  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.
