Mechanisms of spontaneous current generation in an inhomogeneous d -wave superconductor

A boundary between two d-wave superconductors or an s-wave and a d-wave superconductor generally breaks time-reversal symmetry and can generate spontaneous currents due to proximity effect. On the other hand, surfaces and interfaces in d-wave superconductors can produce localized current-carrying states by supporting the $\mathcal{T}$-breaking combination of dominant and subdominant order parameters. We investigate spontaneous currents in the presence of both mechanisms and show that at low temperatures, counterintuitively, the subdominant coupling decreases the amplitude of the spontaneous current due to the proximity effect. Superscreening of spontaneous currents is demonstrated to be present in any ideal $d\ensuremath{-}d$ (but not $s\ensuremath{-}d)$ junction and surface with ${d+id}^{\ensuremath{'}}$ order parameter symmetry. We show that this superscreening is the result of contributions from the local magnetic moment of the condensate to the spontaneous current.