Counterpropagating topological interface states in graphene patchwork structures with regular arrays of nanoholes

Abstract

Triangular and honeycomb lattices are dual to each other -if we puncture holes into a featureless plane in a regular triangular alignment, the remaining body looks like a honeycomb lattice, and vice versa, if the holes are in a regular honeycomb alignment, the remaining body has a feature of triangular lattice. In this work, we reveal that the electronic states in graphene sheets with nanosized holes in triangular and honeycomb alignments are also dual to each other in a topological sense. Namely, a regular hole array perforated in graphene can open a band gap in the energymomentum dispersion of relativistic electrons in the pristine graphene, and the insulating states induced by triangular and honeycomb hole arrays are distinct in topology. In a graphene patchwork with regions of these two hole arrays put side by side counterpropagating topological currents emerge at the domain wall. This observation indicates that the cerebrated atomically thin sheet is where topological physics and nanotechnology meet.Electrons behave as waves in microscopic world, and a regular array of scattering centers causes quantum interference, i.e., Bragg reflection, which governs the electron propagation in terms of energy and momentum. This explains band gaps and band insulators in crystals where ions are regularly aligned. The same principle is effective even if we zoom out a little bit: starting from Bloch waves, with which angstrom-scale structures of the underlying crystal are already taken into account, superstructures in micro-or meso-scopic scales can induce new band gaps and modify the electron propagation. The first example is the superlattice invented by Leo Esaki in order to control properties of semiconductors [1].In this regard, graphene -mono atomic sheet of carbon atoms in honeycomb structure [2] -is a promising playground. First, the honeycomb array of scattering centers is responsible for the most striking feature of graphene, emergent relativistic fermion [3,4] appearing as an isolated gap closing point associated with linear dispersion (Dirac cone) in the band structure. Secondly, graphene is amiable to nano structuring [5][6][7]. One idea is to introduce a regular array of holes, also known as antidot lattice, into graphene, with the remaining body dubbed as graphene nanomesh [8][9][10][11][12][13][14][15][16]. Depending on the hole alignment, the band structure of superstructured graphene can be either gapless or gapped, and in gapped cases the gap size is tunable [8,9,[17][18][19][20][21][22][23][24][25].Historically, gap introduction in a honeycomb lattice model, or mass attachment to emergent relativistic electrons, has been cornerstones in discovering new topological phases of matter. For instance, with an appropriate time reversal symmetry (TRS) breaking term, the honeycomb lattice model can derive the quantum anomalous Hall state [26], which is a typical topological state characterized by the Chern number [27]. When the spinorbit coupling (SOC) is considered in a honeycomb lattice model, one obtains the qua...

Topological magnon modes on honeycomb lattice with coupling textures

“…The p and d magnon modes are shown in Fig. 2 d. The parity of each band at the M point is the same as that at the point, denoting a topologically trivial state 38 , 39 . In the case , the frequencies of p and d modes are inverted at the point as shown in Fig.…”

“…2 c. Now, at the point, all three eigenstates below the gap are of even parity, while at the M point, there are two eigenstates with odd parity and one eigenstate with even parity. The unequal numbers of eigenstates with even parity indicate a topological state 38 , 39 . The band structure of topological state in Fig.…”

Floquet Graphene Antidot Lattices

“…We conclude with some ideas for future research. One immediate question that we have not answered is whether the Floquet graphene antidot lattice features non-trivial topology [71][72][73]. More precisely, this system combines two gapping mechanisms, the time-reversalsymmetry-breaking circularly polarized light and the chiral-symmetry-breaking antidot lattice, both of which retain the particle-hole (charge-conjugation) symmetry.…”

π -fluxes, semimetals, and flat bands in artificial materials

“…Alternative, more rigorous, support for the gapless edge modes at zigzag edges comes from the combination of the chiral (sublattice) symmetry and the mirror symmetry, whose reflection plane is perpendicular to the zigzag edge [31,42]. The gaplessness comes from the degeneracy of the two zero modes having opposite chirality (sublattice polarization) and the opposite parity with respect to the reflection, guaranteed by nontrivial mirror winding numbers.…”