The Anisotropy of the London Penetration Depth and the Upper Critical Field in C-doped $MgB_2$ Single Crystals from Reversible Magnetization

We have studied the anisotropy of the London penetration depth of carbon doped $MgB_2$ single crystals, which was obtained from reversible magnetization measurements with the magnetic field both parallel and perpendicular to the c-axis. Similar to the pure $MgB_2$, the anisotropy of the upper critical field ${\gamma}_H$ decrease with temperature while the anisotropy of the London penetration depth ${\gamma}_{\lambda}$ slowly increases with temperature. However, the temperature dependence of ${\gamma}_H$ is drastically reduced and the value of ${\gamma}_{\lambda}$ becomes nearly ~1 as C is introduced. These results indicate that C substitution increases impurity scattering mainly in the $\sigma$ bands. The temperature dependence of the anisotropies agree well with the theoretical predictions with impurity scattering.