Tuning the Nonlocal Spin-Spin Interaction between Quantum Dots with a Magnetic Field

We describe a device where the nonlocal spin-spin interaction between quantum dots (QDs) can be turned on and off with a small magnetic field. The setup consists of two QDs at the edge of two two-dimensional electron gases (2DEGs). The QDs' spins are coupled through a RKKY-like interaction mediated by the electrons in the 2DEGs. A magnetic field $B_z$ perpendicular to the plane of the 2DEG is used as a tuning parameter. When the cyclotron radius is commensurate with the interdot distance, the spin-spin interaction is amplified by a few orders of magnitude. The sign of the interaction is controlled by finely tuning $B_z$. Our setup allows for several dots to be coupled in a linear arrangement and it is not restricted to nearest-neighbor interaction.

Figure 1

Schematic representation of the proposed device. Both QDs are setup to have an odd number of electrons—this is controlled by the gate voltages $V_1$ and $V_2$. The QDs' spins are then coupled through a RKKY-like interaction mediated by the electrons in the two 2DEGs. This interaction can be tuned by a perpendicular magnetic field $B_z$, being maximum when the cyclotron orbit matches the distance between the two QDs. A third (optional) gate $V_3$ can be used to interrupt one of the electrons' path and cancel out the interaction.

Figure 2

Exchange interaction $J$ as a function of the perpendicular magnetic field $B_z$ for different interdot distances $R=0.5\mu \mathrm{m}$ (a), $1\mu \mathrm{m}$ (b), and $1.5\mu \mathrm{m}$ (c) and $N_0=1$ [$N_0=7$ is shown in (a) (dashed line)]. In the last two cases, $J$ was multiplied by 3 and 6, respectively. The interaction presents large oscillations when the cyclotron radius $r_c$ is such that $2r_c\simeq R/n$ with $n$ an integer number. A fine tuning of $B_z$ allows then to select the magnitude and the sign of the exchange interaction.

Figure 3

Exchange interaction as a function of the interdot distance $R$ for $B_z=0$ (a), 231 mT (b) and 241 mT (c). These values of $B_z$ are indicated in Fig. 2a with arrows. The magnitude of $J$ increases a few orders of magnitude when $R\simeq 2r_c\simeq 500\mathrm{n}\mathrm{m}$ while its sign depends on the precise value of $B_z$. The solid (dashed) lines correspond to $N_0=1$ (7).

Figure 4

Density plot of the exchange interaction $J$ as a function of the perpendicular magnetic field $B_z$ and $R$. The solid lines correspond to different focusing conditions with $n=1$, 2, 3 [cf. Equation (9)]. Note that the magnitude of the interaction is large along those lines while its sign changes (dark and light areas). Inset: $J$ as a function of $R$ along the first hyperbola. The solid line is a fitting to Eq. (10).

Figure 5

Exchange coupling as a function of $B_z$ with the same parameters as in Fig. 2 but with the chemical potential pinned at the partially filled Landau Level. Note that the sign of the interaction tends to be preserved around each focusing field.