Abstract: The properties of a clean superconductor with nanoscale dimensions are governed by quantum confinement of the electrons. This results in a spatially inhomogeneous superconducting condensate and in the formation of new Andreev-type quasiparticle states. These states are mainly located beyond regions where the superconducting condensate is enhanced. A numerical self-consistent solution of the Bogoliubov-de Gennes equations for a cylindrical metallic nanowire shows that these new Andreev-type states decrease the ratio of the energy gap to the critical temperature.

Abstract: From a numerical solution of the Bogoliubov-de Gennes equations, we investigate an interplay of the transverse discrete modes with a longitudinal supercurrent in a metallic cylindrical superconducting nanowire. The superconductor-to-normal transition induced by a longitudinal superflow of electrons is found to occur as a cascade of jumps in the order parameter (supercurrent and superfluid density) as a function of the superfluid velocity for diameters d<1015 nm (for Al parameters) and sufficiently low temperatures T<0.30.4Tc, with Tc the critical temperature. When approaching Tc, the jumps are smoothed into steplike but continuous drops. A similar picture occurs for d>1520 nm. Only when the diameter exceeds 5070 nm the quantum-size cascades are fully washed out, and we arrive at the mesoscopic regime. Below this regime the critical current density jc exhibits the quantum-size oscillations with pronounced resonant enhancements: the smaller the diameter, the more significant is the enhancement. Thickness fluctuations of real samples will smooth out such oscillations into an overall growth of jc with decreasing nanowire diameter.

Abstract: We study the effect, of electron confinement on the superconducting-to-normal phase transition driven by a magnetic field and/or on the current-carrying state of the superconducting condensate in nanowires. Our investigation is based on a self-consistent. numerical solution of the Bogoliubov-de Gennes equations. We show that, in a parallel magnetic field and/or in the presence of supercurrent the transition from superconducting to normal phase occurs as a cascade of discontinuous jumps in the superconducting order parameter for diameters D < 10 divided by 15 nm at T = 0. The critical magnetic held exhibits quantum-size oscillations with pronounced resonant enhancements.

Abstract: We propose an improved image simulation procedure for atomic-resolution annular dark-field scanning transmission electron microscopy (STEM) based on the multislice formulation, which takes thermal diffuse scattering fully into account. The improvement with regard to the classical frozen phonon approach is realized by separating the lattice configuration statistics from the dynamical scattering so as to avoid repetitive calculations. As an example, the influence of phonon scattering on the image contrast is calculated and investigated. STEM image simulation of crystals can be applied with reasonable computing times to problems involving a large number of atoms and thick or large supercells.

Abstract: It is well known that the formation of discrete electron levels strongly influences the pairing in metallic nanograins. Here, we focus on another effect of quantum confinement in superconducting grains that was not studied previously, i.e., spatially nonuniform pairing. This effect is very significant when single-electron levels form bunches and/or a kind of shell structure. We find that, in highly symmetric grains, the order parameter can exhibit variations with position by an order of magnitude. Nonuniform pairing is closely related to a quantum-confinement-induced modification of the pairing-interaction matrix elements and size-dependent pinning of the chemical potential to groups of degenerate or nearly degenerate levels. For illustrative purposes, we consider spherical metallic nanograins and also rectangular shapes. We show that the relevant matrix elements are, as a rule, enhanced in the presence of quantum confinement, which favors spatial variations of the order parameter, compensating the corresponding energy cost. The size-dependent pinning of the chemical potential further increases the spatial variation of the pair condensate. The role of nonuniform pairing is smaller in less symmetric confining geometries and/or in the presence of disorder. However, it always remains of importance when the energy spacing between discrete electron levels δ is approaching the scale of the bulk gap ΔB, i.e., δ>0.10.2 ΔB.

Abstract: We study the effect of electron confinement on the superconducting-to-normal phase transition driven by a magnetic field and/or on the current-carrying state of the superconducting condensate in nanowires. Our investigation is based on a self-consistent numerical solution of the Bogoliubov-de Gennes equations. We show that in a parallel magnetic field and/or in the presence of supercurrent the transition from superconducting to normal phase occurs as a cascade of discontinuous jumps in the superconducting order parameter for diameters D < 10 divided by 15 nm at T = 0. The critical magnetic field exhibits quantum-size oscillations with pronounced resonant enhancements.

Abstract: Electron continement and its effect on the superconducting-to-normal phase transition driven by a magentic field and/or a current is studied in nanowires. Our investigation is based on a self-consistent numerical solution of the Bogoliubov-de Gennes equations. We find that in a parallel magneitc field and/or in the presence of a supercurrent the transition from the superconducting to the normal phase occurs as a cascade of discontinuous jumps in the superconducting order parameter for diameters D < 10 divided by 15 nm at T = 0. The critical magentic field exhibits quantum-size oscillations with pronounced resonant enhancements as a function of the wire radius.