Intrinsic anisotropy of critical current in c-tilted MgB$_{2}$ thin films

The intense investigations of MgB$_{2}$ in recent years have revealed many unusual effects of the two-band superconductivity on the properties of this material. However, given the size and geometry of available single crystals, the critical current along the c-direction, mostly affected by the anisotropy of the two bands, has not been thoroughly investigated. To address the issue of the c-axis current transport, we succeeded to measure J$_{\mathrm{c}}$ along the c axis by growing epitaxial films with tilted c-axis, which offers a unique possibility of probing both the in-plane and the out-of-plane critical currents. We show that anisotropic band parameters determine the anisotropy of vortex pinning with respect to the current direction. This yields a temperature and field dependent anisotropy of the critical current density J$_{\mathrm{c}}$ related to the anisotropic penetration depths at low temperature and low field and to the ratio of $\sigma $ band effective masses along the c and ab crystalline axes at temperatures close to T$_{\mathrm{c}}$ or at applied fields larger than the virtual upper critical field of the $\pi $ band H $\approx $ 0.2T. Our results suggest that the out-of-plane current transport could set the ultimate J$_{\mathrm{c}}$ limit in randomly oriented MgB$_{2}$ polycrystals.