  Investigations of MgB2 and Fe-based superconductors in recent years have
revealed many unusual effects of multiband superconductivity but manifestations
of anisotropic multiband effects in the critical current density Jc have not
been addressed experimentally, mostly because of the difficulties to measure Jc
along the c-axis. To investigate the effect of very different intrinsic
anisotropies of sigma and pi electron bands in MgB2 on current transport, we
grew epitaxial films with tilted c-axis (THETA ~ 19.5{\deg}), which enabled us
to measure the components of Jc both along the ab-plane and the c-axis using
magneto-optical and transport techniques. These measurements were combined with
scanning and transmission electron microscopy, which revealed terraced steps on
the surface of the c-axis tilted films. The measured field and temperature
dependencies of the anisotropic Jc(H) show that Jc,L parallel to the terraced
steps is higher than Jc,T perpendicular to the terraced steps, and Jc of
thinner films (50 nm) obtained from transport experiments at 0.1 T reaches ~10%
of the depairing current density Jd in the ab plane, while magneto-optical
imaging revealed much higher Jc at lower fields. To analyze the experimental
data we developed a model of anisotropic vortex pinning which accounts for the
observed behavior of Jc in the c-axis tilted films and suggests that the
apparent anisotropy of Jc is affected by current pairbreaking effects in the
weaker {\pi} band. Our results indicate that the out-of-plane current transport
mediated by the {\pi} band could set the ultimate limit of Jc in MgB2
polycrystals.
