Berry Curvature and Nonlocal Transport Characteristics of Antidot Graphene

Antidot graphene denotes a monolayer of graphene structured by a periodic array of holes. Its energy dispersion is known to display a gap at the Dirac point. However, since the degeneracy between the $A$ and $B$ sites is preserved, antidot graphene cannot be described by the 2D massive Dirac equation, which is suitable for systems with an inherent $A/B$ asymmetry. From inversion and time-reversal-symmetry considerations, antidot graphene should therefore have zero Berry curvature. In this work, we derive the effective Hamiltonian of antidot graphene from its tight-binding wave functions. The resulting Hamiltonian is a $4\times 4$ matrix with a nonzero intervalley scattering term, which is responsible for the gap at the Dirac point. Furthermore, nonzero Berry curvature is obtained from the effective Hamiltonian, owing to the double degeneracy of the eigenfunctions. The topological manifestation is shown to be robust against randomness perturbations. Since the Berry curvature is expected to induce a transverse conductance, we have experimentally verified this feature through nonlocal transport measurements, by fabricating three antidot graphene samples with a triangular array of holes, a fixed periodicity of 150 nm, and hole diameters of 100, 80, and 60 nm. All three samples display topological nonlocal conductance, with excellent agreement with the theory predictions.

Popular Summary

Graphene is a transparent, flexible conductor with potential for many novel applications from solar cells to wearable electronics. It consists of a sheet, just one atom thick, of carbon atoms laid out in a honeycomb pattern. Some of the unique properties of graphene come from the fact that electrons traveling within the sheet behave as if they have no mass. By creating holes (known as antidots) in the graphene sheet, the electronic properties can be significantly changed. Electrons in such “antidot graphene” recover the mass that appears to be absent in pristine graphene. One key to understanding the behavior of electrons in graphene is a mathematical concept known as Berry curvature. Nonzero Berry curvature, for example, manifests a weak magnetic field that acts on the electrons. In pristine graphene, the Berry curvature is zero; it is widely assumed that the same is true for antidot graphene. We present a mathematical analysis, and experimental verification, that shows the antidot Berry curvature is actually nonzero, which can lead to intriguing electronic effects.

Specifically, we derive, for the first time, an effective Hamiltonian—a mathematical construct that describes the total energy—for antidot graphene in the form of a $4\times 4$ matrix. Based on this Hamiltonian, we predict a nonzero Berry curvature that can give rise to interesting nonlocal transport effects that are absent in pristine graphene. We measure the electronic properties of three samples of antidot graphene, each with a different hole size, and find an excellent match to our theoretical predictions.

Our derived effective Hamiltonian can greatly facilitate the study of electronic properties of antidot graphene precisely at energies where it differs from pristine graphene.

Figure 1

(a) Antidot graphene in the tight-binding model. Red and black dots represent the carbon atoms $A$ and $B$. Here, $L$ denotes the periodicity of the antidot lattice, and $d$ denotes the diameter of the hole. (b) The matrix element magnitude plotted as a function of Bloch wave vector $\mathit{k}$. The term $|H_{12}|$ (black solid squares) is found to vary linearly with $k=|\mathit{k}|$, while the intervalley scattering term $|H_{14}|$ (red solid circles) is almost a constant. (c) Band structures for various $\gamma$ with a fixed periodicity $L=13\mathrm{nm}$. The black circles represent the tight-binding results, and red curves stand for the dispersion relation given by the effective Hamiltonian. The Bloch wave vector $\mathit{k}$ ranges from K to $\mathrm{\Gamma }$, then to M, and finally back to K [here, K, $\mathrm{\Gamma }$, and M points are for the antidot (hole) reciprocal lattice].

Figure 2

Effective velocity $v$ and intervalley scattering strength $m$ are plotted as a function of the geometric factor $\gamma$. The calculated results on velocity $v$ (in units of $v_F$) are shown as black solid squares, and those for $m$ (in units of $t$) are shown as red open circles. These parameters are obtained for a fixed periodicity $L=13\mathrm{nm}$. The dashed curves are linear fittings.

Figure 3

Berry curvature matrix element ${\mathrm{\Omega }}_{11}$ calculated as a function of wave vector $\mathit{k}$ for various values of the geometric factor $\gamma$ [based on Eq. (19)]. The parameter $m/\hslash v$ can be obtained from the tight-binding calculations, where the periodicity $L$ is fixed at 13 nm. Here $a$ denotes the carbon-carbon atomic separation ($=1.42\AA$).

Figure 4

(a) Schematic illustration of an antidot graphene lattice (Lattice B) for calculating the conductance as a function of Fermi energy. The system contains 20 holes in the $x$ direction and 10 holes in the $y$ direction. The inset is a zoom-in image of the antidot lattice with a hole. (b) Calculated conductance plotted as a function of Fermi energy. The black curve represents the conductance of Lattice A, where the zero-conductance plateau indicates the band gap of this system. For Lattice B, the red curve exhibits no zero-conductance regions, implying that there is no band gap. Lattice C differs from Lattice B only at the edges of the holes, where we have added atomic-scale randomness. Ten different randomness profiles were calculated, and the blue curve represents the ensemble-averaged conductance. It is seen that a band gap reappears, and the magnitude of this band gap is the same as that of Lattice A, implying that the transport gap in Lattice C has the same origin as that in Lattice A, i.e., from $K$ to $K^{\prime }$ intervalley scatterings.

Figure 5

Supercells and the simulated band structures. Each supercell comprises nine unit cells, whose hole periodicity is given by $19\sqrt{3}a$. In diagram (a), we have an antidot lattice without any randomness; i.e., periodicity is accurate down to the atomic scale. We can see that there is no band gap, as predicted. (b) Antidot lattice with randomness. For this case, the band gap reappears, accompanied by a small energy splitting for both the conduction and valence bands. In other words, the states in the conduction and valence bands are quasidegenerate.

Figure 6

Cartoon image illustrating different wave states’ evolution under an external electric field under the influence of the Berry curvature. The value of the Berry curvature is shown in the color scale, plotted in the first Brillouin zone of the antidot graphene. The bottom left (right) shows the value of the diagonal matrix term ${\mathrm{\Omega }}_{11}$ (${\mathrm{\Omega }}_{22}$), with the peak value at the $\mathrm{\Gamma }$ point of the antidot reciprocal lattice. The red wave represents the case of $|{\eta }_1^{\prime }{|}^2>1/2$; it travels to the left as $v_x<0$. The blue wave represents the case of $|{\eta }_1^{\prime }{|}^2<1/2$; it travels to the right.

Figure 7

Measurement setup for (a) nonlocal measurements and (b) local measurements. (c) Antidot graphene resistivity is plotted as a function of gate voltage for various temperatures (from 10 to 300 K). Inset: SEM image for the antidot graphene sample. (d) Room-temperature conductivity as a function of gate voltage is shown to be well fitted by the Boltzmann transport theory, which gives the mobility around $1500{\mathrm{cm}}^2/(\mathrm{Vs})$ and the residue carrier density $n_0=2\times {10}^{11}/{\mathrm{cm}}^2$.

Figure 8

(a) The distribution of Berry curvature density $s(x,y)$ obtained by simulations using the comsol package. The decay length $\xi$ is set to be $\xi =W$. The color indicates the magnitude of the Berry curvature density in arbitrary units. (b) The Berry curvature current density $J_S$ is plotted as a function of $x$. The cases of $\xi /W=0.05$, 0.12, 0.3, 1, 2, 4 are shown. In all cases, $J_S$ is seen to decay exponentially with $x$.

Figure 9

(a) Parameters $f$, $\lambda$ and (b) coefficient $\alpha (\xi )$ are plotted as a function of $\xi /W$. The open symbols are simulation results, while the solid curves are from the analytical expression given in Ref. [11]. The simulation results and the predictions of the analytical expression agree well in the range $\xi \gg W$, but they differ when $\xi \le W$. The latter applies to our samples.

Figure 10

Stray current effect simulated by comsol. A voltage drop $\mathrm{V}$ is imposed between the central pair of Hall electrodes. The color scale is in the range of $0.45\mathrm{V}$ to $0.55\mathrm{V}$.

Figure 11

The measured nonlocal resistance $R_{nl}^{\exp }$ is plotted as a function of gate voltage for different temperatures, as shown by the solid curves. The calculated stray current $R_{nl}^{\prime }$ is shown by dashed curves. The magnitude of $R_{nl}^{\prime }$ is amplified by a factor of 7 for 10 K and by a factor of 4 for 20 K.

Figure 12

The peak values (at CNP) of measured nonlocal resistance are plotted as a function of temperature, shown as black solid squares. The calculated stray current effect $R_{nl}^{\prime }\propto \rho (T)$ is shown by the solid red curve. The blue curve denotes the peak values of topological nonlocal resistance calculated, $R_{nl}\propto {\rho }^3(T)$, based on Eq. (28) for a fixed $\xi =450\mathrm{nm}$. The black dashed curve represents $R_{nl}+R_{nl}^{\prime }$. It is clear that there is a crossover from the high-temperature, linear $\rho (T)$ behavior to the low-temperature, ${\rho }^3(T)$ behavior. The measured data are seen to agree with this crossover behavior (dashed curve) extremely well.

Figure 13

The topological nonlocal resistance (defined as $R_{nl}^{\exp }-R_{nl}^{\prime }$) is plotted as a function of carrier density, shown by open circles at temperatures of 10, 20, and 40 K. The fitted nonlocal resistance is shown by the solid black curves. This fitting yields a value for the decay length $\xi$ of 450 nm.

Figure 14

The topological nonlocal resistance (defined as $R_{nl}^{\exp }-R_{nl}^{\prime }$) is plotted as a function of carrier density for sample B (upper panel), with a designed hole diameter of 80 nm, and sample C (bottom panel), with a designed hole diameter of 60 nm, shown by open circles for different samples with different geometric factors. The periodicity is fixed at 150 nm. The solid curves represent the fitted nonlocal resistance.

Figure 15

The black curves represent the I-V curve at 10 K for sample C. The red curve is the derived resistance per square. The resistance at small bias voltage is around $100\mathrm{k}\mathrm{\Omega }$, which is much larger than $h/e^2$. Hence, the existence of a conducting edge state is not likely.