Complex dispersion lines in gapped bilayer graphene: Analytical expressions and shear-displacement effects on monolayer--bilayer--monolayer junction conductance with implications for dynamical AC--DC conversion

Abstract

Analytical treatments of tunneling in bilayer graphene have typically relied on minimal models including only the vertical interlayer hopping $γ_1$ and have been restricted to the weak interlayer bias regime $2\varepsilon \ll γ_1$. These simplifications limit the ability of analytical theories to describe lattice deformations and strong electric-field effects. In this work, we present an analytical theory of evanescent states in electrically gapped bilayer graphene that overcomes both limitations. Specifically, our approach explicitly incorporates the skew interlayer hoppings $γ_3$ and $γ_4$ and remains valid even when the interlayer bias $2\varepsilon$ is comparable to $γ_1$. Focusing on low-energy electronic states near the charge neutrality point, we analytically derive the complex longitudinal wave numbers, the gap width, and the sublattice pseudospin inside the electric-field-induced gap, and systematically analyze their dependence on the interlayer shear displacement $\vecδ=(δ_x,δ_y)$. The analytical expressions quantitatively reproduce exact numerical calculations, demonstrating that skew interlayer hoppings, in particular $γ_3$, play an essential role. Taking the zigzag direction as the longitudinal (transport) $x$ direction, the wave vector becomes complex along $x$ while remaining real along the transverse $y$ direction. For a monolayer--bilayer--monolayer junction with transport along this direction, we find that $δ_y$ has a significantly stronger impact on the conductance than $δ_x$. This anisotropic response is quantitatively explained by the analytical expressions. Furthermore, we identify a shear-induced phase proportional to $δ_y$ that appears universally in the analytical expressions for the gap width, the sublattice pseudospin, and the decay length.