The pure and carbon (C)-doped <TEX>$MgB_2$</TEX> thin films were fabricated on <TEX>$Al_2O_3$</TEX> (0001) substrates at a temperature of <TEX>$650^{\circ}C$</TEX> by using hot-filament-assisted hybrid physical-chemical vapor deposition technique. The <TEX>$T_c$</TEX> value for pure <TEX>$MgB_2$</TEX> film is 38.5 K, while it is between 30 and 35 K for carbon-doped <TEX>$MgB_2$</TEX> films. Expansion in c-axis lattice parameter was observed with increase in carbon doping concentration which is in contrast to carbon-doped <TEX>$MgB_2$</TEX> single crystals. Significant enhancement in the critical current density was obtained for C-doped <TEX>$MgB_2$</TEX> films as compared to the undoped <TEX>$MgB_2$</TEX> film. This enhancement is most probably due to the incorporation of C into <TEX>$MgB_2$</TEX> and the high density of grain boundaries, both help in the pinning of vortices and result in improved superconducting performance.
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