Time-reversal symmetry breaking and spontaneous currents ins-wave/normal-metal/d-wave superconductor sandwiches

We study the physical properties of an $s$-wave–normal-metal–$d$-wave junction in terms of the Andreev bound-state solutions to the Bogoliubov–de Gennes equation in the normal-metal layer. The phase dependence of bound states with different orientations leads to superconducting states with broken time-reversal symmetry for generic orientations of the $d$-wave superconductor crystal. The occurrence of such a state and the associated spontaneous supercurrent along the junction and magnetic field is analyzed also in the framework of Ginzburg-Landau theory.