In-Plane Electric Polarization of Bilayer Graphene Nanoribbons Induced by an Interlayer Bias Voltage

We theoretically show that an interlayer bias voltage in the AB-stacked bilayer graphene nanoribbons with armchair edges induces an electric polarization along the ribbon. Both tight-binding and ab initio calculations consistently indicate that when the bias voltage is weak, the polarization shows opposite signs depending on the ribbon width modulo three. This nontrivial dependence is explained using a two-band effective model. A strong limit of the bias voltage in the tight-binding model shows either one-third or zero polarization, which agrees with the topological argument.

Figure 1

Structure of the $AB$-stacked bilayer GNR with the armchair edges. (a), (b) The black (blue) lines represent the bondings in the lower (upper) layer. (a) $N$ is the width corresponding to a number of rows. The unit cell of the ribbons (red, dashed-line box) contains $4N$ atoms. $nA(A^{\prime })$ and $nB(B^{\prime })$ represent the sublattice in the lower (upper) layer within the $n$th row. (b) The green arrows denote the interlayer hoppings between the $nB$ and $n{A}^{\prime }$ sites forming dimers.

Figure 2

Numerical results of the polarization from the TB model in response to the interlayer bias. The interlayer hopping parameter $t_{\perp }=0.13t$ is fixed. (a) $N=3l$ ($N=27,39$) and (b) $N=3l+1$ ($N=28,40$). The results for $N=39,40$ (green) are vertically offset by 0.2.

Figure 3

Emergence of the electric dipole moment in the strong limit of the interlayer bias voltage $U$. (a) and (b) Top and side views of a part of the nanoribbon. 0 and $-2e$ represent the charge at each layer for large $U$. The dashed and solid red arrows represent the whole and the $y$ component of the dipole moment.

Figure 4

Comparison between the results from the TB model (red solid line) and the 2B model (green dashed line). (a) $N=3l$ ($N=27,39$) and (b) $N=3l+1$ ($N=28,40$). The results for $N=39,40$ are vertically offset by 0.1.

Figure 5

Comparison between the results of DFT calculations and those with the TB model. (a), (b) Polarization obtained by DFT calculations (squares) compared with results with the TB model (lines). The results of the TB with (without) rescaling by the gap size are shown as a solid line (dotted line) (see text). (a) ($N=9,12,15$) are for $N=3l$, and (b) ($N=7,10,13$) are for $N=3l+1$. Note that it is for a spinless system, and the results here should be multiplied by two to compare with experiments. (c) Induced polarization $P$ by DFT calculation divided by $(N+1{)}^2$. (d) Relationship between the out-of-plane electric field $|\mathbf{E}|$ versus on-site potential energy difference $U$. $U$ is scaled by the hopping amplitude $t=2.6\mathrm{eV}$ [7].