Implications of $d_{x^2-y^2}$ Symmetry and Faceting for the Transport Properties of Grain Boundaries in High-$T_{\rm c}$ Superconductors

Grain boundaries in high-$T_{\rm c}$ superconductors have attracted wide interest for their potential in a variety of applications and in fundamental studies of high-$T_{\rm c}$ superconductivity. Two recent experimental results provide a basis for better understanding the grain boundary properties, the mechanisms of which, despite their widespread use, are not yet completely understood. First, it is now well established that the order parameter in many high-$T_{\rm c}$ cuprates has a predominant $d_{x^2-y^2}$ symmetry. Second, microscopy studies have revealed that practical grain boundaries are comprised of facets having various orientations and typical dimensions of the order of 10--100~nm. We analyze the combined effects of faceting and $d_{x^2-y^2}$ symmetry on the transport properties of high-$T_{\rm c}$ grain boundaries. It is found that these effects can partially account for the experimentally observed reduction of the critical current density $J_{\rm c}$ with increasing grain boundary angle $\alpha$. The angular dependence of $J_{\rm c}$ for individual grain boundary facets may deviate considerably from the $J_{\rm c}(\alpha)$ dependence observed in standard measurements that employ macroscopic grain boundaries. This also holds for the product of $J_{\rm c}$ and the normal state resistivity $\rho _n$. The $J_{\rm c}\rho _n$ product measured for standard grain boundary..