Influence of minivalleys and Berry curvature on electrostatically induced quantum wires in gapped bilayer graphene

Abstract

We show that the spectrum of subbands in an electrostatically defined quantum wire in gapped bilayer graphene (BLG) directly manifests the minivalley structure and reflects Berry curvature via the associated magnetic moment of the states in the low-energy bands of this two-dimensional material. We demonstrate how these appear in degeneracies of the low-energy minibands and their valley splitting, which develops linearly in a weak magnetic field. Consequently, magnetoconductance of a ballistic point contact connecting two non-gapped areas of a bilayer through a gapped (top and bottom gated) barrier would reflect such degeneracies by the heights of the first few conductance steps developing upon the increase of the doping of the BLG conduction channel (we consider an adiabatic constriction, where conductance is set by the number of propagating ballistic modes in its narrowest part): 8e 2 /h steps in a wide channel in BLG with a large gap, 4e 2 /h steps in narrow channels, all splitting into a staircase of 2e 2 /h steps upon lifting valley degeneracy by a magnetic field.

Tunable Valley Splitting due to Topological Orbital Magnetic Moment in Bilayer Graphene Quantum Point Contacts

“…Diamond-like feature are numbered with the corresponding conductance values. The height of the large diamonds at G = 6, 10, 14, 18, 22 and 26 e 2 /h has a range from 2 ∼ 5 meV which corresponds to the energy level difference of the valley splitting [6,11,17]. At B ⊥ = 0.5 T, see Fig.…”

“…Next, we discuss how the four-fold degenerate states split in a magnetic field. Zeeman splitting of spin states [6,11,13] occurs for any magnetic field orientation. Valley splitting, on the other hand, is an orbital effect [6], and will therefore only occur for magnetic fields perpendicular to the sample plane.…”

“…Zeeman splitting of spin states [6,11,13] occurs for any magnetic field orientation. Valley splitting, on the other hand, is an orbital effect [6], and will therefore only occur for magnetic fields perpendicular to the sample plane. In analogy to the g s -factor characterizing spin splitting g s µ B B, we introduce the valley g-factor g v , for states that split linearly in a perpendicular magnetic field.…”

“…where the wave functions Ψ(k x ) are obtained from numerical diagonalization of the BLG Hamiltonian [20] including a continuous confinement potential and a spatially modulated gap (akin to References [6,11]). The system parameters have been chosen to correspond to the splittings of the lowest subbands for the lowest modes' energy spacings extracted from Fig.…”

“…Using finite bias spectroscopy we measure a spin g-factor g s ∼ 2 as expected. The valley degree of freedom in bilayer graphene have a nontrivial topology that leads to a Berry curvature and a topological orbital magnetic moment [6]. In our experiment, again using finite bias spectroscopy, we measure a valley splitting which is linear in perpendicular magnetic field.…”