Flat bands in the Weaire–Thorpe model and silicene

Y Hatsugai,K Shiraishi,H Aoki

doi:10.1088/1367-2630/17/2/025009

Abstract

In order to analytically capture and identify peculiarities in the electronic structure of silicene, the Weaire–Thorpe (WT) model, a standard model for treating three-dimensional (3D) silicon, is applied to silicene with a buckled 2D structure. In the original WT model for four hybridized sp3 orbitals on each atom along with inter-atom hopping, the band structure can be systematically examined in 3D, where flat (dispersionless) bands exist as well. For examining silicene, here we re-formulate the WT model in terms of the overlapping molecular-orbital (MO) method which enables us to describe flat bands away from the electron–hole symmetric point. The overlapping MO formalism indeed enables us to reveal an important difference: while in 3D the dipersive bands with cones are sandwiched by doubly-degenerate flat bands, in 2D the dipersive bands with cones are sandwiched by triply-degenerate and non-degenerate (nearly) flat bands, which is consistent with the original band calculation by Takeda and Shiraishi. Thus there emerges a picture for why the whole band structure of silicene comprises a pair of dispersive bands with Dirac cones with each of the bands touching a nearly flat (narrow) band at Γ. We can also recognize that, for band engineering, the bonds perpendicular to the atomic plane are crucial, and that ferromagnetism or structural instabilities are expected if we can shift the chemical potential close to the flat bands.

Highlights

After the physics of graphene was kicked off, originally by a theoretical prediction for a massless Dirac fermion by Wallace[1] back in the 1950’s and by a recent experimental realization[2], interests are extended to wider class of systems
Silicene has a larger degree of freedom than graphene as a target material for applications and theoretical considerations. We focus on this multi-orbital feature of silicene, where we opt for a simplified model, namely we propose to introduce as an extension of the Weaire-Thorpe(WT) model, which was originally conceived for a three-dimensional (3D) silicon with sp3 orbits in a tightbinding model on a diamond lattice[8]
The overlapping MO formulation enables us to pin point, algebraically, an important difference: while in 3D the dipersive bands with cones are sandwiched by doubly-degenerate flat bands, in 2D the dipersive bands with cones are sandwiched by triply-degenerate and non-degenerate flat bands, which is consistent with the original band calculation by Takeda and Shiraishi[4]

Summary

Introduction

After the physics of graphene was kicked off, originally by a theoretical prediction for a massless Dirac fermion by Wallace[1] back in the 1950’s and by a recent experimental realization[2], interests are extended to wider class of systems. The flat bands can have some topological/geometrical origins reflecting the multi-orbital character of a given material as we describe in the present paper. Here we re-formulate the WT model in terms of the overlapping molecular-orbital (MO) method[11] discussed by Hatsugai and Maruyama, which contains WT and enables us to describe flat bands away from the electron-hole symmetric point. We first start with an overlapping molecular orbital theory[11] Applying this to the WT model enables us to generically treat the flat bands away from the electron-hole symmetric point in multi-orbital models while the usual flatband theories[12, 13, 14, 15] focus on those at the electron-hole symmetric point. For the band engineering the bonds perpendicular to the atomic plane are suggested to be crucial

Overlapping molecular orbitals and flat bands

Weaire-Thorpe model

Silicene in a Weaire-Thorpe type model

Conclusion