Since the prediction of a time-reversal symmetry breaking (TRSB) $d+i{d}^{\prime }$ state in twisted bilayer cuprate superconductors by Can et al. [Nat. Phys. 17, 519 (2021)], several experiments have attempted to detect this state, yielding conflicting results. At present, it is not clear which differences in samples or experimental conditions might explain these discrepancies. In this work, we perform a tight-binding lattice model calculation with phenomenological interlayer tunneling, examining the order parameter as a function of twist angle, interlayer tunneling, doping, and temperature. We observe the TRSB state to be correlated to the position of the Van Hove singularity in the normal state, which changes not only as a function of doping but also the tunneling strength. Two such phases are identified as nominally consistent with in-plane $d+i{d}^{\prime }$ and $d+is$ order, but with unexpected transformation properties under bilayer symmetry operations. We calculate the Josephson critical current, in particular examining the angle dependence for various tunneling strengths. Finally, we discuss the existing experiments in the context of our results.

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• Josephson effect
• Superconducting gap
• Superconducting order parameter
• Superconducting phase transition
• Cuprates
• Superconductors
• Twisted bilayers