  Heterostructures of graphene in proximity to magnetic insulators open the
possibility to investigate exotic states emerging from the interplay of
magnetism, strain and charge transfer between the layers. Recent reports on the
growth of self-integrated atomic wires of $\beta$-RuCl$_3$ on graphite suggest
these materials as versatile candidates to investigate these effects. Here we
present detailed first principles calculations on the charge transfer and
electronic structure of $\beta$-RuCl$_3$/heterostructures and provide a
comparison with the work function analysis of the related honeycomb family
members $\alpha$-RuX$_3$ (X = Cl,Br,I). We find that proximity of the two
layers leads to a hole-doped graphene and electron-doped RuX$_3$ in all cases,
which is sensitively dependent on the distance between the two layers.
Furthermore, strain effects due to lattice mismatch control the magnetization
which itself has a strong effect on the charge transfer. Charge accumulation in
$\beta$-RuCl$_3$ strongly drops away from the chain making such
heterostructures suitable candidates for sharp interfacial junctions in
graphene-based devices.
