Gaps tunable by electrostatic gates in strained graphene

We show that when the pseudomagnetic fields created by long-wavelength deformations are appropriately coupled with a scalar electric potential, a significant energy gap can emerge due to the formation of a Haldane state. Ramifications of this physical effect are examined through the study of various strain geometries commonly seen in experiments, such as strain superlattices and wrinkled suspended graphene. Of particular technological importance, we consider setups where this gap can be tunable through electrostatic gates, allowing for the design of electronic devices not realizable with other materials.

Figure 1

Two strained graphene lattice configurations (upper panel) and their corresponding pseudomagnetic field (lower panel) studied in this paper, i.e., (a) strain superlattice and (b) wrinkled graphene.

Figure 2

(a) Sketch of the diagram which describes correlations between the scalar potential and pseudomagnetic field [see Eq. (4)]. (b) Electronic band structure of the graphene superlattice, where each supercell unit [as indicated in Fig. 1a] contains $40\times 40$ graphene lattice units. An out-of plane corrugation amplitude of $h_0=2\hspace{0.5em}\text{Å}$ is used, leading to a nonhomogenoeus pseudomagnetic field [see Fig. 1a] of less than $100\text{T}$. Various scalar potentials as indicated are considered, where $V(\mathrm{r⃗})=c_{0}B(\mathrm{r⃗})$ and $V(\mathrm{r⃗})=c_0{u}_z(\mathrm{r⃗})$ leads to gap opening. $c_0$ is chosen such that max$[V(\mathrm{r⃗})]=0.2\text{V}$.

Figure 3

(a) Finite size strain superlattice of dimensions $200\times 100\text{nm}$ (with supercell as indicated) and its out-of-plane position. (b) The corresponding pseudomagnetic field generated. (c) The spatially resolved current density at Fermi energy ${\epsilon }_f=0$ and zero temperature, under infinitesimally small source-to-drain bias. The scalar potential is defined as $V(\mathrm{r⃗})={\epsilon }_f-{\epsilon }_D$, where ${\epsilon }_D$ is the Dirac energy. Here, we assume $V(\mathrm{r⃗})=0.15\text{V}$. (d) Same as previous, except now the scalar potential varies linearly along the width direction, i.e., $V(y)\propto y$, and $V=\pm 0.2\text{V}$ along the two edges. Conducting channel $2$ is “switched off” with this type of scalar potential.

Figure 4

(a) Schematic of the elongated and wrinkled graphene device, of dimensions $100\times 100\text{nm}$. The transport is along the $x$ direction. In this calculation, the quasi-one-dimensional ripple along $y$ is assumed to have a wavelength of ${\lambda }_w=20\text{nm}$. The gates are aligned underneath the graphene, with alternating potentials of $\pm V_G$ and spaced at distance $d_w=10\text{nm}$ (the same period as the underlying pseudomagnetic field). (b) The gates result in an approximately sinusoidally varying scalar potential in the graphene, with amplitude $V_s$. Energy gaps as a function of $V_s$ are shown for different amount of phase shift between the gates and the underlying pseudomagnetic field, ${\theta }_s=0,\frac{\pi }{4},\frac{\pi }{2}$. (c) and (d) The calculated local density of states as function of $y$ and energy $E$, along an $x$ cut in the middle of the device.

Figure 5

Numerical approach: The graphene ribbon is partitioned into block slices along the $x$ direction (transport) as indicated. Lattice interactions within each block are described by $\alpha$. Nearest neighbor blocks interactions are represented by $\tau$. Device domain ${\Omega }_0$ will include the strains and scalar potential $V(\mathrm{r⃗})$. Left and right lead regions (${\Omega }_L$ and ${\Omega }_R$) are assumed unstrained and electrically doped, due to charge transfer from contacts.