Boundary conditions for Dirac fermions on a terminated honeycomb lattice

Abstract

We derive the boundary condition for the Dirac equation corresponding to a tight-binding model on a two-dimensional honeycomb lattice terminated along an arbitary direction. Zigzag boundary conditions result generically once the boundary is not parallel to the bonds. Since a honeycomb strip with zigzag edges is gapless, this implies that confinement by lattice termination does not in general produce an insulating nanoribbon. We consider the opening of a gap in a graphene nanoribbon by a staggered potential at the edge and derive the corresponding boundary condition for the Dirac equation. We analyze the edge states in a nanoribbon for arbitrary boundary conditions and identify a class of propagating edge states that complement the known localized edge states at a zigzag boundary.Comment: 10 pages, 11 figures (v3, typos corrected, expanded Sec. III

Electron Transport and Quantum-Dot Energy Levels in Z-Shaped Graphene Nanoconstriction with Zigzag Edges

“…The operational simplicity of the device [20], offering a fully-electrostatic control, has inspired designing its counterpart based on a nanoribbon with predominantly zigzag edges [31], similar to these already obtained in well-controlled fabrication processes [7][8][9]. An additional motivation to focus on ribbons with zigzag edges comes from the two theoretical findings: (i) local--density approximation results [32] show that the electronic structure of narrow graphene constrictions can be well described by a simple tight binding model, and (ii) analytical discussion of the tight-binding equations for a semi-infinite honeycomb lattice [33] shows that the zigzag boundary condition applies generically (at low energies) to arbitrary lattice termination, except from the case we have a perfect armchair edge.…”

“…Such a constriction, also known as the valley filter [21], was shown to produce, upon ballistic injection of current, the nonequilibrium valley polarization in a sheet of graphene attached. Motivated by the related analytical result for generic boundary conditions of graphene flakes [33] we have shown numerically [34] that valley polarization is also produced by constrictions with other edges, apart from the perfect armchair edges. These observations are rationalized as follows: For a constriction long enough, the transport becomes one-dimensional, and crystallographic orientation of edges determines the direction of propagation in the first Brillouin zone (see Fig.…”

Exploring the graphene edges with coherent electron focusing

“…If the magnetic length l B = ͱ ប / ͉eB͉ is much larger than the lattice constant of graphene, the general energyindependent boundary condition has the form of a local restriction on the components of the wave function ⌿ at the edge ͑E͒. 19,20 It can be cast into the following form: M ⌿ = ⌿, where the 4 ϫ 4 matrix M may be chosen as Hermitian and unitary matrix: M = M † and M 2 = Î. One can show 20 that demanding: ͑a͒ that the probability current normal to the boundary be zero; ͑b͒ that the boundary should preserve the electron-hole symmetry of the bulk; ͑c͒ and finally, assuming that the boundary conditions do not break the time-reversal symmetry, leads to the following form of the matrix M :…”

“…20 The boundary condition is given by M = ml ͑ z z ͒, where ml = sgn͑m − l͒, where sgn͑ ...͒ is the sign function. After performing analogous calculations as for the armchair edge one finds that…”

van der Waals interaction in magnetic bilayer graphene nanoribbons

“…11 Other edge terminations are possible, but they can be mapped onto three basic types, the armchair being the only one without edge states. 12,13 These localized states close to the Fermi energy are responsible for the magnetic and transport properties of zigzag graphene ribbons, and they are the origin of defect-related interface bands in graphene junctions. 14 Within a simple tightbinding model, armchair graphene nanoribbons (AGNRs) can be either metallic or semiconducting depending on their width, 15 whereas zigzag graphene nanoribbons (ZGNRs) are metallic with edge states.…”