Nonuniform Rashba-Dresselhaus spin precession along arbitrary paths

Electron spin precession in nonuniform Rashba-Dresselhaus two-dimensional electron systems along arbitrary continuous paths is investigated. We derive an analytical formula to describe the spin vectors (expectation values of the injected spin) in such conditions using a contour-integral method. The obtained formalism is capable of dealing with the nonuniformity of the Rashba spin-orbit field due to the inherent random distribution of the ionized dopants, and can be applied to curved one-dimensional quantum wires. Interesting examples are given, and the modification to the spin precession pattern in a Rashba-Dresselhaus channel when taking the random Rashba field into account is shown.

Figure 1

(Color online) Schematic of a continuous curved path sectioned into $N$ pieces.

Figure 2

(Color online) (a) Spin vectors along four ideal 1D half-rings. Rings (i) and (ii) contain only the RSO while in rings (iii) and (iv) the DSO is involved. The spin, polarized parallel either to the radial or the tangential directions, is injected at the lower end of each ring. (b) Corresponding $⟨S_z⟩$ in units of $\hslash ∕2$ as a function of the ring argument $\theta$.

Figure 3

(Color online) Two ideal 1D wires with position-dependent RSO strength. The upper wire shows the effect of a linear $\alpha$ and the lower wire shows a sine-varying $\alpha$. In the main panel, $\alpha$ and $⟨S_z⟩$ as functions of the longitudinal position are plotted.

Figure 4

(Color online) Position-dependent Rashba field $\alpha$ with (a) regular and (b) random dopant distribution, resulting in the corresponding spin precession patterns in (c) and (d), respectively, where point injection of $x$-polarized spins at the left ends, are assumed. The color bars determine the magnitude of $\alpha$ in units of eV nm in (a) and (b), and the $z$ compont of the spin vector $⟨S_z⟩$ in units of $\hslash ∕2$ in (c) and (d). (e) and (f) show $⟨S_x⟩$, $⟨S_y⟩$, $⟨S_z⟩$, and $\alpha$ as functions of longitudinal position $x$ at $y=0$ for the regular and the random cases, respectively.