Tunable valley Hall effect in gate-defined graphene superlattices

We theoretically investigate gate-defined graphene superlattices with broken inversion symmetry as a platform for realizing tunable valley-dependent transport. Our analysis is motivated by recent experiments [C. Forsythe et al., Nat. Nanotechnol. 13, 566 (2018)] wherein gate-tunable superlattice potentials have been induced on graphene by nanostructuring a dielectric in the graphene/patterned-dielectric/gate structure. We demonstrate how the electronic tight-binding structure of the superlattice system resembles a gapped Dirac model with associated valley-dependent transport using an unfolding procedure. In this manner we obtain the valley Hall conductivities from the Berry curvature distribution in the superlattice Brillouin zone, and demonstrate the tunability of this conductivity by the superlattice potential. Finally, we calculate the valley Hall angle relating the transverse valley current and longitudinal charge current and demonstrate the robustness of the valley currents against irregularities in the patterned dielectric.

Figure 1

(a) The superlattice system considered in this work: a graphene sheet (empty and filled circles) gated through a patterned dielectric with triangular zigzag-edged holes yielding an effective superlattice potential (red-to-black gradient). The supercell is marked by the dashed lines (left), alongside the normal (graphene) unit cell (right). The lack of inversion center and the sublattice asymmetric structure of the gated regions induce the valley Hall effect under in-plane electric field. The geometry is characterized by the supercell hexagon side length $L$ and the triangle side length $R$. (b) The corresponding supercell (SBZ) and normal (NBZ) Brillouin zone. The SBZ is shown enlarged four times for clarity. (c) Sketch of the considered graphene/nanostructured-dielectric/gate structure. Here we show nanopatterned hBN with the naturally occurring triangular zigzag edges holes nucleated on boron sites.

Figure 2

(a), (b) Spectral weight (gray surface, $\eta =3\mathrm{meV})$ close to the SBZ $\mathrm{\Gamma }$ point for different values of the constant superlattice potential $V\left({\mathit{r}}_i\right)=1\mathrm{eV}$, 2 eV. The dashed lines in (a) show the $V=0$ (pristine graphene) band structure. (c), (d) Corresponding line cuts of the occupied Berry curvature when the Fermi energy is fixed in the gap at each potential. (e) Supercell Berry curvature in the SBZ with the valence band filled. The pristine system $K$ and $K^{\prime }$ valleys fold to the SBZ $\mathrm{\Gamma }$ point, yielding a sign-changing peak centered on this symmetry point. The horizontal dotted line indicates the cut in $\mathit{k}$ space shown above.

Figure 3

(a) Density of states for different geometries of the superlattice at $V=1\mathrm{eV}$. Inset: Band gaps as a function of the superlattice potential magnitude (full lines) for different geometries of the gated region $\left(L=4\right)$. The band gap widens as the superlattice potential is increased in all considered geometries. Dashed lines show the corresponding shift $\left(E_s\right)$ of the center of the band gap as the superlattice potential is increased. This shift increases linearly with increasing superlattice potential, with the slope determined by the size of the gated region $\left(R\right)$. (b) Line cut of the SBZ Berry curvature in the gap for different geometries of the gated regions $\left(L=4,V=1\mathrm{eV}\right)$. The shape of the Berry curvature distribution broadens for increasing size of the gated region $R$, mirroring the broadening with increasing superlattice potential magnitude $V$. (c) The SBZ Berry curvature when the supercell size is varied instead, demonstrating the opposite scaling in the width.

Figure 4

(a), (b) Line cuts of the unfolded spectral weight (gray surface) close to the NBZ $K$ point for different values of the constant superlattice potential $V\left({\mathit{r}}_i\right)=1\mathrm{eV}$, 2 eV. The result at the $K^{\prime }$ point along this same cut in $\mathit{k}$ space can be found by reflection around the central point $K_{\tau }$, and thus has similar structure. (c), (d) Corresponding line cuts of the unfolded occupied Berry curvature in the $K$ (blue) and $K^{\prime }$ (red) valley with the Fermi energy fixed in the gap at each potential. (e) Unfolded Berry curvature in the NBZ demonstrating equal peaks of opposing signs, indicating the presence of transverse valley currents. The dotted line indicates the cut in $\mathit{k}$ space shown above.

Figure 5

Valley Hall conductivity as a function of filling (full lines) for varying values of the superlattice potential $V$, shown alongside the density of states (dotted lines). Berry curvature accumulated near the band edges causes a saturation of the valley Hall conductivity as the gap is approached, and for small $V$ the quantized $2e^2/h$ value of the massive Dirac model is approached. The inset shows the plateau value in the gap as the superlattice potential is tuned. The valley Hall conductivity decays for larger superlattice potentials, as the supercell bands flatten and the unfolded valley structure is lost.

Figure 6

Valley Hall angle (full lines) and expected nonlocal resistance signal (dashed lines) close to the band edge for two values of the superlattice potential $V=1\mathrm{eV}$, 3 eV. The band gap is indicated by the vertical dashed lines. The valley Hall angle is only finite close to the band edge where ${\sigma }_{xy}\sim {\sigma }_{xx}$, and approaches $\pi /2$ in the gap. The predicted nonlocal resistance close to the band edges is obtained using the expression of Ref. [25]. The peaks in the ratio $R_{NL}/{\rho }_{xx}$ occur exactly at the ${\theta }_v=\pi /4$ point, i.e., when the valley Hall and longitudinal conductivities are equal, ${\sigma }_{xy}^v={\sigma }_{xx}$. These peaks in the nonlocal response shift as the superlattice potential is tuned.

Figure 7

Valley Hall conductivity as a function of filling for different values of the superlattice potential for a smoothly varying potential $\left(u=0.2\right)$, the profile of which is displayed in the inset. The results are similar to the flat-potential case, with some additional structure in the peak structure due to the lifting of degeneracies of bands near the band edge.

Figure 8

Variation of the valley Hall conductivity with respect to irregularities in the edge profile of the superlattice potential, corresponding to irregularities in the dielectric etching. The regular limit for a smoothly varying potential $\left(V=2\mathrm{eV},u=0.2\right)$ is shown in the full black line, alongside the same calculation with random edge profiles at the superlattice potential boundary (gray lines). The average of all such configurations is shown in the red dotted line. The finite valley Hall conductivity does not require a perfectly symmetrical induced potential, and is thus a general prediction in these superlattices.

Figure 9

(a), (b) Band gap variation with the superlattice potential magnitude for different geometries. (c), (d) Band gap $\left(\mathrm{\Delta }\right)$ and shift $\left(E_s\right)$ variation with the supercell size $(L,R=4)$ and extent of the superlattice potential $(L=5,R)$, respectively, shown here for multiple values of the superlattice potential magnitude $\left(V\right)$. The small asterisks indicate the average potential on each site in the supercell, which matches the numerically calculated shift $\left(E_s\right)$. The same general result is obtained for different configurations: The band gap widens for either greater magnitude of the superlattice potential or increasing ratio between gated region and supercell size.

Figure 10

(a) Extended view of the band structure $E_N\left(\mathit{k}\right)$ showing the gap and miniband formation of the supercell. The symmetry points are those of the SBZ. (b) Corresponding density of states, demonstrating the shifted band gap. (c), (d) LDOS, plotted using the radii of black (white) disks to indicate the value at A (B) sites, sampled just above and below the gap at $\omega =0.1,0.29\mathrm{eV}$ [dashed lines in (a)]. The superlattice potential breaks inversion symmetry and causes a splitting of the A/B weight at these sites. (e) Local gap magnitude at each site in the supercell as derived from the local density of states (variations enhanced $\times 5)$, showing a small variation at the potential edge. (f) Corresponding shift in the center of this local gap (variations enhanced $\times 5)$, showing a small difference between A/B sites in the supercell. All plots are for a representative configuration of $\left(L=4,R=3,V=2\mathrm{eV}\right)$.

Figure 11

Spatial variation of the superlattice potential $\left([L,R,u]=[4,4,0.2]\right)$, shown as (a) a color gradient, (b) a contour plot. (c) The equivalent line cuts indicated by the black dotted lines in (b) for different values of the smoothness parameter $u=[0.01,0.2,1]$, which interpolates between the extreme cases of flat and linearly decreasing potentials. (d) Variation of the induced band gap (full lines) and shift (dashed lines) with the smoothness parameter $u$ for the $(L,R)=(4,4)$ geometry outlined above. There is only a small decay in the gap magnitude.