Tunability of qubit Coulomb interaction: Numerical analysis of top-gate depletion in two-dimensional electron systems

We investigate the tunability of electrostatic coupling between solid-state quantum dots as building blocks for quantum bits. Specifically, our analysis is based upon two-dimensional electron gas (2DEG) systems and depletion by top gates. We are interested in whether the Coulomb interaction between qubits can be tuned by electrical means using screening effects. The systems under investigation are analyzed numerically solving the Poisson equation in 3D via relaxation techniques with optimized algorithms for an extended set of boundary conditions. These include an open outer boundary, simulation of 2DEG systems, and dielectric boundaries like the surface of a physical sample. The results show that for currently lithographically available feature sizes, the Coulomb interaction between the quantum bits is weak in general due to efficient screening in the planar geometry of 2DEG and top gates. The evaluated values are on the order of $1\mathrm{\mu }\mathrm{eV}$. Moreover, while it is not possible to turn off the qubit interaction completely, an effective tunability on the order of $50\%$ is clearly realizable while maintaining an intact quantum bit structure.

Figure 1

Linear array of double quantum dots. Charge is exchanged between the dots in each pair but different pairs are only coupled electrostatically. Defining gates are not shown.

Figure 2

(Color online) Electric potential $V\left(z\right)$ in a parallel plate geometry to demonstrate depletion in 2DEG—the material below the surface (to the right of the ${\sigma }_1$ layer) is considered uniform with a dielectric constant of $\epsilon$. The charge densities on the three planes shown are ${\sigma }_1$, ${\sigma }_p$ ($p$ for the positive charge of the donor layer), and ${\sigma }_2$, respectively. $V_0$ is the total potential difference across the whole stack related to the pinning of the Fermi level at the surface.

Figure 3

(Color online) Simulation of 2DEG with top gates and applied gate voltages. The center gates [labeled $V_c$ in panel (a)] extend in between the qubit pair such that the 2DEG underneath the gate $V_c$ can be depleted at will [see panels (b) and (c)]. The color coding for charge distributions is consistently taken such that red corresponds to negative and blue to positive charge density [in gray scale all shades are to be considered red except for the dark gray regions in panel (d)]. (a) Outline of top gates by showing the charge distribution on these gates. The $1\text{−}\mathrm{\mu }\mathrm{m}$ bar shows chosen length scale. (b) Charge distribution in the 2DEG with the center gate at the same voltage as the gates separating the other qubits. (c) Same as (b) but with the voltage on the center gate lifted such that additional charge accumulation is allowed in the 2DEG underneath the gate. (d) Differences in the charge distribution for the setup in (c) resulting from small variations on the plunger gate voltages chosen such that the total change of charge within one dot is $\pm e∕2$. (e) Free electrostatic energy of the different charge configurations shown in (d) taken relative to the first configuration No. 1.

Figure 4

(Color online) Same as Fig. 3 but with the center gate [labeled $V_c$ in panel (a)] extending only halfway through from top to bottom. This structure has also slightly increased quantum dot spacing and size from that in Fig. 3. The color coding for charge distributions is the same, with red for negative and blue for positive charge density [in gray scale all shades are to be considered red except for the dark gray regions in panel (d)]. See the description of panels (a)–(e) in the caption of Fig. 3.