Experimental Demonstration of Spin Geometric Phase: Radius Dependence of Time-Reversal Aharonov-Casher Oscillations

A geometric phase of electron spin is studied in arrays of InAlAs/InGaAs two-dimensional electron gas rings. By increasing the radius of the rings, the time-reversal symmetric Aharonov-Casher oscillations of the electrical resistance are shifted towards weaker spin-orbit interaction regions with their shortened period. We conclude that the shift is due to a modulation of the spin geometric phase, the maximum modulation of which is approximately 1.5 rad. We further show that the Aharonov-Casher oscillations in various radius arrays collapse onto a universal curve if the radius and the strength of Rashba spin-orbit interaction are taken into account. The result is interpreted as the observation of the effective spin-dependent flux through a ring.

Figure 1

Schematic image of the spin geometric phase. The axis label $|{\sigma }_i⟩$ denotes the spinor along the $i$ axis. A 2DEG ring subject to Rashba SOI is assumed to lie in the $x\mathrm{\text{−}}y$ plane; therefore, ${\mathbf{B}}_R$ points along the radial direction. The spin eigenstate is not aligned with ${\mathbf{B}}_R$ but with the precession axis, even in the absence of an external perpendicular field [16].

Figure 2

Magnetoresistance curves of the $40\times 40$ ring array ($r=0.608\mu \mathrm{m}$ for each ring) as a function of gate voltage. The curves are shifted vertically for clarity. Inset: Scanning electron micrograph of the $40\times 40$ ring array. Other fabricated four arrays consist of $50\times 50$ rings with the radius of 0.525, 0.681, 0.857, and $1.05\mu \mathrm{m}$.

Figure 3

(a) AAS oscillation of the $40\times 40$ array ($r=0.608\mu \mathrm{m}$ rings) at $V_g=-2.4\mathrm{V}$. The horizontal axis represents the magnetic field. (b) Gate voltage dependence of the AAS oscillations. The color scale is shown in upper-right margin. (c) AAS amplitude at $B=0$ as a function of gate voltage (time-reversal AC effect). The vertical axis represents the gate voltage. The solid line was calculated by using Eq. (2).

Figure 4

(a) Normalized time-reversal AC oscillations of the electrical resistance plotted against the Rashba SOI strength ($B=0$). According to the Shubnikov–de Haas analysis, the Rashba SOI strength $\alpha$ is proportional to the gate voltage. Solid lines are calculated results by using Eq. (2). (b) Spin geometric phase vs radius for fixed Rashba SOI strengths, together with calculated results (solid lines).

Figure 5

Universal AC oscillations of the electrical resistance at zero magnetic field. The data are the same as those shown in Fig. 4a but plotted against different arguments. Solid lines represent the function in the right-hand side of Eq. (2).