Figure 3: Geometric-phase shift because of an in-plane magnetic field .

In the upper panels (a,c,e), the AC oscillations are shown as a function of the Rashba SO-coupling constant α_R and the in-plane field. In the lower panels (b,d,f), the AC oscillations are shown for four different as a function of the gate voltage, V_g. (a,b) -field dependence of the AC effect (experiment). The AC oscillations shift to weaker SO fields with increasing , which is an indication of a geometric-phase shift. Notations P1-P4 indicate peaks and dips of the AC oscillations. The dashed lines in a are guides for the eye. The dashed lines in b represent the zero-line of the AAS amplitude. (c,d) Perturbation theory for a 1D Rashba ring with a small field. The chain line represents rough estimation of the validity of the perturbation expansion. In c, a suppression factor²⁶ of the AC oscillation amplitude is applied in addition to the phase shift described by the equation (3). The dashed lines in d show the zero-line of the cosine function in the equation (3). (e,f) Numerical calculations of the AC effect in disordered multimode rings. The six transverse modes per spin are considered and the electron mean free path is 2 μm. In all calculations, the g-factor . The dashed lines in f show the baseline, which corresponds to the resistance at α_R=0. Because of the weak antilocalization effect, the AAS oscillations have a negative amplitude against this line and depend on V_g.