Electronic Properties of Bilayer Graphene Strongly Coupled to Interlayer Stacking and an External Electric Field

Bilayer graphene (BLG) with a tunable band gap appears interesting as an alternative to graphene for practical applications; thus, its transport properties are being actively pursued. Using density functional theory and perturbation analysis, we investigated, under an external electric field, the electronic properties of BLG in various stackings relevant to recently observed complex structures. We established the first phase diagram summarizing the stacking-dependent gap openings of BLG for a given field. We further identified high-density midgap states, localized on grain boundaries, even under a strong field, which can considerably reduce the overall transport gap.

Figure 1

(a) Schematic band structures of $AA$, $A{A}^{\prime }$, and $AB$. The solid lines are reflection planes, where the translation vectors ($V_x$, $V_y$) describe the relative displacement between the two layers in the $xy$ plane. (b) Modeling of the two $AB$-stacking boundary. (c) Stacking-dependent potential energy of BLG per unit cell, where the origin corresponds to $AA$ stacking. A lattice Wigner-Seitz cell is highlighted by the solid white line, and the arrow denotes the displacement vector between the two $AB$-stacking domains shown in (b). (d) Minimum energy path between the two $AB$ stackings.

Figure 2

Projected band structures around the $K$ point. Each configuration is represented by the translation vector in the irreducible zone of the lattice Wigner-Seitz cell (triangle), where the lower vertex defines the origin. $k_{\parallel }$ and $k_{\perp }$ are defined for each configuration. Energy ($k$ space) ranges from $-0.5$ ($-0.2{\AA }^{-1}$) to 0.5 eV ($0.2{\AA }^{-1}$) relative to the Fermi level ($K$). Band structures near the high-symmetry stackings are projected onto the $k_{\parallel }$-energy and $k_{\perp }$-energy planes without and with an $E$ field. The inset in the third column of $AA$ stacking highlights a small band gap ($\approx 10\mathrm{meV}$).

Figure 3

(a) A stacking-dependent band gap under a perpendicular $E$ field of $0.5\mathrm{V}/\AA$. A lattice Wigner-Seitz cell is shown by broken lines and an irreducible zone by solid lines. (b) Metal-semiconductor phase boundaries for different electric field strengths are shown for the irreducible zone. (c) Local stacking configurations of simulated structure are represented by colors in the triangle at the left. (d) Local densities of states of the spotted region in (c) are plotted from the left with (dotted line) and without (solid line) an $E$ field.