Multiple unpinned Dirac points in group-Va single-layers with phosphorene structure

Yunhao Lu,Shengyuan A Yang,Guoqing Chang,Yoshiyuki Kawazoe,Yinzhu Jiang,Shan Guan,Jianzhong Jiang,Di Zhou,Hsin Lin,Xue-Sen Wang,Weiguang Chen,Yuan Ping Feng

doi:10.1038/npjcompumats.2016.11

Abstract

AbstractEmergent Dirac fermion states underlie many intriguing properties of graphene, and the search for them constitutes one strong motivation to explore two-dimensional (2D) allotropes of other elements. Phosphorene, the ultrathin layers of black phosphorous, has been a subject of intense investigations recently, and it was found that other group-Va elements could also form 2D layers with similar puckered lattice structure. Here, by a close examination of their electronic band structure evolution, we discover two types of Dirac fermion states emerging in the low-energy spectrum. One pair of (type-I) Dirac points is sitting on high-symmetry lines, while two pairs of (type-II) Dirac points are located at generic k-points, with different anisotropic dispersions determined by the reduced symmetries at their locations. Such fully-unpinned (type-II) 2D Dirac points are discovered for the first time. In the absence of spin-orbit coupling (SOC), we find that each Dirac node is protected by the sublattice symmetry from gap opening, which is in turn ensured by any one of three point group symmetries. The SOC generally gaps the Dirac nodes, and for the type-I case, this drives the system into a quantum spin Hall insulator phase. We suggest possible ways to realise the unpinned Dirac points in strained phosphorene.

Highlights

Recent years have witnessed a surge of research interest in the study of Dirac fermions in condensed matter systems, ranging from graphene and topological insulator surfaces in two-dimensions (2D) to Dirac and Weyl semimetals in 3D,1–4 which possess many intriguing physical properties owing to their relativistic dispersion and chiral nature
2D Dirac fermion states have been extensively discussed in honeycomb lattices, commonly shared by group-IVa elements with graphene as the most prominent example,[5,6,7,8,9] for which Dirac points are pinned at the two inequivalent high-symmetry points K and K0 of the hexagonal Brillouin zone (BZ), around which the dispersion is linear and isotropic
Xu et al, unpublished) and Bi29–32 grown on suitable substrates, and been predicted for As as well.[22]. Motivated by these previous experimental and theoretical works, and in view of the ubiquitous presence of the Dirac fermions and the associated interesting physics, one may wonder: Is it possible to have Dirac fermion states hosted in such 2D puckered lattices? A simple consideration shows that here any possible Dirac point cannot occur at high-symmetry points

Summary

INTRODUCTION

Recent years have witnessed a surge of research interest in the study of Dirac fermions in condensed matter systems, ranging from graphene and topological insulator surfaces in two-dimensions (2D) to Dirac and Weyl semimetals in 3D,1–4 which possess many intriguing physical properties owing to their relativistic dispersion and chiral nature. When SOC is included, the Dirac nodes would generally be gapped.[36] For type-I points, treating SOC as a perturbation, its leading-order symmetry-allowed form is HSOC 1⁄4 τΔσzsz, where sz is Pauli matrix for real spin. This is similar to the intrinsic SOC term in graphene,[37] which opens a gap of 2|Δ| at the Dirac points. Breaking all three symmetries i, c2y and c2z can generate a trivial gap term mσz at the Dirac points, which competes with the SOC gap This happens when each atomic plane forms additional buckling structure.[32] as long as the Figure 4. Stress that the type-II points here are distinct in that they are realized in a native crystalline structure with relatively high

DISCUSSION

MATERIALS AND METHODS