Quasiclassical Green’s function approach to mesoscopic superconductivity

Abstract

Recent experiments on mesoscopic normal-metal-superconductor heterostructures resolve properties on length scales and at low temperatures such that the temperature is below the Thouless energy k B T ≤ E Th . We describe the properties of these systems within the framework of quasiclassical many-body techniques. Diffusive and ballistic systems are covered, both in equilibrium and nonequilibrium situations. Thereby we demonstrate the common physical basis of various subtopics.

Proximity effect in normal-metal quasiparticle traps

“…IV to estimate the influence of the proximity effect on qubit lifetime. The calculations are based on the quasiclassical approach to superconductivity, which we briefly discuss in Appendix A and is presented in more details in a number of reviews and textbooks [34][35][36][37]. In this formalism, the properties of a superconductor can be encoded in the pairing angle θ S ( ,r) which, for disordered superconductors, obeys an equation known as Usadel equation [38].…”

“…4, we compare the density of states obtained from a self-consistent numerical solution of the Usadel equations (28) and (29) to an approximate semianalytic formula which we arrive at by substituting Eqs. (36) and (37) [with θ Su ( ) found by numerically solving Eqs. (15) and (16) In the next section we will be interested in the spatial evolution of the normalized density of states and pair amplitude away from the normal-metal trap.…”

“…(50) for the distribution function and the latter into the definitions of the spectral functions (9) and (10); in those equations, we use the semianalytic results for the pairing angle to determine the density of states and the pair amplitude [see Eq. (36) and the text below Eq. (37)] and perform the final integration over energy numerically.…”

“…According to Eq. (36), in the uncovered section of the superconductor the pairing angle θ L is the sum of the BCS angle θ BCS and a correction. Assuming the correction to be small, |θ BCS − θ Su | θ BCS , the corrections to n and p are then …”

“…13 we show the density of states near the trap edge x/ξ = −1, obtained in three ways: (1) from the numerical solution of the Usadel and self-consistent equations (28), (29), and (14); (2) using the semianalytical expression in Eq. (36) in which θ Su ( ) is calculated numerically; (3) plotting the analytical formulas in Eqs. (38) and (40).…”