ARTICLES PUBLISHED ONLINE: 13 JULY 2014 | DOI: 10.1038/NNANO.2014.128
Direct determination of spin–orbit interaction coefficients and realization of the persistent spin helix symmetry A. Sasaki1, S. Nonaka1, Y. Kunihashi1†, M. Kohda1, T. Bauernfeind2, T. Dollinger2, K. Richter2 and J. Nitta1 * The spin–orbit interaction plays a crucial role in diverse fields of condensed matter, including the investigation of Majorana fermions, topological insulators, quantum information and spintronics. In III–V zinc-blende semiconductor heterostructures, two types of spin–orbit interaction—Rashba and Dresselhaus—act on the electron spin as effective magnetic fields with different directions. They are characterized by coefficients α and β, respectively. When α is equal to β, the so-called persistent spin helix symmetry is realized. In this condition, invariance with respect to spin rotations is achieved even in the presence of the spin–orbit interaction, implying strongly enhanced spin lifetimes for spatially periodic spin modes. Existing methods to evaluate α/β require fitting analyses that often include ambiguity in the parameters used. Here, we experimentally demonstrate a simple and fitting parameter-free technique to determine α/β and to deduce the absolute values of α and β. The method is based on the detection of the effective magnetic field direction and the strength induced by the two spin–orbit interactions. Moreover, we observe the persistent spin helix symmetry by gate tuning.
A
s an essential feature of the spin–orbit interaction (SOI), the spin of an electron moving in an electric field experiences, in its rest frame, an effective magnetic field that couples to the spin's magnetic moment, even in the absence of any external magnetic field. The SOI is ubiquitous and appears in many different areas of physics and applications1–3. The SOI can be engineered by material design or by tuning an electric field4,5 to explore new spin-related phenomena and spin functionalities. In semiconductor-based spintronics, for example, the effective magnetic field Beff can be used for the generation6, manipulation7–12 and detection13 of spins solely by electrical means. However, the SOI inevitably gives rise to spin relaxation14, thus disrupting long spin coherence times. This crucial problem can be overcome by utilizing two different types of SOIs—Rashba15 and Dresselhaus16 SOIs. Spin SU(2) symmetry, an invariance with respect to specific rotations of electron spin, is realized and spin relaxation is suppressed correspondingly when both SOIs are of equal strength, giving rise to the persistent spin helix symmetry17–21. As a result, precise evaluation and control of α/β (where α and β denote the strengths of the Rashba and Dresselhaus SOIs, respectively) will pave the way for future applications in spintronics and quantum information22–24. However, in transport measurements, the simultaneous evaluation of Rashba and Dresselhaus SOIs usually involves large ambiguities due to model-dependent fitting25, and in optical measurements, fitting analyses are also required26–29. As a consequence there has been a longstanding quest for a simple, robust and reliable technology to evaluate α/β. Here, we demonstrate a fitting-free in situ determination of α/β in transport measurements with coexisting Rashba and Dresselhaus SOIs. In narrow wires, application of an in-plane magnetic field Bin modulates the amplitude of the magneto-conductance, that is, weak localization (WL), through additional dephasing arising from spin-induced time-reversal symmetry breaking30.
Theory predicts the maximum amplitude of WL to exist when Bin and Beff are parallel to one another31. This enables us to evaluate α/β from the anisotropic WL amplitude, without relying on any fitting. By applying the proposed concept to InGaAs-based wire structures, we realize gate-controlled persistent spin helix symmetry and find a novel fingerprint of this symmetry. In addition to the evaluation of the ratio α/β, and going beyond the theoretical proposal31, we show that the absolute values of α and β can be deduced from the minimum WL amplitude when Bin and Beff have equal strengths. As a proof of concept we provide measurements in InGaAs wires. Our demonstration of the proposed metrological concept31 can be extended to a wide range of materials, because the direction and strength of the effective magnetic fields in a confined structure are directly related to α and β. The precise determination of α and β is indispensable in exploring new spin-related phenomena and functionalities.
Concept of the present measurements According to the proposed metrological concept31, the evaluation of α/β is achieved by detecting the direction θeff of Beff = BR + BD , where BR and BD are the effective magnetic fields induced by the Rashba and linear Dresselhaus SOIs. For a quantum well grown in the [001] direction of III–V zinc-blende semiconductor heterostructures, the corresponding Hamiltonians are given by α (1) HR = − σ x py − σ y px h and β HD = − σ x px − σ y py h
(2)
1
Graduate school of Engineering, Tohoku University, 6-6-02 Aramaki-Aza Aoba, Aoba-ku, Sendai 980-8579, Japan, 2 Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany. † Present address: NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan. *e-mail: [email protected]
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Figure 1 | Weak localization anisotropy to the relative angle between Beff and Bin. a,c, The directions of total magnetic field Btotal = Bin + Beff are shown for ±p momentum directions. b,d, Calculated weak localizations (WLs) are shown at α/β = 2/3 in a [100] wire with |Bin| = 0.75 T for Bin || Beff (d, θin = 146°) and Bin ∦ Beff (b, θin = 25°), where θin is the angle of the in-plane magnetic field defined with respect to the [100] direction. The magnetoconductance is plotted against the magnetic flux in a unit cell of the numerical grid, Φs = a 2|Bp| (a is the lattice constant and Bp is the perpendicular magnetic field) in units of the magnetic flux quantum Φ0 = h/e (e is the electron charge). In a,b, when Bin ∦ Beff , Btotal is randomized for ±p electron directions, suppressing the WL amplitude. In c,d, when Bin || Beff , the total magnetic field Btotal along the ±p directions remains unidirectional, showing the deepest WL amplitude.
where ħ = h/2π (h is Planck's constant), α and β are coefficients of the Rashba and Dresselhaus SOIs, which determine the strengths of BR and BD , respectively, and σi and pi (i = x, y) are the Pauli spin matrices and the electron momentum, respectively. The x and y directions are parallel to [100] and [010], respectively. Let us assume a one-dimensional wire along [100], where electron momentum is restricted to ±p along the wire direction. The direction of Beff acting on the electron spins is therefore fixed as unidirectional32,33. Owing to the perpendicular orientation of BR and BD , the relationship between the SOI parameters and the effective field orientation θeff is simply given by α = −tan θeff β
(3)
It should be noted that the cubic Dresselhaus SOI vanishes along the [100] axis. Hence, through the detection of θeff in the one-dimensional wire, direct evaluation of the α/β ratio is possible. The central concept to the determination of θeff through transport measurements is shown schematically in Fig. 1. For a narrow wire with W ≪ Lso (W, wire width; Lso , spin precession length, where Lso = ħ2/2m*α or ħ 2/2m*β, where m* is an effective mass), the randomization of spin is suppressed by the quasi-one-dimensional confinement, which fixes the electron momentum direction and therefore the spin precession axis Beff. Therefore, only WL is observable, according to theory32,33 and experiment34,35. The Cooperon conductance contribution, composed of triplet and singlet terms of opposite sign, exhibits a positive conductance correction (weak anti-localization, WAL) in the case of spin mixing, when triplet terms are suppressed, and a negative conductance 704
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correction if the spin is not randomized (in weak localization), because the triplet terms overcompensate the singlet. Magnetic flux through correlated path pairs leads to randomization of the phases entering the Cooperon, thereby destroying WL and WAL. This applies to a two-dimensional system as well as to a quasione-dimensional wire with W≪Lso , in which the magnetic flux may be enclosed within scattering trajectories. The application of a static in-plane magnetic field Bin that is not parallel to Beff further induces an additional spin relaxation by breaking the unidirectional alignment of Beff (Fig. 1a). It should be noted that the influence of different angles of an in-plane magnetic field on a persistent spin helix state in the two-dimensional case has been studied in ref. 20. The additional spin relaxation induces dephasing in the time-reversal related pairs of electron paths by mixing spin up and down phases. This dephasing leads to weaker quantum interference, resulting in suppression of the WL amplitudes (Fig. 1b). However, when Bin is applied in parallel to Beff , the total magnetic field Btotal = Bin + Beff remains unidirectional for ±p (Fig. 1c), preserving the long spin relaxation length and resulting in an enhanced WL amplitude (Fig. 1d). Hence, the WL amplitude becomes anisotropic as a function of the relative angle between Bin and Beff. This concept enables us to determine θeff by rotating the direction of Bin and therefore to obtain the α/β ratio by using equation (3) without any fittings.
Numerical simulations of weak localization anisotropy To confirm the presented concept we conduct numerical calculations of the magneto-conductance of wires with different α/β values including the persistent spin helix configuration. In this calculation, to compare with the result for the wire in the [100] direction, we also include results for wires along the [110] and [−110] directions, which provide parallel and antiparallel configurations between BR and BD , respectively. The numerical calculations are based on the recursive Green's function approach36 (for details see Supplementary Section I). We theoretically calculate the magnetoconductance of a narrow wire by changing the perpendicular magnetic field Bp under fixed Bin , whose angle θin is defined with respect to the [100] direction. The three different wire directions along the [100], [110] and [−110] axes are referred to as [100], [110] and [−110] wires, respectively. It should be noted that the wire width is set to 20 nm, which does not give rise to purely one-dimensional confinement, but is smaller than the spin precession length. Figures 1b,d present the calculated results for the [100] wire at θin = 25° (Fig. 1b) and 146° (Fig. 1d) (Bin = 0.75 T), and shows clear WL signals with different conductance amplitude. We define the amplitude of WL as ΔG (Fig. 1d) and produce a polar plot as a function of θin for different α/β ratios and wire directions (Fig. 2a–c). The WL amplitude shows two-fold symmetries for all wire directions. Furthermore, its symmetry axis depends on the crystal direction. Such dependencies are understood in terms of the relative angle between Bin and Beff. Because the Bin || Beff configuration does not enhance spin relaxation, the angle of the maximum WL amplitude corresponds to the Beff direction. On the other hand, the Bin ∦ Beff configuration induces spin relaxation, resulting in a reduced WL amplitude. For [110] and [−110] wires, because Beff always appears along the 135° and 45° directions, respectively, the WL amplitude becomes maximal at θin = 135° and 45° and is thus unaffected by the different α/β ratios (Fig. 2b,c). For the [100] wire, however, the Beff direction changes with the α/β ratio, resulting in an angular shift of the maximum WL amplitude (Fig. 2a). As shown in Fig. 2d, the angle of the maximum WL amplitude (θeff ) is perfectly aligned to equation (3). For the persistent spin helix symmetry, that is, at α/β = 1 (red plot in Fig. 2a–c), although the maximum WL peak still appears at θin = 135° for the [110] wire, interestingly, the anisotropy of the WL amplitude is quenched for the [−110] wire. This is due to the mutual compensation of BR and BD
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(Fig. 2e). This numerically observed quench of the WL anisotropy is a fingerprint of the persistent spin helix symmetry.
Evaluation of α/β ratio and persistent spin helix symmetry To utilize this concept in a real device we fabricated parallel wires along the [100], [110] and [−110] directions in series, as shown in Fig. 3 (for more information on the device structure and fabrication see Methods), and covered the entire structure with an Al2O3/Cr/Au top-gate electrode to control α through the gate voltage, Vg. Magneto-conductance measurements were performed simultaneously for three wire directions by sweeping Bp under constant Bin and Vg , at T = 1.7 K. Figure 4a presents the magneto-conductance for the [100] wire at θin = 24° and 142° with |Bin| = 1.0 T and Vg = −5.0 V. The restricted electron momentum in a narrow wire suppresses the spin relaxation, leading to the appearance of WL instead of WAL. In addition, the modulation of the WL amplitude Δσ between θin = 24° and 142° indicates anisotropic dephasing due to additional spin relaxation. To reveal the effect of Bin on WL amplitude, we measured the magneto-conductance in the range 0° ≤ θin ≤ 180° (Fig. 4b) and extracted Δσ as a function of θin (Fig. 4c). The WL amplitude changes continuously, as shown in Fig. 4b, and the oscillatory behaviour of Δσ in Fig. 4c indicates the modulation of electron dephasing induced by spin relaxation. To make a comparison with the theoretical prediction (Fig. 2a–d), we include a polar plot of Δσ' (subtracted Δσ from its minimum)
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Figure 3 | Schematic illustration of measurement configuration. The set-up shown allows the simultaneous performance of conductance measurements in all three considered wire orientations along the [100], [110] and [−110] axes. The conductance for each wire orientation is an average over multiple wires to minimize signatures from universal conductance corrections. Lower left: Scanning electron microscope images of a typical example of InGaAs wire structure, showing a top view (upper) and side view (lower). W, wire width. Scale bars in upper and lower images are 5 µm and 2 µm, respectively.
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Figure 4 | Weak localization anisotropy in the [100] direction wire at V g = −5 V. a, Measured magneto-conductance traces for in-plane magnetic field angles θin = 24° and 142°. Red and blue dashed lines are minimum weak localization (WL) amplitude levels for θin = 24° and 142°, respectively. Δσ is the WL amplitude extracted at |Bp| = 5 mT. b, Colour plot of WL amplitude measured in the range 0° ≤ θin ≤ 180°. Cross-sections through the red and blue dashed lines provide the shape of the magneto-conductance data for θin = 24° and 142°, respectively, shown in a. The cross-section through the white dashed line corresponds to the WL amplitude shown in c. The zero point is chosen as the minimum of the magneto-conductance (σ at |Bp| = 0 mT). c, WL amplitude Δσ for different θin. Red and blue dashed lines indicate angles for minimum and maximum WL amplitudes, respectively. The black dotted line is the minimum WL amplitude level. Values of Δσ' shown by black arrows are the magneto-conductance difference from minimum Δσ. d, Polar plot of Δσ' in the range 0° ≤ θin ≤ 360° for |Bin| = 1.0 T. We extended the result for 0° ≤ θin ≤ 180° to 180° ≤ θin ≤ 360° by considering spatial symmetry. Blue dashed line: direction of maximum WL amplitude. The green arrow indicates a major peak angle corresponding to the effective magnetic field direction that directly provides the α/β ratio. e, Polar plots with different amplitudes of in-plane field, |Bin| = 0.2–1 T. The blue dashed line shows a direction of maximum WL amplitude that does not depend on the amplitude of |Bin|. The green arrow shows the major peak angle guaranteeing high accuracy of direct α/β detection. 706
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as a function of θin (Fig. 4d). Due to the spatial symmetry of the configuration, we safely extend the result to 180° ≤ θin ≤ 360°. This plot represents how an additional spin relaxation is induced by the relative angle between Bin and Beff. Because Δσ' shows a two-fold symmetry that is perfectly consistent with the theoretical prediction, θin for the maximum Δσ' corresponds to θeff. The second minor maximum WL amplitude around an angle of 50° is also well reproduced. According to the theoretical study, for the case of Bin ⊥ Beff , although the total Hamiltonian HQ1D mixes ˆ indi˜ Q1D = Cˆ −1 H ˜ Q1D fulfils H ˜ Q1D C, spins, its spin rotated version, H cating the orthogonal symmetry class37. Here, HQ1D is a single-particle Hamiltonian of a disordered quantum wire and Ĉ is the operator of complex conjugation (the definition of HQ1D is given in Supplementary Section I). This leads to the recovery of a WL amplitude and the appearance of the minor peaks. The orthogonal symmetry class can describe the system only when the electron momentum perpendicular to the wire is negligible. Hence, the appearance of these minor peaks is strong evidence for dimensional control of the electron spin and the validity of the α/β evaluation technique in our measurement set-up. Also, to confirm the dependence of our method on the strength of Bin , we include additional polar plots for various values of Bin ranging from 0.2 to 1.0 T, while leaving the other parameters unaltered (Fig. 4e). According to theory31, the detection technique for α/β cannot be correctly applicable when |Bin| ≫|Beff| because Btotal becomes strongly aligned in the direction of Bin and electron dephasing is suppressed for any θin. This potentially changes the major peak angle. However, no shift of the major peak is observed at different values of Bin , which guarantees the direct detection of α/β with high accuracy. (The |Bin| dependencies for other wire directions are discussed in Supplementary Section II.) To further confirm the validity of this concept, we plot Δσ' for all considered wire directions with different Vg (Fig. 5). For the [110] and [−110] wires, the Δσ' maxima correspond to 135° and 45°, respectively, consistent with the predicted angle θeff depicted in Fig. 2b,c. Because Beff is always oriented perpendicular to the [110] and [−110] wire directions (purple wires in Fig. 5b,c) for different Vg , θin at the Δσ' maximum does not change according to the α/β ratio. From these considerations, we directly probe the precise value of θeff by measuring the WL maximum. Moreover, we can extract the α/β ratio from the anisotropy of the [100] wire. As shown in Fig. 5a, the Δσ' maximum is shifted towards 135° by decreasing Vg from 0 V to −9 V, showing α/β modulation through the gate. As α is increased with reducing carrier density Ns (2.44 × 1012 cm−2 at Vg = 0 V to 2.01 × 1012 cm−2 at Vg = −9 V) and is smaller than β at Vg = 0 V, the α/β ratio is approaching unity at Vg = −9 V. By using equation (3), we determine the α/β ratio as α/β = −tan(157° ± 1°) = 0.42 ± 0.02 at Vg = 0 V, α/β = −tan(142° ± 1°) = 0.78 ± 0.03 at Vg = −5 V and α/β = −tan(133° ± 1°) = 1.07 ± 0.04 at Vg = −9 V. The condition of persistent spin helix symmetry is satisfied for the α/β ratio at Vg = −9 V. As shown in Fig. 2e, under the persistent spin helix symmetry Beff vanishes in the [−110] configuration due to the mutual cancellation of BR and BD. Only at Vg = −9 V in the [−110] wire do we observe quenched anisotropy (Fig. 5c). This is caused by the dominant strength of Bin compared to Beff , which gives rise to a homogeneous dephasing rate regardless of the value of θin , and the shape of the polar plot becomes a circle when we add the offset of the magneto-conductance (Vg = −9 V in Fig. 5c). This quenching of the WL anisotropy agrees with the numerical result. It can be regarded as a characteristic fingerprint to support the realization of the persistent spin helix symmetry revealed by the present method.
Evaluation of absolute values of α and β We confirm that the WL amplitude shows a minimum when |Bin| is close to |Beff| (refs 31,37). By detecting |Beff| from the |Bin|
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Figure 6 | Numerically calculated and experimental weak localization amplitude as a function of |Bin|/|Beff| ratio. a, Calculated result. The calculation is performed for W ≪ Lso without cubic Dresselhaus SOI in a [100] quasi-one-dimensional wire. We show the dependence of the weak localization (WL) amplitude ΔG on the ratio |Bin|/|Beff|. To observe ΔG(|Bin| = 0) in the basis 0, offset −ΔG(|Bin| = 0) is added. The θ-dependence (the angle between Bin and Beff) is shown for θ = 30° (red squares), 60° (blue circles), 120° (light blue inverted triangles) and 150° (pink triangles). We observe clear-cut minima of the WL conductance if the ratio |Bin|/|Beff| is close to 1. b–d, Experimentally obtained WL amplitudes G' = Δσ(θ ≠ 0°) − Δσ(θ = 0°) at Vg = −5 V (carrier density Ns = 2.16 × 1012 cm−2) plotted for the [100] (b), [110] (c) and [−110] (d) wires. Each angle θ (≠0°) is shown as θ = 30° (red squares), 60° (blue circles), 120° (light blue inverted triangles) and 150° (pink triangles). Thick green lines indicate dip positions of G'. NATURE NANOTECHNOLOGY | VOL 9 | SEPTEMBER 2014 | www.nature.com/naturenanotechnology
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dependence of the WL amplitude, we can estimate the absolute values of α and β to an accuracy of up to a factor of two. This concept is based on the idea that maximum spin relaxation caused by the most randomizing field Btotal = Bin + Beff , with |Bin| ≈ |Beff|, leads to a minimum WL amplitude. This idea is consistent with our analytical toy model describing the spin interference term of two one-dimensional time-reversal symmetry related paths (Supplementary Section III). Moreover, the numerical calculation shows minima of the WL amplitude if the ratio |Bin|/|Beff| is close to 1 (Fig. 6a). We experimentally observe the |Bin| dependence of the ‘difference of the WL amplitudes G', Δσ(θ ≠ 0°) − Δσ(θ = 0°)’, at Vg = −5 V (carrier density Ns = 2.16 × 10−12 cm−2), where Δσ is the WL amplitude and θ is the angle between Bin and Beff (Fig. 6b–d). We used the WL amplitude at θ = 0° as a reference value for the remaining WL amplitude. The characteristic line shapes of the numerical (Fig. 6a) and experimental (Fig. 6b–d) data sets are remarkably similar. Moreover, when we compare the traces in each wire, |Bin| ≈ |Beff| at the dip position is largest in the [110] wire and smallest in the [−110] wire. This result agrees with the relation between |Beff| and |BR|, |BD| in each crystal direction: |Beff| = |BR| + |BD| ([110] wire) > |Beff| = √(|BR|2 + |BD|2) ([100] wire) > |Beff| = |BR| – |BD| ([−110] wire). Along the [100] axis in particular, cubic Dresselhaus SOI is negligible, so α and β can be deduced by using the estimated |Beff| and the α/β ratio obtained from WL anisotropy measurements in the [100] wire. We estimate α and β approximately as α ≈ 2.0 × 10−13 eV m and β ≈ 3.7 × 10−13 eV m by using the evaluated α/β = −tan(θeff ≈ 152°) ≈ 0.53 and |Beff| ≈ 1.55 T. The small value of α of the order of 1 × 10−13 eV m is reasonable, because our heterostructure design corresponds to a symmetric quantum well.
Conclusions We have demonstrated a novel metrological concept and measurement technique to evaluate the α/β ratio, which can be applied in different material systems. The present concept enables us to determine the effective magnetic field direction originating from the SOIs with high accuracy, and also allows for the evaluation of the α/β ratio, including the correct sign, without fitting. We have also realized the gate-controlled persistent spin helix symmetry and have found a new fingerprint indicating the realization of the persistent spin helix symmetry. This method can determine the sign of the α/β ratio, so it is possible to detect an inverted persistent spin helix symmetry with −α/β by tuning the gate voltage. For example, the gate-controlled transition between the persistent spin helix and inverted persistent spin helix symmetries provides possibilities for future spintronic devices requiring large spin coherence times24. Furthermore, we have proposed a new technique to estimate the absolute values of α and β by detecting the strength of Beff. These results pave the way to explore new spin-related phenomena and to facilitate the design of future spintronic and quantum information devices based on both Rashba and Dresselhaus SOIs.
Methods The following heterostructure was used: In0.52Al0.48As (200 nm, buffer layer)/ In0.52Al0.48As (6 nm, carrier supply layer; Si doping concentration of 2 × 1018 cm−3)/ In0.52Al0.48As (6 nm, spacer layer)/In0.7Ga0.3As (10 nm, quantum well)/ In0.52Al0.48As (6 nm, spacer layer)/In0.52Al0.48As (6 nm, carrier supply layer; Si doping concentration of 2 × 1018 cm−3)/In0.52Al0.48As (25 nm, cap layer). We made a symmetric quantum well by doping on both sides of the well. Because of the symmetric shape of the quantum well, α was adjusted to a small value to match β. This structure was grown epitaxially on a (001) InP substrate by metal–organic chemical vapour deposition. The epitaxial wafer was processed into narrow wire structures using electron-beam lithography and reactive-ion etching. The structural parameters of the wire are as follows: wire length L = 200 µm, effective width W = 750 nm and number of parallel wires N = 100 (to average out universal conductance fluctuations). The wire width of 750 nm was set to be shorter than the spin precession length Lso = ħ2/2m *α ≈ 1.79 µm at a sheet carrier density of Ns = 2.0 × 1012 cm−2 in the InGaAs 2DEG channel. Ns and the electron mobility μ at Vg = 0 V are 2.44 × 1012 cm−2 and 6.68 m2 V−1 s−1, respectively. To apply Vg , we deposited 200 nm Al2O3 and 150 nm Cr/Au as an insulator and a top gate by atomic 708
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layer deposition and electron-beam deposition. The measurement was carried out using an a.c. lock-in technique for all wire sets at 1.7 K. Ns was deduced from the fast Fourier transformation of the Shubnikov–de Haas oscillations. The perpendicular magnetic field was swept at ±15 mT, and a fixed in-plane magnetic field was used that ranged from 0.2 to 6.0 T.
Received 22 January 2014; accepted 27 May 2014; published online 13 July 2014
References 1. Mourik, V. et al. Signatures of majorana fermions in hybrid superconductor– semiconductor nanowire devices. Science 336, 1003–1007 (2012). 2. Chen, Y. L. et al. Experimental realization of a three-dimensional topological insulator, Bi2Te3. Science 325, 178–181 (2009). 3. Datta, S. & Das, B. Electronic analog of the electro-optic modulator. Appl. Phys. Lett. 56, 655–657 (1990). 4. Nitta, J., Akazaki, T., Takayanagi, H. & Enoki, T. Gate control of spin–orbit interaction in an inverted In0.53Ga0.47As/In0.52Al0.48As heterostructure. Phys. Rev. Lett. 78, 1335–1338 (1997). 5. Engels, G., Lange, J., Schäpers, T. & Lüth, H. Experimental and theoretical approach to spin splitting in modulation-doped InxGa1−xAs/InP quantum wells for B → 0. Phys. Rev. B 55, R1958–R1961 (1997). 6. Kohda, M. et al. Spin–orbit induced electronic spin separation in semiconductor nanostructures. Nature Commun. 3, 1038 (2012). 7. König, M. et al. Direct observation of the Aharonov–Casher phase. Phys. Rev. Lett. 96, 076804 (2006). 8. Koga, T., Sekine, Y. & Nitta, J. Experimental realization of a ballistic spin interferometer based on the Rashba effect using a nanolithographically induced square loop array. Phys. Rev. B 74, 041302(R) (2006). 9. Bergsten, T., Kobayashi, T., Sekine, Y. & Nitta, J. Experimental demonstration of the time reversal Aharonov–Casher effect. Phys. Rev. Lett. 97, 196803 (2006). 10. Nowack, K. C., Koppens, F. H. L., Nazarov, Y. V. & Vandersypern, L. M. K. Coherent control of a single electron spin with electric fields. Science 318, 1430–1433 (2007). 11. Frolov, S. M. et al. Ballistic spin resonance. Nature 458, 868–871 (2009). 12. Sanada, H. et al. Manipulation of mobile spin coherence using magnetic-fieldfree electron spin resonance. Nature Phys. 9, 280–283 (2013). 13. Brüne, C. et al. Evidence for the ballistic intrinsic spin Hall effect in HgTe nanostructures Nature Phys. 6, 448–454 (2010). 14. D'yakonov, M. I. & Perel', V. I. Spin relaxation of conduction electrons in noncentrosymmetric semiconductors. Sov. Phys. Solid State 13, 3023–3026 (1971). 15. Rashba, E. I. Properties of semiconductors with an extremum loop. Sov. Phys. Solid State 2, 1224–1238 (1960). 16. Dresselhaus, G. Spin–orbit coupling effects in zinc blende structures. Phys. Rev. 100, 580–586 (1955). 17. Schliemann, J. & Loss, D. Anisotropic transport in a two-dimensional electron gas in the presence of spin–orbit coupling. Phys. Rev. B. 68, 165311 (2003). 18. Bernevig, B. A., Orenstein, J. & Zhang, S-C. Exact SU(2) symmetry and persistent spin helix in a spin–orbit coupled system. Phys. Rev. Lett. 97, 236601 (2006). 19. Koralek, J. D. et al. Emergence of the persistent spin helix in semiconductor quantum wells. Nature 458, 610–613 (2009). 20. Walser, M. P., Reichl, C., Wegscheider, W. & Salis, G. Direct mapping of the formation of a persistent spin helix. Nature Phys. 8, 757–762 (2012). 21. Kohda, M. et al. Gate-controlled persistent spin helix state in (In,Ga)As quantum wells. Phys. Rev. B 86, 081306(R) (2012). 22. Cartoixà, X., Ting, D. Z-Y. & Chang, Y-C. A resonant spin lifetime transistor. Appl. Phys. Lett. 83, 1462–1464 (2003). 23. Schliemann, J., Egues, J. C. & Loss, D. Nonballistic spin-field-effect transistor. Phys. Rev. Lett. 90, 146801 (2003). 24. Kunihashi, Y. et al. Proposal of spin complementary field effect transistor. Appl. Phys. Lett. 100, 113502 (2012). 25. Miller, J. B. et al. Gate-controlled spin–orbit quantum interference effects in lateral transport. Phys. Rev. Lett. 90, 076807 (2003). 26. Ganichev, S. D. et al. Experimental separation of Rashba and Dresselhaus spin splittings in semiconductor quantum wells. Phys. Rev. Lett. 92, 256601 (2004). 27. Meier, L. et al. Measurement of Rashba and Dresselhaus spin–orbit magnetic fields. Nature Phys. 3, 650–654 (2007). 28. Studer, M., Salis, G., Ensslin, K., Driscoll, D. C. & Gossard, A. C. Gate-controlled spin–orbit interaction in a parabolic GaAs/AlGaAs quantum well. Phys. Rev. Lett. 103, 027201 (2009). 29. Ishihara, J., Ohno, Y. & Ohno, H. Direct imaging of gate-controlled persistent spin helix state in a modulation-doped GaAs/AlGaAs quantum well. Appl. Phys. Exp. 7, 013001 (2014). 30. Meijer, F. E., Morpurgo, A. F., Klapwijk, T. M. & Nitta, J. Universal spin-induced time reversal symmetry breaking in two-dimensional electron gases with Rashba spin–orbit interaction. Phys. Rev. Lett. 94, 186805 (2005).
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31. Scheid, M., Kohda, M., Kunihashi, Y., Richter, K. & Nitta, J. All-electrical detection of the relative strength of Rashba and Dresselhaus spin–orbit interaction. Phys. Rev. Lett. 101, 266401 (2008). 32. Kettemann, S. Dimensional control of antilocalization and spin relaxation in quantum wires. Phys. Rev. Lett. 98, 176808 (2007). 33. Mal'shukov, A. G. & Chao, K. A. Waveguide diffusion modes and slowdown of D'yakonov–Perel' spin relaxation in narrow two-dimensional semiconductor channels. Phys. Rev. B 61, R2413–R2416 (2000). 34. Schäpers, Th. et al. Suppression of weak antilocalization in GaxIn1–xAs/InP narrow quantum wires. Phys. Rev. B 74, 081301(R) (2006). 35. Kunihashi, Y., Kohda, M. & Nitta, J. Enhancement of spin lifetime in gate-fitted InGaAs narrow wires. Phys. Rev. Lett. 102, 226601 (2009). 36. Wimmer, M. & Richter, K. Optimal block-tridiagonalization of matrices for coherent charge transport. J. Comput. Phys. 228, 8548–8565 (2009). 37. Scheid, M., Adagideli, İ., Nitta, J. & Richter, K. Anisotropic universal conductance fluctuation in disordered quantum wires with Rashba and Dresselhaus spin–orbit interaction and applied in-plane magnetic field. Semicond. Sci. Technol. 24, 064005 (2009).
Acknowledgements The authors acknowledge support from the Strategic Japanese–German Joint Research Program. K.R. thanks the DFG for support within Research Unit FOR 1483.
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T.D. acknowledges support by the DFG within research project SFB 689. This work was financially supported by Grants-in-Aid from the Japan Society for the Promotion of Science (JSPS; no. 22226001).
Author contributions A.S., S.N. and Y.K. performed device fabrication and measurements. T.B., T.D. and K.R. performed numerical calculations. A.S. and M.K. wrote the main part of the manuscript. T.D. and K.R. wrote the theoretical part. All authors discussed the results and worked on the manuscript at all stages. M.K., K.R. and J.N. planned the project. J.N. directed the research.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to J.N.
Competing financial interests
The authors declare no competing financial interests.
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Direct determination of spin–orbit interaction coefficients and realization of the persistent spin helix symmetry A. Sasaki1, S. Nonaka1, Y. Kunihashi1†, M. Kohda1, T. Bauernfeind2, T. Dollinger2, K. Richter2 and J. Nitta1 * The spin–orbit interaction plays a crucial role in diverse fields of condensed matter, including the investigation of Majorana fermions, topological insulators, quantum information and spintronics. In III–V zinc-blende semiconductor heterostructures, two types of spin–orbit interaction—Rashba and Dresselhaus—act on the electron spin as effective magnetic fields with different directions. They are characterized by coefficients α and β, respectively. When α is equal to β, the so-called persistent spin helix symmetry is realized. In this condition, invariance with respect to spin rotations is achieved even in the presence of the spin–orbit interaction, implying strongly enhanced spin lifetimes for spatially periodic spin modes. Existing methods to evaluate α/β require fitting analyses that often include ambiguity in the parameters used. Here, we experimentally demonstrate a simple and fitting parameter-free technique to determine α/β and to deduce the absolute values of α and β. The method is based on the detection of the effective magnetic field direction and the strength induced by the two spin–orbit interactions. Moreover, we observe the persistent spin helix symmetry by gate tuning.
A
s an essential feature of the spin–orbit interaction (SOI), the spin of an electron moving in an electric field experiences, in its rest frame, an effective magnetic field that couples to the spin's magnetic moment, even in the absence of any external magnetic field. The SOI is ubiquitous and appears in many different areas of physics and applications1–3. The SOI can be engineered by material design or by tuning an electric field4,5 to explore new spin-related phenomena and spin functionalities. In semiconductor-based spintronics, for example, the effective magnetic field Beff can be used for the generation6, manipulation7–12 and detection13 of spins solely by electrical means. However, the SOI inevitably gives rise to spin relaxation14, thus disrupting long spin coherence times. This crucial problem can be overcome by utilizing two different types of SOIs—Rashba15 and Dresselhaus16 SOIs. Spin SU(2) symmetry, an invariance with respect to specific rotations of electron spin, is realized and spin relaxation is suppressed correspondingly when both SOIs are of equal strength, giving rise to the persistent spin helix symmetry17–21. As a result, precise evaluation and control of α/β (where α and β denote the strengths of the Rashba and Dresselhaus SOIs, respectively) will pave the way for future applications in spintronics and quantum information22–24. However, in transport measurements, the simultaneous evaluation of Rashba and Dresselhaus SOIs usually involves large ambiguities due to model-dependent fitting25, and in optical measurements, fitting analyses are also required26–29. As a consequence there has been a longstanding quest for a simple, robust and reliable technology to evaluate α/β. Here, we demonstrate a fitting-free in situ determination of α/β in transport measurements with coexisting Rashba and Dresselhaus SOIs. In narrow wires, application of an in-plane magnetic field Bin modulates the amplitude of the magneto-conductance, that is, weak localization (WL), through additional dephasing arising from spin-induced time-reversal symmetry breaking30.
Theory predicts the maximum amplitude of WL to exist when Bin and Beff are parallel to one another31. This enables us to evaluate α/β from the anisotropic WL amplitude, without relying on any fitting. By applying the proposed concept to InGaAs-based wire structures, we realize gate-controlled persistent spin helix symmetry and find a novel fingerprint of this symmetry. In addition to the evaluation of the ratio α/β, and going beyond the theoretical proposal31, we show that the absolute values of α and β can be deduced from the minimum WL amplitude when Bin and Beff have equal strengths. As a proof of concept we provide measurements in InGaAs wires. Our demonstration of the proposed metrological concept31 can be extended to a wide range of materials, because the direction and strength of the effective magnetic fields in a confined structure are directly related to α and β. The precise determination of α and β is indispensable in exploring new spin-related phenomena and functionalities.
Concept of the present measurements According to the proposed metrological concept31, the evaluation of α/β is achieved by detecting the direction θeff of Beff = BR + BD , where BR and BD are the effective magnetic fields induced by the Rashba and linear Dresselhaus SOIs. For a quantum well grown in the [001] direction of III–V zinc-blende semiconductor heterostructures, the corresponding Hamiltonians are given by α (1) HR = − σ x py − σ y px h and β HD = − σ x px − σ y py h
(2)
1
Graduate school of Engineering, Tohoku University, 6-6-02 Aramaki-Aza Aoba, Aoba-ku, Sendai 980-8579, Japan, 2 Institut für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany. † Present address: NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi, Kanagawa 243-0198, Japan. *e-mail: [email protected]
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Figure 1 | Weak localization anisotropy to the relative angle between Beff and Bin. a,c, The directions of total magnetic field Btotal = Bin + Beff are shown for ±p momentum directions. b,d, Calculated weak localizations (WLs) are shown at α/β = 2/3 in a [100] wire with |Bin| = 0.75 T for Bin || Beff (d, θin = 146°) and Bin ∦ Beff (b, θin = 25°), where θin is the angle of the in-plane magnetic field defined with respect to the [100] direction. The magnetoconductance is plotted against the magnetic flux in a unit cell of the numerical grid, Φs = a 2|Bp| (a is the lattice constant and Bp is the perpendicular magnetic field) in units of the magnetic flux quantum Φ0 = h/e (e is the electron charge). In a,b, when Bin ∦ Beff , Btotal is randomized for ±p electron directions, suppressing the WL amplitude. In c,d, when Bin || Beff , the total magnetic field Btotal along the ±p directions remains unidirectional, showing the deepest WL amplitude.
where ħ = h/2π (h is Planck's constant), α and β are coefficients of the Rashba and Dresselhaus SOIs, which determine the strengths of BR and BD , respectively, and σi and pi (i = x, y) are the Pauli spin matrices and the electron momentum, respectively. The x and y directions are parallel to [100] and [010], respectively. Let us assume a one-dimensional wire along [100], where electron momentum is restricted to ±p along the wire direction. The direction of Beff acting on the electron spins is therefore fixed as unidirectional32,33. Owing to the perpendicular orientation of BR and BD , the relationship between the SOI parameters and the effective field orientation θeff is simply given by α = −tan θeff β
(3)
It should be noted that the cubic Dresselhaus SOI vanishes along the [100] axis. Hence, through the detection of θeff in the one-dimensional wire, direct evaluation of the α/β ratio is possible. The central concept to the determination of θeff through transport measurements is shown schematically in Fig. 1. For a narrow wire with W ≪ Lso (W, wire width; Lso , spin precession length, where Lso = ħ2/2m*α or ħ 2/2m*β, where m* is an effective mass), the randomization of spin is suppressed by the quasi-one-dimensional confinement, which fixes the electron momentum direction and therefore the spin precession axis Beff. Therefore, only WL is observable, according to theory32,33 and experiment34,35. The Cooperon conductance contribution, composed of triplet and singlet terms of opposite sign, exhibits a positive conductance correction (weak anti-localization, WAL) in the case of spin mixing, when triplet terms are suppressed, and a negative conductance 704
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correction if the spin is not randomized (in weak localization), because the triplet terms overcompensate the singlet. Magnetic flux through correlated path pairs leads to randomization of the phases entering the Cooperon, thereby destroying WL and WAL. This applies to a two-dimensional system as well as to a quasione-dimensional wire with W≪Lso , in which the magnetic flux may be enclosed within scattering trajectories. The application of a static in-plane magnetic field Bin that is not parallel to Beff further induces an additional spin relaxation by breaking the unidirectional alignment of Beff (Fig. 1a). It should be noted that the influence of different angles of an in-plane magnetic field on a persistent spin helix state in the two-dimensional case has been studied in ref. 20. The additional spin relaxation induces dephasing in the time-reversal related pairs of electron paths by mixing spin up and down phases. This dephasing leads to weaker quantum interference, resulting in suppression of the WL amplitudes (Fig. 1b). However, when Bin is applied in parallel to Beff , the total magnetic field Btotal = Bin + Beff remains unidirectional for ±p (Fig. 1c), preserving the long spin relaxation length and resulting in an enhanced WL amplitude (Fig. 1d). Hence, the WL amplitude becomes anisotropic as a function of the relative angle between Bin and Beff. This concept enables us to determine θeff by rotating the direction of Bin and therefore to obtain the α/β ratio by using equation (3) without any fittings.
Numerical simulations of weak localization anisotropy To confirm the presented concept we conduct numerical calculations of the magneto-conductance of wires with different α/β values including the persistent spin helix configuration. In this calculation, to compare with the result for the wire in the [100] direction, we also include results for wires along the [110] and [−110] directions, which provide parallel and antiparallel configurations between BR and BD , respectively. The numerical calculations are based on the recursive Green's function approach36 (for details see Supplementary Section I). We theoretically calculate the magnetoconductance of a narrow wire by changing the perpendicular magnetic field Bp under fixed Bin , whose angle θin is defined with respect to the [100] direction. The three different wire directions along the [100], [110] and [−110] axes are referred to as [100], [110] and [−110] wires, respectively. It should be noted that the wire width is set to 20 nm, which does not give rise to purely one-dimensional confinement, but is smaller than the spin precession length. Figures 1b,d present the calculated results for the [100] wire at θin = 25° (Fig. 1b) and 146° (Fig. 1d) (Bin = 0.75 T), and shows clear WL signals with different conductance amplitude. We define the amplitude of WL as ΔG (Fig. 1d) and produce a polar plot as a function of θin for different α/β ratios and wire directions (Fig. 2a–c). The WL amplitude shows two-fold symmetries for all wire directions. Furthermore, its symmetry axis depends on the crystal direction. Such dependencies are understood in terms of the relative angle between Bin and Beff. Because the Bin || Beff configuration does not enhance spin relaxation, the angle of the maximum WL amplitude corresponds to the Beff direction. On the other hand, the Bin ∦ Beff configuration induces spin relaxation, resulting in a reduced WL amplitude. For [110] and [−110] wires, because Beff always appears along the 135° and 45° directions, respectively, the WL amplitude becomes maximal at θin = 135° and 45° and is thus unaffected by the different α/β ratios (Fig. 2b,c). For the [100] wire, however, the Beff direction changes with the α/β ratio, resulting in an angular shift of the maximum WL amplitude (Fig. 2a). As shown in Fig. 2d, the angle of the maximum WL amplitude (θeff ) is perfectly aligned to equation (3). For the persistent spin helix symmetry, that is, at α/β = 1 (red plot in Fig. 2a–c), although the maximum WL peak still appears at θin = 135° for the [110] wire, interestingly, the anisotropy of the WL amplitude is quenched for the [−110] wire. This is due to the mutual compensation of BR and BD
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Figure 2 | Numerical results for conductance in quasi-one-dimensional quantum wires. The calculation is performed for W ≪ Lso without the cubic Dresselhaus SOI. We fix the ratio α/β at 1/3 (blue dotted line), 2/3 (green dashed line) and 1 (red solid line). Purple bars indicate the wire direction. a–c, Polar plots of WL amplitudes as a function of θin along [100], [110] and [−110] directions. Schematic insets: Angle θeff of the effective magnetic field for each wire orientation. d, Detected θeff (red filled circle) is perfectly fitted by the relation θeff = arctan(−α/β), shown by the dashed line. e, Orientation of Beff in momentum space under the persistent spin helix symmetry (α = β). The arrows indicate the direction of the effective magnetic field Beff describing spin–orbit coupling. The length of the shown arrows indicates the strength of Beff. Left: Rashba and Dresselhaus effective magnetic fields (BR and BD , respectively) are shown. Right: Total effective magnetic field Beff = BR + BD is shown. The persistent spin helix symmetry induces unidirectional alignment of Beff.
(Fig. 2e). This numerically observed quench of the WL anisotropy is a fingerprint of the persistent spin helix symmetry.
Evaluation of α/β ratio and persistent spin helix symmetry To utilize this concept in a real device we fabricated parallel wires along the [100], [110] and [−110] directions in series, as shown in Fig. 3 (for more information on the device structure and fabrication see Methods), and covered the entire structure with an Al2O3/Cr/Au top-gate electrode to control α through the gate voltage, Vg. Magneto-conductance measurements were performed simultaneously for three wire directions by sweeping Bp under constant Bin and Vg , at T = 1.7 K. Figure 4a presents the magneto-conductance for the [100] wire at θin = 24° and 142° with |Bin| = 1.0 T and Vg = −5.0 V. The restricted electron momentum in a narrow wire suppresses the spin relaxation, leading to the appearance of WL instead of WAL. In addition, the modulation of the WL amplitude Δσ between θin = 24° and 142° indicates anisotropic dephasing due to additional spin relaxation. To reveal the effect of Bin on WL amplitude, we measured the magneto-conductance in the range 0° ≤ θin ≤ 180° (Fig. 4b) and extracted Δσ as a function of θin (Fig. 4c). The WL amplitude changes continuously, as shown in Fig. 4b, and the oscillatory behaviour of Δσ in Fig. 4c indicates the modulation of electron dephasing induced by spin relaxation. To make a comparison with the theoretical prediction (Fig. 2a–d), we include a polar plot of Δσ' (subtracted Δσ from its minimum)
Bp
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Figure 3 | Schematic illustration of measurement configuration. The set-up shown allows the simultaneous performance of conductance measurements in all three considered wire orientations along the [100], [110] and [−110] axes. The conductance for each wire orientation is an average over multiple wires to minimize signatures from universal conductance corrections. Lower left: Scanning electron microscope images of a typical example of InGaAs wire structure, showing a top view (upper) and side view (lower). W, wire width. Scale bars in upper and lower images are 5 µm and 2 µm, respectively.
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Figure 4 | Weak localization anisotropy in the [100] direction wire at V g = −5 V. a, Measured magneto-conductance traces for in-plane magnetic field angles θin = 24° and 142°. Red and blue dashed lines are minimum weak localization (WL) amplitude levels for θin = 24° and 142°, respectively. Δσ is the WL amplitude extracted at |Bp| = 5 mT. b, Colour plot of WL amplitude measured in the range 0° ≤ θin ≤ 180°. Cross-sections through the red and blue dashed lines provide the shape of the magneto-conductance data for θin = 24° and 142°, respectively, shown in a. The cross-section through the white dashed line corresponds to the WL amplitude shown in c. The zero point is chosen as the minimum of the magneto-conductance (σ at |Bp| = 0 mT). c, WL amplitude Δσ for different θin. Red and blue dashed lines indicate angles for minimum and maximum WL amplitudes, respectively. The black dotted line is the minimum WL amplitude level. Values of Δσ' shown by black arrows are the magneto-conductance difference from minimum Δσ. d, Polar plot of Δσ' in the range 0° ≤ θin ≤ 360° for |Bin| = 1.0 T. We extended the result for 0° ≤ θin ≤ 180° to 180° ≤ θin ≤ 360° by considering spatial symmetry. Blue dashed line: direction of maximum WL amplitude. The green arrow indicates a major peak angle corresponding to the effective magnetic field direction that directly provides the α/β ratio. e, Polar plots with different amplitudes of in-plane field, |Bin| = 0.2–1 T. The blue dashed line shows a direction of maximum WL amplitude that does not depend on the amplitude of |Bin|. The green arrow shows the major peak angle guaranteeing high accuracy of direct α/β detection. 706
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as a function of θin (Fig. 4d). Due to the spatial symmetry of the configuration, we safely extend the result to 180° ≤ θin ≤ 360°. This plot represents how an additional spin relaxation is induced by the relative angle between Bin and Beff. Because Δσ' shows a two-fold symmetry that is perfectly consistent with the theoretical prediction, θin for the maximum Δσ' corresponds to θeff. The second minor maximum WL amplitude around an angle of 50° is also well reproduced. According to the theoretical study, for the case of Bin ⊥ Beff , although the total Hamiltonian HQ1D mixes ˆ indi˜ Q1D = Cˆ −1 H ˜ Q1D fulfils H ˜ Q1D C, spins, its spin rotated version, H cating the orthogonal symmetry class37. Here, HQ1D is a single-particle Hamiltonian of a disordered quantum wire and Ĉ is the operator of complex conjugation (the definition of HQ1D is given in Supplementary Section I). This leads to the recovery of a WL amplitude and the appearance of the minor peaks. The orthogonal symmetry class can describe the system only when the electron momentum perpendicular to the wire is negligible. Hence, the appearance of these minor peaks is strong evidence for dimensional control of the electron spin and the validity of the α/β evaluation technique in our measurement set-up. Also, to confirm the dependence of our method on the strength of Bin , we include additional polar plots for various values of Bin ranging from 0.2 to 1.0 T, while leaving the other parameters unaltered (Fig. 4e). According to theory31, the detection technique for α/β cannot be correctly applicable when |Bin| ≫|Beff| because Btotal becomes strongly aligned in the direction of Bin and electron dephasing is suppressed for any θin. This potentially changes the major peak angle. However, no shift of the major peak is observed at different values of Bin , which guarantees the direct detection of α/β with high accuracy. (The |Bin| dependencies for other wire directions are discussed in Supplementary Section II.) To further confirm the validity of this concept, we plot Δσ' for all considered wire directions with different Vg (Fig. 5). For the [110] and [−110] wires, the Δσ' maxima correspond to 135° and 45°, respectively, consistent with the predicted angle θeff depicted in Fig. 2b,c. Because Beff is always oriented perpendicular to the [110] and [−110] wire directions (purple wires in Fig. 5b,c) for different Vg , θin at the Δσ' maximum does not change according to the α/β ratio. From these considerations, we directly probe the precise value of θeff by measuring the WL maximum. Moreover, we can extract the α/β ratio from the anisotropy of the [100] wire. As shown in Fig. 5a, the Δσ' maximum is shifted towards 135° by decreasing Vg from 0 V to −9 V, showing α/β modulation through the gate. As α is increased with reducing carrier density Ns (2.44 × 1012 cm−2 at Vg = 0 V to 2.01 × 1012 cm−2 at Vg = −9 V) and is smaller than β at Vg = 0 V, the α/β ratio is approaching unity at Vg = −9 V. By using equation (3), we determine the α/β ratio as α/β = −tan(157° ± 1°) = 0.42 ± 0.02 at Vg = 0 V, α/β = −tan(142° ± 1°) = 0.78 ± 0.03 at Vg = −5 V and α/β = −tan(133° ± 1°) = 1.07 ± 0.04 at Vg = −9 V. The condition of persistent spin helix symmetry is satisfied for the α/β ratio at Vg = −9 V. As shown in Fig. 2e, under the persistent spin helix symmetry Beff vanishes in the [−110] configuration due to the mutual cancellation of BR and BD. Only at Vg = −9 V in the [−110] wire do we observe quenched anisotropy (Fig. 5c). This is caused by the dominant strength of Bin compared to Beff , which gives rise to a homogeneous dephasing rate regardless of the value of θin , and the shape of the polar plot becomes a circle when we add the offset of the magneto-conductance (Vg = −9 V in Fig. 5c). This quenching of the WL anisotropy agrees with the numerical result. It can be regarded as a characteristic fingerprint to support the realization of the persistent spin helix symmetry revealed by the present method.
Evaluation of absolute values of α and β We confirm that the WL amplitude shows a minimum when |Bin| is close to |Beff| (refs 31,37). By detecting |Beff| from the |Bin|
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Figure 6 | Numerically calculated and experimental weak localization amplitude as a function of |Bin|/|Beff| ratio. a, Calculated result. The calculation is performed for W ≪ Lso without cubic Dresselhaus SOI in a [100] quasi-one-dimensional wire. We show the dependence of the weak localization (WL) amplitude ΔG on the ratio |Bin|/|Beff|. To observe ΔG(|Bin| = 0) in the basis 0, offset −ΔG(|Bin| = 0) is added. The θ-dependence (the angle between Bin and Beff) is shown for θ = 30° (red squares), 60° (blue circles), 120° (light blue inverted triangles) and 150° (pink triangles). We observe clear-cut minima of the WL conductance if the ratio |Bin|/|Beff| is close to 1. b–d, Experimentally obtained WL amplitudes G' = Δσ(θ ≠ 0°) − Δσ(θ = 0°) at Vg = −5 V (carrier density Ns = 2.16 × 1012 cm−2) plotted for the [100] (b), [110] (c) and [−110] (d) wires. Each angle θ (≠0°) is shown as θ = 30° (red squares), 60° (blue circles), 120° (light blue inverted triangles) and 150° (pink triangles). Thick green lines indicate dip positions of G'. NATURE NANOTECHNOLOGY | VOL 9 | SEPTEMBER 2014 | www.nature.com/naturenanotechnology
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dependence of the WL amplitude, we can estimate the absolute values of α and β to an accuracy of up to a factor of two. This concept is based on the idea that maximum spin relaxation caused by the most randomizing field Btotal = Bin + Beff , with |Bin| ≈ |Beff|, leads to a minimum WL amplitude. This idea is consistent with our analytical toy model describing the spin interference term of two one-dimensional time-reversal symmetry related paths (Supplementary Section III). Moreover, the numerical calculation shows minima of the WL amplitude if the ratio |Bin|/|Beff| is close to 1 (Fig. 6a). We experimentally observe the |Bin| dependence of the ‘difference of the WL amplitudes G', Δσ(θ ≠ 0°) − Δσ(θ = 0°)’, at Vg = −5 V (carrier density Ns = 2.16 × 10−12 cm−2), where Δσ is the WL amplitude and θ is the angle between Bin and Beff (Fig. 6b–d). We used the WL amplitude at θ = 0° as a reference value for the remaining WL amplitude. The characteristic line shapes of the numerical (Fig. 6a) and experimental (Fig. 6b–d) data sets are remarkably similar. Moreover, when we compare the traces in each wire, |Bin| ≈ |Beff| at the dip position is largest in the [110] wire and smallest in the [−110] wire. This result agrees with the relation between |Beff| and |BR|, |BD| in each crystal direction: |Beff| = |BR| + |BD| ([110] wire) > |Beff| = √(|BR|2 + |BD|2) ([100] wire) > |Beff| = |BR| – |BD| ([−110] wire). Along the [100] axis in particular, cubic Dresselhaus SOI is negligible, so α and β can be deduced by using the estimated |Beff| and the α/β ratio obtained from WL anisotropy measurements in the [100] wire. We estimate α and β approximately as α ≈ 2.0 × 10−13 eV m and β ≈ 3.7 × 10−13 eV m by using the evaluated α/β = −tan(θeff ≈ 152°) ≈ 0.53 and |Beff| ≈ 1.55 T. The small value of α of the order of 1 × 10−13 eV m is reasonable, because our heterostructure design corresponds to a symmetric quantum well.
Conclusions We have demonstrated a novel metrological concept and measurement technique to evaluate the α/β ratio, which can be applied in different material systems. The present concept enables us to determine the effective magnetic field direction originating from the SOIs with high accuracy, and also allows for the evaluation of the α/β ratio, including the correct sign, without fitting. We have also realized the gate-controlled persistent spin helix symmetry and have found a new fingerprint indicating the realization of the persistent spin helix symmetry. This method can determine the sign of the α/β ratio, so it is possible to detect an inverted persistent spin helix symmetry with −α/β by tuning the gate voltage. For example, the gate-controlled transition between the persistent spin helix and inverted persistent spin helix symmetries provides possibilities for future spintronic devices requiring large spin coherence times24. Furthermore, we have proposed a new technique to estimate the absolute values of α and β by detecting the strength of Beff. These results pave the way to explore new spin-related phenomena and to facilitate the design of future spintronic and quantum information devices based on both Rashba and Dresselhaus SOIs.
Methods The following heterostructure was used: In0.52Al0.48As (200 nm, buffer layer)/ In0.52Al0.48As (6 nm, carrier supply layer; Si doping concentration of 2 × 1018 cm−3)/ In0.52Al0.48As (6 nm, spacer layer)/In0.7Ga0.3As (10 nm, quantum well)/ In0.52Al0.48As (6 nm, spacer layer)/In0.52Al0.48As (6 nm, carrier supply layer; Si doping concentration of 2 × 1018 cm−3)/In0.52Al0.48As (25 nm, cap layer). We made a symmetric quantum well by doping on both sides of the well. Because of the symmetric shape of the quantum well, α was adjusted to a small value to match β. This structure was grown epitaxially on a (001) InP substrate by metal–organic chemical vapour deposition. The epitaxial wafer was processed into narrow wire structures using electron-beam lithography and reactive-ion etching. The structural parameters of the wire are as follows: wire length L = 200 µm, effective width W = 750 nm and number of parallel wires N = 100 (to average out universal conductance fluctuations). The wire width of 750 nm was set to be shorter than the spin precession length Lso = ħ2/2m *α ≈ 1.79 µm at a sheet carrier density of Ns = 2.0 × 1012 cm−2 in the InGaAs 2DEG channel. Ns and the electron mobility μ at Vg = 0 V are 2.44 × 1012 cm−2 and 6.68 m2 V−1 s−1, respectively. To apply Vg , we deposited 200 nm Al2O3 and 150 nm Cr/Au as an insulator and a top gate by atomic 708
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layer deposition and electron-beam deposition. The measurement was carried out using an a.c. lock-in technique for all wire sets at 1.7 K. Ns was deduced from the fast Fourier transformation of the Shubnikov–de Haas oscillations. The perpendicular magnetic field was swept at ±15 mT, and a fixed in-plane magnetic field was used that ranged from 0.2 to 6.0 T.
Received 22 January 2014; accepted 27 May 2014; published online 13 July 2014
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Acknowledgements The authors acknowledge support from the Strategic Japanese–German Joint Research Program. K.R. thanks the DFG for support within Research Unit FOR 1483.
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T.D. acknowledges support by the DFG within research project SFB 689. This work was financially supported by Grants-in-Aid from the Japan Society for the Promotion of Science (JSPS; no. 22226001).
Author contributions A.S., S.N. and Y.K. performed device fabrication and measurements. T.B., T.D. and K.R. performed numerical calculations. A.S. and M.K. wrote the main part of the manuscript. T.D. and K.R. wrote the theoretical part. All authors discussed the results and worked on the manuscript at all stages. M.K., K.R. and J.N. planned the project. J.N. directed the research.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to J.N.
Competing financial interests
The authors declare no competing financial interests.
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