Spin-Orbit Coupling, Antilocalization, and Parallel Magnetic Fields in Quantum Dots

We investigate antilocalization due to spin-orbit coupling in ballistic GaAs quantum dots. Antilocalization that is prominent in large dots is suppressed in small dots, as anticipated theoretically. Parallel magnetic fields suppress both antilocalization and also, at larger fields, weak localization, consistent with random matrix theory results once orbital coupling of the parallel field is included. In situ control of spin-orbit coupling in dots is demonstrated as a gate-controlled crossover from weak localization to antilocalization.

Figure 1

Average conductance $⟨g⟩$ (squares) and variance of conductance $\mathrm{v}\mathrm{a}\mathrm{r}(g)$ (triangles) calculated from $\sim 200$ statistically independent samples (see text) as a function of perpendicular magnetic field $B_{\perp }$ for (a) $8.0\mathrm{\mu }{\mathrm{m}}^{\mathrm{2}}$ dot, (b) $5.8\mathrm{\mu }{\mathrm{m}}^{\mathrm{2}}$ center-gated dot, and (c) $1.2\mathrm{\mu }{\mathrm{m}}^{\mathrm{2}}$ dot at $T=0.3\mathrm{K}$, along with fits to RMT (solid curves). In (b), the center gate is fully depleted. Vertical lines indicate the fitting range; error bars of $⟨g⟩$ are about the size of the squares.

Figure 2

(a) Difference of average conductance from its value at large $B_{\perp }$, $\delta g(B_{\perp },B_{\parallel })$, as a function of $B_{\perp }$ for several $B_{\parallel }$ for the $8.0\mathrm{\mu }{\mathrm{m}}^{\mathrm{2}}$ dot at $T=0.3\mathrm{K}$ (squares) with RMT fits (curves). (b) Sensitivity of $\delta g(0,B_{\parallel })$ to ${\nu }_{\mathrm{s}\mathrm{o}}$ for the $8.0\mathrm{\mu }{\mathrm{m}}^{\mathrm{2}}$ dot, $1\le {\nu }_{\mathrm{s}\mathrm{o}}\le 2$ (shaded), ${\nu }_{\mathrm{s}\mathrm{o}}=1.4$ (solid line), and ${\nu }_{\mathrm{s}\mathrm{o}}=0.8$ (dashed line). (c) $\delta g(0,B_{\parallel })$ (markers) with RMT predictions (dashed curves) and one parameter (solid curves) or two parameter fits (dotted curves) using RMT including a suppression factor due to orbital coupling of $B_{\parallel }$; see text.

Figure 3

(a) Difference of average conductance from its value at large $B_{\perp }$, $\delta g(B_{\perp },0)$, for various temperatures with $B_{\parallel }=0$ for the $8.0\mathrm{\mu }{\mathrm{m}}^{\mathrm{2}}$ dot (squares), along with RMT fits (solid curves). (b) Spin-orbit lengths ${\lambda }_{\mathrm{s}\mathrm{o}}$ (circles) and phase coherence times ${\tau }_{\phi }$ (triangles) as a function of temperature, from data in (a).

Figure 4

Difference of average conductance $⟨g⟩$ from its value at $B_{\perp }=0$ as a function of $B_{\perp }$ for various center gate voltages $V_g$ in the center-gated dot (squares), along with fits to RMT [5]. Good fits are obtained though the theory assumes homogeneous SO coupling. Error bars are the size of the squares. Inset: ${\lambda }_{\mathrm{s}\mathrm{o}}$ and ${\kappa }_{\parallel }$ as a function of $V_g$ extracted from RMT fits; see text.