In the seminal work by G. Eilenberger [Z. Phys. 214, 195 (1968)], the quasiclassical expression for the free energy of spin-singlet superconductor has been suggested. Starting from the Luttinger-Ward formulation we derive the Eilenberger free energy and find its generalization for superconductor or superfluid with spin-triplet correlations. Besides ordinary superconductors with various scattering mechanisms, the obtained free energy functional can be used for systems with spin-triplet pairing such as superfluid $^3$He and superconducting systems with spatially-inhomogeneous exchange field or spin-orbit coupling. Using this general result we derive the simplified expression for the free energy in the diffusive limit in terms of the momentum-averaged propagators.
Generalized quasiclassical theory of the long-range proximity effect and spontaneous currents in superconducting heterostructures with strong ferromagnets
We consider the steady-state nonequilibrium behavior of mesoscopic superconducting wires connected to normal-metal reservoirs. Going beyond the diffusive limit, we utilize the quasiclassical theory and perform a self-consistent calculation that guarantees current conservation through the entire system. Going from the ballistic to the diffusive limit, we investigate several crucial phenomena such as charge imbalance, momentum-resolved nonequilbrium distributions, and the current-to-superflow conversion. Connecting to earlier results for the diffusive case, we find that superconductivity can bre…
Superconducting gap function in the organic superconductor (TMTSF)2ClO4 with anion ordering; First-principles calculations and quasiclassical analysis for angle-resolved heat capacity
The Eilenberger-Larkin-Ovchinnikov-Eliashberg quasiclassical theory of superconductivity is a powerful method enabling studies of a wide range of equilibrium and non-equilibrium phenomena in conventional and unconventional superconductors. We introduce here a finite element method, based on a discontinuous Galerkin approach, to self-consistently solve the underlying transport equations for general device geometries, arbitrary mean free path and symmetry of the superconducting order parameter. We present results on i) the influence of scalar impurity scattering on phase crystals in $d$-wave sup…
In recent years, substantial progress has been made regarding the application of quasiclassical theory on superconducting hybrid structures. This theoretical framework is reliant on a proper set of boundary conditions in order to describe multilayered systems. With the advent of the field of superconducting spintronics, systems which combine heavy metal layers, in which there is large spin--orbit coupling, with ferromagnets have received a great deal of attention, due to their potential for generating long range triplet superconductivity. In contrast to interfaces of strongly spin polarized ma…
We use the quasiclassical Eilenberger theory to study the topological superconducting proximity effects between a segment of a nanowire with a p-wave order parameter and a metallic segment. This model faithfully represents key qualitative features of an experimental setup, where only a part of a nanowire is in immediate contact with a bulk superconductor, inducing topological superconductivity. It is shown that the Eilenberger equations represent a viable alternative to the Bogoliubov-de Gennes theory of the topological superconducting heterostructures and provide a much simpler quantitative d…
The odd-frequency superconducting state appears generally in hybrid structures consisting of conventional superconductors and other materials, and features electrons that form temporally non-local Cooper pairs. The quasiclassical theory of superconductivity has been extensively used to model such systems, finding in many cases excellent agreement with experimental measurements. Therefore, it is of interest to study effective models of odd-frequency superconductivity to predict new phenomena associated with this form of unconventional pairing. We establish necessary criteria that the quasiclass…
We investigate the third-harmonic generation (THG) of s-wave superconductors under microwave pulse irradiation. We consider the effect of paramagnetic impurities on the THG intensity of dirty superconductors. The nonlinear response function is calculated using the method of the quasiclassical Green function. It is shown that the amplitude mode is included as the vertex correction and makes a predominant contribution to the THG intensity. When the effect of paramagnetic impurities is weak, the THG intensity shows a peak at the temperature at which the superconducting gap is about the same as th…
General boundary conditions for the quasiclassical kinetic equations in the presence of magnetism and unconventional superconductivity
University of Jyväskylä（于韦斯屈莱大学）；Nagoya University（名古屋大学）；Centro de Física de Materiales (CFM-MPC) Centro Mixto CSIC-UPV/EHU（材料物理中心（CFM-MPC）CSIC-巴斯克国立理工大学联合中心）；Donostia International Physics Center (DIPC)（圣塞巴斯蒂安国际物理中心）
The recent Comment by Vorontsov [arXiv:2007.13696] claims that surface pair-density-wave superconductivity with critical temperature higher than the bulk FFLO critical temperature is not supported by microscopic theory. The conclusion is reached by using an approximate semi-microscopic quasiclassical approach. Here we show that a fully microscopic approach unambiguously demonstrates the existence of surface pair-density-wave superconductivity.
We report a theoretical study on the phase transition between superconducting states with and without spontaneous surface current. The phase transition takes place due to the formation of surface Andreev bound states in unconventional superconductors. Based on the quasiclassical theory of superconductivity, we examine the influence of atomic-scale surface roughness on the surface phase transition temperature $T_s$. To describe the surface effect, the boundary condition for the quasiclassical Green's function is parameterized in terms of specularity (the specular reflection probability in the n…
We theoretically study the local density of states in superconducting proximity structure where two superconducting terminals are attached to a side surface of a normal-metal wire. Using the quasiclassical Green's function method, the energy spectrum is obtained for both of spin-singlet $s$-wave and spin-triplet $p$-wave junctions. In both of the cases, the decay length of the proximity effect at the zero temperature is limited by a depairing effect due to inelastic scatterings. In addition to the depairing effect, in $p$-wave junctions, the decay length depends sensitively on the transparency…
We study the transport properties of a diffusive Josephson junction between two spin-split superconductors made of superconductor-ferromagnetic insulator bilayers (FIS) on top of a 3D topological insulator (TI). We derive the corresponding Usadel equation describing the quasiclassical Green's functions in these systems and first solve the equation analytically in the weak-proximity case. We demonstrate the appearance of an anomalous phase in the absence of an external magnetic field. We also explore non-reciprocal electronic transport. Specifically, we calculate the junction's diode efficiency…
Field-angle-dependent low-energy excitations around a vortex in the superconducting topological insulator CuxBi2Se3
研究了超导拓扑绝缘体Cu $_{x}$ Bi $_{2}$ Se $_{3}$中单涡旋附近的准粒子激发，重点研究了点节点超导态.受最近Knight位移测量的启发，我们提出了两种方法来检测点节点的位置，在准经典处理中使用Kramer-Pesch近似的显式态密度公式。平行于$c$轴的涡旋周围的零能局域态密度具有双重形状，并且随着能量的增加沿着节方向分裂;这些行为可以通过扫描隧道显微镜检测到。当磁场平行于点节点方向时，在$a$ - $b$平面上旋转磁场下态密度的角度依赖性具有深极小值，这可以通过角分辨热容和热导率测量来检测。所有的理论预测都可以通过标准的实验技术在磁场中检测到。
Static spin susceptibility of superconductors with spin-orbit relaxation has been calculated in the seminal work of A.A. Abrikosov and L.P. Gor'kov [Sov. Phys. JETP, {\bf 15}, 752 (1962)]. Surprisingly the generalization of this result to finite frequencies has not been done despite being quite important for the modern topic of superconducting spintronics. The present paper fills this gap by deriving the analytical expression for spin susceptibility. The time-dependent spin response is shown to be captured by the quasiclassical Eilenberger equation with collision integrals corresponding to the…
I present a review of the theory and basic equations for charge transport in superconducting alloys starting from the Keldysh formulation of the quasiclassical transport equations developed by Eilenberger, Larkin and Ovchinnikov and Eliashberg. This formulation is the natural extension of Landau's theory of normal Fermi liquids to the superconducting state of strongly correlated metals. For dirty metals the transport equations reduce to equations for charge diffusion, with the current response given by the Drude conductivity at low temperatures. The extension of the diffusion equation for the …
The magnetization in a superconductor induced due to the inverse proximity effect is investigated in hybrid bilayers containing a superconductor and a ferromagnetic insulator or a strongly spin-polarized ferromagnetic metal. The study is performed within a quasiclassical Green function framework, wherein Usadel equations are solved with boundary conditions appropriate for strongly spin-polarized ferromagnetic materials. A comparison with recent experimental data is presented. The singlet to triplet conversion of the superconducting correlations as a result of the proximity effect with a ferrom…
When the impurity mean free path is short, only spin-polarized Cooper pairs which are non-locally and antisymmetrically correlated in time may exist in a half-metallic ferromagnet. As a consequence, the half-metal acts as an odd-frequency superconducting condensate. We demonstrate both analytically and numerically that quantum vortices can emerge in half-metals despite the complete absence of conventional superconducting correlations. Because these metals are conducting in only one spin band, we show that a circulating spin supercurrent accompanies these vortices. Moreover, we demonstrate that…
