Boundary conditions in quasiclassical theory of superconductivity are of crucial importance for describing proximity effects in heterostructures between different materials. Although they have been derived for the ballistic case in full generality, c
orresponding boundary conditions for the diffusive limit, described by Usadel theory, have been lacking for interfaces involving strongly spin-polarized materials, such as e.g. half-metallic ferromagnets. Given the current intense research in the emerging field of superconducting spintronics, the formulation of appropriate boundary conditions for the Usadel theory of diffusive superconductors in contact with strongly spin-polarized ferromagnets for arbitrary transmission probability and arbitrary spin-dependent interface scattering phases has been a burning open question. Here we close this gap and derive the full boundary conditions for quasiclassical Green functions in the diffusive limit, valid for any value of spin polarization, transmission probability, and spin mixing angles (spin-dependent scattering phase shifts). It allows also for complex spin textures across the interface and for channel off-diagonal scattering (a necessary ingredient when the numbers of channels on the two sides of the interface differ). As an example we derive expressions for the proximity effect in diffusive systems involving half-metallic ferromagnets. In a superconductor/half-metal/superconductor Josephson junction we find $phi_0$ junction behavior under certain interface conditions.
The usually negligibly small thermoelectric effects in superconducting heterostructures can be boosted dramatically due to the simultaneous effect of spin splitting and spin filtering. Building on an idea of our earlier work [Phys. Rev. Lett. $textbf
{110}$, 047002 (2013)], we propose realistic mesoscopic setups to observe thermoelectric effects in superconductor heterostructures with ferromagnetic interfaces or terminals. We focus on the Seebeck effect being a direct measure of the local thermoelectric response and find that a thermopower of the order of $sim200$ $mu V/K$ can be achieved in a transistor-like structure, in which a third terminal allows to drain the thermal current. A measurement of the thermopower can furthermore be used to determine quantitatively the spin-dependent interface parameters that induce the spin splitting. For applications in nanoscale cooling we discuss the figure of merit for which we find enormous values exceeding 1 for temperature $lesssim 1$K.
We study vortex structure in a two-band superconductor, in which one band is ballistic and quasi-two-dimensional (2D), and the other is diffusive and three-dimensional (3D). A circular cell approximation of the vortex lattice within the quasiclassica
l theory of superconductivity is applied to a recently developed model appropriate for such a two-band system [Tanaka et al 2006 Phys. Rev. B 73, 220501(R); Tanaka et al 2007 Phys. Rev. B 75, 214512]. We assume that superconductivity in the 3D diffusive band is weak, i.e., mostly induced, as is the case in MgB$_2$. Hybridization with the weak 3D diffusive band has significant and intriguing influence on the electronic structure of the strong 2D ballistic band. In particular, the Coulomb repulsion and the diffusivity in the weak band enhance suppression of the order parameter and enlargement of the vortex core by magnetic field in the strong band, resulting in reduced critical temperature and field. Moreover, increased diffusivity in the weak band can result in an upward curvature of the upper critical field near the transition temperature. A particularly interesting feature found in our model is the appearance of additional bound states at the gap edge in the strong ballistic band, which are absent in the single-band case. Furthermore, coupling with the weak diffusive band leads to reduced band gaps and van Hove singularities of energy bands of the vortex lattice in the strong ballistic band. We find these intriguing features for parameter values appropriate for MgB$_2$.
