Among the many anticipated applications of graphene, some - such as transistors for Si microelectronics - would greatly benefit from the possibility to deposit graphene directly on a semiconductor grown on a Si wafer. We report that Ge(001) layers on Si(001) wafers can be uniformly covered with graphene at temperatures between 800{deg}C and the melting temperature of Ge. The graphene is closed, with sheet resistivity strongly decreasing with growth temperature, weakly decreasing with the amount of deposited C, and reaching down to 2 kOhm/sq. Activation energy of surface roughness is low (about 0.66 eV) and constant throughout the range of temperatures in which graphene is formed. Density functional theory calculations indicate that the major physical processes affecting the growth are: (1) substitution of Ge in surface dimers by C, (2) interaction between C clusters and Ge monomers, and (3) formation of chemical bonds between graphene edge and Ge(001), and that the processes 1 and 2 are surpassed by CH$_{2}$ surface diffusion when the C atoms are delivered from CH$_{4}$. The results of this study indicate that graphene can be produced directly at the active region of the transistor in a process compatible with the Si technology.
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