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

We study spontaneous time-reversal symmetry breaking at Josephson tunnel ju nctions 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 sp litting of midgap states (MGS) by spontaneous establishment of a phase diff erence phi=+/-pi /2 across the junction. This occurs for transparencies D m uch greater than xi (0)/lambda and temperatures k(B)T much less thanD 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 juncti ons with D much less than 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 (B)T(TRSB)=(1/6)(xi (0)/lambda)Delta (0) and show that the spatial shape of the gap is unimportant.