Boundary conditions for Dirac fermions on a terminated honeycomb lattice

By A. Akhmerov, C. Beenakker. 2007

372 citations recorded by the source corpus.

We derive the boundary condition for the Dirac equation corresponding to a tight-binding model on a two-dimensional honeycomb lattice terminated along an arbitrary 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.

Subfields: Atomic and Molecular Physics, and Optics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/5c8a4a81744791e5904679664f8ea8608533cea7

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Atomic and Molecular Physics, and Optics

1. E. Rossi
2. E. Hwang
3. S. Adam
4. N. M. R. Peres
5. Franco Nori
6. A. Rozhkov
7. O. Yazyev
8. Rudolf Podgornik

In Materials Chemistry

1. N. M. R. Peres
2. E. Rossi
3. E. Hwang
4. S. Adam
5. O. Yazyev
6. Franco Nori
7. A. Rozhkov
8. Liying Jiao

Citations in the corpus

• Spatially resolving edge states of chiral graphene nanoribbons (2011, background): “From left to right, the chiralities of experimentally measured GNRs are (13, 1), (3, 1), (4, 1), (5, 2), and (8, 1) respectively, corresponding to a range of chiral angle 3.”
• Materials perspective on Casimir and van der Waals interactions (2015, background): “…for GNRs, on the other hand, is obtained by requiring the wave function be periodic along the GNR axis and vanish at the edges, which introduces edge dependent (zigzag or armchair) phase factors in the energy spectra (Akhmerov and Beenakker, 2008; Brey and Fertig, 2006; Sasaki et al., 2011).”
• Electronic transport in two-dimensional graphene (2010, background): “ulation, while Son et al. (2006a) showed that the density functional results could be obtained from the tightbinding model with some added edge disorder. By considering arbitrary boundary conditions, Akhmerov and Beenakker (2008) demonstrated that the behavior of the zigzag edge is the most generic ”
• Colloquium: The transport properties of graphene: An introduction (2010, methodology): “We consider that at r = R the wave function satisfies the zig-zag boundary conditions (Akhmerov and Beenakker, 2008; Dominguez-Adamé, 1990), ψ1,m(kR)/ψ2,m(kR) = 0, where ψi,m(kr), with i = 1, 2, is the i−component of the Dirac spinor.”
• Electronic properties of mesoscopic graphene structures: Charge confinement and control of spin and charge transport (2011, methodology): “The easiest way to detect the presence of edge states is to use the Weyl–Dirac equation with the boundary condition appropriate for the zigzag edge [34,36,50].”