Theory of quasiparticle tunneling between dissimilar superconductors

Quasiparticle tunneling in the Superconductor-Insulator-Superconductor geometry is studied theoretically by means of the Bogoliubov-de Gennes equations. A model is developed which preserves particle-hole symmetry throughout the system. The metals are treated in the effective-mass approximation, and each metal is characterized by its effective mass, Fermi energy, work function, pair potential, and coherence length, each of which is permitted to be different for the two metals. One of the metals is presumed to be monocrystalline and is modelled with an ellipsoidal Fermi surface characterized by its orientation and the three eigenvalues of its inverse effective-mass tensor. The pair potential in each metal varies continuously near the interfaces with a characteristic length given by the coherence length in that metal. In the insulator, account is taken of the intrinsic energy gap, applied potential, and the potential due to a static space charge. The physical effect of the image potential on the metal-insulator interfaces is discussed, and a resisting resonance phenomenon is analyzed. The Bogoliubov-de Gennes equations are solved in the metals and the solutions are matched, using appropriate boundary conditions, to solutions in narrow metallic regions of the insulator which are created near the interfaces by the image potential. Solutions on opposite sides of the insulator barrier are related by the normal-phase WKB connection formulas. The resulting system wave functions are used to calculate the electric current transmission and reflection coefficents for quasiparticles. The importance of the model parameters is studied by numerically calculating the tunneling coefficients under various conditions. The proper selection rule for tunneling is found to be that the component of the wave vector parallel to the interface vanish