Gate-controlled anisotropy in Aharonov-Casher spin interference: Signatures of Dresselhaus spin-orbit inversion and spin phases

The coexistence of Rashba and Dresselhaus spin-orbit interactions (SOIs) in semiconductor quantum wells leads to an anisotropic effective field coupled to carriers’ spins. We demonstrate a gate-controlled anisotropy in Aharonov-Casher (AC) spin interferometry experiments with InGaAs mesoscopic rings by using an in-plane magnetic field as a probe. Supported by a perturbation-theory approach, we find that the Rashba SOI strength controls the AC resistance anisotropy via spin dynamic and geometric phases and establish ways to manipulate them by employing electric and magnetic tunings. Moreover, assisted by two-dimensional numerical simulations, we identify a remarkable anisotropy inversion in our experiments attributed to a sign change in the renormalized linear Dresselhaus SOI controlled by electrical means, which would open the door to new possibilities for spin manipulation.

Figure 1

Schematic illustration of spin geometric and dynamic phases in an AC spin interferometer. (Left) In the moving electron's rest frame, the SOI field subtends a solid angle (blue) in a round trip around the interference ring. The solid angle is proportional to the spin geometric phase. Only when the Lamor frequency of spin precession ${\omega }_s$ is fast enough compared with the orbital frequency ${\omega }_c$, the SOI field is in $x\text{−}y$ plane (adiabatic limit). Spin precession around SOI field $B_{\mathrm{total}}$ is associated with the dynamical phase. The angle $\theta$ is given by the relation $tan\theta ={\omega }_s/{\omega }_c$. (Right) The in-plane field modulates the geometric phase by changing the solid angle subtended by the total effective field.

Figure 2

(a) Scanning electron microscope image of an array of 40 × 40 InGaAs-based rings. The radius of each ring is 610 nm. (b) Calculated potential energies relative to the Fermi energy (left scale) and the squared wave functions (right scale) for the samples used in this paper. The wave functions are confined in the InGaAs quantum well. The potential gradient becomes larger with increasing negative gate voltage, resulting in an enhancement of the Rashba SOI strength.

Figure 3

(a) AAS oscillations with fixed Rashba SOI strength $\alpha =-1.5\times {10}^{-12}\mathrm{eV}\mathrm{m}$ measured by varying the in-plane magnetic field direction for a constant field strength $B_{\parallel }=1\mathrm{T}$. (b) Filtered AAS oscillations. The amplitude of the AAS oscillations shows an angle dependence, with the maximum and minimum at $\phi =\pi /4$ and $\phi =3\pi /4$, respectively.

Figure 4

(a) AAS oscillations corresponding to a fixed Rashba SOI strength $\alpha =-2.8\times {10}^{-12}\mathrm{eV}\mathrm{m}$ measured by varying the in-plane magnetic field direction for a constant field strength $B_{\parallel }=1\mathrm{T}$. (b) Filtered AAS oscillations. The extrema of the AAS oscillation amplitude are switched when compared with the case shown in Fig. 3, with the maximum at $\phi =3\pi /4$ and the minimum at $\phi =\pi /4$.

Figure 5

(a) Angle dependence of the AAS amplitudes at $B_{\perp }=0$ in the presence of an in-plane field $B_{\parallel }=1\mathrm{T}$. For a Rashba SOI strength $\alpha \approx -1.5\times {10}^{-12}\mathrm{eV}\mathrm{m}$, the maximum and minimum appear around $\phi =\pi /4$ and $\phi =3\pi /4$, respectively. The anisotropic response inverts for $\alpha \approx -2.8\times {10}^{-12}\mathrm{eV}\mathrm{m}$. (b) Corresponding 2D numerical simulations with realistic parameters. The results are in good agreement with the experimental data.

Figure 6

Gate-voltage dependence of the AAS oscillation amplitude at $B_{\perp }=0$ under in-plane fields $B_{\parallel }=1\mathrm{T}$ (left) and $B_{\parallel }=2\mathrm{T}$ (right) applied along different directions. The AAS amplitude modulation by the Rashba SOI strength arises from the AC spin interference. An anisotropic response is observed at in-plane field angles $\phi =\pi /4$ (triangle) and $\phi =3\pi /4$ (circle). For an in-plane field $B_{\parallel }=1\mathrm{T}$, the anisotropy is reversed by tuning the Rashba SOI strength $\alpha$ while keeping its sign unchanged, as shown by arrows.

Figure 7

(a) AC resistance difference (anisotropy) between in-plane field orientations $\phi =\pi /4$ and $\phi =3\pi /4$ as a function of the Rashba SOI parameter $\alpha$. The anisotropy shows an oscillatory behavior as a function of $\alpha$. (b) Corresponding 2D numerical simulations. A remarkable discrepancy appears around $\alpha =-1.2\sim -1.7\times {10}^{-12}\mathrm{eV}\mathrm{m}$, interpreted as a (gate-controlled) sign change of the renormalized linear Dresselhaus SOI (see text).

Figure 8

Renormalized linear Dresselhaus SOI ${\beta }^{\prime }$ including the strain induced term as a function of the carrier density. The red dashed line does not include the strain term.

Figure 9

Gate-voltage dependence of the SdH oscillations. From top to bottom, the corresponding carrier densities are 0.8, 1.0, 1.2, 1.35, 1.55, and $1.7\times {10}^{16}{\mathrm{m}}^{-2}$.

Figure 10

Relation between Rashba SOI parameter and carrier density.