Tuning the topological insulator states of artificial graphene

We develop a robust, nonperturbative approach to study the band structure of artificial graphene. Artificial graphene, as considered here, is generated by imposing a superlattice structure on top of a two-dimensional hole gas in a semiconductor heterostructure, where the hole gas naturally possesses large spin-orbit coupling. Via tuning of the system parameters we demonstrate how best to exploit the spin-orbit coupling to generate time reversal symmetry protected topological insulator phases. Our major conclusion relates to a second set of topological Dirac bands in the band structure (with spin Chern number $C=3$), which were not reliably obtainable in previous perturbative approaches to artificial graphene. Importantly, the second Dirac bands host more desirable features than the previously studied first set of Dirac bands (with $C=1$). Moreover, we find that upon tuning of the system parameters, we can drive the system to the highly desirable regime of the topological flat band. We discuss the possibilities this opens up for exotic, strongly correlated phases.

Figure 1

Schematic view of artificial graphene. (a) A top view of the superlattice etched into the the metallic top gate. Blue dots represent positive potentials (antidots). $L$ is the lattice spacing. (b) A cross section of the AlGaAs-GaAs-AlGaAs heterostructure. $d$ is the quantum well confined length; $z_0$ is the separation length between the superlattice top gate and the 2DHG.

Figure 2

(a) The dispersion of holes in the semiconductor quantum well: Solid lines are the dispersions obtained from the multiband Luttinger Hamiltonian approach (3). The dashed line corresponds to the single-band, quadratic approximation, i.e., using ${\mathcal{E}}_k=k^2/\left(2m^{*}\right)$. The projections of the total angular momentum are denoted $\sigma$, which at $k=0$ are identically the physical spin projections. (b) A plot of the relative weight of the projection onto each physical spin component $S_z$, and their dependence on $\mathit{k}$, in the lowest-band Kramers spin state $|l=0,{\sigma }_l=+3/2,\mathit{k}\rangle \equiv |\uparrow ,\mathit{k}\rangle$.

Figure 3

(a) Spectrum of 2DHG in the triangular well confinement of Eq. (7). (b) Rashba splitting: The energy splitting between the two lowest ($l=0$) dispersion bands in (a).

Figure 4

Brillouin zone. ${\mathit{G}}_i$ are vectors connecting zone corners ${\mathit{K}}_j$. ${\mathit{K}}_j^{\prime }$ are parity reflections of ${\mathit{K}}_j$.

Figure 5

Evolution of the miniband spectrum upon tuning $d/L$ and $W$. All bands are doubly degenerate due to $\mathcal{T}$ and $\mathcal{P}$ symmetry; each band possesses a pair of Kramers spins, which we can refer to as spin $\uparrow$ and $\downarrow$. (a) Corresponds to the Dirac regime, with small spin-orbit interaction set by the small ratio $d/L=1/8$. We call the lower (upper) two minibands in the vicinity of the Brillouin zone corners $\mathit{K},{\mathit{K}}^{\prime }$, the first (second) Dirac points. The superlattice potential (9) has strength $W=2E_L$, with $E_L$ from (10), chosen to produce steep second Dirac cones. (b) The parameters, $d/L=1.75/8$ and $W=E_L$, are chosen to produce a prominent TI gap at the second Dirac cones. (c) Demonstration of a nearly flat, topologically nontrivial band; the second highest in energy miniband is seen to become nearly flat upon choosing parameters $d/L=1.75/8$ and $W=3E_L$, while the Chern numbers for each of the Kramers spins in this band are $C_{\uparrow ,\downarrow }=\pm 3$. Note: The topological band gaps at the first Dirac points are nonzero, yet smaller than the thickness of the lines; see Fig. 6.

Figure 6

Comparison of the first Dirac points with parameters (a) $d/L=1/4$, (b) $d/L=1/2$, with the strength of the potential the same for both cases, $W=2E_{L=4}$. The black dotted lines connecting the upper and lower bands in each figure are a schematic representation of the edge modes. The arrows indicate the Kramers spin of the edge modes.

Figure 7

Effect of Rashba ($\mathcal{P}$ breaking) on the band structure. The bulk (solid lines) and edge (dashed lines) bands are evaluated in the vicinity of (a) the $\mathit{K}$ points, and (b) the ${\mathit{K}}^{\prime }$ points. We take the parameters $\eta =\gamma =v=1$ in Eq. (16). The arrows indicate the Kramers spin of the edge modes.