Models of c-axis twist Josephson tunneling

We calculate the critical current density $J_c^J\left({\phi }_0\right)$ for Josephson tunneling between identical high-temperature superconductors twisted an angle ${\phi }_0$ 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 $J_c^J(\pi /4)=0.\mathrm{}$ This general result is inconsistent with the data of Li et al. [Phys. Rev. Lett. 83, 4160 (1999)] on ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ (Bi2212), which showed $J_c^J$ to be independent of ${\phi }_0.$ If the momentum parallel to the barrier is conserved in the tunneling process, $J_c^J$ should vary substantially with the twist angle ${\phi }_0$ 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 $J_c^J\left({\phi }_0\right)$ 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 $T<~T_c.$ 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.