A topological insulator and helical zero mode in silicene under an inhomogeneous electric field

Abstract

Silicene is a monolayer of silicon atoms forming a two-dimensional (2D) honeycomb lattice and shares almost all the remarkable properties of graphene. The low-energy structure of silicene is described by Dirac electrons with relatively large spin-orbit interactions owing to its buckled structure. A key observation is that the band structure can be controlled by applying an electric field to a silicene sheet. In particular, the gap closes at a certain critical electric field. Examining the band structure of a silicene nanoribbon, we show that a topological phase transition occurs from a topological insulator to a band insulator with an increase of electric field. We also show that it is possible to generate helical zero modes anywhere in a silicene sheet by adjusting the electric field locally to this critical value. The region may act as a quantum wire or a quantum dot surrounded by topological and/or band insulators. We explicitly construct the wave functions for some simple geometries based on the lowenergy effective Dirac theory. These results are also applicable to germanene, which is a 2D honeycomb structure of germanium.

Spin magnetic susceptibility of ferromagnetic silicene in the presence of Rashba spin-orbit coupling

“…9,11 This slight buckling is responsible for a strong intrinsic spin-orbit coupling (SOC), resulting in a band gap near the Dirac points which provides a mass to the Dirac electrons. 9,11,12 Moreover, a potential difference of ∆ z = E z d appears between the two sublattices while exposed to an external electric field E z which is vertically applied to the sheet, as presented in Fig. 1.…”

“…In absence of any external perturbation and in presence of those which ∆ z < ∆ so Dirac fermions in silicene are in the TI phase but when ∆ z > ∆ so a transition to the BI phase is expected. 9,11,12 The particular case of ∆ z = ∆ so leads silicene into the VSPM phase where the lowest band gap at each valley closes concluded by a Dirac-like point. 19 The MS of a matter yields its degree of magnetization in response to an external magnetic field.…”

Zigzag nanoribbons of two-dimensional silicene-like crystals: magnetic, topological and thermoelectric properties

“…[38][39][40][41][42] However, the role of Coulomb interaction and topological edge states in the gap, has not been studied thoroughly enough yet. In a recent paper 43 we have analyzed the influence of topological states on the thermoelectric properties, with the main focus on the role of a staggered exchange field [5][6][7] and of an electric field perpendicular to the atomic plane. Here we consider in more details the role of Coulomb interaction, which has a significant impact on the topological properties.…”

Superconducting proximity effect in silicene: Spin-valley-polarized Andreev reflection, nonlocal transport, and supercurrent

“…As a result, silicene is under the right circumstances a quantum spin Hall insulator with topologically protected edge states. In fact, it is possible [5] to achieve a rich variety of topological states in silicene due to a unique feature: the buckled structure causes the sublattices to respond differently to an applied electric field, which in turn induces a fermion mass-gap which is tunable. Closing and reopening this gap allows for a transition between different topological phases at a critical field value |E z | = E c as shown in Fig.…”

“…5 in the Supplementary Information). Using a tight-binding formalism, one obtains the following lattice Hamiltonian [5,10,11]:…”