Critical de Broglie wavelength in superconductors

There are growing numbers of experimental evidences that the self-field critical currents, J(c)(sf,T), are a new instructive tool to investigate fundamental properties of superconductors ranging from atomically thin films [M. Liao et al., Nat. Phys. 6 (2018), https://doi.org/10.1038/s41567-017-0031-6; E. F. Talantsev et al., 2D Mater. 4 (2017) 025072; A. Fete et al., Appl. Phys. Lett. 109 (2016) 192601] to millimeter-scale samples [E. F. Talantsev et al., Sci. Rep. 7 (2017) 10010]. The basic empirical equation which quantitatively accurately described experimental J(c)(sf,T) was proposed by Talantsev and Tallon [Nat. Commun. 6 (2015) 7820] and it was the relevant critical field (i.e. thermodynamic field, B-c, for type-I and lower critical field, B-c1, for type-II superconductors) divided by the London penetration depth, lambda(L). In this paper, we report new findings relating to this empirical equation. It is that the critical wavelength of the de Broglie wave, lambda(dB,c) , of the superconducting charge carrier which within a numerical pre-factor is equal to the largest of two characteristic lengths of Ginzburg-Landau theory, i.e. the coherence length, xi , for type-I superconductors or the London penetration depth, lambda(L), for type-II superconductors. We also formulate a microscopic criterion for the onset of dissipative transport current flow: p(s) . 2.lambda(L)/ln(1+root 2.(lambda(L)/xi)) >= 1/2 . (h/2 pi), where p(s) is the charge carrier momentum, h is Planck's constant and the inequality sign "<" is reserved for the dissipation-free flow.