Electron-Electron Interactions in Artificial Graphene

Recent advances in the creation and modulation of graphenelike systems are introducing a science of “designer Dirac materials”. In its original definition, artificial graphene is a man-made nanostructure that consists of identical potential wells (quantum dots) arranged in an adjustable honeycomb lattice in the two-dimensional electron gas. As our ability to control the quality of artificial graphene samples improves, so grows the need for an accurate theory of its electronic properties, including the effects of electron-electron interactions. Here we determine those effects on the band structure and on the emergence of Dirac points.

Figure 1

Left: Rectangular unit cell in real space. Right: Orthorhombic irreducible Brillouin zone (rectangle $\Gamma XSY$) compared with the conventional one ($\Gamma MK$) for the honeycomb lattice. The basis vectors ${\mathbf{B}}_x$ and ${\mathbf{B}}_y$ are defined in the text. The hexagonal high symmetry points are $\mathbf{K}\equiv \frac{1}{3}{\mathbf{B}}_y={\mathbf{K}}^{\prime }$, $\mathbf{M}\equiv \mathbf{\Gamma }={\mathbf{M}}^{\prime }$. The mapping from the hexagonal to the orthorhombic cell is performed as follows: the $\Gamma -M$ line corresponds to the $\Gamma -X-\Gamma$ path in the orthorhombic cell; the $M-K$ segment corresponds to $\Gamma -K^{\prime }$, and the $K-\Gamma$ segment corresponds to the two segment path $K^{\prime }-K^{\prime \prime }-M^{\prime }$, with ${\mathbf{K}}^{\prime \prime }\equiv \frac{1}{2}({\mathbf{B}}_x+\frac{1}{3}{\mathbf{B}}_y)$. Inset: Part of the lattice in real space.

Figure 2

Energy bands calculated for noninteracting (a) and interacting electrons ($N=1$ per dot) in density-functional theory (b). Clear Dirac points at the Fermi level shifted to zero (dashed lines) and a linear dispersion relation can be found in both cases (circles).

Figure 3

Electron densities in artificial graphene ($N=1$ per dot) calculated for two depths of the potential wells, $V_0=-0.2\mathrm{meV}$ (up) and $V_0=-0.5\mathrm{meV}$ (down). The left and right panels show the results without and with the $e\mathrm{\text{−}}e$ interactions, respectively.

Figure 4

Energy gap at the $M$ point in the Brillouin zone as a function of the quantum-dot depth. Results without (squares) and with $e\mathrm{\text{−}}e$ interactions (circles) are shown for $N=1$ per dot. The inset shows a zoom of the regime where the gap opens.

Figure 5

Energy bands close to the Fermi level shifted to zero (dashed lines) calculated for noninteracting (a) and interacting electrons when there are four electrons in each quantum dot. The two lowest bands (not shown) are located at about $-0.9$ and $-0.8\mathrm{meV}$ in (a) and (b), respectively.