Complex dispersion lines in gapped bilayer graphene: Analytical expressions and shear-displacement effects on monolayer--bilayer--monolayer junction conductance

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.

Figures (10)

• Figure 1: (a) Schematic three-dimensional view and (b) side view of a mono--bi--mono junction. A layer $-$ of length $(N-2)a/2$ is AB-stacked and interlayer-coupled to a layer $+$ connecting the source and drain electrodes, where $N$ is an integer and $a$ is the lattice constant. (c) Conceptual illustration of a pump--probe scheme: a $y$-polarized pump field, independently controlled and arriving first, induces a shear displacement $\delta_y$ along the $y$ direction, while a delayed, independently controlled $x$-polarized probe field induces the probability current $I_x^{(0)}$ along the $x$ direction for electrons incident from the monolayer region. Cases where $\delta_y$ and $I_x^{(0)}$ are (i) in phase and (ii) out of phase are shown. The horizontal axis represents time $t$. See Sec. VIII for details.
• Figure 2: Atomic positions are labeled by integers $j_x$ and $j_y$ along the $x$ and $y$ directions. The vector $\vec{\delta}=(\delta_x,\delta_y)$ denotes the shear displacement of layer $-$ relative to layer $+$. (a) Wave-function amplitudes $(\circ_+,\bullet_+,\circ_-,\bullet_-)$ at $j_x=j_y=0$ for $\delta_x=0$ and $b \gg \delta_y > 0$. The layer index $\pm$ is suppressed for clarity, and $b=a/\sqrt{3}$ is the in-plane bond length. Dashed and solid lines denote layer $+$ and layer $-$, respectively. (b) Wave-function amplitudes $(\circ,\bullet)\lambda^{j_x}\omega^{j_y}$ in each layer for $|j_x|\le 1$ and $|j_y|\le 1$. (c) Interlayer hopping parameters $\gamma_1$, $\gamma_3$, and $\gamma_4$. The top panel shows a schematic three-dimensional view, while the bottom panel shows a projection onto the $(x,y)$ plane for $\delta_x=\delta_y=0$.
• Figure 3: The relation between $(k_x^{\mathrm{r}} a, -k_x^{\mathrm{i}} a)$ and the energy $E$ for $\varepsilon=0.35~\mathrm{eV}$, $k_y=0$, and $\vec{\delta}=0$. Here, $k_x^{\mathrm{r}}=\mathrm{Re}(k_x)$ and $k_x^{\mathrm{i}}=\mathrm{Im}(k_x)$. Each DL is labeled by $(\tau,l)$; for $k_x^{\mathrm{i}} \neq 0$, the labels are shown only for those with $|\lambda|=\exp(-k_x^{\mathrm{i}} a/2)<1$. See the text for the definition of $(\tau,l)$, $m$ and $m'$.
• Figure 4: The DLs in the $K$ valley for $\vec{\delta}=0$ at (a) $k_y=0$, (b) $3k_yb=0.03\pi$, and (c) $3k_yb=0.032\pi$, for $\varepsilon=0.15$ eV. Solid lines represent exact results, while circles correspond to the $\widetilde{\Delta}\xi$ approximation. For the circles, calculations are performed at $E=-0.5, -0.48, \ldots, 0.5$ eV with an energy spacing of $0.02$ eV, except in the region where $|q|^2 < \gamma_0^4 \times 10^{-5}$. The RDLs are labeled in ascending order of energy as $E_1$, $E_2$, $E_3$, and $E_4$. See the text for the definitions of the points $m$, $m'$, $2$, $2'$, $2"$, $3$, $3'$, and $3"$.
• Figure 5: CDLs in the $K$ valley for $\varepsilon=0.15$ eV and three values of $3k_yb/2$. The top, middle, and bottom panels correspond to $\vec{\delta}=(0,0)$, $(0,1)$ pm, and $(0,-1)$ pm, respectively. Solid, dotted, and dashed lines represent exact results, whereas squares, circles, and triangles denote results obtained within the $\widetilde{\Delta}\xi$ approximation. The approximation results are evaluated at $E=-0.3, -0.29, \ldots, 0.3$ eV with an energy spacing of $0.01$ eV, except in the region where $|q|^2 < \gamma_0^4 \times 10^{-5}$. Different line styles and symbols correspond to different values of $k_y$, as indicated in the legend. The left and right panels show projections onto the $(k_x^{\mathrm{r}}a/\pi, -k_x^{\mathrm{i}} a)$ and $(E, -k_x^{\mathrm{i}} a)$ planes, respectively. Vertical lines at $E=0$ in the right panels indicate the correspondence with the points $m$ and $m'$ in the left panels.