Periodically Gated Bilayer Graphene as an Electronic Metamaterial

This paper is a contribution to the joint Physical Review Applied and Physical Review Materials collection titled Two-Dimensional Materials and Devices.

We study ballistic transport in periodically gated bilayer graphene as a candidate for a two-dimensional electronic metamaterial. Our calculations use the equilibrium Green’s-function formalism and take into account quantum corrections to charge density changes induced by a periodically modulated top-gate voltage. Our results reveal an intriguing interferencelike pattern, similar to that of a Fabry-Perot interferometer, in the resistance map as a function of the voltage $V_{\mathrm{BG}}$ applied to the extended bottom gate and $V_{\mathrm{TG}}$ applied to the periodic top gate.

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Figure 1

A schematic view of a periodically gated bilayer-graphene (BLG) device with a top gate formed by a 2D nanowire array and a bottom gate extending across the entire device.

Figure 2

The relevant regions and the electrostatic potential in a periodically gated BLG device. (a) A schematic side view of the device with BLG in the channel. The BLG is at ground potential and is separated from the top gate by an $h_T$-thick $h$-BN sheet and from the bottom gate by an $h_B$-thick protective $h$-BN sheet. We distinguish the central region I between the top and bottom gates from region II outside the top gate, but above the bottom gate. (b) The electrostatic potential $U(x,z)$ in the central region I of the device, calculated for $V_{\mathrm{BG}}=-0.76$ V and $V_{\mathrm{TG}}=3.48$ V.

Figure 3

The electronic band structure of gated (a) BLG and (b) MLG. The presented $E(k)$ results are obtained for BLG with $V_{\mathrm{TG}}=4.76$ V and $V_{\mathrm{BG}}=-1.34$ V in (a) and MLG with $V_{\mathrm{TG}}=4.42$ V and $V_{\mathrm{BG}}=-1.36$ V in (b). $k_x$ is given in units of $2\pi /L$ and $k_y$ in units of $2\pi /(\sqrt{3}d)$, where $d=1.42\AA$ is the interatomic distance in graphene. In the left panels of (a) and (b), the black solid lines denote bands with $k_x=0$ and the red dashed lines denote bands with $k_x=0.5$ at the Brillouin zone boundary. The green solid lines denote the $E=E_F=0$ energy level and the green dashed lines denote the gate-dependent charge-neutrality level in each system. The two right panels in (a) and (b) show band dispersion along $k_x$ for two values of $k_y$.

Figure 4

The calculated resistance map of the BLG device. (a) The calculated source-drain resistance $R$ within the central region I as a function of the top-gate voltage $V_{\mathrm{TG}}$ and the bottom-gate voltage $V_{\mathrm{BG}}$. (b) The predicted resistance map of the entire device containing both regions I and II. Two vertical lines at $V_{\mathrm{BG}}=0$ and $V_{\mathrm{BG}}\approx -1.5$ V associated with first- and second-generation Dirac points in region II are superimposed onto the resistance gap of region I in (a).

Figure 5

The calculated resistance maps of the BLG device at temperatures (a) $T=10$ K, (b) $T=20$ K, (c) $T=30$ K, and (d) $T=40$ K.

Figure 6

The calculated resistance maps of BLG devices with different values of the width $W$ of each wire and the interwire distance $L$ within the periodic top gate. Results are presented for (a) $W=25$ nm, $L=240$ nm, (b) $W=50$ nm, $L=120$ nm, (c) $W=75$ nm, $L=120$ nm, and (d) $W=100$ nm, $L=120$ nm. The numerical data in the respective panels are convoluted by Gaussians with (a) $\sigma =0.12$ V, (b) $\sigma =0.06$ V, (c) $\sigma =0.04$ V, and (d) $\sigma =0.04$ V.

Figure 7

(a) A schematic cross section and (b) the calculated resistance map of the MLG device. The $R(V_{\mathrm{TG}},V_{\mathrm{BG}})$ results are convoluted by a Gaussian with a full width at half maximum of $0.05$ V.