Spin relaxation dynamics of quasiclassical electrons in ballistic quantum dots with strong spin-orbit coupling

We performed path integral simulations of spin evolution controlled by the Rashba spin-orbit interaction in the semiclassical regime for chaotic and regular quantum dots. The spin polarization dynamics have been found to be strikingly different from the D’yakonov-Perel’ (DP) spin relaxation in bulk systems. Also an important distinction has been found between long time spin evolutions in classically chaotic and regular systems. In the former case the spin polarization relaxes to zero within relaxation time much larger than the DP relaxation, while in the latter case it evolves to a time independent residual value. The quantum mechanical analysis of the spin evolution based on the exact solution of the Schrödinger equation with Rashba SOI has confirmed the results of the classical simulations for the circular dot, which is expected to be valid in general regular systems. In contrast, the spin relaxation down to zero in chaotic dots contradicts what has to be expected from quantum mechanics. This signals on the importance of quantum effects missed in the semiclassical simulations for long time spin dynamics.

Figure 1

(Color online) (a) Electron motion inside a quantum dot. The trajectory consists of three straight segments ${\gamma }_1$, ${\gamma }_2$, and ${\gamma }_3$. (b) The corresponding spin evolution on the $s_x{s}_y$ plane, which is projected from (c). (c) The spin evolution induced by the Rashba spin-orbit interaction on the three-dimensional unit sphere.

Figure 2

(Color online) Electrons trajectories (solid lines) for short time intervals: (a) in bulk, (b) circular quantum dot, (c) triangular quantum dot, and (d) Sinai quantum dot.

Figure 3

Solid curves $C_1$, $C_2$, and $C_3$ represent the time dependent polarization $P_c^z\left(t\right)$ for 2124 particles in an unbounded QW with $L_{\mathrm{so}}=10$, 6, 2, and the mean free path $l_m=1$. The particles were initially placed inside a circular area of the radius $R=1$ and polarized in the $z$ direction. The dashed curves depict the DP relaxation calculated from Eq. (28). For comparison, curve $C_4$ shows $P_c^z\left(t\right)$ for 2124 particles confined inside a circular dot of the radius $R=1$ and $L_{\mathrm{so}}=10$.

Figure 4

(Color online) Time dependence of $P_c^z\left(t\right)$ for $L_{\mathrm{so}}=10$ in the smooth circular dot (curve $C_5$), the circular dot with the rough boundary (curve $C_6$), the triangular dot (curve $C_7$), and the Sinai billiard (curve $C_8$).

Figure 5

(Color online) Time dependence of $P_c^z\left(t\right)$ for $L_{\mathrm{so}}=2$ in the smooth circular dot (curve $C_5$), the circular dot with the rough boundary (curve $C_6$), the triangular dot (curve $C_7$), and the Sinai billiard (curve $C_8$).

Figure 6

The residual polarization $P_c^z$ vs the spin rotation length $L_{\mathrm{so}}$ for a smooth circular dot.

Figure 7

Electron trajectories on the $xy$ plane [(a)–(h)] and respective spin evolution patterns on the $s_x{s}_y$ plane [(i)–(p)] for $L_{\mathrm{so}}=5$. (a), (b), and (c) periodic triangular, hexahedral, and starlike trajectories in the smooth circular dot. (d) A nonperiodic trajectory in the smooth circular dot. (e) A stochastic trajectory in the rough circular dot. (f) and (g) Two periodic trajectories in the triangular dot. (h) A nonperiodic trajectory in the triangular dot.

Figure 8

(Color online) The residual spin polarization $P^z$ vs $L_{\mathrm{so}}$ with $k=20$, 30, 40, (dashed), $k=5$ (dotted), $k=1$ (solid), and $k=0.1$ (dash-dotted). The dashed curve coincides with the curve from Fig. 6.

Figure 9

The function $Z_{\nu }\left(\epsilon \right)$ for $\nu =-1∕2$, $1∕2$, and $3∕2$. This function is singular at $\epsilon =0$ for $\nu =-1∕2$.