The proximity-induced spin-orbit coupling (SOC) in heterostructures of twisted graphene and topological insulators (TIs) Bi2Se3 and Bi2Te3 is investigated from first principles. To build commensurate supercells, we strain graphene and correct thus resulting band offsets by applying a transverse electric field. We then fit the low-energy electronic spectrum to an effective Hamiltonian that comprises orbital and spin-orbit terms. For twist angles 0°≤ Θ 20°, we find the dominant spin-orbit couplings to be of the valley-Zeeman and Rashba types, both a few meV strong. We also observe a sign change in the induced valley-Zeeman SOC at Θ ≈ 10°. Additionally, the in-plane spin structure resulting from the Rashba SOC acquires a non-zero radial component, except at 0°o r 30°. At 30°the graphene Dirac cone interacts directly with the TI surface state. We therefore explore this twist angle in more detail, studying the effects of gating, TI thicknesses, and lateral shifts on the SOC parameters. We find, in agreement with previous results, the emergence of the proximitized Kane-Mele SOC, with a change in sign possible by electrically tuning the Dirac cone within the TI bulk band gap.

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The valley Zeeman physics of excitons in monolayer transition metal dichalcogenides provides valuable insight into the spin and orbital degrees of freedom inherent to these materials. Being atomically-thin materials, these degrees of freedom can be influenced by the presence of adjacent layers, due to proximity interactions that arise from wave function overlap across the 2D interface. Here, we report 60 T magnetoreflection spectroscopy of the A- and B- excitons in monolayer WS$_2$, systematically encapsulated in monolayer graphene. While the observed variations of the valley Zeeman effect for the A- exciton are qualitatively in accord with expectations from the bandgap reduction and modification of the exciton binding energy due to the graphene-induced dielectric screening, the valley Zeeman effect for the B- exciton behaves markedly different. We investigate prototypical WS$_2$/graphene stacks employing first-principles calculations and find that the lower conduction band of WS$_2$ at the $K/K'$ valleys (the $CB^-$ band) is strongly influenced by the graphene layer on the orbital level. Specifically, our detailed microscopic analysis reveals that the conduction band at the $Q$ point of WS$_2$ mediates the coupling between $CB^-$ and graphene due to resonant energy conditions and strong coupling to the Dirac cone. This leads to variations in the valley Zeeman physics of the B- exciton, consistent with the experimental observations. Our results therefore expand the consequences of proximity effects in multilayer semiconductor stacks, showing that wave function hybridization can be a multi-step energetically resonant process, with different bands mediating the interlayer interactions. Such effects can be further exploited to resonantly engineer the spin-valley degrees of freedom in van der Waals and moiré heterostructures.

The proximity-induced spin-orbit coupling (SOC) in heterostructures of twisted graphene and topological insulators (TIs) Bi 2 Se 3 and Bi 2 Te 3 is investigated from first principles. To build commensurate supercells, we strain graphene and correct thus resulting band offsets by applying a transverse electric field. We then fit the low energy electronic spectrum to an effective Hamiltonian that comprises orbital and spin-orbit terms. For twist angles 0 • 20 • , we find the dominant spin-orbit couplings to be of the valley-Zeeman and Rashba types, both a few meV strong. We also observe a sign change in the induced valley-Zeeman SOC at ≈ 10 • . Additionally, the in-plane spin structure resulting from the Rashba SOC acquires a nonzero radial component, except at 0 • or 30 • . At 30 • the graphene Dirac cone interacts directly with the TI surface state. We therefore explore this twist angle in more detail, studying the effects of gating, TI thicknesses, and lateral shifts on the SOC parameters. We find, in agreement with previous results, the emergence of the proximitized Kane-Mele SOC, with a change in sign possible by electrically tuning the Dirac cone within the TI bulk band gap.