Control of the two-electron exchange interaction in a nanowire double quantum dot

The two-electron exchange coupling in a nanowire double quantum dot (DQD) is shown to possess Moriya's anisotropic superexchange interaction under the influence of both the Rashba and Dresselhaus spin-orbit couplings (SOCs) and a Zeeman field. We reveal the controllability of the anisotropic exchange interaction via tuning the SOC and the direction of the external magnetic field. The exchange interaction can be transformed into an isotropic Heisenberg interaction, but the uniform magnetic field becomes an effective inhomogeneous field whose measurable inhomogeneity reflects the SOC strength. Moreover, the presence of the effective inhomogeneous field gives rise to an energy-level anticrossing in the low-energy spectrum of the DQD. By fitting the analytical expression for the energy gap to the experimental spectroscopic detections [S. Nadj-Perge et al., Phys. Rev. Lett. 108, 166801 (2012)], we obtain the complete features of the SOC in an InSb nanowire DQD.

Figure 1

(a) Schematic diagram of the nanowire double quantum dot (DQD) with a strong spin-orbit coupling (SOC). The gates LB, CB, and RB define the barriers that form the DQD, and the plunger gates LP and RP control the electron occupancy of the individual QDs. The SOC vector $\widehat{\mathbf{a}}=(cos\theta ,sin\theta ,0)$ and the magnetic-field direction $\widehat{\mathbf{n}}=(cos\phi ,sin\phi ,0)$ are shown in the $x-y$ plane. (b) The confinement potential along the wire direction (the $x$ axis). Each dot contains only a single electron, and because of the SOC the localized eigenstates of the QDs are quasispin states.

Figure 2

The energy spectrum of the nanowire DQD versus the Zeeman-field splitting ${\mathrm{\Delta }}_{\mathrm{z}}$ for different values of the inhomogeneity $\mathrm{\Delta }B$. The dashed (red) curves represent the case when $\mathrm{\Delta }B=0$. The solid (blue) curves are for $\mathrm{\Delta }B=0.21$, where $2{\mathrm{\Delta }}_{\mathrm{SO}}^{\mathrm{DD}}$ denotes the energy gap between the levels $|{\mathrm{\Phi }}_3\rangle$ and $|{\mathrm{\Phi }}_4\rangle$ at the anticrossing point when ${\mathrm{\Delta }}_{\mathrm{z}}\simeq J_0$.

Figure 3

(a) Components of the DM vector $\widehat{v}$ as a function of the angle $\phi -\theta$, for $x_0=30$ nm and $x_{\mathrm{so}}=200$ nm. (b) Schematic diagram of the DM vector $\widehat{v}$ in the ${\mathbf{e}}_x-{\mathbf{e}}_z$ plane, with the azimuthal angle $\gamma =arccos\left[cos(\phi -\theta )/f\right]$.

Figure 4

The magnitude of the energy gap ${\mathrm{\Delta }}_{\mathrm{SO}}^{\mathrm{DD}}$ as a function of the magnetic-field direction angle $\phi$. The solid (red) line corresponds to the experimental data in Ref. [46]. The other three lines represent the theoretical results based on Eq. (11) under different values of $\theta$, with $J_0\simeq 24.6\mu \mathrm{eV},2\mathrm{d}=50$ nm, and $x_{\mathrm{so}}=165\mathrm{nm}$.