Upper critical field and anisotropy in carbon alloyed MgB$_2$ thin films

We have studied the upper critical field and its anisotropy $\gamma$ of C-alloyed MgB$_2$ thin films grown on (0001) SiC single crystal substrates by hybrid physical-chemical vapor deposition (HPCVD). Different concentrations of C were introduced during the deposition, which increased the residual resistivity systematically but did not affect $T_c$ seriously. The upper critical field was found to increase to above 60 T for $H$ parallel to the ab plane and up to $\sim$ 40T for $H$ parallel to the $c$ axis with even moderate amounts of C doping. Moreover, we show that $H_{c2}$ stays at these levels in a wide range of C doping. The $H_{c2}$ anisotropy was found to decrease as the C concentration increases. The $H_{c2}(T)$ curves for both $H \parallel ab$ and $H \parallel c$ directions were explained by a theoretical model of dirty limit two band superconductivity, which takes into account different scattering rates in $\pi$ and $\sigma$ bands, as well as interband scattering. The differences in the $H_{c2}(T)$ and $\gamma (T)$ for different samples can be explained by the differences of the relative scattering rates in each band which make it possible to adjust the $\pi$ and $\sigma$ scattering such that $H_{c2}$ perpendicular to $ab$ can attain almost 20T at 20K.