Odd-frequency S-wave Research Articles

Even- and Odd-Frequency Superconductivity in Q1D Organic Superconductors Under Magnetic Field

One of the quasi-one-dimensional organic superconductors $$(\hbox {TMTSF})_2X$$ , called Bechgaard salts, has attracted a lot of interest since its discovery in 1980. These compounds have high critical magnetic field $$H_{\mathrm {c}2}$$ in the direction parallel to the layers, and their pairing symmetries seem to be different between low-field phase and high-field phase. In addition, recently, the possibility of the realization of odd-frequency superconductivity was pointed out. Therefore, we study pairing symmetries of Bechgaard salts under magnetic field with frequency-dependent pairing interaction due to spin and charge fluctuation. The model is given by the extended Hubbard model on a quasi-one-dimensional square lattice, and the spin susceptibility and charge susceptibility for the effective pairing interactions are derived by random-phase approximation under magnetic field. The odd-frequency s-wave and even-frequency f-wave regions expand as the magnetic field is increased.

Current fluctuations in unconventional superconductor junctions with impurity scattering

The order parameter of bulk two-dimensional superconductors is classified as nodal, if it vanishes for a direction in momentum space, or gapful if it does not. Each class can be topologically nontrivial if Andreev bound states are formed at the edges of the superconductor. Non-magnetic impurities in the superconductor affect the formation of Andreev bound states and can drastically change the tunneling spectra for small voltages. Here, we investigate the mean current and its fluctuations for two-dimensional tunnel junctions between a normal-metal and unconventional superconductors by solving the quasi-classical Eilenberger equation self-consistently, including the presence of non-magnetic impurities in the superconductor. As the impurity strength increases, we find that superconductivity is suppressed for almost all order parameters since (i) at zero applied bias, the effective transferred charge calculated from the noise-current ratio tends to the electron charge $e$ and (ii) for finite bias, the current-voltage characteristics follows that of a normal state junction. There are notable exceptions to this trend. First, gapful nontrivial (chiral) superconductors are very robust against impurity scattering due to the linear dispersion relation of their surface Andreev bound states. Second, for nodal nontrivial superconductors, only p_x-wave pairing is almost immune to the presence of impurities due to the emergence of odd-frequency s-wave Cooper pairs near the interface. Owing to their anisotropic dependence on the wave vector, impurity scattering is an effective pair breaking mechanism for the rest of nodal superconductors. All these behaviors are neatly captured by the noise-current ratio, providing a useful guide to find experimental signatures for unconventional superconductivity.

Effects of vorticity and impurity on NMR relaxation rate in chiral p-wave superconductors

In order to study site-selective NMR in chiral p-wave superconductors, we calculate local nuclear relaxation rate T1−1 in the vortex lattice state by Eilenberger theory with and without non-magnetic impurity scattering in the Born limit and unitary limit. The local T1−1 in the NMR resonance line shape is different between two chiral states p±, depending on whether the chirality is parallel or anti-parallel to the vorticity. In the p−-wave, anomalous suppression of local T1−1 occurs around the vortex core due to the negative coherence term coming from odd-frequency s-wave Cooper pair induced around the vortex. We especially examine the site dependence of the anomalous suppression of local T1−1, including the applied magnetic field dependence and the impurity effects.

Site-selective NMR for odd-frequency Cooper pairs around vortex in chiralp-wave superconductors

In order to identify the pairing symmetry with chirality, we study site-selective NMR in chiral p-wave superconductors. We calculate local nuclear relaxation rate 1/T_1 in the vortex lattice state by Eilenberger theory, including the applied magnetic field dependence. We find that 1/T_1 in the NMR resonance line shape is different between two chiral states p_{pm}(=p_x{pm}ip_y), depending on whether the chirality is parallel or anti-parallel to the vorticity. Anomalous suppression of 1/T_1 occurs around the vortex core in the chiral p_- wave due to the negative coherence term coming from the odd-frequency s-wave Cooper pair induced around the vortex with Majorana state.

Odd-frequency Cooper-pair amplitude around a vortex core in a chiralp-wave superconductor in the quantum limit

Solving the Bogoliubov-de Gennes equation, we study the spatial structure of odd-frequency s-wave Cooper pair amplitudes at each quantized energy level of vortex bound states in chiral p-wave superconductors. For zero energy Majorana states, the odd-frequency s-wave pair amplitude has the same spatial structure as that of the local density of states even in atomic length scale. This relation also holds in finite energy bound states for single vortex winding anti-parallel to the chirality, but not for parallel vortex winding. The double winding vortex case is also studied.

Proximity Effect between a Dirty Fermi Liquid and Superfluid 3He

The proximity effect in superfluid 3He partly filled with high porosity aerogel is discussed. This system can be regarded as a dirty Fermi liquid/spin-triplet p-wave superfluid junction. Our attention is mainly paid to the case when the dirty layer is in the normal state owing to the impurity pair-breaking effect by the aerogel. We use the quasiclassical Green’s function to determine self-consistently the spatial variations of the p-wave order parameter and the impurity self-energy. On the basis of the fully self-consistent calculation, we analyze the spatial dependence of the pair function (anomalous Green’s function). The spin-triplet pair function has in general even-frequency odd-parity and odd-frequency even-parity components. We show that the admixture of the even- and odd-frequency pairs occurs near the aerogel/superfluid 3He-B interface. Among those Cooper pairs, only the odd-frequency s-wave pair can penetrate deep into the aerogel layer. As a result, the proximity-induced superfluidity in a thick aerogel layer is dominated by the Cooper pair with the odd-frequency s-wave symmetry. We also analyze the local density of states and show that it has a characteristic zero-energy peak reflecting the existence of the odd-frequency s-wave pair, in agreement with previous works using the Usadel equation.

Triplet supercurrents in clean and disordered half-metallic ferromagnets

Interfaces between materials with differently ordered phases present unique opportunities to study fundamental problems in physics. One example is the interface between a singlet superconductor and a half-metallic ferromagnet, where Cooper pairing occurs between electrons with opposite spin on one side, while the other displays 100% spin polarisation. The recent surprising observation of a supercurrent through half-metallic CrO_2 therefore requires a mechanism for conversion between unpolarised and completely spin polarised supercurrents. Here we suggest a conversion mechanism based on electron spin precession together with triplet pair rotation at interfaces with broken spin-rotation symmetry. In the diffusive limit the triplet supercurrent is dominated by inter-related odd-frequency s-wave and even-frequency p-wave pairs. In the crossover to the ballistic limit additional symmetry components become relevant. The interface region exhibits a superconducting state of mixed-spin pairs with highly unusual symmetry properties that opens up new perspectives for exotic Josephson devices.

Symmetries of Pairing Correlations in Superconductor–Ferromagnet Nanostructures

Using selection rules imposed by the Pauli principle, we classify pairing correlations according to their symmetry properties with respect to spin, momentum, and energy. We observe that inhomogeneity always leads to mixing of even- and odd-energy pairing components. We investigate the superconducting pairing correlations present near interfaces between superconductors and ferromagnets, with focus on clean systems consisting of singlet superconductors and either weak or half-metallic ferromagnets. Spin-active scattering in the interface region induces all of the possible symmetry components. In particular, the long-range equal-spin pairing correlations have odd-frequency s-wave and even-frequency p-wave components of comparable magnitudes. We also analyze the Josephson current through a half-metal. We find analytic expressions and a universality in the temperature dependence of the critical current in the tunneling limit.

Quasiparticles and pairing in doped triangular antiferromagnets

The motion of holes and the possibility of superconducting pairing in a doped triangular antiferromagnet are discussed. The exchange of RPA paramagnons leads to pairing in form of odd-frequency s-wave triplets competing with d-wave even-frequency singlets.

Magnetic transitions and superconductivity in the t-J model

With the use of the spin-wave and Born approximations the energy spectrum of the two-dimensional t-J model is self-consistently calculated in the range of hole concentrations, 0\ensuremath{\leqslant}x\ensuremath{\lesssim}0.3. The anomalous magnon Green's functions, which arise due to the hole-magnon interaction, are taken into consideration. They lead to a sharp transition from short-range antiferromagnetic order to a completely disordered paramagnetic state at x\ensuremath{\approx}0.19, in addition to the transition from long-range to short-range antiferromagnetic order at x\ensuremath{\approx}0.02--0.04. In the region of hole concentrations 0.04\ensuremath{\lesssim}x\ensuremath{\lesssim}0.19 the obtained shape of the Fermi surface, the hole dispersion near the Fermi level, and the density of states on it are in satisfactory agreement with experiment in ${\mathrm{La}}_{2\mathrm{\ensuremath{-}}\mathrm{x}}$${\mathrm{Sr}}_{\mathrm{x}}$${\mathrm{CuO}}_{4}$ and Bi2212. The Eliashberg formalism is used for calculating ${\mathrm{T}}_{\mathrm{c}}$. The hole-magnon interaction is found to be unable alone to give rise to superconductivity. By adding a moderate interaction with apex oxygen vibrations high ${\mathrm{T}}_{\mathrm{c}}$'s are obtained for even-frequency ${\mathrm{d}}_{{\mathrm{x}}^{2}\mathrm{\ensuremath{-}}{\mathrm{y}}^{2}}$ pairing in the range 0.04\ensuremath{\lesssim}x\ensuremath{\lesssim}0.19. For larger hole concentrations the odd-frequency s-wave solution becomes the leading one which can lead to s-wave superconductivity in the overdoped regime with the participation of a hole-phonon interaction of the respective symmetry.

Anomalous Fermi liquid and strong-coupling superconductivity in cuprates

The hole spectrum of the two-dimensionalt-J model is shown to change drastically due to transitions from the long-range to short-range antiferromagnetic order at the hole concentrationx≈0.04 and from the short-range order to a paramagnetic state atx≈0.19. At 0.05≲x≲0.19 the hole band has large flat regions similar to those observed in photoemission. These regions lead to an anomalous 2D Fermi liquid with a 2D Fermi surface. In spite of a sharp maximum produced by the flat regions in the density of states near the Fermi level, the hole-magnon interaction alone was found to be unable to give rise to superconductivity. By adding a moderate interaction with apex oxygen vibrations highTc are obtained fordx2−y2 pairing. Strong odd-frequency s-wave superconducting fluctuations appear near the transition into the paramagnetic state.