*6.1. Spin-Independent Function C<sup>α</sup>*R,*α*D

The function *C<sup>α</sup>*R,*α*D has symmetries, *C<sup>α</sup>*1,*α*2 = *C<sup>α</sup>*2,*α*1 , as well as *C<sup>α</sup>*1,*α*2 = *C*−*α*1,*α*<sup>2</sup> , *C<sup>α</sup>*1,*α*2 = *C<sup>α</sup>*1,−*α*<sup>2</sup> . Moreover, it also obeys the relation *C<sup>α</sup>*R,*α*D |*ν* = − *C<sup>α</sup>*R,*α*D | *π*2 −*ν*. Therefore, the angle *ν* = *π*/4 is rather special. At this angle, C*<sup>α</sup>*R,*α*D is identically zero and hence no pumping. One can check this since *F*1 = *F*2 and hence *c*1 = *c*2 and *s*1 = *s*2, then

$$\mathbf{r}\_x = \mathbf{r}\_1 \mathbf{s}\_1 (\sin \xi\_1 - \sin \xi\_4) = 2c\_1 s\_1 \cos \beta \sin(\theta + \frac{\pi}{4}),\tag{47}$$

$$
\tau\_y = -\mathbf{c}\_1 \mathbf{s}\_1 (\cos \zeta\_3 \mathbf{s} + \cos \zeta\_2 \mathbf{s}) = -2\mathbf{c}\_1 \mathbf{s}\_1 \cos \beta \cos(\theta - \frac{\pi}{4}).\tag{48}
$$

Therefore, the relation *τx* = <sup>−</sup>*τy* holds for any *β* and *θ* and C*<sup>α</sup>*R,*α*D = 0.

Because of its symmetric property, we focus on the function *C<sup>α</sup>*R,*α*D in the range 0 ≤ *α*R, *α*D ≤ *π*. As an example, we chose *β* = *π*/5 and the results for *ν* = *π*/2 and *ν* = 3*π*/8 are shown in Figure 3. The result for *ν* = *π*/4 is uniformly zero as noted above and that for *ν* = *π*/8 is similar to that for *ν* = 3*π*/8 with reversing the sign of the function. There are areas where the absolute value of C*<sup>α</sup>*R,*α*D is enhanced near (*<sup>α</sup>*R, *<sup>α</sup>*D)=( *π*2 , <sup>0</sup>),(0, *π*2 ), which can be understood from Equation (36) since |*<sup>τ</sup>z*| is very close to one. If we choose *β* = *π*/4, the scattering states "flips" at *π*/2 when *α*R is increased from zero to *π* with *α*D = 0 [3]. Then the behavior of *<sup>C</sup><sup>α</sup>*R,*α*D=<sup>0</sup> becomes quite singular, which may need further investigation (not being discussed here).

**Figure 3.** Contour plot of the function C*<sup>α</sup>*R,*α*D depending on the Rashba, *α*R, and Dresselhaus, *α*D, SOI strength parameters. We chose the geometric angles *β* = *π*/5 and *ν* = *π*/2 (**left**) and *ν* = 3*π*/8 (**right**).
