Role of Orbital Dynamics in Spin Relaxation and Weak Antilocalization in Quantum Dots

We develop a semiclassical theory for spin-dependent quantum transport to describe weak (anti)localization in quantum dots with spin-orbit coupling. This allows us to distinguish different types of spin relaxation in systems with chaotic, regular, and diffusive orbital classical dynamics. We find, in particular, that for typical Rashba spin-orbit coupling strengths, integrable ballistic systems can exhibit weak localization, while corresponding chaotic systems show weak antilocalization. We further calculate the magnetoconductance and analyze how the weak antilocalization is suppressed with decreasing quantum dot size and increasing additional in-plane magnetic field.

Figure 1

Average modulation factor $\overline{\mathcal{M}}(L)$ as a function of orbit length $L$ in units of bounce length $L_b$ for the quarter-circle billiard (curve 1), the desymmetrized Sinai billiard (curve 2), the desymmetrized diamond billiard (curve 3), and for an unbounded diffusive system with mean-free path equal to $L_b$ (analytical curve 4). The relative strength of spin-orbit interaction is ${\theta }_R/2\pi =0.2$. Inset: $\overline{\mathcal{M}}(L)$ for the desymmetrized Sinai billiard at different values of ${\theta }_R$.

Figure 2

Relative quantum correction to the reflection $\delta \mathcal{R}/\delta {\mathcal{R}}^{(0)}$ versus spin-orbit interaction ${\theta }_R$ for $\mathbf{B}=0$ in the desymmetrized Sinai (solid curve), diamond (dashed curve), and quarter-circle (dashed curve with circles) billiards with ${\mathsf{P}}_c/(w+w^{\prime })=90$. Lower left inset: $\delta \mathcal{R}/\delta {\mathcal{R}}^{(0)}$ versus Zeeman interaction ${\theta }_Z$ for the desymmetrized Sinai billiard. The in-plane field is directed parallel (solid curve) and perpendicular (dashed curve) to the long side. Upper right inset: $\delta \mathcal{R}/\delta {\mathcal{R}}^{(0)}$ versus perpendicular magnetic flux for the same billiard with ${\theta }_Z=0$.