Higher-order contributions to the Rashba-Bychkov effect with application to the Bi/Ag(111) surface alloy

Abstract

In order to explain the anisotropic Rashba-Bychkov effect observed in several metallic surfacestate systems, we use k·p perturbation theory with a simple group-theoretical analysis and construct effective Rashba Hamiltonians for different point groups up to third order in the wavenumber. We perform relativistic ab initio calculations for the (• Bi/Ag(111) surface alloy and from the calculated splitting of the band dispersion we find evidence of the predicted third-order terms. Furthermore, we derive expressions for the corresponding third-order Rashba parameters to provide a simple explanation to the qualitative difference concerning the Rashba-Bychkov splitting of the surface states at Au(111) and Bi/Ag(111).

Spin-resolved band structure of a densely packed Pb monolayer on Si(111)

“…The apparently larger splitting δk y ≈ 0.1Å −1 of the peaks a, b and e, f compared to the K and M directions is due to the intersection under a finite angle (≈60 • ) with respect to the high-symmetry directions. Moreover, the S z components at these k points are nonzero in consistency with the Rashba model including third-order terms within C 3v symmetry [43,44]. In other words, the existence of finite S z components corroborates that higher-order Rashba terms are necessary for the description of the band structure.…”

Strongly anisotropic spin-orbit splitting in a two-dimensional electron gas

“…Indeed, even the dispersion of the topologically protected surface states on these materials can be viewed in the context of anisotropic Rashba-type splitting [17,18]. For several systems, it has been found that the simple Rashba effect which is linear in k is insufficient to describe the spin-orbit splitting and further correction terms such as a cubic anisotropy [17,18] and a cubic isotropic correction were introduced [19].…”

“…Indeed, even the dispersion of the topologically protected surface states on these materials can be viewed in the context of anisotropic Rashba-type splitting [17,18]. For several systems, it has been found that the simple Rashba effect which is linear in k is insufficient to describe the spin-orbit splitting and further correction terms such as a cubic anisotropy [17,18] and a cubic isotropic correction were introduced [19].Here we show that strongly Rashba-split 2DEGs with intriguing new properties can be created by surfacedoping of the topological insulator Bi 2 Te 2 Se. The band splitting in these 2DEGs is anisotropic to a degree that the bands are strongly spin-split in one direction and almost degenerate in another, possibly paving the way towards new anisotropic transport phenomena.…”

“…An anisotropic spin splitting is only predicted when geometric anisotropy of the potential is taken into account [22] or higher order Rashba terms are included in a k·p model Hamiltonian [17][18][19]. For the present case of the C 3v point group, the symmetry-constrained energies for the splitting of a free electron-like state are given by [17,18] …”

“…With this, the splitting has the highest complexity permitted up to the third order given the symmetry constraints of the system [19]. Anisotropicity and non-parabolicity are taken into account by expanding the effective mass in the kinetic energy term as a polynomial of k as…”