Artificial graphene in a strong magnetic field: Bulk current distribution and quantum phase transitions

We present calculations of the equilibrium current density and Chern numbers for a two-dimensional electron gas in a sinusoidal periodic potential with infinite strip geometry and a perpendicular magnetic field. We consider a triangular lattice of antidots with large ($a=120$ nm) lattice spacing. Such a system is known as artificial graphene (AG). To compute the current density we numerically diagonalize the AG Hamiltonian over a set of Landau level basis states; this takes into account coupling between different Landau levels. Our calculations show that, at magnetic fields typical for quantum Hall measurements, extended streams of current are present in the bulk of the sample when the chemical potential lies within a bulk band gap. We investigate the scaling of these streams with potential strength. Knowledge of the AG energy levels allows us to compute the Chern number associated with each energy gap. We demonstrate that in tuning the height of the potential modulation the Chern number can undergo a transition between two different values.

Figure 1

A long stripe of width $L_x=10a$ used in the calculations, $L_y\gg L_x$.

Figure 2

Calculated dispersions within the $n=0$ Landau level for the following values of magnetic flux: (a) $\varphi /{\varphi }_0=2$ (two subbands), (b) $\varphi /{\varphi }_0=3$ (three subbands), (c) $\varphi /{\varphi }_0=7/2$ (seven subbands), and (d) $\varphi /{\varphi }_0=4$ (four subbands). The potential strength is $W=0.4$ meV and the sample width is $L_x=30a$ for a superlattice period $a=120$ nm. The edge-state dispersions are shown in color, with the left edge blue and the right edge magenta.

Figure 3

Density of states as a function of the chemical potential for (a) $\varphi /{\varphi }_0=3$ and (c) $\varphi /{\varphi }_0=2.96$, given in arbitrary units. Hall conductivity as a function of the chemical potential for (b) $\varphi /{\varphi }_0=3$ and (d) $\varphi /{\varphi }_0=2.96$, computed using Eq. (13). As in Fig. 2 we have $W=0.4\mathrm{meV}$ and $L_x=30a$. The Hall conductivity is computed via Eq. (13), which is valid within energy gaps (see discussion at the end of Sec. 2).

Figure 4

Color map of ${\sigma }_{xy}$ computed numerically using Eq. (13) plotted as a function of the chemical potential and flux per unit cell. As in Fig. 2 the potential strength is $W=0.4$ meV and only the lowest Landau level is shown.

Figure 5

Maps of ${\sigma }_{xy}$ over the $(\mu ,B)$ plane in the region around gap 2 at $\varphi /{\varphi }_0=4$. Each plot shows this region at a different potential strength $W$. (a) $W=0.5\mathrm{meV}$, (b) $W=0.6\mathrm{meV}$, and (c) $W=0.7\mathrm{meV}$.

Figure 6

Schematic of the evolution in band structure on the $(\mu ,B)$ plane which leads to a change in the Chern number. Black bands represent bulk states and white and blue regions represent connected regions of edge states. As $W$ is increased the central energy gap (at $\varphi /{\varphi }_0=q/p$) closes and then reopens with a different Chern number. Chern numbers in each energy gap are labeled ${\sigma }_i$. We note that if the central gap occurs at a flux $\varphi /{\varphi }_0=p/q$, then the difference in the Chern number between the two gaps shown is an integer multiple of $q$.

Figure 7

Overview of computed $j_y$ data at each value of the chemical potential shown in Fig. 2. The first column contains color maps of $j_y(x,y)$ at $W=0.4\mathrm{meV}, a=120\mathrm{nm}$, and $\varphi /{\varphi }_0=3$ for (a) $\mu ={\mu }_1$, (c) $\mu ={\mu }_2$, and (e) $\mu ={\mu }_3$ [see Fig. 2]. Solid blue lines indicate the points for which $j_y(x,y)=0$ and white circles indicate antidot lattice sites. Each plot in the second column [(b), (d), (f)] shows a cut of the data to its left taken at $y=0.5a$ [i.e., $j_y(x,0.5a)$]. In addition, we include the data for $W=0.1$, 0.2, 0.3, and 0.5 meV. The labels $O$ and $X$ in (a) and (b) indicate the circulating and extended parts of the current, respectively. In each calculation we fix $L_x=10a=1.2\mu \mathrm{m}$.