Coexistence of Rashba and Dirac dispersions on the surface of centrosymmetric topological insulator decorated with transition metals

Abstract

The Dirac cone originates from the bulk topology, yet its primary contribution comes from the surface since spatially, the Dirac state emerges at the boundary between the trivial and topological phases. At the same time, the Rashba states emerge in regions where inversion symmetry is broken. On the surface of the centrosymmetric topological insulators, both Rashba and Dirac bands are present and their hybridization produces the giant Rashba effect, modifying both Rashba's and Dirac's bands. Therefore, pure Rashba and Dirac fermions are inherently incompatible on the surface of centrosymmetric topological insulators if the material is homogeneous. Inspired by recent experiments, we focused on the (111) polar surface of PbSe, which becomes a topological crystalline insulator under compressive strain, and we established the conditions under which a topological system can simultaneously host pure Rashba and Dirac surface states close to the Fermi level. The coexistence of pure Dirac and Rashba dispersions is only possible in a non-homogeneous centrosymmetric topological insulator, where the spatial origins of the two bands are effectively separated. In the experimentally observed case of PbSe, we demonstrate that a metallic overlayer induces a strong electrostatic potential gradient in the subsurface region, which in turn generates the electric field responsible for Rashba splitting in the subsurface layers. Consequently, in PbSe(111), the Rashba states arise from subsurface layers, while the Dirac states live mainly on the surface layers. Finally, we compare the properties of the Rashba in the trivial and topological phases; the calculated Rashba coefficient agrees qualitatively with the experimental results.

Figures (10)

• Figure 1: (a) Crystal structure of the TCI grown along the (111) direction with the Se termination. The TCI is covered by a single layer with the 3$d$ transition metals on top of Se atoms. The black, green and blue atoms represent the Pb, Se and Cu atoms, respectively. a, b and c are the real-space lattice vectors. (b) Three-dimensional Brillouin zone of the bulk TCI with three vectors of reciprocal lattice a*, b* and c*. In red and blue, there are the 6 and 2 L-points of the three-dimensional Brillouin zone, respectively. The projections of the reciprocal space on the (111) surface of the TCI with the surface Dirac cones at $\overline{\Gamma}$ (plotted in green) and $\overline{M}$ (plotted in purple) are shown.
• Figure 2: RSS and DSS of an asymmetric slab in the limit of large thickness. a) Pure RSS on the surface of normal insulators (NI). b) Interplay between RSS and DSS on the surface of topological insulators. c) Pure DSS on the surface of topological insulators with negligible spin-orbit or in which the RSS is absent. d) Coexistence of pure Rashba and Dirac fermions in the surface band structure of the TCI. The Dirac point was put close to the valence band, as found in the experimental results. Possible options to have the coexistence of pure Rashba and Dirac fermions: e) the surface becomes trivial and the Dirac bands lie in the subsurface and f) an electric field ($\vec{E}$) in the bulk pushes the Rashba states to lie in the subsurface. The Rashba coefficient is defined as the ratio $\alpha_R$=$\frac{2E_R}{k_0}$ where E$_R$ and k$_0$ are represented in panel a) and described in the text.
• Figure 3: Band structure of a symmetric uncovered slab in the trivial phase (left panel). Band structure of a symmetric uncovered slab in the topological phase (right panel). The Fermi level is set at zero energy.
• Figure 4: Band structure of an asymmetric uncovered slab in the trivial phase (left panel). Band structure of an asymmetric uncovered slab in the topological phase (right panel). The Fermi level is set at zero energy.
• Figure 5: Band structure of an asymmetric slab covered by the 3$d$ metal in the hollow position and the topological phase. The Fermi level is set at zero energy.