Quartic energy band engineering in artificial semiconductor honeycomb lattices

TL;DR

This work addresses engineering quartic energy dispersion at the band edge in artificial graphene-like lattices. It combines analytical tight-binding theory with numerical Bloch-wave simulations for two geometries—staggered and planar honeycomb lattices—using a Gaussian-dot confinement model. The key contributions are the identification of three quartic regimes, $\{\text{Mexican-hat-shaped}, \text{purely quartic}, \text{non-MHS quartic}\}$, governed by the hopping ratio $ξ=t_2/t_1$, sublattice bias $\delta V$, and geometry, with a critical line $ξ_c = \frac{1}{6\sqrt{1 + (\delta V / 6 t_{1})^2}}$ marking transitions via $E - E_v = -\alpha (k^2 + s k_{c}^{2})^{2}$ and $s \in \{-1,0,1\}$. The results show that staggered lattices can realize all three quartic classes, while planar lattices largely cannot realize MHS, and provide a phase diagram in parameter space to guide design of quartic band edges. This work offers practical design principles for engineering quartic band structures in artificial quantum simulators, enabling exploration of van Hove singularity-driven phenomena in controllable platforms. $ξ = t_2/t_1$, $ξ_c = rac{1}{6\,\sqrt{1 + (\delta V / 6 t_{1})^2}}$, and $E - E_v = -\alpha (k^2 + s k_{c}^{2})^{2}$ are central relations.

Abstract

Artificially engineered lattices provide a flexible platform for reproducing and extending the electronic behavior of atomic-scale materials. Artificial graphene systems, in particular, mimic graphene-like linear dispersion with tunable Dirac cones and offer a route to realizing more exotic band structures. Here we examine the emergence of quartic energy dispersion in artificial graphene heterostructures using analytical modeling and numerical solutions of the effective Hamiltonian. We identify three distinct quartic band types: Mexican-hat-shaped (MHS), purely quartic, and non-MHS quartic bands, and determine the conditions under which each arises. We find that a staggered honeycomb lattice supports all three classes of quartic dispersion, whereas its planar counterpart yields only purely quartic and non-MHS forms. These results demonstrate the feasibility of engineering quartic band edges in artificial lattices and clarify how lattice geometry can be used to tailor their characteristics.

Figures (5)

• Figure 1: (a) Staggered honeycomb structure, with $a_{dd}$ and $\Delta$ being inter-dot distance and relative displacement of sublattices in the out-of-plane direction, respectively. (b) Radius and length of the dots is represented by $\rho$ and $L$. (c) Energy dispersion for the staggered lattice with $a_{dd}=$ 50 nm, $\rho=20$ nm, $L=10$ nm and $V_{0}=-12$ meV. (d) Planar structure with repulsive potentials at the bridge sites of the honeycomb structure are shown as gray circles.
• Figure 2: Energy dispersions for various parameters, as the dot radius is varied between 20 and 22 nm (a), the confinement voltage changes between $-10.5$ and $-11.5$ meV (b), and as the dot heights are changed from 8 to 12 nm (c).
• Figure 3: Color map of the extracted coefficient $s\,k_{c}^{2}$ as a function of $\rho$ and potential depth $V_{0}$. Negative values ($s\,k_{c}^{2} < 0$) indicate Mexican-hat-shaped dispersion with a ring-like minimum, while positive values correspond to conventional parabolic bands centered at $\Gamma$. The dashed line marks the $s\,k_{c}^{2} = 0$ condition, indicating purely quartic dispersion. The diagram highlights three regimes: gapless (band touching), MHS (quartic with indirect gap), and non-MHS (parabolic). This phase map guides the design of quartic band structures.
• Figure 4: Solid lines represent the energies of classical two-dot unit cells. Attractive potentials 70 meV and 100 meV energies demonstrated by dotted-dashed lines.
• Figure 5: Effects of sublattice potential differences as calculated for the staggered (a), and planar (b) geometries.