  We use a combination of experimental techniques to demonstrate a general
occurrence of spin-orbit interaction (SOI) in graphene on transition metal
dichalcogenide (TMD) substrates. Our measurements indicate that SOI is
ultra-strong and extremely robust, despite it being merely
interfacially-induced, with neither graphene nor the TMD substrates changing
their structure. This is found to be the case irrespective of the TMD material
used, of the transport regime, of the carrier type in the graphene band, and of
the thickness of the graphene multilayer. Specifically, we perform weak
antilocalization measurements as the simplest and most general diagnostic of
SOI, and show that the spin relaxation time is very short in all cases
regardless of the elastic scattering time. Such a short spin-relaxation time
strongly suggests that the SOI originates from a modification of graphene band
structure. We confirmed this expectation by measuring a gate-dependent beating,
and a corresponding frequency splitting, in the low-field Shubnikov-de Haas
magneto-resistance oscillations in high quality bilayer graphene on WSe$_2$.
These measurements provide an unambiguous diagnostic of a SOI-induced splitting
in the electronic band structure, and their analysis allows us to determine the
SOI coupling constants for the Rashba term and the so-called spin-valley
coupling term, i.e., the terms that were recently predicted theoretically for
interface-induced SOI in graphene. The magnitude of the SOI splitting is found
to be on the order of 10 meV, more than 100 times greater than the SOI
intrinsic to graphene. Both the band character of the interfacially induced
SOI, as well as its robustness and large magnitude make graphene-on-TMD a
promising system to realize and explore a variety of spin-dependent transport
phenomena, such as, in particular, spin-Hall and valley-Hall topological
insulating states.
