Tunneling in the Normal-Metal-Insulator-Superconductor Geometry Using the Bogoliubov Equations of Motion

Abstract

The quasiparticle (or thermal current) transmission probability $W\left(E\right)\mathrm{}$ for excitations going from a normal to a superconducting region through an oxide layer ($N-I-S$ geometry) is calculated from first principles using the Bogoliubov—de Gennes equations of motion. We also work out the transmission probability $\overline{W}\mathrm{}\left(E\right)\mathrm{}$ for electrical currents. For thick oxide layers ($W_{\mathrm{NN}}\ll 1$), we find that both $W\left(E\right)\mathrm{}$ and $\overline{W}\mathrm{}\left(E\right)\mathrm{}$ are given by $\frac{W_{\mathrm{NN}}E}{{(E^2-{\Delta }^2)}^{\frac{1}{2}}}$ for $E>\mathrm{}\Delta$. This is in agreement with the tunneling Hamiltonian approach. In the opposite limit of no oxide layer ($W_{\mathrm{NN}}=1\mathrm{}$), we find that $W\left(E\right)\mathrm{}$ goes smoothly over into the expression obtained by Andreev for $N-S$ junctions. On the other hand, $\overline{W}\mathrm{}\left(E\right)\mathrm{}$ reduces to unity, as expected. All our results are for sharp interfaces.