Common-path interference and oscillatory Zener tunneling in bilayer graphene p-n junctions

Abstract

Interference and tunneling are two signature quantum effects that are often perceived as the yin and yang of quantum mechanics: a particle simultaneously propagating along several distinct classical paths versus a particle penetrating through a classically inaccessible region via a single least-action path. Here we demonstrate that the Dirac quasiparticles in graphene provide a dramatic departure from this paradigm. We show that Zener tunneling in gapped bilayer graphene, which governs transport through p-n heterojunctions, exhibits common-path interference that takes place under the tunnel barrier. Due to a symmetry peculiar to the gapped bilayer graphene bandstructure, interfering tunneling paths form conjugate pairs, giving rise to high-contrast oscillations in transmission as a function of the gate-tunable bandgap and other control parameters of the junction. The common-path interference is solely due to forward-propagating waves; in contrast to Fabry-Pérot-type interference in resonant-tunneling structures, it does not rely on multiple backscattering. The oscillations manifest themselves in the junction I-V characteristic as N-shaped branches with negative differential conductivity. The negative dI/dV, which arises solely due to under-barrier interference, can enable new high-speed active-circuit devices with architectures that are not available in electronic semiconductor devices.

Fig. 1.

Zener tunneling and common-path interference in BLG in the uniform-field model. Interference of two least-action tunneling paths results in oscillations, n = 1,2, .... Shown is transmission at normal incidence, p_y = 0, as a function of bandgap size, in units Δ₀ = [(Fℏ)²/2m]^1/3 (semilog scale). Numerical results (red symbols), obtained by integrating Eq. 8, agree with the WKB result, Eqs. 1 and 4 (blue curve) in the entire range of Δ, large and small. Inset shows schematic setup of p-n junction: The bandgap E_g = 2Δ, the linear barrier potential V(x) = -Fx (see Eq. 2), and a pair of interfering tunneling paths.

Fig. 2.

Evolution of a two-level system slowly driven through an avoided level crossing, Eq. 8. Nonadiabatic transitions between different levels, corresponding to Zener tunneling, take place in the Larmor precession region -p_Δ ≲ p ≲ p_Δ, where . Shown are adiabatic energy levels of the Hamiltonian, Eq. 8 (blue line) and schematic partition into regions of adiabatic evolution and Larmor precession.

Fig. 3.

The I–V characteristic of a BLG p-n junction combines features of Esaki diode (N-shaped branches with negative differential conductivity) and Zener diode (a breakdown-type behavior). Thus a single p-n junction can serve as an active-circuit element with multiple functionality. Valleys in the I–V dependence correspond to n = 1 node of the oscillations in transmission in Fig. 1. Shown is the I–V dependence given by Eq. 11 for parameter values: U/V₀ = 0.1,0.2,0.3 (curves 1, 2, and 3, respectively). Units are V₀ = Δ(L/ℓ_Δ) and , where W is the lateral width of the junction and N = 4 is the spin/valley degeneracy in BLG. Inset shows junction schematic, with U the built-in potential induced by doping or by gates, and E_g = 2Δ the bandgap.