Enhancement of the irreversibility field by carbon substitution in single crystal MgB$_2$

We report the detailed study of the irreversibility field {\hirr} of single crystals of Mg(B$_{1-x}$C$_{x}$)$_2$ ($x$=0, 2, 3.5, 5 and 7%) in strong magnetic fields of up to 40 T. High sensitive torque measurements revealed that the {\hirr} is greatly enhanced by a factor of two ($\mu_{0}${\hirr}(0)$\simeq$33 T in $x$=5%) by carbon substitution compared to the pristine {\mgb} owing to a reduction in the inplane coherence length. We also observed the temperature-dependent {\hirr} anisotropy in all the carbon contents. This fact strongly suggests that the two-gap superconductivity of {\mgb} is less influenced by a small amount of carbon substitution. We discuss these results focusing on the impact of carrier doping and impurity scattering on two-gap superconductor.