Asymmetric exchange between electron spins in coupled semiconductor quantum dots

We obtain a microscopic description of the interaction between electron spins in bulk semiconductors and in pairs of semiconductor quantum dots. Treating the $\mathbf{k}\mathbf{\bullet }\widehat{\mathbf{p}}$ band mixing and the Coulomb interaction on the same footing, we obtain in the third order an asymmetric contribution to the exchange interaction arising from the coupling between the spin of one electron and the relative orbital motion of the other. This contribution does not depend on the inversion asymmetry of the crystal and does not conserve the total spin. We find that this contribution is $\sim {10}^{-3}$ of the isotropic exchange, and is of interest in quantum information. Detailed evaluations of the asymmetric exchange are given for several quantum dot systems.

Figure 1

(Color online) (a) Dependence of the asymmetric exchange $\beta ∕J$ on the separation $d$ along the growth axes between two cylindrical vertically coupled dots with a lateral offset $b=4\mathrm{nm}$, for a smaller dot size $a_{-}=5\mathrm{nm}$ and differences between radii of dots $\Delta a=a_{+}-a_{-}∊[5,6]\mathrm{nm}$. (b) dependence of asymmetric exchange on the lateral offset $b$ at a separation $d=10\mathrm{nm}$ for $a_{-}=5\mathrm{nm}$ and varying $\Delta a$. The insets in [(a)/(b)] are lateral/vertical views of the geometry. (c) Asymmetric exchange $\beta ∕J$ obtained here (solid line) compared with the asymmetric exchange ${\beta }_D∕J$ from Dresselhaus coupling (dashed line) as a function of separation $d$, for $\Delta a=0.22\mathrm{nm}$, $a_{-}=5\mathrm{nm}$, and $b=4\mathrm{nm}$.

Figure 2

(Color online) Dependence of modulus of asymmetric exchange $\beta ∕J$ on angle $\gamma$ between the axes of two elliptical vertically coupled dots of equal major and minor axes $a_{-,x}=a_{+,y}=3.5\mathrm{nm},a_{-,y}=a_{+,x}=6.5\mathrm{nm}$, at $b=4\mathrm{nm}$. The inset gives a projection of the dot contours on a plane perpendicular to the growth axis, using the harmonic potential parameters to represent the dot sizes.

Figure 3

(Color online) Dependence of the modulus of the asymmetric exchange $\beta ∕J$ on angle $\gamma$ between the axes of two elliptical laterally coupled dots of equal potential sizes $A_{-,x}=A_{+,y}=6\mathrm{nm}$ and $A_{-,y}=A_{+,x}=9\mathrm{nm}$. The inset shows the two QDs that lie in the same plane, seen along the growth axis, using the Gaussian potential parameters to define their sizes.