Aharonov-Casher Effect in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ Square-Ring Interferometers

Electrical control of spin dynamics in ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ was investigated in ring-type interferometers. Aharonov-Bohm and Altshuler-Aronov-Spivak resistance oscillations against a magnetic field, and Aharonov-Casher resistance oscillations against the gate voltage were observed in the presence of a Berry phase of $\pi$. A very large tunability of spin precession angle by the gate voltage has been obtained, indicating that ${\mathrm{Bi}}_2{\mathrm{Se}}_3$-related materials with strong spin-orbit coupling are promising candidates for constructing novel spintronic devices.

Figure 1

(a) Scanning electron microscopy (SEM) image of the as-grown single-crystalline ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ nanoplates and nanoribbons on Si substrate. (b) High-resolution transmission electron microscopy image of a nanoplate. (c) Electron diffraction pattern of a nanoplate with growth direction along $[11\overline{2}0]$. (d) SEM image of the square-ring devices. The whitish colloid on top of the rings is the residual PMMA mask. (e) X-ray powder diffraction pattern consistent with PDF cards No. 33–214 confirms the ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ phase. (f) $R_{xx}$ and $R_{xy}$ curves of a Hall-bar fabricated on the same nanoplate connecting to device #1. The data were taken at 0.3 K.

Figure 2

(a) Magnetoresistance (MR) of device #1 measured at 0.3 K and $V_g=0$. Inset: MR around zero field and at several different temperatures. (b) and (c): Details of MR fluctuations in low and high magnetic fields, at the two windows marked in (a), respectively. (d) and (e): MR in (b) and (c) after the UCF background is subtracted, labeled as $\Delta R$. (f) Fast Fourier transformation (FFT) of $\Delta R$ in a field range of 0.2 to 8 T. The red line and arrows are guidance for the eyes, showing the peak position and the upper and lower bounds expected from the innermost, middle, and outermost electron trajectories of the square rings.

Figure 3

2D plots of $\Delta R$ as a function of magnetic field $B$ and back-gate voltage $V_g$ measured at 0.3 K for devices #1 (a) and #2 (b). The long and short arrows in both plots illustrate the $B$-dependent AB and AAS oscillation periods, respectively. The white lines in (b) indicate the $\pi$ phase shift of the $V_g$-dependent AC oscillations in the AB region. The $V_g$-dependent oscillation frequency doubles in the AAS region. The rectangle area in (b) is magnified and shown at the upper right corner.

Figure 4

(a) Illustration of AB interference (red trajectories) and AAS interference (solid and dottd green loops) for charges in a square ring. The existence of SOC creates an effective magnetic field $B_{\mathrm{so}}$ pointing towards/ or from the center of the ring for counterclockwise (CCW)/clockwise (CW) propagation modes, which influences spin precession and generates an AC phase in addition to the AB and AAS phases. (b) $A_1(\theta )$ and $A_2(\theta )$ as a function of $\theta$ (see text). The former varies roughly at twice frequency of the latter. (c) $\Delta R\mathrm{\text{−}}{V}_g$ curves taken from Fig. 3b at fixed fields marked by the lines and arrows of corresponding colors. The spin precession angle is modulated by $4\pi$ by varying $V_g$ from the interval 2.16 to 2.77 V marked by the two stars. The dashed lines help illustrating the opposite phases between the green and blue curves in the AAS region.