Persistent spin helix in Rashba-Dresselhaus two-dimensional electron systems

A persistent spin helix (PSH) in spin-orbit-coupled two-dimensional electron systems was recently predicted to exist in two cases: [001] quantum wells (QWs) with equal coupling strengths of the Rashba and the Dresselhaus interactions (RD), and Dresselhaus-only [110] QWs. Here we present supporting results and further investigations, using our previous results [Phys. Rev. B 72, 153305 (2005)]. Refined PSH patterns for both RD [001] and Dresselhaus [110] QWs are shown, such that the feature of the helix is clearly seen. We also discuss the time dependence of spin to reexamine the origin of the predicted persistence of the PSH. For the RD [001] case, we further take into account the random Rashba effect, which is much more realistic than the constant Rashba model. The distorted PSH pattern thus obtained suggests that such a PSH may be more observable in the Dresselhaus [110] QWs, if the dopants cannot be regularly enough distributed.

Figure 1

Schematic sketch of a 2DES, where an electron spin described by a given state ket $\mid s{⟩}_{{\vec{r}}_i}$ is ideally injected at ${\vec{r}}_i$ with polar and azimuthal angles ${\theta }_s$ and ${\varphi }_s$, respectively.

Figure 2

(Color online) Effective magnetic fields drawn in the $k_x$-$k_y$ coordinates for (a) the RD model, and (c) the Dresselhaus [110] model. (b) and (d) show the corresponding PSH patterns in a $1\times 1\mathrm{\mu }{\mathrm{m}}^2$ $\mathrm{InAs}$-based 2DES, with assumed point injection of spins indicated by the bold arrows. Color bar calibrates $⟨S_z⟩$ in (b) and $⟨S_x⟩$ in (d), in units of $\hslash ∕2$, with yellow (bright) and red (dark) colors meaning positive and negative values, respectively.

Figure 3

Band structure of a spin-orbit-coupled 2DES. Vertical projection fixes $k$ as $k_F$, leading to time-resolved spin precession, while horizontal projection fixes energy as $E_F$, leading to spatial spin precession.

Figure 4

(Color online) PSH patterns in RD [001] QWs with (a) regularly and (b) randomly distributed dopants. Bold arrows indicate the assumed point injection of spin. Color bar calibrates $⟨S_z⟩$ in units of $\hslash ∕2$ with yellow (bright) and red (dark) colors meaning positive and negative values, respectively. Note that the $x$ axis here is pointing to [110].