Andreev scattering, Josephson currents, and coupling energy in clean superconductor-semiconductor-superconductor junctions

Superconductor-semiconductor-superconductor junctions with superconducting Nb banks coupled by the degenerate III-V semiconductor n-type InAs are described by the Bogoliubov-de Gennes equations (BdGE) with spatially and energy-dependent effective masses, abruptly changing conduction-band edges, interface barriers, and vanishing pair potential in the semiconducting (Sm) layer. Phase coherence between the superconducting (S) banks is mediated by Andreev scattering of ballistic quasiparticles, which is the only mechanism considered for Cooper pair transfer. The bound-state subbands, broadened by scattering from the mismatches at the S-Sm interfaces, split off at finite phase differences Φ between the pair potentials in either S region. At arbitrary temperatures T below the critical temperature ${\mathit{T}}_{\mathit{c}}$ of Nb, the Josephson-current density j(Φ), computed numerically from the solutions of the BdGE, can be simulated very well by j(Φ)=${\mathit{j}}_{\mathit{c}}$sin[Φ-${\mathit{L}}_{\mathrm{kin}}$j(Φ)], where the kinetic-inductance parameter ${\mathit{L}}_{\mathrm{kin}}$ decreases with increasing temperature and decreasing Sm layer thickness 2a. The maximum coupling energy per unit area is ${\mathit{E}}_{\mathit{J}}$(Φ=π)=${\mathit{j}}_{\mathit{c}}$ħ/e for all ${\mathit{L}}_{\mathrm{kin}}$. The critical Josephson-current density ${\mathit{j}}_{\mathit{c}}$ is 5.6×${10}^5$ A ${\mathrm{cm}}^{\mathrm{-}2}$ at T=0 K, n=${10}^{19}$ ${\mathrm{cm}}^{\mathrm{-}3}$, 2a=0.3 μm, and vanishing interface barrier strength Z. ${\mathit{j}}_{\mathit{c}}$ decreases with increasing 2a, T, and Z; the decrease with temperature becomes more and more pronounced as the electron concentration n in the Sm layer decreases; the decrease with Z can be understood by the Z dependence of the Andreev scattering probability. The Josephson currents computed from Andreev scattering are so large that they should destroy any pair potential possibly induced in the Sm layer by the proximity effect.