We report the effects of carbon incorporation on the structural and magnetic properties of epitaxial Mn${}_{5}$Ge${}_{3}$C${}_{x}$ films grown on Ge(111) by the solid phase epitaxy method. This variation of molecular beam epitaxy favors the diffusion process of carbon atoms. We show that up to a carbon molar concentration $x$ of \ensuremath{\sim}0.6--0.7, the atoms are incorporated in the interstitial sites of the Mn${}_{5}$Ge${}_{3}$ lattice. Such a process results in a linear increase of the Curie temperature ${T}_{C}$ of the alloy, which can reach a value as high as \ensuremath{\sim}430 K [${T}_{C}\ensuremath{\approx}460$ K at $M({T}_{C})=0$]. Above this carbon content, ${T}_{C}$ is found to decrease. Structural characterizations reveal that Mn${}_{5}$Ge${}_{3}$C${}_{x}$ films are in perfect epitaxy when $x$ \ensuremath{\le} 0.6, whereas cluster formation in the grown layers is detected above that threshold. The clusters can be attributed to manganese carbide (MnC) compounds which are formed when the carbon content exceeds the saturation value of 0.6 by consuming previously deposited carbon. Theoretical calculations accurately reproduce the main trend of ${T}_{C}$ variation as well as the cluster formation for $x$ larger than the saturation content. In addition, we also show that after post-thermal annealing, the carbon-doped Mn${}_{5}$Ge${}_{3}$C${}_{x}$ alloys remain magnetically and structurally stable up to a temperature as high as 850 \ifmmode^\circ\else\textdegree\fi{}C. The results are very promising for integrating Mn${}_{5}$Ge${}_{3}$C${}_{x}$ into ferromagnetic-semiconductor heterostructures, the ultimate goal being the realization of spintronic devices.
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