Scalable tight-binding model for strained graphene

Abstract

We generalize the scalable tight-binding model for graphene, which allows for efficient quantum transport simulations in the Dirac regime, to account for elastic strain. We show that the original scalable model with scaling factor $s$ is readily applicable to strained graphene, provided that the displacement fields corresponding to the deformed graphene lattice are properly scaled. In particular, we show that the long-wavelength theory remains invariant when the strain tensor is scaled by $s$. This is achieved in practice by scaling the in-plane displacement fields by $s$ while the out-of-plane displacements have to be scaled by $\sqrt{s}$. We confirm these scaling laws by extensive numerical simulations, starting with the pseudomagnetic field and the local density of states for different scaled lattices. The latter allows us to study pseudo-Landau levels as well as hybrid Landau levels in the presence of an external magnetic field. Finally, we consider quantum transport simulations motivated by a recent experiment, where a uniaxial strain barrier is engineered in monolayer graphene by vertically misaligned gates. Our work generalizes the scalable tight-binding model to allow for efficient modeling of quantum transport in large-scale strained graphene devices.

Figures (7)

• Figure 1: Examples of strained graphene devices: (a) uniaxial strain applied to a graphene ribbon, (b) shear strain applied to a graphene ribbon with one end fixed, and (c) triaxial strain applied to a hexagonal graphene flake. The region marked by a small blue rectangle in (c) is magnified in (d) and (e), considering an unscaled ($s=1$) and scaled ($s=2$) graphene lattice, respectively, without scaling the in-plane displacement field $\mathbf u$, where empty (solid) dots represent lattice points in the graphene sheet before (after) the strain is applied. (f) Calculated PMF profiles of a $D_h=20\mathord{\rm nm}$ flake considering $s=1$ with $\Delta=0.1\mathord{\rm nm}$ (left) and $s=2$ with $\Delta=0.2\mathord{\rm nm}$ (right); $D_h$ and $\Delta$ are defined in (c). (g) Same as (f) but with a larger $\Delta$ as indicated.
• Figure 2: Illustration of (a) an unstrained graphene lattice with nearest-neighbor bond vectors $\{ \mathbf{d}_1^0, \mathbf{d}_2^0, \mathbf{d}_3^0 \}$ and (b) a strained graphene lattice with nearest-neighbor bond vectors $\{ \mathbf{d}_1, \mathbf{d}_2, \mathbf{d}_3 \}$.
• Figure 3: Local density of states at the center of a scaled and strained zigzag graphene flake, $D$. Shown in (a) for $s=3$ without strain (gray square) and with strain (deformed red square). The corresponding lattice sites are shown in the bottom inset. (b) $D(E)$ for unstrained graphene scaled by $s=1,2,3,4$ for an external magnetic field $B_z=10\mathord{\rm T}$. The inset shows $D/s^2$ near the 4th LL. The red curve in (b) for $s=3$ gives the line cut marked in (c), showing identical $D(B_z,E)$ on both sublattices [defined in the right inset of (a)]. Panels (d) and (e) show $D(B_s,E)$ at sites A and B, respectively, for $B_z=0$ and the same range of the PMF $B_s$ as for $B_z$ in (c). Panels (f) and (g) show $D(B_z,E)$ at site A and B, respectively, for $B_s=10\mathord{\rm T}$. Panels (h) and (i) show $D(B_s,E)$ at site A and B, respectively, for $B_z=10\mathord{\rm T}$. Panels (j) and (k) are enlarged (and rotated) low-field maps marked by the purple boxes on (f) and (h), respectively. Here $D \in [0,0.15]$ for all maps such that white [blue] and black [red] correspond to zero and the maximum, respectively, in panels (c)--(i) [(j)--(k)].
• Figure 4: Transport simulations for a graphene nanoslide Zhang2022DeBeule2025a illustrated in (a), using (b)--(d) periodic and (e)--(f) open boundary conditions. The width of the ribbon is $W=500\mathord{\rm nm}$. (b) Inverse conductance $1/G$ versus carrier densities $n_1$ and $n_2$ in the left and right leads, respectively, for an unscaled ($s=1$) lattice. Inset: Carrier density profile used in the transport simulations. The conductance along the dashed line is shown in (c) and compared to results with properly and improperly scaled displacements. This illustrates the square-root scaling of the out-of-plane displacement field (see the legend). (d) Scaling factor dependence of the conductance with correct scaling at the carrier densities marked by the vertical dashed lines in (c). (e) Inverse conductance map for open boundary conditions computed with $(s,\Delta_z)=(4,2\mathord{\rm nm})$. (f) shows $G$ along the dashed line in (e) with $(s,\Delta_z)=(1,\sqrt{1}\mathord{\rm nm}),(2,\sqrt{2}\mathord{\rm nm}),\cdots,(8,\sqrt{8}\mathord{\rm nm})$, presented as individual curves (upper part) and a color map (lower part). Schematics showing lattices under the (g) periodic and (h) open boundary conditions along the $y$ direction.
• Figure 5: (a) Local density of states normalized by the squared scaling factor, $D/s^2$, as a function of the external magnetic field $B_z$ at zero pseudomagnetic field $B_s=0$ (upper row) and finite pseudomagnetic field $B_s=10\mathord{\rm T}$ (lower row). (b) Same as (a) but with $B_z$ and $B_s$ interchanged.