Landau levels in spin-orbit coupling proximitized graphene: Bulk states

Abstract

We study the magnetic-field dependence of Landau levels in graphene proximitized by large spin-orbit coupling materials, such as transition-metal dichalcogenides or topological insulators. In addition to the Rashba coupling, two types of intrinsic spin-orbit interactions, uniform (Kane-Mele type) and staggered (valley Zeeman type), are included to resolve their interplay with magnetic orbital effects. Employing a continuum model approach, we derive analytic expressions for low-energy Landau levels, which can be used to extract local orbital and spin-orbit coupling parameters from scanning probe spectroscopy experiments. We compare different parameter regimes to identify fingerprints of relative and absolute magnitudes of intrinsic spin-orbit coupling in the spectra. The inverted band structure of graphene proximitized by WSe 2 leads to an interesting crossing of Landau states across the bulk gap at a crossover field, providing insights into the size of Rashba spin-orbit coupling. Landau-level spectroscopy can help to resolve the type and signs of the intrinsic spin-orbit coupling by analyzing the symmetry in energy and number of crossings in the Landau fan chart. Finally, our results suggest that the strong response to the magnetic field of Dirac electrons in proximitized graphene can be associated with extremely large self-rotating magnetic moments.

Edge states in proximitized graphene ribbons and flakes in a perpendicular magnetic field: emergence of lone pseudohelical pairs and pure spin-current states

“…We employ a tight-binding model supplemented with Peierl's substitution to study the electronic structure of graphene zigzag nanoribbons and finite flakes. The Landau levels calculated by the tight-binding approach are in excellent agreement with the bulk Landau level predictions [42].…”

“…A detailed derivation of the Hamiltonian of the Landau levels for bulk proximitized graphene is presented in Ref. [42]. Graphene with spin-orbit coupling in magnetic fields was studied from different perspectives by other groups as well [45,60].…”

“…On the other hand, in the Kane-Mele model, a bulk band gap forms between the zero Landau levels, which is stable under an external magnetic field. In the systems with valley-Zeeman spinorbit coupling, the bulk band gap closes and reopens at a crossover value of the external magnetic field equal to [42]…”

“…To compare with the bulk [42], we also plot bulk Landau levels at K and K . For small magnetic fields (1.5 mT), the Landau levels start to form and bands at the K and K valleys lose their dispersion.…”

“…Zeeman effects in proximitized graphene were investigated earlier, demonstrating the appearance of the quantum anomalous Hall effect [39] and chiral Majorana modes [40]. Magnetic orbital response of helical edge states to magnetic fields was studied in previous works [7,[41][42][43][44][45][46][47]. Rather surprisingly, the quantum spin Hall edge states, which are generated by uniform intrinsic (Kane-Mele) spin-orbit coupling, are not necessarily destroyed by the cyclotron effect [48,49], which can theoretically be used to switch between the quantum spin Hall and quantum Hall regimes by gating.…”