Gapped state of a carbon monolayer in periodic magnetic and electric fields

Abstract

When smooth, zero-on-average, periodic magnetic and electric fields are applied to a carbon mono-layer (graphene), a gap between the valence and conduction band is introduced. Here this gapped state is studied analytically. It is found that it does not correspond to a band insulator: a constant electric field induces a quantized Hall current even though the magnetic flux through the sample is zero and there are no Landau levels. The phenomenon is of the same type as that discovered by Haldane for a graphene sample in a periodic magnetic field that is not smooth, i.e. varies rapidly on the scale of the graphene lattice constant. The effect can be explained in terms of the topological theory of Thouless, Kohmoto, Nightingale and den Nijs. For the system studied in this paper, an explanation in terms of simple physical principles is also presented. Thus some of the mystery is taken out of the apparently strange phenomenon of a Hall effect without magnetic flux. Furthermore, Haldane's model requires control over external magnetic fields on length scales less than an angstrom and is therefore hard to realize experimentally. For the model studied here, control over external fields on length scales that are larger by two orders of magnitude or more is sufficient. The model is therefore more amenable to experimental realization.

Magnetic superlattice and finite-energy Dirac points in graphene

“…Surprisingly enough, despite the strong anisotropy of the magnetic profile, the dispersion presents a Dirac cone with an isotropically renormalized velocity. [12][13][14] To see this, we notice that Tr Ω(E = 0, k y , d) = 2 cosh(2dk y ), which can be easily checked by the explicit calculation of the zero-energy states, and by further expanding the trace to lowest order in E and k y we obtain…”

Gaps tunable by electrostatic gates in strained graphene

“…Recently, a novel state of matter, a quantum Hall insulator without a macroscopic magnetic field (Haldane state 5 ), has spawned the interest in unusual topological properties of band structures, leading to the prediction of topological insulators in two and three dimensions [6][7][8][9][10] . It was understood afterwards that such Haldane state can be realized in a graphene superlattice by a suitable combination of scalar and vector electromagnetic potentials 11 . A gap opens in the electronic spectrum, turning graphene into a quantum Hall insulator with protected chiral edge states.…”

“…A periodic magnetic field, when correlated with a scalar potential leads to a gap whose signs are opposite in the two valleys 11 . A combination of this gap and the gap due to strains leads to gaps of different values in the two valleys, allowing for the control of the valley and sublattice degrees of freedom.…”

Kronig-Penney model on bilayer graphene: Spectrum and transmission periodic in the strength of the barriers

“…Such superlattices (SLs) are commonly used to alter the band structure of nanomaterials. In singlelayer graphene already a number of papers relate their work to the theoretical understanding of such periodic structures [8][9][10][11][12][13][14] . Much less experimental and theoretical work has been done on bilayer graphene 14,15 .…”