Band topology and the quantum spin Hall effect in bilayer graphene

Abstract

We consider bilayer graphene in the presence of spin orbit coupling, to assess its behavior as a topological insulator. The first Chern number n for the energy bands of single and bilayer graphene is computed and compared. It is shown that for a given valley and spin, n in a bilayer is doubled with respect to the monolayer. This implies that bilayer graphene will have twice as many edge states as single layer graphene, which we confirm with numerical calculations and analytically in the case of an armchair terminated surface. Bilayer graphene is a weak topological insulator, whose surface spectrum is susceptible to gap opening under spin-mixing perturbations. We also assess the stability of the associated topological bulk state of bilayer graphene under various perturbations. Finally, we consider an intermediate situation in which only one of the two layers has spin orbit coupling, and find that although individual valleys have non-trivial Chern numbers, the spectrum as a whole is not gapped, so that the system is not a topological insulator.

The electronic properties of bilayer graphene

“…Taken with the quadratic termĥ 0 in the Hamiltonian (38),ĥ SO produces a gap of magnitude 2λ SO in the spectrum of bilayer graphene, but the two low-energy bands remain spin and valley degenerate (as in a monolayer): E = ± λ 2 SO + v 4 p 4 /γ 2 1 . However, there are gapless edge excitations and, like monolayer graphene [99], bilayer graphene in the presence of intrinsic spin-orbit coupling is a topological insulator with a finite spin Hall conductivity [110,111].…”

“…Note that the quantum spin Hall state produces a term equivalent to that of intrinsic spin-orbit coupling, equation (46), which describes a topological insulator state [99,110,111]. By way of contrast, the gapless nematic phase has a qualitatively similar effect on the spectrum of bilayer graphene as lateral strain [249], described in section VII, producing an additional term in the two-band Hamiltonian of the form of equation (109), with parameter w taking the role of an order parameter.…”

Dynamical conductivity of AA-stacked bilayer graphene

“…AA-STACKING WITH SPIN ORBIT COUPLING Finally, to cover a variety of possible scenarios, we examine the effect of spin orbit coupling (SOC) on AAstacked bilayer graphene. Such effects in AA-and ABstacked bilayers were considered by Prada et al 43 in the context of studying systems which may manifest a topological insulating phase. These authors have studied both the case of SOC in each plane and SOC in only one plane, the latter case taken to be a toy model for spin-orbit proximity effect.…”

“…We also contrast the AA-stacked case with that for AAA-stacking. Finally, because of the connection between the Dirac nature of graphene and topological insulators (TIs), we also follow-up on a toy-model 43 by providing the conductivity for two AA-stacked sheets with spin orbit coupling in one or both planes, potentially mimicking weakly coupled TIs or a TI in proximity to a metallic sheet.…”

Intrinsic Spin Hall and Spin Nernst Effects in Single-Layer Graphene: Tight-Binding vs. Effective Model

“…Using the k · p approximation one can derive the well known eective Hamiltonian (known as the Kane Hamiltonian) for the states near the K and K ′ points of the Brillouin zone [9,12]. This Hamiltonian for the K point has the form [9]: …”