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 a door to new possibilities for spin manipulation.


I. INTRODUCTION
Spintronics and spin-based quantum computing rely on the precise manipulation of spin orientations and related spin phases. Electron spins may couple directly to a magnetic field (Zeeman interaction) as well as to an electric field via spin-orbit interaction (SOI), resulting in a momentum-dependent effective magnetic field acting on itinerant spins. In particular, the electric-field-controllable Rashba SOI [1,2,3] is a prominent resource for spin-orbitronics [4], i.e., for the generation [5,6,7], manipulation [8,9], and detection [10,11] of spins by electrical means only. The direction of the effective Rashba field is perpendicular to the momentum of the spin carriers, but its strength is isotropic. In III-V compound semiconductors, the Dresselhaus SOI [12] induced by bulk inversion asymmetry also plays an important role in spin dynamics [13]. The direction of the effective Dresselhaus field has a different symmetry from Rashba's one. Therefore, the combination of Rashba and Dresselhaus SOIs gives rise to an anisotropic, momentum-dependent field.
A spin interferometer is an invaluable tool to probe the spin-phase information carried by electrons via the Aharonov-Casher (AC) effect [14,15], the electromagnetic dual of the Aharonov-Bohm (AB) effect [16,17]. The role played by Rashba and Zeeman fields on spin phases has been widely investigated in this context [18]- [22]. In contrast, the effects of introducing Dresselhaus SOI on spin phases are not yet well understood. Since hybrid-field engineering is a prerequisite to attain spin manipulation at the nanoscale, the electric control of the Dresselhaus SOI strength and sign appears as a challenging goal that would supply us with new tools for efficient spin control.
In this paper, we use Aharonov-Casher (AC) spin interferometry to extract information about the spinorbit fields and related spin phases. We study the anisotropic response of AC resistance measurements in an array of InGaAs-based mesoscopic rings subject to in-plane magnetic fields oriented along different directions. The experiment shows that the sign of the AC resistance anisotropy changes as a function of the Rashba SOI strength. Perturbation-theory calculations indicate that the AC resistance anisotropy is modulated by the Rashba SOI strength via spin dynamic and geometric phases as well as by the direction of the in-plane Zeeman field [Section II & Appendix A2]. In addition, we find that the reported data are to a great extent reproduced by numerical results performed at constant Dresselhaus SOI strength [23]. There is, however, a remarkable discrepancy: the experiment reveals an extra sign inversion in the anisotropy which is not reproduced by the numerical calculations. This is consistently explained by a sign change of the renormalized linear Dresselhaus SOI emerging from strain effects in the working material, which is controlled electrically. Our results provide crucial information about the SO fields and show how different spin-phase contributions can be manipulated, demonstrating a potential for applications in spintronics and spin-based quantum technologies.
In the following Sec. II, the concept of spin dynamic and geometric phases in magnetic textures is introduced. In Sec. III, we show the anisotropic response of these phases when perturbed by additional Dresselhaus and Zeeman terms. The analytical details on the perturbation theory is described in Appendix A. In Sec. IV, we describe the gate-controlled anisotropy in Aharonov-Casher (AC) spin interferometry experiments with InGaAs mesoscopic rings by using an in-plane magnetic field as a probe. In Sec. V, we discuss a sign change of the renormalized linear Dresselhaus SOI. Section VI summarizes the paper.

II. SPIN DYNAMICS IN MAGNETIC TEXTURES
A magnetic texture is a magnetic field, either of real or effective (e.g., spin-orbit) origin, with nonuniform orientation. The spin dynamics of a carrier travelling through a magnetic texture is determined by the ratio of two characteristic frequencies: the Lamor frequency of spin precession around the local magnetic field, , and an orbital frequency accounting for the change of direction of the magnetic field from the point of view of the spin carrier, [24]. The spin dynamics is said to be adiabatic if the carrier's spin can stay (anti)align with the local magnetic field all across the magnetic texture. This corresponds to the regime where the spin precession frequency is much larger than the orbital frequency, ≫ . In the adiabatic limit, spin states have been shown to acquire phase contributions of geometric nature in addition to the usual dynamic quantum phases [25]. These geometric (or Berry) phases are identified with the solid angle Ω subtended by spins after a round trip in the Bloch sphere (spin texture). However, the adiabatic limit is difficult to achieve in usual experimental setups, where both frequencies tend to be of comparable magnitude and the spin dynamics is non-adiabatic. Still, geometric phases can be generalized to non-adiabatic situations with identical interpretation in terms of spin solid angles [26] even when the non-adiabatic spin texture does not coincide with the magnetic texture. Complementary dynamic spin phases are identified with the projection of the spin texture on the magnetic texture. See Fig. 1 for an illustration in the case of AC rings from the point of view of the spin carrier's rest frame.

INTERFERENCE
The coexistence of Rashba and Dresselhaus SOIs leads to an anisotropic effective magnetic field ( 0 ) with two-fold symmetry given by where is the 2 nd -order-perturbation Zeeman phase shift reported in [22], which was demonstrated to be of purely geometric origin. In contrast, the corresponding 2 nd -order-perturbation Dresselhaus phase shift shows a hybrid geometric/dynamic origin [Appendix A2]. This is also the case for the 3 rd -order-perturbation spin phase [Appendix A2], which is responsible for the anisotropic response of the conductance to the in-plane Zeeman field's direction , defined with respect to the [100] direction. Moreover, its linear dependence on shows that the anisotropic is sensitive to a sign inversion of the Dresselhaus SOI.
The AC conductance anisotropy can be studied by defining Α = ( /4) − (3 /4) , the conductance difference for in-plane Zeeman fields ∕∕ oriented along different symmetry axes. The resulting expression in this approximation is The anisotropy Α oscillates as a function of , , and ∕∕ as the corresponding phases increase.
However, within this perturbative regime, only the dominating Rashba AC phase √1 + � 2 − 1 is expected to induce a sign inversion of Α as the Rashba SOI strength changes (a sign inversion due to the Zeeman field beyond the perturbative approach was confirmed by numerical analysis in [23]).
Moreover, � shows that an additional sign inversion is expected in Α in case the Dresselhaus SOI changes sign. Also notice that Eq. (4) implies that the anisotropic response is originated from the joint action of the Dresselhaus and Zeeman perturbations on the Rashba system.
Additionally, a phenomenological discussion on the role of disorder in the conductance and, particularly, resistance (better suited in experiments) can be found in Appendix A3.

IV. EXPERIMENTS
The experimental setup consist of a top-gate-attached 40×40 ring array (ring radius r= 610 nm) fabricated by electron beam lithography and reactive ion etching. A scanning electron microscope image of the array is shown in Fig. 2 A common strategy is to investigate the gate-voltage dependence of the Al'tshuler-Aronov-Spivak (AAS) [17] oscillations amplitude, originated from the interference of time-reversal (TR) paths in the absence of magnetic flux (i.e., for vanishing perpendicular magnetic field ⊥ = 0 ). The phase contribution from the orbital part of the wave function to TR-path interference is always constructive at ⊥ = 0. Therefore, the AAS amplitude dependence on voltage reflects a phase contribution from the spin part of the wave function. This gives access to the AC spin-interference effect independently from the orbital phases at any gate-voltage value.
Ensemble averaging in the ring-array structure leads to clear AAS-interference patterns in transport measurements. We focused on AC spin interference under in-plane magnetic fields of variable strength ∕∕ and direction , defined with respect to the [100] axis. The magnetoresistance (MR) for fixed gate voltage and ∕∕ = 1 T was measured for different orientations . is explained by the narrowing of the effective channel width for decreasing carrier densities [28]. Most importantly, for = -2.8 x 10 -12 eVm ( Fig. 4(b)) the AAS amplitude shows its minimum at = π/4 and its maximum at = 3π/4, a response opposite to the one observed at = -1.5 x 10 -12 eVm ( Fig. 3(b)). This demonstrates the Rashba-SOI-induced anisotropy inversion without changing the sign of the Rashba SOI.
To study the observed inversion of the anisotropic response, in Fig. 5 (a) we show detailed experimental data on the Zeeman field angle dependence of the AAS amplitude for a field strength ∕∕ =1 T at two different Rashba SOI strengths. We find that the angle-dependent pattern inverts as changes from -1.5 x 10 -12 eVm to -2.8 x 10 -12 eVm while α's sign remains constant. This is well accounted by perturbation theory, Eq. (4), where the anisotropy inversion is attributed to the AC phase √1 + � 2 − 1 in Α , sharing geometric and dynamic phase contributions [20], [29]. The AAS amplitude dependence on the Zeeman field angle (for a given ) has also a hybrid geometric/dynamic phase origin via [Appendix A2]. We notice that a purely geometric spin-phase tuning by the Zeeman field's strength is possible at magic angles =0, π (where vanishes) through [22].
In order to account for realistic conditions in our models beyond the limitations of perturbation theory, we resort to 2D numerical simulations of disordered multi-mode rings. We use the Kwant code [30] with a disorder potential corresponding to a mean-free path of 1.8 µm, which is shorter that the ring circumference 3.8 µm. This disorder is crucial to develop dominating AAS interference paths [17].
The calculation details are described in [23]. We assume a ring radius of 610 nm and a ring channel including 5 modes, with carrier density Ns= 1.52 x 10 16 m -2 . The in-plane Zeeman energy is set to g ∕∕ =0.17 meV, with g= 3 and ∕∕ = 1T, while is fixed to 0.3 x 10 -12 eVm. These parameters are very similar to those of the InGaAs QW used in the present experiment. The results, depicted in Fig. 5 (b), show that the maximum and minimum AAS amplitudes appear around = π/4 and = 3π/4 for = -1.5 x 10 -12 eVm, while this anisotropy is reversed for = -2.8 x 10 -12 eVm.
This is in quite good agreement with the experimental results shown in Fig. 5 (a).
In Fig. 6 we present the AAS amplitude measured as a function of the gate voltage corresponding to two different in-plane field angles = π/4 (red) and = 3π/4 (blue) and field strengths ∕∕ = 1 T (Fig. 6 (left)) and ∕∕ =2 T ( Fig. 6 (right)) for ⊥ =0. The oscillatory response as a function of is due to the AC effect induced by spin phases in TR-path interference. The observed period is well reproduced by perturbation theory, Eq. (2), once the gate voltage dependence of is taken into account. We find that the AC oscillation amplitude decreases by increasing ∕∕ from 1 T to 2 T. This is explained by the spin-induced dephasing effect, as discussed in Ref. [27] and experimentally confirmed in [31]. This discrepancy is remarkable. The most plausible reason for such an additional anisotropy reversal is a sign change in the Dresselhaus SOI, as expected from the two-fold symmetry of the effective field of Eq. (1) and the perturbation theory in Eq. (4). By taking into account higher order and strain induced Dresselhaus effects, one notices that the sign of the resulting renormalized linear Dresselhaus SOI can be controlled by modifying the carrier density [32]. The Dresselhaus SOI Hamiltonian HD including an additional strain term is given by
The value of ′ is controlled electrically by the carrier density through = �2 . The confinement wave vector 〈 2 〉 = ∫ Ψ * ( ) (− 2 / 2 )Ψ( ) in the InGaAs QW is estimated to be  [35], the renormalized linear Dresselhaus SOI ′ including the strain term is plotted as a function of the carrier density in Fig. 8. From Fig. 8, we obtain the critical carrier density 1.95 × 10 16 m −2 at which ′ changes its sign. It is difficult to explain our result without considering the strain term as shown by the red dashed line. It should be emphasized that the critical density is not changed if the ratio between and � /ℏ is preserved. This critical carrier density is consistent with the one corresponding to the additional anisotropy reversal in Fig. 7 (a) and supports the conclusion of a sign change of ′ in our experiment by electric means.

VI. CONCLUSIONS
Our The Hamiltonian for spin carriers with effective mass m* confined in a Rashba 1D ring of radius r (parametrized by the azimuthal angle ) is given by [19] (S1) with frequencies . (S2) The main contributions to (S1) are the kinetic energy (first term) and the Rashba spin-orbit coupling (second term), corresponding to an effective (momentum-dependent) magnetic field pointing along the radial direction. The third term is the Meijer's correction [19] that guarantees the hermiticity of the Hamiltonian. The latter can be neglected in the semiclassic limit of large Fermi momentum, typically satisfied in mesoscopic semiconductors.
We perturb the radial magnetic texture in ℋ 0 by introducing an in-plane Zeeman term Δℋ 1 and a Dresselhaus spin-orbit term Δℋ 2 of the form By following the standard perturbation theory for nondegenerate systems [36] we find the first signs of anisotropy in the perturbed eigenenergies only after a 3rd-order expansion in Δℋ = Δℋ 1 + Δℋ 2 . This procedure leads to The anisotropic response of the perturbed eigenenergies (S10) to the Zeeman field orientation appears at the 1 st order in the Dresselhaus coupling strength and at the 2 nd order in Zeeman one, 16 � 2 sin(2 ).
showing that the anisotropy can discriminate the sign of the Dresselhaus term. The perturbed eigenstates � , , ⟩ need to be expanded only up to 2 nd order in Δℋ = Δℋ 1 + Δℋ 2 to show the first anisotropic features due to the joint Dresselhaus-Zeeman action. Up to a normalization factor, they read where the sums in (S11) run such that the denominators do not vanish. The perturbative corrections to the first term in (S11) lead to 2 and 2 contributions, only, while the last term shows additional joint contributions. We point out that the results (S10) and (S11) hold for 1 ≪ � ≪ 2 , where degeneracy mixing is avoided and the perturbative approach is sound.

Anisotropic conductance and the role of geometric/dynamic spin phases.
We calculate the AAS (Al'tshuler-Aronov-Spivak) corrections to the conductance of a two-terminal AC 1D ring originated from the interference of time-reversed paths at the lowest order (i.e., semiclassical paths describing single windings around the ring corresponding to strongly coupled contacts) by following a procedure similar to our previous works on Rashba rings [20,22], where the phase difference gathered by counter-propagating spin carriers in found by solving = for noninteger orbital numbers , with the Fermi energy. As a result, the AAS conductance takes the We then find (S12) with (S13) (S14) . (S15) These contributions represent a Zeeman phase shift , a Dresselhaus phase shift , and an anisotropic phase shift . The latter depends explicitly on , showing the two-fold symmetry anticipated in Eq. (1) with opposite extreme values at /4 and 3 /4. Notice that and derive from the quadratic contributions to the perturbed eigenenergies (S10). Hence, according to perturbation theory [36,37], they originate from the linear contributions to the perturbed eigenstates (S11). As for , it is a consequence of the cubic contributions to (S10) and the quadratic one to (S11).
Each of the phases (S13-S15) can be of either pure or hybrid geometric/dynamic origin. The geometric-phase contribution to the conductance (S12) can be evaluated from the perturbed eigenstates (S11) as [22,25] = ∫ � , , ������� | | plane field [20,22]. The additional contributions to the geometric phase in (S16) are interpreted as perturbations ΔΩ to Ω 0 . The share of these geometric-phase contributions in the conductance (S12) depend on the corresponding weight factors appearing in (S16). The Zeeman contribution to (S12) results to be of purely geometric origin (as reported in [22]) as a consequence of a weight factor 1 (absolute value) in (S16). The Dresselhaus contribution to (S12), with a weight factor ½ in (S16), turns out to be only 50% geometric (the other 50% is of dynamic origin), likely due to the different symmetry class of Zeeman and Dresselhaus perturbations. As for the geometric-phase contribution to the anisotropic phase , the corresponding weight factor ½ in (S16) indicates a 50% share. Namely, has an hybrid geometric/dynamical origin.

Role of disorder.
The role of disorder can be effectively accounted by introducing a classical conductance 0 and a quantum-correction amplitude ≪ 1 such that The resistance = 1/ , better suited in experiments, then reads with 0 = 1/ 0 the classical resistance.
By noticing that the anisotropic phase is much smaller than the unperturbed AC phase √1 + � 2 − 1, we rewrite the resistance as 0 ≈ 1 + 2 sin(2 ) (S19) The Eq. (S19) shows an anisotropic response of the resistance to the Zeeman field's direction .
Moreover, the sign of the anisotropy can be independently modulated by the Rashba strength � but its response is isotropic.

APPENDIX B: CARRIER DENSITY DEPENDENCE OF RASHBA SOI STRENGTH
The gate fitted Hall bar (70 µm x 280 µm) was fabricated on the same chip on which the spin interferometer (40 x 40 ring array) was put. The relation between carrier density and Rashba SOI strength was obtained from the analysis of Shubnikov-de Haas (SdH) oscillations as shown in Fig. 9.
The SdH oscillations show a beating pattern because of spin splitting due to the strong Rashba SOI.
The Rashba SOI strength is given by Here, ↑ and ↓ are the spin split densities, which can be obtained from the fast Fourier Transform (FFT) spectra of SdH oscillations. The electron effective mass * = 0.05 can be estimated by analyzing the temperature dependence of SdH oscillation amplitude. The relation between the Rashba SOI parameter α and the carrier density is plotted in Fig. 10. In the above analysis, we assumed that the Dresselhaus SOI strength is negligible since = 〈 2 〉 is one order of magnitude smaller than the Rashba SOI strength. (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 is fast enough compared with the orbital frequency , the SOI field is in x-y plane (adiabatic limit). Spin precession around SOI field Btotal is associated with the dynamical phase. The angle is given by the relation tan = / .
(Right) The in-plane field modulates the geometric phase by changing the solid angle subtended by the total effective field.