Degenerate Perturbation in Band-Gap Opening of Graphene Superlattice

Abstract

We study the graphene band-gap engineering by introducing different defects, namely the defects breaking the inversion symmetry and the ones periodically patterning graphene into superlattice such as the regularly arranged antidots, etc. Comparing to the primitive unit cell of graphene, the pseudo-graphene superlattice, referred to the pristine graphene supercell, modulates the boundary condition accordingly. According to the energy band-folding picture, these superlattices can be categorized into two groups on the basis of the Dirac cone position. In some cases, the Dirac points K and K′ in primitive cell are folded to the Γ point of pseudo-superlattices. The coincidence of Dirac points with Γ point results in the fourfold degeneracy. In these systems, a band gap at Γ point can be opened by introducing periodically arranged defects such as the antidots, which could be easily utilized in experiment, for example, by making the graphene nanomesh through lithography technique. In the other cases, the twofold degenerate Dirac points remain nonequivalent with Γ point in pseudo-superlattice, and the method to break the inversion symmetry could open their band gaps.

Inducing a Finite In-Plane Piezoelectricity in Graphene with Low Concentration of Inversion Symmetry-Breaking Defects

“…In the other case, the twofold degenerate Dirac points do not fold into Γ and a band gap opening can be induced by breaking the inversion symmetry. 55 We will discuss below how these two cases can be clearly discriminated also from the analysis of the elastic and piezoelectric response of BNG structures. …”

Engineering the Electronic Structure of Two-Dimensional Materials with Near-Field Electrostatic Effects of Self-Assembled Organic Layers

“…All SAMs modulate the potential on the scale of 100meV with patterns decided by the size and shape of molecules: Muffin-tin type for benzene/graphene, Kronig-Penney type for C8-BTBT/graphene. We note that the use of such periodic potentials on comparable lengthscales has previously been proposed as a means to tune the electronic properties of graphene and other 2D materials [29,30,35,36,[55][56][57]. In Fig.…”

“…In the case of bulk 3D materials, electrostatic modification occurs through far-field effects resulting from the monopole and areal dipole of the material/molecule interface [1][2][3][8][9][10][11][12][13], with consequences on observables such as work functions [14][15][16], band offsets [17] and superconducting transitions [18]. While also impacted by far-field effects [5,6,19], 2D electronic states such as surface states [20][21][22] and electronic states in 2D materials [18,[23][24][25][26][27][28][29][30] can also be impacted by near-field effects resulting from higher moments of the surface/interface electronic density, typically observed within ≈ 10 Å [31][32][33] of the interface. In particular, for these systems, near-field modulation of the electrostatic potential has been proposed as a means to engineer band structures, wave functions, and topological properties [20,21,[34][35][36][37][38][39][40].…”

Energy Gap Tuning of Graphene Layers with Single Molecular F2 Adsorption

“…The rectangular supercell creates four-fold degeneracy at Γ point. 33 As a first remark, the band structure of two physisorbed adsorption sites, that is, center and bridge of in-plane sites, is The Journal of Physical Chemistry C Article similar to that of isolated graphene, which includes two sets of degenerate π bands with presenting a linear dispersion at Dirac point; however, for chemisorbed sites, the degeneracy at cross site partially shifted and that for top site totally shifted in conclusion of mixing of states because of the chemisorbed mechanism. In contrast, for perpendicular F 2 orientation, there is a nearly flat band and band dispersion near the Dirac point, originated from local electronic states of the F atom, which is located on top of the graphene layer.…”