Boundary resistance of the superconducting-normal interface

Abstract

The resistances of superconductor—normal-metal—superconductor sandwiches have been measured in which the mean free path of the superconductor was greater than or comparable with the coherence length. Below about 0.5 $T_c$, the resistance was nearly independent of temperature and, to within the experimental error, equal to the estimated resistance of the normal layer, indicating that there was relatively little interface contamination. At temperatures above about 0.9 $T_c$ the resistance rose rapidly as the temperature was increased towards $T_c$. A theory for this rise, based on the normal metal-insulator-superconductor tunneling theory of Tinkham and Clarke, is used to calculate the quasiparticle charge imbalance, $Q^{*}$, injected into the superconductor from the normal layer. The resultant additional boundary voltage per unit current is expressed as a boundary resistance $R_b=\frac{Z\left(T\right)\mathrm{}{\left(D{\tau }_{Q^{*}}\right)}^{\frac{1}{2}}{\rho }_S}{A}$, where $Z\left(T\right)\mathrm{}$ is a universal function of temperature, and $D$, ${\tau }_{Q^{*}}$, ${\rho }_S$, and $A$ are the electron diffusion coefficient, the charge relaxation time, the normal-state resistivity, and the cross-section area of the superconductor. Above 0.9 $T_c$, the data are an excellent fit to the theory if one takes ${\tau }_{Q^{*}}=\frac{4k_{B}T{\tau }_{E=0\mathrm{}}\left(T_c\right)}{\pi {\Delta }_{\infty }\mathrm{}\left(T\right)\mathrm{}}$, where ${\tau }_{E=0\mathrm{}}\left(T_c\right)$ is the inelastic-scattering time at the Fermi surface at $T_c$, and ${\Delta }_{\infty }\mathrm{}\left(T\right)\mathrm{}$ is the energy gap far from the interface. The inferred values of ${\tau }_{E=0\mathrm{}}\left(T_c\right)$ in ${\mathrm{Pb}}_{0.99}$ ${\mathrm{Bi}}_{0.01}$, Sn, ${\mathrm{Sn}}_{0.99}$ ${\mathrm{In}}_{0.01}$, and In, 0.25 × ${10}^{-10\mathrm{}}$ s, 2.6 × ${10}^{-10\mathrm{}}$ s, 1.1 × ${10}^{-10\mathrm{}}$ s, and 1.1 × ${10}^{-10\mathrm{}}$ s, respectively, are generally in good agreement with the computed values of Kaplan et al.