Integer quantum Hall effect in gapped single-layer graphene

Abstract

Analytical expressions for the Hall conductivity σ yx and the longitudinal resistivity ρ xx are derived in gapped, single-layer graphene using linear response theory. The gap 2 , described by a mass term, is induced by a substrate made of hexagonal boron nitride (h-BN) and produces two levels at ± . It is shown that σ yx has the same form as for a graphene sample supported by a common substrate without a mass term. The differences are a shift in the energy spectrum, which is not symmetric with respect to the Dirac point for either valley due to the gap, the absence of a zero-energy Landau level, and the nonequivalence of the K and K valleys. In addition, the dispersion of the energy levels, caused by electron scattering by impurities, modifies mostly plateaus due to the levels at ± . It is shown that the resistivity ρ xx exhibits an oscillatory dependence on the electron concentration. The main difference with the usual graphene samples, on SiO 2 substrates, occurs near zero concentration, as the energy spectra differ mostly near the Dirac point.

Edge states in graphene-like systems

“…This model is mathematically equivalent to the JaynesCummings model in quantum optics describing a two level system coupled to a bosonic mode. The resulting eigenfunctions and eigenvalues have the following properties 45,76,77 . First, there is a spectrum of states with wave functions with non-zero φ A and φ B , with energies:…”

Theory of substrate, Zeeman, and electron-phonon interaction effects on the quantum capacitance in graphene

“…The corresponding two dimensional Dirac-like Hamiltonian [18][19][20][21][22] for an electron in the Landau gauge is H ¼ vðgr x p x þ r y p y Þ þ Dr z . Here, r ¼ ðr x ; r y ; r z Þ is the vector of Pauli matrices, g ¼ þ1/À1 denotes the K=K 0 valley, D is an energy term representing the substrate induced inversion symmetry breaking, v is the velocity of the Dirac states, and p ¼ (p x , p y ) is the two-dimensional canonical momentum.…”

“…[15][16][17][18][19][20] Being smooth on the atomic scale and endowed with a lattice constant close to graphene, h-BN has great advantages over other substrates, like SiO 2 . Also, it is typically free of dangling bonds and charge traps.…”

Quantum Hall effect in gapped graphene heterojunctions

“…28,29 This relation permits us to anticipate that the quantum Hall effect of massive Dirac fermions is much closer to that of massless Dirac fermions than to that of nonrelativistic electrons. 30 The situation becomes more interesting when the gap, or mass, is not homogeneous. This could be the case, for instance, of a heterostructure made of two graphene or graphenelike materials with different gaps 1 and 2 , such as the atomic layers of hybridized BN and graphene domains, 31,32 or if the gap is substrate driven and, due to lattice mismatch, features amplitude modulations larger than the graphene unit cell.…”