Crystal-Field Engineering of Spin-Orbit Coupling in Graphene Heterostructures*

Spin-orbit interaction is an essential ingredient to the manifestation of interesting phenomena like spin-Hall effect and nontrivial topological phases. Graphene has an extremely weak spin-orbit coupling. However, different substrates are able to generate a variety of spin-orbit couplings by proximity effect, depending on the symmetries that are broken in graphene.

The influence of a crystal lattice environment on states of an atom is called crystal-field effect, which is widely present in condensed matter and is essential to understand the physics of solids. In this work, we use a tight-binding approach to obtain an effective low-energy Hamiltonian for graphene when it is stacked on a substrate. We consider that atoms of the substrate are under the effect of low symmetric crystal field. We than include, perturbatively, the spin-orbit coupling on atoms of the substrate and show how to generate different spin-orbit couplings in graphene, depending on the crystal-field of the substrate.These results open a novel route to engineer spin-orbit coupling in graphene.