Quantum cascades in nano-engineered superconductors: geometrical, thermal and paramagnetic effects - Chen, Yajiang; Shanenko, A A; Croitoru, M D; Peeters, F M (Journal of physics. Condensed matter, 2012)
The effect of a parallel magnetic field on the orbital motion of electrons in high-quality superconducting nanowires resulting in a superconductor-to-normal transition which occurs through a cascade of jumps in the order parameter as a function of the magnetic field. Such cascades originate from the transverse size quantization that splits the conduction band into a series of subbands. Here, based on a numerical solution of the Bogoliubov-de Gennes equations for a hollow nanocylinder, we investigate how the quantum-size cascades depend on the confining geometry, i.e., by changing the cylinder radius R and its thickness d we cover the range from the nanowire-like to the nanofilm-like regime. The cascades are shown to become much less pronounced when increasing R d, i.e., when the nanofilm-like regime is approached. When the temperature is non-zero they are thermally smoothed. This includes the spin-magnetic-field interaction which reduces the critical (depairing) parallel magnetic field Hc, but does not have any qualitative effect on the quantum cascades. From our calculations it is seen that the paramagnetic limiting field Hpar significantly exceeds Hc, even in extremely narrow nanocylinders, i.e., when R,d are down to a few nanometers, and Hc, is only about 10% larger when switching-off the spin-magnetic-field interaction in this case. Both characteristic fields, Hc, and Hpar, exhibit pronounced quantum-size oscillations. We demonstrate that the quantum cascades and the quantum-size oscillations survive in the presence of surface roughness.