Abstract
Since the formulation of the geometric phase by Berry, its relevance has been demonstrated in a large variety of physical systems. However, a geometric phase of the most fundamental spin1/2 system, the electron spin, has not been observed directly and controlled independently from dynamical phases. Here we report experimental evidence on the manipulation of an electron spin through a purely geometric effect in an InGaAsbased quantum ring with Rashba spinorbit coupling. By applying an inplane magnetic field, a phase shift of the Aharonov–Casher interference pattern towards the small spinorbitcoupling regions is observed. A perturbation theory for a onedimensional Rashba ring under small inplane fields reveals that the phase shift originates exclusively from the modulation of a pure geometricphase component of the electron spin beyond the adiabatic limit, independently from dynamical phases. The phase shift is well reproduced by implementing two independent approaches, that is, perturbation theory and nonperturbative transport simulations.
Introduction
Geometric phases arise in wave systems^{1} where the parameters of the wave function are cycled around a circuit. Such phase factors can be observed via interference of waves traversing different paths. Berry^{1} has shown that the electron spin wave function in an adiabatic evolution acquires a geometric phase that depends only on the geometry of the traversed path in the parameter space. As a consequence, the geometric phase is robust against dephasing. This is in contrast to the timedependent dynamical phase of a particle^{2,3,4}. Experimentally, geometricphase factors have been observed in various systems^{5,6}. However, the geometric phase of an electronic spin has never been directly observed and manipulated independently of the dynamical phase before.
Here we demonstrate the control of the geometric phase of an electron spin in a mesoscopic semiconductor device in which an array of rings forms interference paths. The experiment shows geometric phases beyond the adiabatic limit^{7}. A geometricphase shift of the Aharonov–Casher (AC) effect^{8,9,10} is induced with an inplane magnetic field and measured in an interference pattern of the electron current. Our findings show manipulation of the geometric phase independently of the dynamical phase without introducing additional geometricphase factors such as the Aharonov–Bohm phase.
Results
Concept of present experiment
Spins in a magnetic field acquire a dynamical phase, which physically corresponds to phase accumulation because of the spin precession around the magnetic field. In a mesoscopic ring system with an inhomogeneous magnetic field, the spins acquire also a geometric phase that depends on the path of the spin. As an example, consider a onedimensional (1D) ring in a textured magnetic field^{11} as shown in Fig. 1a. The magnetic field seen by a propagating electron along the ring subtends, from its rest frame, a solid angle in the magneticfield space. The spin acquires not only a dynamical phase but also a geometric phase, which is proportional to the solid angle (Fig. 1b).
The Bychkov–Rashba spinorbit (SO) coupling^{12} can be used to create an effective inplane magneticfield texture (Fig. 1c). The coupling is induced at an interface between semiconductor layers having different energy gaps by an electric field perpendicular to the layers (say along the z axis). It is a relativistic effect where an electron travelling through an electric field feels an effective SO magnetic field B_{SO}, the direction of which is perpendicular both to the electric field and a wave vector k, that is, B_{SO}∝α_{R}(k × e_{z}). Here the electric field enters through the Rashba SOcoupling constant α_{R} that is therefore electrically tunable with a topgate electrode^{13,14}, and e_{z} denotes a unit vector in z direction. This effective Rashba field couples to the electron spin, which acquires dynamical and geometric phases in the transport around the ring^{15,16,17}. Interference of these spinrelated phases gives rise to conductance oscillations when α_{R} is varied^{18,19,20}. Recently, the SOinduced geometric phase has been extracted^{21} from a detailed analysis of spininterference effects as a function of the SOcoupling constant α_{R} and the ring radius r.
In a 1D Rashba ring placed in the x–y plane, a travelling electron has a nonzero wave vector k only in the inplane direction because of the electrostatic confinement along e_{z}. Consequently, the effective field B_{SO} seen by the electron always points in the radial direction (see Fig. 1c). However, if the revolution frequency ω_{0} around the ring is comparable to the spinprecession frequency ω_{SO}, spin transport is nonadiabatic. As a result the electron spin does not exactly precess around B_{SO} (see refs 22, 23); spin eigenstates are no longer parallel to B_{SO} and the corresponding polar angle, θ, satisfies , with being the effective mass of an electron and ħ the reduced Planck’s constant, as if there was an effective magnetic field normal to the ring’s plane besides B_{SO} as depicted in Fig. 1d. This is in analogy to NMR experiment where an rf magnetic field yields a static outofplane field in a rotating frame of reference. Hence, an electron spin in the Rashba ring acquires a geometric phase, which corresponds to the solid angle subtended by the spin eigenstates.
Now, consider a Rashba ring with an external magnetic field applied parallel to the ring plane as shown in Fig. 1e: the total magnetic field seen by the electron is uniformly tilted towards the applied field direction, which changes the solid angle in the magneticfield space (Fig. 1f). As we will show, this allows for controlled modulation of the pure geometric phase independently of the dynamical phase. We emphasize that the separation of the geometric and dynamical phases has not been achieved in mesoscopic spin systems so far: the SOcoupling constant α_{R} and the ring radius r modulate both the geometric and dynamical phases simultaneously through the degree of adiabaticity, Q_{R}.
Inplane field dependence of the AC effect
In our experiment, an array of 40 × 40 rings (Fig. 2a) and a Hall bar have been fabricated lithographically from a shallow InGaAs quantum well (see ref. 24 for the basic properties). The Hall bar and the ring array are in the same current path and they have been covered with the same topgate electrode. The electrical resistance has been measured with a standard ac lockin technique at a temperature of 1.5 K with a liquid helium cryostat where we can apply magnetic fields both perpendicular and parallel to the ring plane. Figure 2b shows the Altshuler–Aronov–Spivak (AAS) effect in the ring array at a gate voltage of 1.1V as a function of the perpendicular magneticfield strength.
We study the control of the spin geometric phase via the AC effect^{8,9,10}, the electromagnetic dual of the Aharonov–Bohm effect. Specifically, in our system, the AC effect is realized as the gate modulation of spin interference. The AAS amplitude oscillates with the gate voltage, as shown in Fig. 2c, indicating the gate modulation of the spin phase. This oscillatory behaviour of the magnetoresistance at zero (perpendicular) external field is visualized in Fig. 2d. Because of disorder in the samples and the ensemble average over the ring array, pairs of timereversed (TR) paths, that is, the path pairs travelling in opposite directions around the ring, contribute predominantly to the interference pattern in the magnetoresistance. The AC effect in the TR paths gives rise to the resistance modulation given by^{25,23}
where \text{\delta}{R}_{{\alpha}_{\text{R}}\ne 0} and \text{\delta}{R}_{{\alpha}_{\text{R}}=0} are, respectively, the electrical resistance with and without Rashba SO coupling in a ring. Equation (2), plotted as the red dashed line in Fig. 2d, is in agreement with experiment. The phase of the cosine function in equation (2) shows that the AC phase comprises two components, namely, a dynamical component 2πQ_{R} sin θ and a geometric one −2π(1−cos θ). The latter exactly represents the solid angle subtended by the spin eigenstates in the Rashba ring (Fig. 1d).
The dependence of the AC effect on the inplane magnetic field in experiment is shown in Fig. 3a,b. In Fig. 3a, the gate voltage is translated to the Rashba SOcoupling constant α_{R} by using the relation obtained from the Shubnikov–de Haas analysis^{13} in the Hall bar. As is applied, the amplitude of the AC effect is suppressed because of spininduced TR symmetry breaking^{26} (see the Supplementary Note 1). Further, the AC oscillations exhibit a quadratic shift with towards positive gate voltages, that is, weaker Rashba SOcoupling strengths. In the following, we explain the observed shift by the sole modulation of the geometric phase as shown in Fig. 1f.
Perturbation theory and numerical transport calculations
To evaluate the effect of inplane magnetic fields on the geometric phases in the limit of small , we employ perturbation theory for a 1D Rashba ring (see the Methods section). The phase shift is calculated to first order in . We find that the AC interference between TR pairs of paths leads to conductance oscillations
where and . Further, k_{F} is the Fermi wave number, is the Landé gfactor and μ_{B} is the Bohr magneton. By comparing equations (1) and (3), we find that provides an additional phase 2π φ quadratic in . As plotted in Fig. 3c,d, equation (3) reproduces the observed phase shift within the energy range of small Zeeman energy (due to ) compared with the kinetic and Rashba SOcoupling energies. A similar quadratic shift in the interference peak positions has been recently calculated also for weakly coupled rings^{27}.
To determine the origin of the ACphase shift, we calculate a geometricphase contribution φ_{AA} using eigenstates calculated to first order in the inplane field:
with ϕ being the azimuthal angle in the ring plane and j being an integer. By comparing equations (1, 2, 3) and (6), we confirm that the perturbation term φ has a purely geometric origin, at least to first order in the inplane field strength. This is because electron spins in the TR interference paths experience a uniaxial field, which does not yield a dynamical phase difference through the spin precession. Hence, our central result is that the observed shift of the AC phase is induced by the modulation of the pure geometric phase of an electron spin.
Note that the wave number enters into the denominator of the geometricphase shift term φ, equation (4). In our samples the conducting channel contains at least six transverse modes, each having a unique wave number. To take into account the effect of the multiple transport modes and for an adequate treatment of field strengths beyond the perturbative limit, we employ the numerical recursive Green’s function method^{28} (see the Supplementary Note 2). Results calculated at fixed carrier density 1.0 × 10^{16} m^{−2} are shown in Fig. 3e,f. The geometric shift of the AC oscillations occurs even in the presence of multiple modes (see the Supplementary Note 3).
Discussion
Figure 4 shows the comparison of the geometric shift in experiment (filled symbols) with the perturbation theory (solid lines) and the nonperturbative numerical calculations (open symbols) at peak and dip positions of the AC oscillations (P1–P4; see Fig. 3). The carrier density in the numerical calculations is increased with gate voltage (see the Supplementary Note 2). The perturbation theory for a single transverse mode is in good agreement with experiment in the regime of strong SO coupling (P1 and P2). This indicates that the phase shift is dominated only by a few modes or just the lowest one, presumably because of stronger decoherence for higher transverse modes, which have the smaller wave numbers along the ring. The discrepancy is significant at weak SOcoupling regions (P3 and P4) that are beyond the limit of validity of the perturbation expansion. We note that wave numbers associated with the modes in multimode numerical calculations depend on the energy level spacing in the confining channel of the ring. Numerical calculations assume a squareshaped potential, which may lead to the overestimation of the phase shift at P1. Further, in the numerical model a transition from weak antilocalization to weak localization with changing Rashba SO coupling results in the increased background resistance at small SO fields (see Fig. 3f). As a result, the positions of P3 and P4 may shift. However, almost all the results show the peaks and dips shift towards weak Rashba SO coupling, that is, positive gate voltages. We are thus able to reasonably simulate and explain the experimentally observed spinphase shift by both numerical calculations and the perturbation theory.
For further evidence, we have measured the radius dependence of the induced phase shift. Figure 5 shows the comparison of the shift in samples of rings with r=0.6 and 1.1 μm. We see that both in experiment and in the perturbation theory, geometricphase shift increases with the ring radius. This confirms an important prediction of our theory contained in the equations (3) and (4). From the comparison between experiment and the theory, the diamagnetic shift of energy is negligible in our samples because the diamagnetic effect should not have the radius dependence. To the best of our knowledge, this is the first experimental demonstration of the pure geometricphase control of an electron spin in solidstate devices. We conclude that this multiparameter control of the spin geometric phase may pave the way for future spintronic applications, such as geometricgate operation^{29} of a solidstate flying qubit or control of persistent spin currents in a mesoscopic ring^{11,30}, which may have potential applications for nonvolatile memory devices.
Methods
Sample fabrication and measurements
The sample was epitaxially grown on a (001) InP substrate by metal organic chemical vapour deposition. It consists of, from the bottom, In_{0.52}Al_{0.48}As (200 nm, buffer layer)/In_{0.52}Al_{0.48}As (6 nm, carrier supply layer; Sidoping concentration of 4 × 10^{18} m^{−3})/In_{0.52}Al_{0.48}As (15 nm, spacer layer)/In_{0.53}Ga_{0.47}As (2.5 nm, quantum well)/In_{0.70}Ga_{0.30}As (10 nm, quantum well)/In_{0.53}Ga_{0.47}As (2.5 nm, quantum well)/InP (5 nm, stopper layer)/In_{0.52}Al_{0.48}As (20 nm, barrier layer)/AlAs (1.5 nm, barrier layer)/In_{0.52}Al_{0.48}As (5 nm, cap layer). An array of 40 × 40 rings and a Hall bar (70 × 280 μm^{2}) have been fabricated by means of ebeam lithography and reactiveion etching. To form an ohmic contact, AuGeNi was used (annealing temperature of 275 °C, 7 min). As a topgate insulator, a 200nmthick Al_{2}O_{3} layer was deposited by atomic layer deposition. A Cr/Au topgate electrode was fabricated on the insulator layer.
A fourterminal standard lockin technique has been used to measure the perpendicular magnetoresistance of the ring array as a function of the gate voltage and the inplane magnetic field. At a given gate voltage, the Rashba SOcoupling constant and the carrier density in the ring structure can be estimated by measuring the Shubnikov–de Haas oscillations in the Hall bar.
Perturbation theory
We introduce a perturbative approach for studying the effect of small inplane magnetic fields in the conductance of 1D rings subject to Rashba SO coupling. We demonstrate that the inplane field acts on the geometrical spin phase alone without contributing to dynamical phases at the lowest perturbative order, allowing the controlled manipulation of geometrical phases. The Hamiltonian reads
with
Here H_{0} is the unperturbed Hamiltonian of a 1D Rashba ring of radius r lying in the x–y plane parametrized by the azimuthal angle ϕ. We have defined ω_{0}=ħ/mr^{2} and ω_{SO}=2α_{R}/ħr as the characteristic frequencies of the kinetic and SO Rashba terms, respectively, together with the polar Pauli matrices σ_{r}=cosϕ σ_{x}+sinϕ σ_{y} and σ_{ϕ}=−sinϕ σ_{x}+cosϕ σ_{y}. The third term in equation (8) guarantees H_{0} to be Hermitian^{22}, losing relevance for large momenta. The ΔH in equation (9) is a perturbation on H_{0} by an inplane magnetic field along the x direction, with the corresponding Larmor frequency. The unperturbed eigenstates n, λ, s›_{0} of H_{0} (with spin s=↑,↓, travel sense λ=±1, and orbital wave numbers n≥0) are^{23}
where θ is the inclination of the local spinquantization axis with respect to the z axis given by tanθ=ω_{SO}/ω_{0}≡Q_{R}. Notice that θ does not depend on any quantum number, being shared by all eigenstates. The corresponding eigenenergies are
Where ↑ spins are defined to maximize the eigenenergies. We notice that states with opposite travel sense n,−,s›_{0} and n−1,+,s›_{0} are degenerated (that is, ). In addition, we find that , namely, the perturbation ΔH does not mix degenerate states. This means that we can apply a nondegenerate perturbation theory. We further notice that an additional degeneracy () arises in the unperturbed system for vanishing Rashba coupling (Q_{R}=0), meaning that the perturbative approach we present here is not valid for .
We calculate the perturbed eigenstates n,λ,s› and eigenenergies to the lowest order in ω_{B}. Taking into account that _{0}‹n±1, λ, sΔHn, λ, s›_{0}=λs(ħω_{B}/4)sinθ and _{0}‹n±1, λ, −sΔHn, λ, s›_{0}=ħω_{B}(cosθ±s)/4, we find (by following, for example, the textbook by Sakurai^{31})
with normalization constant , and
We calculate the ring conductance by following the procedure introduced in (refs 23, 25). This is based on the 1D Landauer formula , where the conductance G is determined by the quantum probabilities of transmission T^{σσ′}=t^{σσ′}^{2} (with amplitudes t^{σσ′}) for incoming (outgoing) spins σ′ (σ). We consider strongly coupled contacts at diametrically opposed locations ϕ=0 and ϕ=π. Because of the energy splitting, incoming spin carriers with Fermi energy E_{F} entering at ϕ=0 can propagate coherently through the ring along four possible channels with (noninteger) wave numbers depending on spin (s) and travel sense (λ). The are determined by solving in equation (16) and the relative weight of each propagation channel is given by the projection of the incoming spin state on the corresponding spin eigenstates, equation (15), at the entrance point. We assume that, because of strong coupling to the contacts, the carriers propagate directly from the entrance (ϕ=0) to the exit (ϕ=π) point along the shortest paths without any additional winding (see Fig. 6a). After interference, we find by summing over all spin indices, with
where we assumed a large electron density corresponding to in equations (15) and (16), in agreement with the experimental conditions. The conductance then reads
The result of equation (20) holds for single ballistic rings. For an ensemble of rings subject to shape fluctuations and/or disorder, instead, the main contribution to the sampleaveraged conductance comes from TR paths leading to the AAS oscillations (see Fig. 6b). By following a similar procedure (taking into account that TR paths contribute to the transmission probability T through the reflection probability R by unitarity T+R=1), we find an AAS conductance
which applies to the reported experiment. The φ of equation (19) corresponds to a phase shift in the AC conductance introduced by the inplane magnetic field. It is quadratic in the field’s strength ω_{B} and proportional to . The latter illustrates the known fact that spin dynamics under Zeeman coupling depends on the electrons velocity^{32}, in contrast to Rashba coupling. We further notice that φ is an increasing function of the ring’s radius r.
To demonstrate the geometric nature of φ, we first notice in equations (20) and (21) that , where and are the unperturbed contributions to the dynamical and geometrical phases, correspondingly, with Ω^{0} the solid angle described by the unperturbed spin eigenstates over the ring (see ref. 23). The limit of adiabatic spin dynamics, where purely geometrical Berry phases arise^{1}, corresponds to Q_{R}→∞(θ→π/2). For a finite Q_{R} (0<θ<π/2), the spin dynamics is nonadiabatic and the geometrical phase (depending explicitly on Q_{R}) is referred to as the Aharonov–Anandan phase^{7}. For the perturbed eigenstates, by defining , the geometricphase contribution φ_{AA} can be extracted as
There we find that the magnetic shift φ is purely geometric. This means that, at the lowest order of perturbation, the inplane magnetic field modifies the eigenstate spin texture without contributing to the dynamical spin phase. More explicitly, from equation (24) we find ΔΩ=−2π φ showing that the inplane field perturbs the spin eigenstates by reducing the solid angle with respect to the unperturbed case. This means that the perturbation hinders adiabatic spin dynamics: In the regime (with unperturbed spinors along θ→π/2) spins are lifted from the ring’s plane by the inplane magnetic field. The full Aharonov–Anandan geometric phase in the perturbed system reads (mod 2π) γ=−π[(1−cosθ)−ϕ] for single ballistic rings (Fig. 6a) and γ=−2π[(1−cosθ)−ϕ] for rings ensembles in AAS configuration (Fig. 6b).
Additional information
How to cite this article: Nagasawa, F. et al. Control of the spin geometric phase in a semiconductor quantum ring. Nat. Commun. 4:2526 doi: 10.1038/ncomms3526 (2013).
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Acknowledgements
We acknowledge support from Strategic JapaneseGerman Joint Research Program. H.S. and K.R. thank DFG for support within Research Unit FOR 1483. D.F. acknowledges support from the Ramón y Cajal program, from the Spanish Ministry of Science’s project No. FIS201129400, and from the Junta de Andaluca’s Excellence Project No. P07FQM3037. F.N. and J.N. are grateful to M.Kohda for valuable discussions. J.N. thanks N.Nagaosa for valuable discussions. This work was financially supported by GrantsinAid from the Japan Society for the Promotion of Science (JSPS) No. 22226001.
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F.N. performed measurements, analysed the results and wrote the manuscript. D.F. derived the theoretical model and wrote part of the manuscript. H.S. performed the numerical calculations and analysed them with K.R. H.S. wrote part of the Supplementary Information. All authors discussed the results and worked on the manuscript at all stages. J.N. directed the research.
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Supplementary Figures S1S6, Supplementary Notes 13 and Supplementary References (PDF 161 kb)
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Nagasawa, F., Frustaglia, D., Saarikoski, H. et al. Control of the spin geometric phase in semiconductor quantum rings. Nat Commun 4, 2526 (2013). https://doi.org/10.1038/ncomms3526
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DOI: https://doi.org/10.1038/ncomms3526
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