Transport between twisted graphene layers

Abstract

Commensurate-incommensurate transitions are ubiquitous in physics and are often accompanied by intriguing phenomena. In few-layer graphene (FLG) systems, commensurability between honeycomb lattices on adjacent layers is regulated by their relative orientation angle θ, which is in turn dependent on sample preparation procedures. Because incommensurability suppresses inter-layer hybridization, it is often claimed that graphene layers can be electrically isolated by a relative twist, even though they are vertically separated by a fraction of a nanometer. We present a theory of interlayer transport in FLG systems which reveals a richer picture in which the specific conductance depends sensitively on θ, single-layer Bloch state lifetime, in-plane magnetic field, and bias voltage. We find that linear and differential conductances are generally large and negative near commensurate values of θ, and small and positive otherwise.Experimental advances in the fabrication of graphenebased structures [1,2] have now provided researchers with a multitude of systems that have strikingly distinct electronic properties. By engineering the substrate underlying exfoliated samples [3][4][5], identifying exfoliated fragments with folds[6], or controlling epitaxial growth conditions [7,8], the size and shape of the honeycomb lattice arrays [9,10] and the number of graphene layers and their orientations can all be varied. This structural diversity nourishes hopes for a future carbon-based electronics[11] with band-structure and transport characteristics that can be tailored for different types of applications.FLG has advantages over single-layer-graphene because it has a larger current-carrying capacity and because its electronic properties are sensitive to more engineerable system parameters [12]. In nature it appears in a variety of stacking arrangements, the most common being Bernal and rhombohedral sequences which can form three dimensional lattices. It has been understood for some time [13] [16]. The present work is motivated primarily by the need to achieve a more complete understanding of transport in these graphitic nanostructures, which currently appear to provide the most promising platform for applications.In a bilayer system, the relative rotation angle θ can be classified as either commensurate or incommensurate [17]. In the former case the misaligned bilayer system still forms a crystal, albeit one with larger lattice vectors and more than four atoms per unit cell. Commensurability occurs at a countably infinite set of orientations; but the probability that a randomly selected orientation angle is commensurate vanishes. The energy bands of commensurate twisted multilayers disperse approximately linearly with momentum [18][19][20], except at energies very close to the Dirac point. However, the Dirac velocity is reduced FIG. 1: Interlayer (RC) equilibration rate as a function of twist angle θ. These results were calculated for two layers with equal carrier densities (n = 5 × 10 12 cm −2 ) and ǫFτ = 3, where ǫF is the Fermi e...

Long-range atomic ordering and variable interlayer interactions in two overlapping graphene lattices with stacking misorientations

“…From a theoretical perspective, twisted bilayer graphene (TBG) has been examined extensively using both continuum [6][7][8][9] and ab-initio [10][11][12][13][14] approaches. These studies have shown that the individual monolayers become significantly decoupled by the rotational faults.…”

Resonant tunneling and intrinsic bistability in twisted graphene structures

“…It is also implicit in Eq. (12) that the top and bottom graphene lattices have a common lattice point [17]; a rigid horizontal translation between lattices adds an additional overall phase to the matrix T j [18]. We stress, however, that relative phases in T j do not alter in any significant way the physics of tunneling in Eq.…”

“…Indeed, incommensurability between graphene lattices already suppresses interlayer hybridization, regardless of the layer separation being a fraction of a nanometer, thus enabling the sequential tunneling regime [18]. Furthermore, the massless Dirac spectrum, and thus Eq.…”

“…rapidly on the reciprocal lattice vector G 1,2 scale [17,18]. This leads to the tunneling Hamiltonian…”

“…Here H 1,2 are the free-particle terms describing massless Dirac particles in each graphene layer, and T 12 describes the interlayer tunnel coupling [17][18][19]. The free particle terms are…”