Spin qubits in antidot lattices

We suggest and study designed defects in an otherwise periodic potential modulation of a two-dimensional electron gas as an alternative approach to electron spin based quantum information processing in the solid-state using conventional gate-defined quantum dots. We calculate the band structure and density of states for a periodic potential modulation, referred to as an antidot lattice, and find that localized states appear, when designed defects are introduced in the lattice. Such defect states may form the building blocks for quantum computing in a large antidot lattice, allowing for coherent electron transport between distant defect states in the lattice, and for a tunnel coupling of neighboring defect states with corresponding electrostatically controllable exchange coupling between different electron spins.

Figure 1

(Color online) (a) Schematic illustration of a periodic antidot lattice; antidots may, e.g., be fabricated using local oxidation of a $\mathrm{Ga}\left[\mathrm{Al}\right]\mathrm{As}$ heterostructure. (b) Geometry of the periodic antidot lattice with the Wigner-Seitz cell marked in gray and the antidot diameter $d$ and lattice constant $\Lambda$ indicated. (c) A designed defect leads to the formation of defect states in which an electron with spin $\mathbf{S}$ can reside. (d) Tunnel coupled defects. The coupling can be controlled using a split-gate with an effective opening denoted $w$.

Figure 2

Band structures and densities of states $g\left(ϵ\right)$ of the periodic antidot lattice for three different values of the relative antidot diameter $d∕\Lambda$. Notice the different energy scales for the three cases. On each band structure the gap ${\vartheta }_{\mathrm{eff}}$ is indicated, below which no states exist for the periodic lattice. The band gaps and the gap below ${\vartheta }_{\mathrm{eff}}$ are highlighted as hatched regions. Also shown is the periodic lattice structure with the Wigner–Seitz cell indicated in gray, as well as the first Brillouin zone (FBZ) with the three high-symmetry points and the irreducible FBZ indicated.

Figure 3

(Color online) Energy spectrum for a single defect. The (dimensionless) eigenvalues corresponding to localized states are shown as a function of the relative antidot diameter $d∕\Lambda$. For a given choice of $\Lambda$, the eigenvalues can be converted to meV using Eq. (3). (a) Energy spectrum for defect states residing in the gap below ${\vartheta }_{\mathrm{eff}}$. The full line indicates the height ${\vartheta }_{\mathrm{eff}}$ of the effective potential in which the localized states reside. The dotted lines are the approximate expressions given by Eqs. (4, 6, 7). The approximate results for ${ϵ}_1$ are in almost perfect agreement with the numerical calculations. (b) Energy spectrum for the defect states residing in the lowest band gap region. The full lines indicate the band gap edges of the periodic structure, ${ϵ}_3^{\left(K\right)}$ and ${ϵ}_2^{\left(\Gamma \right)}$, giving upper and lower limits to the existence of bound states. The inset in both figures show the localized states corresponding to the two lowest energy eigenvalues indicated by the dashed vertical lines. The absolute square is shown.

Figure 4

(Color online) (a) The (dimensionless) tunnel coupling $\mid \tau \mid$ as a function of the relative split gate constriction width $w∕\Lambda$ for three different values of $d∕\Lambda$ in the single-level regime. For a given choice of $\Lambda$, the tunnel couplings can be converted to meV using Eq. (3). (b) Time propagation of an electron initially prepared in the left defect state for $d∕\Lambda =0.4$ and $w∕\Lambda =0.6$. The absolute square of the initial wave function is shown in the upper left panel. The following panels show the state after a time span of $T∕8$, $2T∕8$, and $3T∕8$, respectively, where $T$ is the oscillation period.

Figure 5

(Color online) (a) The structure considered for resonant coupling of distant defect states; two defects separated by a central line of $N=3$ antidots, with a central back gate $V_g$ controlling the potential square well in the region marked with dashed lines. A simple three-level model of the system is illustrated below. (b) The eigenvalue spectrum of the three-level model. The dashed line marks the point of resonance.

Figure 6

(Color online) Energy eigenvalues as a function of the magnitude $\mid V_g\mid$ of the back gate for the structure illustrated in Fig. 5 for $d∕\Lambda =0.5$ and a central line of $N=7$ antidots separating the two defects. The resonances are marked with dotted lines and characterized by a symmetric splitting of the eigenvalues.

Figure 7

(Color online) Numerical time propagation of an electron initially prepared in the left defect of the structure illustrated in Fig. 5a and corresponding to the results of Fig. 6 with $\mid V_{\mathrm{g}}\mid \simeq 16.54$. The charge densities $\rho (x,y)$ are shown in the upper panels, while the lower panels show $\int dy\rho (x,y)$. The oscillation period is denoted $T$.

Figure 8

(Color online) Exchange coupling $J$ for a double defect structure. (a) Exchange coupling as a function of the relative split gate constriction width $w∕\Lambda$ for two different values of the relative antidot diameter and a lattice constant $\Lambda =45\mathrm{nm}$. (b) Exchange coupling as a function of the lattice constant $\Lambda$ for three different values of the relative split gate constriction width.

Figure 9

(Color online) Exchange coupling $J$ for a double defect structure as a function of ${\omega }_c∕{\omega }_0$, where ${\omega }_c=eB∕m^{*}c$ and ${\omega }_0=\hslash ∕\left(2m^{*}{\Lambda }^2\right)$. Results are shown for a relative split gate constriction width $w∕\Lambda =2$, and two different values of the relative antidot diameter $d∕\Lambda$. The lattice constant is (a) $\Lambda =30\mathrm{nm}$ and (b) $\Lambda =45\mathrm{nm}$.