuss two possible singlet-triplet superconductors: CeRh$_2$As$_2$ a…
We discuss the possibility of spin-triplet superconductivity in a two-dimensional Hubbard model on a triangular lattice within the third-order perturbation theory. When we vary the symmetry in the dispersion of the bare energy band from D_2 to D_6, spin-singlet superconductivity in the D_2-symmetric system is suppressed and we obtain spin-triplet superconductivity in near the D_6-symmetric system. In this case, it is found that the vertex terms, which are not included in the interaction mediated by the spin fluctuation, are essential for realizing the spin-triplet pairing. We point out the pos…
We review magnetic and superconducting instabilities in the t-t' Hubbard model on the two-dimensional square lattice as obtained with functional one-loop renormalization group techniques. Special emphasis is put on ferromagnetic and triplet superconducting tendencies that could be relevant to the triplet superconductor Sr2RuO4.
Despite being of fundamental importance and potential interest for topological quantum computing, spin-triplet superconductors remain rare in solid state materials after decades of research. In this work, we present a general mechanism for spin-triplet superconductivity in multi-band systems, where a non-retarded pairing interaction between conduction electrons is produced by their electronic repulsion to a third-electron undergoing a virtual interband transition. Our theory is analytically controlled by an interband hybridization parameter, and explicitly demonstrated in doped band insulators…
Thin film junctions and bilayers of the doped topological insulator $\rm Bi_2Se_3$ and the s-wave superconductor NbN were found to exhibit conductance spectra with prominent zero bias and coherence peaks. Various tunneling models with different pair potentials have failed to fit our data, except for the triplet $p_x+ip_y$ pair potential, which breaks time reversal symmetry, but yielded reasonably good fits. This provides evidence for proximity induced triplet superconductivity in the $\rm Bi_2Se_3$ layer near the interface with the NbN film.
We demonstrate that in multilayered superconductor-ferromagnet structures a non-collinear alignment of the magnetizations of different ferromagnetic layers generates a triplet superconducting condensate which is odd in frequency. This triplet condensate coexists in the superconductors with the conventional singlet one but decays very slowly in the ferromagnet, which should lead to a large Josephson effect between the superconductors separated by the ferromagnet. Depending on the mutual direction of the ferromagnetic moments the Josephson coupling can be both of 0 and $π$ type.
We study pairing symmetry of noncentrosymmetric superconductors based on the extended Hubbard model on square lattice near half-filling, using the random phase approximation. We show that d+f-wave pairing is favored and the triplet f-wave state is enhanced by Rashba type spin-orbit coupling originating from the broken inversion symmetry. The enhanced triplet superconductivity stems from the increase of the effective interaction for the triplet pairing and the reduction of the spin susceptibility caused by the Rashba type spin-orbit coupling which lead to the increase of the triplet component a…
We propose that screened pairing interactions mediated by phonons could give rise to a spin triplet superconductivity in ferromagnetic compounds such as UGe_2. It is pointed out that the pairing interactions include anisotropic components such as those of p, d, f-waves in addition to dominant s-wave component due to the momentum dependence. Since the ferromagnetic long-range order coexists, there is a large splitting of the Fermi surfaces of up and down spin electrons, which suppresses singlet pairing. Therefore, triplet pairing occurs at last due to the sub-dominant anisotropic interactions, …
Atomic layers deposited on semiconductor substrates introduce a platform for the realization of the extended electronic Hubbard model, where the consideration of electronic repulsion beyond the onsite term is paramount. Recently, the onset of superconductivity at 4.7K has been reported in the hole-doped triangular lattice of tin atoms on a silicon substrate. Through renormalization group methods designed for weak and intermediate coupling, we investigate the nature of the superconducting instability in hole-doped Sn/Si(111). We find that the extended Hubbard nature of interactions is crucial t…
The unique Fermi surfaces and their nesting properties of Sr2RuO4 are considered. The existence of unconventional superconductivity is shown microscopically, for the first time, from the magnetic interactions (due to nesting) and the phonon-mediated interactions. The odd-parity superconductivity is favored in the $α$ and $β$ sheets of the Fermi surface, and the various superconductivities are possible in the $γ$ sheet. There are a number of possible odd-parity gaps, which include the gaps with nodes, the breaking of time-reversal symmetry and $\vec{d}\parallel \hat{z}$.
We propose theoretically that a magnetic field can realize spin-triplet superconductivity in repulsively interacting electron systems having strong ferromagnetic spin fluctuations. We confirm the general idea for the low-density Hubbard model on a triangular lattice, whose Fermi surface consists of disconnected pieces, by calculating the pairing susceptibility in a moderate magnetic field with the quantum Monte-Carlo method combined with the dynamical cluster approximation.
The density-matrix renormalization group method is used to study the ground state of the two-chain zigzag-bond Hubbard model at quarter filling. We show that, with a proper choice of the signs of hopping integrals, the ring exchange mechanism yields ferromagnetic spin correlations between interchain neighboring sites, and produces the attractive interaction between electrons as well as the long-range pair correlations in the spin-triplet channel, thereby leading the system to triplet superconductivity. We argue that this novel mechanism may have possible relevance to observed superconductivity…
In conventional superconductors, electrons of opposite spins are bound into Cooper pairs. However, when the superconductor is in contact with a non-uniformly ordered ferromagnet, an exotic type of superconductivity can appear at the interface, with electrons bound into three possible spin-triplet states. Triplet pairs with equal spin play a vital role in low-dissipation spintronics. Despite the observation of supercurrents through ferromagnets, spectroscopic evidence for the existence of equal-spin triplet pairs is still missing. Here we show a theoretical model that reveals a characteristic g…
