Spin-spin coupling in electrostatically coupled quantum dots

We study the spin-spin coupling between two single-electron quantum dots due to the Coulomb and spin-orbit interactions, in the absence of tunneling between the dots. We find an anisotropic $XY$ spin-spin interaction that is proportional to the Zeeman splitting produced by the external magnetic field. This interaction is studied both in the limit of weak and strong Coulomb repulsion with respect to the level spacing of the dot. The interaction is found to be a nonmonotonic function of interdot distance $a_0$ and external magnetic field and, moreover, vanishes for some special values of $a_0$ and/or magnetic field orientation. This mechanism thus provides a new way to generate and tune spin interaction between quantum dots. We propose a scheme to measure this spin-spin interaction based on the spin-relaxation-measurement technique.

Figure 1

The figure shows a sketch of the model system which consists of two identical quantum dots in the $xy$ plane, separated by distance $a_0$ (measured from dot center to dot center). ${\vec{S}}_i$ denotes the spin of electron $i=1,2$, $\lambda$ is the dot radius, and $\vec{B}$ is the external magnetic field. The respective orbital wave functions of electron 1 and 2 are assumed to have no overlap (i.e., tunneling between the dots is excluded). The remaining purely electrostatic Coulomb interaction between the electron charges leads, via spin-orbit interaction, to an effective coupling between their spins. This spin-spin interaction depends sensitively on the orientation of $\vec{B}$, with no component along it, and is proportional to ${\vec{B}}^2$.

Figure 2

(Color online) The function $G$ occurring in Eq. (42) plotted as a function of the geometric distance $a_0$ between the dot centers scaled by the dot radius $\lambda$ for different magnetic field orientations. The dashed line represents the dipolar approximation of $G$ for a perpendicular magnetic field $(\theta =0)$ which scales as $a_0^{-3}$.

Figure 3

(a) The effective distance $a$ as a function of the Coulomb interaction strength $\lambda ∕a_B$ for $a_0∕\lambda =2$. The full line represents the variational result from Eq. (57). The dashed line corresponds to the one obtained from the classical equilibrium solution of Eq. (53). (b) Effective distance $a∕\lambda$ as a function of the geometrical one $a_0∕\lambda$ for $\lambda ∕a_B=3$. All distances are scaled with the dot radius $\lambda$. The dotted line is $a_0∕\lambda$ which is shown for comparison.

Figure 4

The logarithm of infidelity $1-F$ from Eq. (62) as a function of the geometric distance $a_0∕\lambda$ (scaled with the dot radius $\lambda$) for two different values of Coulomb interaction strength $\lambda ∕a_B$.

Figure 5

The function $I$ from Eq. (68) as a function of the dimensionless geometric distance $a_0∕\lambda$. (a) The case of perpendicular magnetic $(\theta =0)$ field for three different Coulomb strength parameters $\lambda ∕a_B$. (b) The case of in-plane magnetic field $(\theta =\pi ∕2)$, for two values of the Coulomb strength $\lambda ∕a_B$, for three angles between the inter-dot distance vector ${\mathit{a}}_0$ and the magnetic field. The groups of lines 1, 2, and 3 correspond to $\Phi =0$, $\Phi =\pi ∕4$, and $\Phi =\pi ∕2$, respectively.

Figure 6

Scheme to measure the coupling constant $J_{\mathrm{eff}}$ in quantum dots without tunnel coupling. In the initialization step (a), the left dot is at equilibrium with one electron in the lower Zeeman sublevel and the right dot is empty. At the start of the coherent evolution step (b), the right dot is filled with one electron in either upper or lower Zeeman level during a short time $\hslash ∕\Gamma ⪡\hslash ∕J_{\mathrm{eff}}$, and the dots are deep in the Coulomb blockade valley ${\Gamma }^2∕U_{\pm }⪡J_{\mathrm{eff}}$. Further, the two spins evolve coherently due to the spin-spin interaction $J_{\mathrm{eff}}$ during a fixed time $\tau \gtrsim \hslash ∕J_{\mathrm{eff}}$. In the read-out step (c), the left dot is brought up to the Fermi surface, so that the electron can tunnel to the lead only if it is in the upper Zeeman sublevel. The latter event is recorded by a charge detector nearby the left dot.

Figure 7

Residual oscillations in the averaged probability ${\overline{P}}_{L\downarrow }\left(\tau \right)$ for values of $A∕\sqrt{N}$ that exceed the spin-spin interaction strength $4J_{\mathrm{eff}}$. The period of oscillations is not affected by the hyperfine interaction and is given by ${\tau }_0=\pi \hslash ∕4J_{\mathrm{eff}}$. With increasing the hyperfine strength $A∕\sqrt{N}$, the amplitude of oscillations decreases as $\propto \sqrt{N}∕A$. As a function of the waiting time $\tau$, the envelope of oscillations decays as $\propto 1∕\sqrt{\tau }$.