Time-reversal symmetry breaking at Josephson tunnel junctions of purely d-wave superconductors

We study spontaneous time-reversal symmetry breaking at Josephson tunnel junctions of d-wave superconductors in the absence of subdominant components of the order parameter. For tunnel junctions, when the orientation is close to 0/45 (for which a gap lobe points towards the junction on one side and a gap node on the other), the mechanism of the symmetry breaking is the splitting of midgap states (MGS) by spontaneous establishment of a phase difference $\phi=\pm \pi/2$ across the junction. This occurs for transparencies $D\gg\xi_0/\lambda$ and temperatures $k_BT\ll D\Delta_0$, where $\xi_0$ is the coherence length, $\lambda$ is the penetration depth, and $\Delta_0$ is the maximum energy gap. On the other hand. tunnel junctions with $D\ll\xi_0/\lambda$ effectively behave as surfaces, for which the mechanism of symmetry breaking is self-induced Doppler shifts of MGS. For this instability, we calculate the phase transition temperature $k_BT_{TRSB}=(1/6)(\xi_0/\lambda)\Delta_0$ and show that the spatial shape of the gap is unimportant.