Analytical treatments of tunneling in bilayer graphene have typically relied on minimal models including only the vertical interlayer hopping ${\gamma }_1$ and have been restricted to the weak interlayer-bias regime ($2\varepsilon \ll {\gamma }_1$). Consequently, they cannot adequately describe lattice deformations or strong electric-field effects. In this work, we present an analytical theory of evanescent states in electrically gapped bilayer graphene that overcomes both limitations. Our approach explicitly incorporates the skew interlayer hoppings ${\gamma }_3$ and ${\gamma }_4$ and remains valid even when the interlayer bias $2\varepsilon$ is comparable to ${\gamma }_1$. 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 within the electric-field-induced gap. We then systematically analyze the dependence of these quantities on the interlayer shear displacement $\vec{\delta }=({\delta }_x,{\delta }_y)$, and find that skew interlayer hoppings, in particular ${\gamma }_3$, play an essential role. For transport along the zigzag ($x$) direction, the longitudinal wave vector becomes complex, whereas the transverse wave vector remains real. For a monolayer/bilayer/monolayer junction with transport along the zigzag direction, we find that ${\delta }_y$ has a significantly stronger impact on the conductance than ${\delta }_x$. Furthermore, we identify a shear-induced phase proportional to ${\delta }_y$ that appears universally in the analytical expressions for the gap width, the sublattice pseudospin, and the decay length. These results establish a unified framework for shear- and bias-controlled evanescent tunneling in bilayer graphene and suggest broader relevance to nonequilibrium transport phenomena in layered materials.

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• Electrical conductivity
• Quantum transport
• Transport phenomena
• 2-dimensional systems
• Graphene
• Landauer formula
• Transfer matrix method