Models ofc-axis twist Josephson tunneling

We calculate the critical current density JcJ(φ₀) for Josephson tunneling between identical high-temperature superconductors twisted an angle φ₀ about the c axis. Regardless of the shape of the two-dimensional Fermi surface and for very general tunneling matrix elements, an order parameter (OP) with general d-wave symmetry leads to JcJ(π/4)=0. This general result is inconsistent with the data of Li et al. [Phys. Rev. Lett. 83, 4160 (1999)] on Bi₂Sr₂CaCu₂O_8+δ (Bi2212), which showed JcJ to be independent of φ₀. If the momentum parallel to the barrier is conserved in the tunneling process, JcJ should vary substantially with the twist angle φ₀ when the tight-binding Fermi surface appropriate for Bi2212 is taken into account, even if the OP is completely isotropic. We quantify the degree of momentum nonconservation necessary to render JcJ(φ₀) constant within experimental error for a variety of pair states by interpolating between the coherent and incoherent limits using five specific models to describe the momentum dependence of the tunneling matrix element squared. From the data of Li et al., we conclude that the c-axis tunneling in Bi2212 must be very nearly incoherent, and that the OP must have a nonvanishing Fermi-surface average for T<~Tc. We further show that the apparent conventional sum-rule violation observed by Basov et al. [Science 283, 49 (1999)] can be consistent with such strongly incoherent c-axis tunneling.