Stretchable Persistent Spin Helices in GaAs Quantum Wells

Stretchable Persistent Spin Helices in GaAs Quantum Wells PHYSICAL REVIEW X 7, 031010 (2017) 1 2,† 3,‡ 1 2 Florian Dettwiler, Jiyong Fu, Shawn Mack, Pirmin J. Weigele, J. Carlos Egues, 3,4 1,* David D. Awschalom, and Dominik M. Zumbühl Department of Physics, University of Basel, CH-4056 Basel, Switzerland Instituto de Física de São Carlos, Universidade de São Paulo, 13560-970 São Carlos, São Paulo, Brazil California NanoSystems Institute, University of California, Santa Barbara, California 93106, USA Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA (Received 16 February 2017; revised manuscript received 21 May 2017; published 18 July 2017) The Rashba and Dresselhaus spin-orbit (SO) interactions in 2D electron gases act as effective magnetic fields with momentum-dependent directions, which cause spin decay as the spins undergo arbitrary precessions about these randomly oriented SO fields due to momentum scattering. Theoretically and experimentally, it has been established that by fine-tuning the Rashba α and renormalized Dresselhaus β couplings to equal fixed strengths α ¼ β, the total SO field becomes unidirectional, thus rendering the electron spins immune to decay due to momentum scattering. A robust persistent spin helix (PSH), i.e., a helical spin-density wave excitation with constant pitch P ¼ 2π=Q, Q ¼ 4mα=ℏ , has already been experimentally realized at this singular point α ¼ β, enhancing the spin lifetime by up to 2 orders of magnitude. Here, we employ the suppression of weak antilocalization as a sensitive detector for matched SO fields together with independent electrical control over the SO couplings via top gate voltage V and back gate voltage V to extract all SO couplings when combined with detailed numerical simulations. We demonstrate for the first time the gate control of the renormalized β and the continuous locking of the SO fields at α ¼ β; i.e., we are able to vary both α and β controllably and continuously with V and V , while T B keeping them locked at equal strengths. This makes possible a new concept: “stretchable PSHs,” i.e., helical spin patterns with continuously variable pitches P over a wide parameter range. Stretching the PSH, i.e., gate controlling P while staying locked in the PSH regime, provides protection from spin decay at the symmetry point α ¼ β, thus offering an important advantage over other methods. This protection is limited mainly by the cubic Dresselhaus term, which breaks the unidirectionality of the total SO field and causes spin decay at higher electron densities. We quantify the cubic term, and find it to be sufficiently weak so that the extracted spin-diffusion lengths and decay times show a significant enhancement near α ¼ β. Since within the continuous-locking regime quantum transport is diffusive (2D) for charge while ballistic (1D) for spin and thus amenable to coherent spin control, stretchable PSHs could provide the platform for the much heralded long-distance communication ∼8–25 μm between solid-state spin qubits, where the spin diffusion length for α ≠ β is an order of magnitude smaller. DOI: 10.1103/PhysRevX.7.031010 Subject Areas: Condensed Matter Physics, Quantum Information, Spintronics The inextricable coupling between the electron spatial and spin degrees of freedom—the spin-orbit (SO) interaction— underlies many fundamental phenomena such as the spin Hall effects—quantum and anomalous [1]—and plays a crucial role in newly discovered quantum materials hosting dominik.zumbuhl@unibas.ch Permanent address: Department of Physics, Qufu Normal University, Qufu, Shandong, 273165, China. Present address: Naval Research Laboratory, Washington, D.C. 20375, USA. FIG. 1. Stretchable PSHs. Illustration of spin helices at different values of α¼β accessible in the measurements. The position x for Published by the American Physical Society under the terms of one 2π rotation (dashed curve) is changing for the gate-locked the Creative Commons Attribution 4.0 International license. regime α¼β. The gray box highlights how the spin rotation can be Further distribution of this work must maintain attribution to controlled (in situ) at fixed position∼4.8 μmby∼π=2 over the same the author(s) and the published article’s title, journal citation, ˆ ˆ and DOI. range of α¼β.Thex ∥½110 andx ∥½110 axes define the 2Dplane. þ − 2160-3308=17=7(3)=031010(8) 031010-1 Published by the American Physical Society FLORIAN DETTWILER et al. PHYS. REV. X 7, 031010 (2017) Majorana [2] and Weyl fermions [3]. In nanostructures, the injection of spin polarization; see, e.g., Refs. [12,13]. Other SO coupling strength can be varied via gate electrodes [4,5]. spin communication modes can be envisaged with this As recently demonstrated [6], this enables controlled spin setup. The distance is limited mainly by the deviation modulation [7] of charge currents in nonmagnetic (quasibal- from α ¼ β and by the cubic Dresselhaus term, which is listic) spin transistors. small in this range, as we quantify later on, and leads to spin The SO coupling in a GaAs quantum well has two decay with spin-diffusion lengths λ ∼ 8–25 μm over eff dominant contributions: the Rashba [8] and the Dresselhaus which the spin dephases by 1 rad. Note that this type of [9] effects, arising from the breaking of the structural spin manipulation and spin transfer is not possible for a and crystal inversion symmetries, respectively. When the helix with α ≠ β, since λ quickly drops below the helix eff Rashba α and Dresselhaus β SO couplings match at pitch as the SO couplings are deviating from the sym- α ¼ β [10,11], the direction of the combined Rashba- metry point. Dresselhaus field becomes momentum independent, thus The full electrical control of the SO couplings demon- suppressing D’yakonov-Perel spin-flip processes, provided strated in our 9.3-nm-wide quantum well can tune from that the cubic Dresselhaus term be small. The significantly α ¼ β ¼ 5 meV Å to 4 meV Å, thus enabling stretchable enhanced spin lifetime at α ¼ β enables nonballistic spin PSHs with pitches P stretching from 3.5 to 4.4 μm; see transistors and persistent spin helices (PSHs) [10,11]. Fig. 1. Within the shortest spin-diffusion length λ ∼ 8 μm eff However, despite substantial efforts, so far this symmetry for our 9.3-nm well, controlled spin rotations by an angle point has been achieved only at isolated points with finely θ ¼ Qx ¼ 2πx =P can be performed under spin protec- þ þ tuned system parameters [12–14], which is too difficult to tion on any spin sitting at a position x along the stretchable be reliably attained on demand as required for a useful PSH by varying P in the range above. For example, a spin technology. at x ∼ 4.8 μm can be rotated by Δθ ∼ π=2 as P varies in the Stretchable persistent spin helices.—Here, we overcome range above; see gray box shading in Fig. 1. Thus, this outstanding obstacle by (i) using a technique that stretchable helices could provide a platform for long- allows independent control of the SO couplings via a top distance spin communication. gate voltage V and a back gate voltage V while Additional results.—WAL is also used to identify other T B (ii) simultaneously measuring the suppression of weak regimes such as the Dresselhaus regime α ¼ 0 in a more antilocalization (WAL) in an external magnetic field as a symmetrically doped sample. Combined with numerical sensitive probe for matched SO couplings. While gate simulations, we extract the SO couplings α and β, the tuning of the renormalized Dresselhaus coefficient β was bulk Dresselhaus parameter γ, the spin-diffusion lengths, already theoretically described in 1994 [15], we demon- and spin-relaxation times over a wide range of system strate this for the first time here in an experiment, and parameters. We also quantify the detrimental effects employ this tunability to show robust continuous locking of of the third harmonic of the cubic Dresselhaus term, which the Rashba and Dresselhaus couplings at αðV ;V Þ¼ T B mainly limits spin protection. Interestingly, our spin- βðV ;V Þ over a wide range of densities n, i.e., a T B diffusion lengths and spin-relaxation times are significantly “symmetry line” (not a point) in the (V , V ) plane. T B enhanced within the locked α ¼ β range, thus attesting that This allows us to introduce the concept of the “stretchable our proposed setup offers a promising route for spin persistent spin helix,” see Fig. 1, with spin density protection and manipulation. s ∼ sinðQx Þ, s ¼ 0, and s ∼ cosðQx Þ and variable x þ x z þ In what follows, we first explain tuning of the Rashba þ − pitch P ¼ 2π=Q, Q ¼ 4mα=ℏ . The stretchable PSH coupling, then the essential density dependence of the Dresselhaus coupling β that enables the continuous locking makes possible gate control of the spin precession over long distances due to strong protection from spin of the SO fields, how it also leads to spin decay at higher decay by up to 2 orders of magnitude enhanced densities, followed by the relevant weak-localization– spin lifetimes at the symmetry point α ¼ β—without weak-antilocalization (WL-WAL) detection scheme, mea- requiring in-plane electric fields to induce drift [16], surements, and simulations. A full account of our approach, and without relying on micron-width channels to suppress including additional data and details of the model and decay [17]. simulations, is presented in the Appendix and the SM [18]. Long-distance spin communication.—Within the range Controlling the Rashba coupling α.—The Rashba coef- ficient [8] α can be tuned with the wafer and doping profile of the continuously matched-locked SO couplings α ¼ β, quantum transport in the well is diffusive for charge (2D) [12] as well as in situ using gate voltages [4,5] at constant while essentially ballistic (1D) for spins [see Supplemental density n and thus independent of the Dresselhaus term; see Material (SM), Sec. V [18]]. A stretchable PSH could thus below. A change of top gate voltage V can be compen- be used to coherently couple, e.g., spin qubits over sated by an appropriate, opposing change of back gate unprecedented long distances. Figure 1 illustrates how voltage V [see Fig. 2(a)] to keep n fixed [19,20] while spin information can be conveyed between spins via a changing the gate-induced electric field δE in the quantum stretchable PSH. These stretchy waves can be excited upon well, where z⊥2D plane. Another Rashba term due to 031010-2 6.5 5.5 α ≈ β 4.5 3.5 STRETCHABLE PERSISTENT SPIN HELICES IN GAAS … PHYS. REV. X 7, 031010 (2017) 9.3-nm QW, asymmetrically doped the Rashba α term. An additional term with the same (a) (b) 0.4 0 form—the interface Dresselhaus term [24]—could also 1 n =4.5 1 play a role; see SM [18]. 0.2 3 To a very good approximation, the coefficient β ≃ pffiffiffiffiffiffiffiffi -2 γk =4, where the Fermi vector k ≃ 2πn and n is the 5 3 F F 0.0 carrier density of the 2D gas. This neglects the tiny angular 500 μm anisotropy in the Fermi wave vector due to the competition -0.2 -4 between the Rashba and Dresselhaus effects (especially in -2.0 -1.0 0.0 1.0 V [V] GaAs wells). Note that by approximating β ≃ γπn=2, both B 3 the first-harmonic and the third-harmonic parts of the (c) α α n =4.5 w cubic-in-k Dresselhaus term actually become linear in k -6 β α α (see SM for details [18]) and, more importantly, become g +d 6 4 density dependent. We can now group the linear-in-k 3 5 Dresselhaus term β together with the first-harmonic -8 7 contribution β into a single renormalized Dresselhaus term by defining β ¼ β − β . It is this density-dependent 1 3 3 2 1 0 -1 -0.2 0.0 0.2 renormalized coefficient β that can be tuned with a gate δE [V/μm] B [mT] voltage to match the Rashba α coupling continuously. This matching leads to a k-independent spinor (or, equivalently, FIG. 2. Weak localization (WL) as an α ¼ β detector, gate to a k-independent effective SO field), whose direction is control of Rashba α at constant density. (a) Measured charge immune to momentum scattering. In this way we achieve density n (color) versus top gate voltage V and back gate voltage independent, continuous control of the Rashba and V (9.3-nm well). Contours of constant density ð3.5–7.5Þ × 11 −2 Dresselhaus terms by using top gate and back gate voltages. 10 cm are shown. Inset: Optical micrograph of typical Hall This is an unprecedented tunability of the SO terms within bar, with contacts (yellow), gate (center), and mesa (black lines). a single sample. (b) Normalized longitudinal conductivity Δσ=σ ¼½σðB Þ − 0 Z σð0Þ=σð0Þ versus B ⊥2D plane. Curves for gate configurations Spin decay at higher densities.—The strength of the 11 −2 1–7 along constant n ¼ 4.5 × 10 cm are shown (offset third-harmonic contribution of the Dresselhaus term is also vertically), also labeled in (a) and (c). (c) Simulated Rashba α described by the coefficient β . This term, however, is and Dresselhaus β coefficients (see text) against gate-induced detrimental to spin protection as it breaks the angular 11 −2 field change δE , shown for constant n ¼ 4.5 × 10 cm . The symmetry of the other linear SO terms and makes the spinor δE axis—decreasing from left to right—corresponds exactly to k dependent and susceptible to in-plane momentum scat- the V abscissa of (a) for a covarying V , such that n ¼ 4.5 × B T tering, even for matched couplings α ¼ β. As we show, the 11 −2 10 cm constant. Sketches of the well potential at 1, 4, and 6 detrimental effect of the third-harmonic contribution does illustrate the change of α with δE . Note that αðδE ¼ 0Þ ≠ 0 Z Z not prevent our attaining the continuous locking over a since the external E field (see SM [18]) is not zero at δE ¼ 0. relevant wide range of electron densities. Gate-tunable range of the Dresselhaus coupling β.—For donor electric fields [21,22] is negligible in our structures; the narrow quantum wells we use here, β is essentially gate see SM [18]. independent since the wave function spreads over the full 2 2 Linear and cubic Dresselhaus terms in 2D.—Because of width of the well. This also implies hk i ≪ ðπ=WÞ (the the well confinement along the z direction (growth), the infinite well limit), see Fig. 3(d), due to wave function cubic-in-momentum bulk (3D) Dresselhaus SO interaction penetration into the finite barriers. Thus, a change of gives rise to, after the projection into the lowest quantum density by a factor of ∼2.5 changes β =β ¼ πn=h2k i 3 1 z well subband eigenstates, distinct terms that are linear and by the same factor, resulting in a gate-tunable range of cubic in k, the 2D electron wave vector. The linear-in-k 0.08 ≲ β =β ≲ 0.2. In addition, quantum wells of width 3 1 term has a coefficient β ¼ γhk i and turns out practically W ¼ 8, 9.3, 11, and 13 nm were used [12,25], resulting in a 1 z independent of density in the parameter range of interest change of β by roughly a factor of 2. here. The cubic-in-k term, on the other hand, is density Detection scheme for matched SO couplings.—WAL is a dependent and has yet two components with distinct well-established signature of SO coupling in magneto- angular symmetries: (i) the first-harmonic contribution conductance σðB Þ [15,23,26–29] exhibiting a local maxi- proportional to sin ϕ and cos ϕ and (ii) the third-harmonic mum at zero field. In the α ¼β regime, the resulting contribution proportional to sin 3ϕ and cos 3ϕ; here, ϕ is internal SO field is uniaxial, and spin rotations commute the polar angle in 2D between k and the [100] direction and are undone along time-reversal loops. Therefore, WAL (see SM [18]). Interestingly, the first-harmonic contribution is suppressed and the effectively spinless situation display- with coefficient β has the same angular symmetry as both ing weak localization [i.e., σðB Þ exhibiting a local mini- 3 Z the linear-in-k Dresselhaus β term (see Refs. [15,23]) and mum at B ¼ 0] is restored [10,11,14,23]. Away from the 031010-3 V [V] α, β [meV Å] -3 Δσ/σ [x10 ] 0 2.5 6.5 5.5 4.5 3.5 FLORIAN DETTWILER et al. PHYS. REV. X 7, 031010 (2017) (a) 3 (b) 9.3-nm QW small corrections [24,30] even up to room temperature. data fit QW [nm] γ [eV Å ] α α 0.6 8 11.0 ± 1 6 e Also, note that Shubnikov–de Haas oscillations do not 9.3 11.6 ± 1 α α g+d w 11 11.8 ± 1 show any spin-orbit splitting here (see SM [18]) given the 13 10.9 ± 1.6 4 0.4 strength of SO coupling in GaAs, making it clear that locked 2 the quantum corrections in WAL and their suppression at the symmetry point present a very sensitive detector for SO 0.2 coupling. β (= α) Continuous locking α ¼ β.—We proceed to demonstrate β β 1 3 0.0 gate locking of the SO couplings α, β. Figure 2(b) displays σðB Þ of the 9.3-nm well for top gate and back gate -0.2 configurations labeled 1–7, all lying on a contour of constant density; see Fig. 2(a). Along this contour, β is -0.4 0 held fixed since the density is constant (β is essentially 3 4 5 6 -2 -1 0 1 gate independent), while α is changing as the gate voltages 11 -2 V [V] B n [x10 cm ] are modifying the electric field δE perpendicular to the (d) (c) 0.15 quantum well. Across these gate configurations, the con- inf QW calc ductance shows a transition from WAL (configuration 1 0.10 and 2) to WL (4 and 5) back to WAL (7). Selecting the most 0.05 pronounced WL curve allows us to determine the symmetry point α ¼ β. This scheme is repeated for a number of 0.00 8 10 12 14 8 10 12 densities, varying n by a factor of 2, yielding the symmetry W [nm] W [nm] point α ¼ β for each density n [see Fig. 3(a), blue markers], thus defining a symmetry line in the ðV ;V Þ plane. Along T B FIG. 3. Tuning and continuously locking α ¼ β. (a) The this line, β is changing with density, as previously described, markers indicate α ≈ β for four different well widths (asymmetric and α follows β, remaining “continuously” locked at α ¼ β. doping) and various densities (gray contours of constant n, 11 −2 As mentioned earlier, this is a very interesting finding, labeled in units of 10 cm ) in the V and V plane. Error T B as it should allow the creation of persistent spin helices with bars result from the finite number of conductance traces in the ðV ;V Þ space. Theory fits (solid lines) are shown for each well, gate-controllable pitches, as illustrated in Fig. 1. B T with γ as the only fit parameter (inset table, error bars dominated Simulations and fitting of γ.—Self-consistent calcula- by systematic error; see below). The dashed blue line indicates tions combined with the transport data can deliver all SO the slope of constant α ¼ β , neglecting β , which is inconsistent 1 3 parameters. The numerical simulations [31] (see Appendix with the data. (b) Simulation of locked α ¼ β versus density n and SM [18]) can accurately calculate α and hk i. This along solid blue line from (a), showing the various SO contri- leaves only one fit parameter, γ, the bulk Dresselhaus butions (see text). (c) Values of γ from fits for each well width W. coefficient, which can now be extracted from fits to the Red dashed line is the average γ ¼ 11.6  1 eV Å (excluding density dependence of the symmetry point, see solid blue W ¼ 13 nm due to its larger error), gray area is the ∼9% error, line in Fig. 3(a), giving excellent agreement with the data stemming mostly from the systematic uncertainty in the input (blue markers). This procedure can be repeated for a set of parameters of the simulations (see Appendix). (d) hk i as a wafers with varying quantum well width and thus varying function of well width W for realistic (markers) and infinite (blue) potential. β . This shifts the symmetry point α ¼ β, producing nearly parallel lines, as indicated with colors in Fig. 3(a) corre- sponding to the various wafers as labeled. As shown, matched regime, the SO field is not uniaxial, spin rotations locking α ¼ β over a broad range is achieved in all wafers. do not commute, and trajectories in time-reversal loops Since gate voltages can be tuned continuously, any and all interfere destructively upon averaging [26] due to the SO points on the symmetry lines α ¼ β can be reached. Again phases picked up along the loops, thus leading to WAL. performing fits over the density dependence of the sym- Hence, this suppression is a sensitive detector for α ¼β. metry point for each well width, we obtain very good At high β , this detection scheme becomes approximate, giving α ¼β [14]. We note that the WL dip—often used agreement, see Fig. 3(a), and extract γ ¼ 11.6  1 eV Å to determine phase coherence—sensitively depends on the consistently for all wells [Fig. 3(c)]. We emphasize that γ is SO coupling [e.g., curves 3–6 in Fig. 2(b)], even before notoriously difficult to calculate and measure [28,29,32]; WAL appears. Negligence of SO coupling could thus lead the value we report here agrees well with recent studies to spurious or saturating coherence times. At higher [13,32,33]. Obtaining consistent values over wide ranges of temperatures, when quantum coherence is lost, this detec- densities and several wafers with varying well widths tion scheme becomes inoperable, while it is expected that provides a robust method to extract γ. the mechanism for tuning both Rashba and effective Beyond γ, the simulations reveal important information Dresselhaus coefficients continues to function with only about the gate tuning of the SO parameters. The Rashba 031010-4 α ≈ β α ≈ β α ≈ β α ≈ β V [V] γ [eV Å ] 2 -2 <k > [nm ] β [meV Å] α [meV Å] 9 7.5 4.5 STRETCHABLE PERSISTENT SPIN HELICES IN GAAS … PHYS. REV. X 7, 031010 (2017) 11-nm QW, symmetrically doped coefficient is modeled as α ¼ α þ α þ α in the gþd w e (a) (b) 2 1 simulation, with gate and doping term α p n = 6 , quantum well gþd α≈ β 2 0.4 structure term α , and Hartree term α . Along a contour of 0 w e constant density, the simulations show that mainly α and r gþd 0.2 -2 α are modified, while α and β remain constant; see w e t 2 4 -4 Fig. 2(c). The density dependence for locked α ¼ β, on the 0.0 α≈ 0 other hand, shows that while β is nearly constant, β is 1 3 6 -6 -0.2 linearly increasing with n, thus reducing β ¼ β − β ; see 1 3 -8 Fig. 3(b). Hence, to keep α ¼ β locked, α has to be reduced B B Z 2 Z 1 α = 0 8 -0.4 correspondingly. The Hartree term α , however, increases n = 9 -1 -3 -2 -1 0 1 for growing n. Thus, on the α ¼ β line, the other α terms— V [V] q mainly the gate-dependent α —are strongly reduced, gþd -2 (c) maintaining locked α ¼ β, as shown in Fig. 3(b).We α α≈ β n = 6 emphasize that neglecting the density dependence of β 6 3 β -3 and fixing α ¼ β þ const results in a line with slope indicated by the blue dashed line in Fig. 3(a), which is -4 clearly inconsistent with the data. Thus, the density- 0 dependent β enabling gate tunability of the Dresselhaus 4 2 0 -2 -0.1 0.1 δE [V/μm] B [mT] term is crucial here. Z Z Dresselhaus regime.—We now show that α can be tuned FIG. 4. The Dresselhaus and the cubic regime. (a) Locked through β and through zero in a more symmetrically doped regime α ≈ β (black or gray symbols) and Dresselhaus regime wafer, opening the Dresselhaus regime β ≫ α. We intro- α ≈ 0 (red symbols) from the broadest WAL minima (maximal duce the magnetic field B , where the magnetoconduc- SO B ) in the V and V plane for a more symmetrically doped SO T B tance exhibits minima at B ≈−B . These minima Z1 Z2 11-nm well. The solid black line displays the α ¼ β simulation, describe the crossover between WAL and WL, where the while the dashed red line marks the simulated α ¼ 0 contour. Aharonov-Bohm dephasing length and the SO diffusion Open black markers (leftmost V ) are entering the nonlinear gate length are comparable. Beyond the WAL-WL-WAL tran- regime, causing a slight deviation from theory, which assumes sition [Fig. 4(b), upper panel], B is seen to peak and linear gate action. The rightmost V points (gray) are obtained SO B from the minimal B in the presence of WAL. (b) Sequence at decrease again (dashed curve). The gate voltages with SO 11 −2 11 −2 n ¼ 6 × 10 cm (upper panel) and n ¼ 9 × 10 cm (lower maximal B are added to Fig. 4(a) for several densities SO panel), shifted vertically for clarity. Each brown or blue marker in (red markers). We surmise that these points mark α ≈ 0: (a) corresponds to a trace in (b), as labeled by numerals or letters. B signifies the crossover from WL to WAL-like con- SO B is indicated as a guide for the eye by black dashed curves for SO ductance, thus defining an empirical measure for the effects negative B . B increases and peaks (indicating α ¼ 0) before Z SO of SO coupling (larger B , stronger effects). For α ¼ 0, the SO decreasing again (upper panel). Broken spin symmetry regime full effect of β on the conductance becomes apparent (lower panel): WAL is no longer suppressed here due to without cancellation from α, giving a maximal B . SO symmetry breaking from the cubic term at large n. Still, α ≈ β Indeed, the simulated α ¼ 0 curve [dashed red line in can be identified with the narrowest WAL peak. (c) Simulation of 11 −2 Fig. 4(a)] cuts through the experimental points, also α and β along n ¼ 6 × 10 cm . α traverses both β (black reflected in Fig. 4(c) by a good match with the simulated arrow) and for smaller δE also zero (red arrow). α ¼ 0 crossing point (red arrow). Diverging spin-orbit lengths.—For a comparison of vice versa for α ¼ −β. An in-plane rotation of the PSH by a experiment and simulation, we convert the empirical B fixed angle π=2 from α ¼þβ to α ¼ −β was recently SO pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi demonstrated [34]. to a “magnetic length” λ ¼ ℏ=2eB , which we later SO SO Figure 5 shows the theoretical spin diffusion length λ on interpret as a spin-diffusion length, where e> 0 is the eff (see Appendix) and the ballistic λ ,togetherwith the electron charge and the factor of 2 accounts for time- experimental λ , all agreeing remarkably well. Since at reversed pairs of closed trajectories. We also introduce the SO α ¼ β spin transport is ballistic despite charge diffusion, ballistic SO lengths λ ¼ ℏ =ð2m jα  βjÞ. These lengths λ and its diffusive counterpart λ (small β ) are essentially correspond to a spin rotation of 1 rad, as the electrons travel − eff 3 equivalent, as shown in SM [18]. The enhanced λ around along x ˆ and x ˆ , respectively, with spins initially aligned SO þ − α=β ¼ 1 corresponds to an increased spin relaxation perpendicular to the corresponding SO field [e.g., for an time τ ¼ λ =ð2DÞ.Notethat maxðλ ; λ Þ quantifies the electron moving along the x ˆ , its spin should point along SO þ − þ SO x ˆ or z ˆ so spin precession can occur; see SM, Eq. (S20), for deviation from the uniaxial SO field away from α ¼ β,and an expression of the SO field [18]]. For α ¼þβ, thus the extent to which spin rotations are not undone in a closed trajectory due to the non-Abelian nature of spin λ diverges (no precession, indicating that an electron ˆ rotations around noncollinear axes. This leads to WAL, a traveling along x does not precess) while λ is finite, and − þ 031010-5 α = β V [V] α, β [meV Å] -3 -3 x10 Δσ/σ [x10 ] 0 FLORIAN DETTWILER et al. PHYS. REV. X 7, 031010 (2017) semiconductor nanostructures such as quantum wires, quantum dots, and electron spin qubits. Moreover, our work relaxes the stringency (i.e., the “fine-tuning”) of the α ¼ β symmetry condition at a particular singular point (gate) by introducing a “continuous locking” of the SO couplings αðV ;V Þ¼ βðV ;V Þ over a wide range of T B T B voltages, which should enable new experiments exciting persistent spin helices with variable pitches in GaAs wells [12,13], i.e., stretchable PSHs. Further, this concept is also applicable to a range of other III-V semiconductors in various suitable configurations. Another possibility is the generation of a Skyrmion lattice (crossed spin helices) with variable lattice constants, as recently proposed in Ref. [35]. Finally, we stress that within the continuously locked regime of SO couplings we demonstrate in our study, FIG. 5. Experimental and theoretical SO lengths and SO times. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SO-coupled quantum transport in our samples shows a very Experimental λ ¼ ℏ=2eB (markers, densities as labeled, SO SO distinctive feature: it is diffusive (2D) for charge while 11 −2 in units of 10 cm ) as a function of the dimensionless ratio ballistic (1D) for spins, thus providing a unique setting for α=β (from SO simulation). The ballistic λ (blue and red dashes) coherent spin control. This ultimately adds a new function- and effective λ (black dashed curve) are only weakly n eff ality to the nonballistic spin transistor of Ref. [10]; i.e., it can dependent (small β ) when plotted against α=β. Thus, curves 11 −2 now be made to operate as the ideal (ballistic) Datta-Das spin for only one density (n ¼ 6 × 10 cm ) are shown. The transistor—but in a realistic 2D diffusive system, with yet experimental uncertainty on λ is captured by the spread given SO by the three slightly different densities. The coherence length controlled spin rotations protected from spin decay. L ≈ 7 μm is added for illustration (obtained from WL curves), setting the visibility of SO effects on the conductance and thus the ACKNOWLEDGMENTS width of the WAL-WL-WAL transition. Inset: Experimental spin We would like to thank A. C. Gossard, D. Loss, D. L. relaxation time τ ¼ λ =ð2DÞ (circles) as a function of α=β for SO SO Maslov, and G. Salis for valuable inputs and stimulating two densities as indicated. Theory curves τ (dashed) now eff discussions. This work was supported by the Swiss include the symmetry-breaking third-harmonic term, preventing divergence at α=β ¼ 1, while λ (main panel) does not. Nanoscience Institute (SNI), NCCR QSIT, Swiss NSF, eff ERC starting grant, EU-FP7 SOLID, and MICROKELVIN, finite B ,and λ ≃ maxðλ ; λ Þ,asobserved(seeFig. 5). SO SO þ − U.S. NSF DMR-1306300 and NSF MRSEC DMR- Unlike the corresponding time scales, the SO lengths are only 1420709 and ONR N00014-15-1-2369, Brazilian grants weakly dependent on density and mobility when plotted FAPESP (SPRINT program), CNPq, PRP/USP (Q- against α=β, allowing a comparison of various densities. NANO), and natural science foundation of China (Grant The third-harmonic contribution of cubic-in-k term No. 11004120). causes spin relaxation even at α ¼ β and becomes visible F. D., J. F., P. J. W., J. C. E., and D. M. Z. designed the at large densities: WAL is present in all traces and through experiments, analyzed the data. and co-wrote the paper. All α ¼ β [Fig. 4(b), lower panel], because the SO field can no authors discussed the results and commented on the longer be made uniaxial, thus breaking spin symmetry and manuscript. S. M. and D. D. A. designed, simulated, and reviving WAL. A partial symmetry restoration is still carried out the molecular beam epitaxy growth of the apparent, where—in contrast to the α ¼ 0 case—a minimal heterostructures. F. D. processed the samples and with B is reached (dashed curves) consistent with α ¼ β [gray SO P. J. W. performed the experiments. J. F. and J. C. E. devel- markers Fig. 4(a) at large n]. We include the cubic β in the oped and carried out the simulations and theoretical work. spin-relaxation time τ (see Appendix), shown in the inset eff of Fig. 5 for two densities, finding good agreement with the APPENDIX: MATERIALS AND METHODS experimental τ ¼ λ =ð2DÞ, where D is the diffusion SO SO 1. GaAs quantum well materials constant. Over the whole locked regime of Fig. 3(b),WAL is absent, and τ is enhanced between 1 and 2 orders of The wells are grown on an n-doped substrate (for details SO magnitude compared to α ¼ 0. Finally, the coherence see SM [18]) and fabricated into Hall bar structures [see length L sets an upper limit for the visibility of SO inset of Fig. 2(a)] using standard photolithographic meth- effects: WAL is suppressed for λ ≫ L , setting the width ods. The 2D gas is contacted by thermally annealed eff φ of the WAL-WL-WAL transition (see SM [18]). GeAu=Pt Ohmic contacts, optimized for a low contact Final remarks and outlook.—This work is laying the resistance while maintaining high back gate tunability (low leakage currents) and avoiding short circuits to the back foundation for a new generation of experiments benefiting from unprecedented command over SO coupling in gate. On one segment of the Hall bar, a Ti=Au top gate with 031010-6 STRETCHABLE PERSISTENT SPIN HELICES IN GAAS … PHYS. REV. X 7, 031010 (2017) dimensions of 300 × 100 μm is deposited. The average time-reversed trajectories: Δφ ¼ 2eAB=ℏ, where A is the gate-induced E-field change in the well is defined as loop area. Here, we take A ¼ λ ¼ 2Dτ as a characteristic SO SO δE ¼ 1=2ðV =d − V =d Þ, with effective distance “diffusion area” probed by our WL or WAL experiment, with Z T T B B d from the well to the top gate or back gate, respectively, τ being the spin-dephasing time, and λ the spin-diffusion T=B SO SO extracted using a capacitor model, consistent with the full length. By taking Δφ ¼ 1 (rad) at B ¼ B , we can extract SO quantum description (see SM [18]). Contours of constant the spin-diffusion length λ and spin-dephasing time τ SO SO pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi density follow δV =d ¼ −δV =d . Deviations from lin- T T B B from the minima of the WAL curves from λ ¼ ℏ=2eB SO SO ear behavior appear at most positive or negative gate −1 and τ ¼ ℏð4eDB Þ , respectively. The factor of 4 here SO SO voltages due to incipient gate leakage and hysteresis. stems from the two time-reversed paths and the diffusion length. 2. Low-temperature electronic measurements The experiments are performed in a dilution refrigerator 5. Effective SO times and lengths with base temperature 20 mK. We use a standard four-wire Theoretically, we determine τ via a spin random walk SO lock-in technique at 133 Hz and 100 nA current bias, process [D’yakonov-Perel (DP)]. The initial electron spin chosen to avoid self-heating while maximizing the signal. in a loop can point (with equal probability) along the s , The density is determined with Hall measurements in the s , and s axes (analogous to x , x , and z, respectively), x z þ − classical regime, whereas Shubnikov–de Haas oscillations þ which have unequal spin-dephasing times τ , τ , are used to exclude occupation of the second subband, DP;s DP;s x x which is the case for all the data we discuss. The WAL and τ . For unpolarized, independent spins, we take DP;s −3 signature is a small correction (10 ) to total conductance. the average τ ¼ðτ þ τ þ τ Þ=3, which leads eff DP;s DP;s DP;s x x z pffiffiffiffiffiffiffiffiffiffiffiffiffi To achieve a satisfactory signal-to-noise ratio, longitudinal to an effective spin-difusion length λ ¼ 2Dτ . eff eff conductivity traces Δσ=σ ¼½σðBÞ − σð0Þ=σð0Þ are mea- Actually, λ is defined from the average variance eff sured at least 10 times and averaged. 2 2 λ ¼ σ¯ ¼ 2Dτ , obtained by averaging the spin- eff eff 2 2 dependent variances σ ¼2Dτ , σ ¼2Dτ , and s DP;s s DP;s x x x x − − þ þ 3. Numerical simulations σ ¼ 2Dτ over the spin directions s , s , and s (this s DP;s x x z z z þ − The simulations calculate the Rashba coefficient α and is equivalent to averaging over the τ’s and not over 1=τ’s). hk i based on the bulk semiconductor band parameters, the In the SM [18], we discuss the spin random walk and well structure, the measured electron densities, and the provide expressions for the DP times including corrections measured gate lever arms. We solve the Schrödinger and due to the cubic β term. Figure 5 shows curves for the Poisson equations self-consistently (“Hartree approxima- spin-dephasing times and lengths presented here. In the tion”), obtain the self-consistent eigenfunctions, and then main panel, the cubic β is neglected in λ since for 3 eff determine α via appropriate expectation values [31]. The 11 −2 n ≤ 7 × 10 cm , WL appears at α ¼ β (small β ). In Dresselhaus coefficient γ is extracted from fits of the contrast, the cubic term is included in τ in the inset since eff simulation to the experiment, which detects the absence 11 −2 at the higher density, n ¼ 9 × 10 cm , WAL persists 2 2 2 of WAL at α ¼ β ¼ γðhk i − k =4Þ. Thus, given α and hk i z F z (sufficiently strong β ). 2 3 from the simulation and the measured n ¼ k =ð2πÞ,we obtain γ ¼ 11.6  1 eV Å consistently for all asymmetri- cally doped wells. Taking into account the uncertainties of the band parameters, the experimental errors, and a [1] J. Sinova, S. O. Valenzuela, J. Wunderlich, C. Back, and negligible uncertainty on hk i, an overall uncertainty of 3 T. Jungwirth, Spin Hall Effects, Rev. Mod. Phys. 87, 1213 about 9%–10% or about 1 eV Å on γ results. About (2015). 1%–2% error originates from the experimental uncertainty [2] V. Mourik, K. Zuo, S. M. Frolov, S. R. Plissard, E. P. A. M. of determining α ¼ β. The doping distribution (above or Bakkers, and L. P. 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[18] See Supplemental Material at http://link.aps.org/ 98, 226802 (2007). supplemental/10.1103/PhysRevX.7.031010 for further de- [33] M. P. Walser, U. Siegenthaler, V. Lechner, D. Schuh, S. D. tails on the wafer structure, temperature dependence, Ganichev, W. Wegscheider, and G. Salis, Dependence of the numerical simulations, other SO terms, effective SO fields, Dresselhaus Spin-Orbit Interaction on the Quantum Well diffusive SO times and lengths, and Shubnikov-de Haas Width, Phys. Rev. B 86, 195309 (2012). oscillations. [34] K. Yoshizumi, A. Sasaki, M. Kohda, and J. Nitta, [19] S. J. Papadakis, E. P. De Poortere, H. C. Manoharan, M. Gate-Controlled Switching between Persistent and Inverse Shayegan, and R. Winkler, The Effect of Spin Splitting on Persistent Spin Helix States, Appl. Phys. Lett. 108, 132402 the Metallic Behavior of a Two-Dimensional System, (2016). Science 283, 2056 (1999). [35] J. Fu, P. H. Penteado, M. O. Hachiya, D. Loss, and J. C. [20] D. Grundler, Large Rashba Splitting in InAs Quantum Wells Egues, Persistent Skyrmion Lattice of Noninteracting Elec- due to Electron Wave Function Penetration into the Barrier trons with Spin-Orbit Coupling, Phys. Rev. Lett. 117, Layers, Phys. Rev. Lett. 84, 6074 (2000). 226401 (2016). 031010-8 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Physical Review X American Physical Society (APS)
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PHYSICAL REVIEW X 7, 031010 (2017) 1 2,† 3,‡ 1 2 Florian Dettwiler, Jiyong Fu, Shawn Mack, Pirmin J. Weigele, J. Carlos Egues, 3,4 1,* David D. Awschalom, and Dominik M. Zumbühl Department of Physics, University of Basel, CH-4056 Basel, Switzerland Instituto de Física de São Carlos, Universidade de São Paulo, 13560-970 São Carlos, São Paulo, Brazil California NanoSystems Institute, University of California, Santa Barbara, California 93106, USA Institute for Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA (Received 16 February 2017; revised manuscript received 21 May 2017; published 18 July 2017) The Rashba and Dresselhaus spin-orbit (SO) interactions in 2D electron gases act as effective magnetic fields with momentum-dependent directions, which cause spin decay as the spins undergo arbitrary precessions about these randomly oriented SO fields due to momentum scattering. Theoretically and experimentally, it has been established that by fine-tuning the Rashba α and renormalized Dresselhaus β couplings to equal fixed strengths α ¼ β, the total SO field becomes unidirectional, thus rendering the electron spins immune to decay due to momentum scattering. A robust persistent spin helix (PSH), i.e., a helical spin-density wave excitation with constant pitch P ¼ 2π=Q, Q ¼ 4mα=ℏ , has already been experimentally realized at this singular point α ¼ β, enhancing the spin lifetime by up to 2 orders of magnitude. Here, we employ the suppression of weak antilocalization as a sensitive detector for matched SO fields together with independent electrical control over the SO couplings via top gate voltage V and back gate voltage V to extract all SO couplings when combined with detailed numerical simulations. We demonstrate for the first time the gate control of the renormalized β and the continuous locking of the SO fields at α ¼ β; i.e., we are able to vary both α and β controllably and continuously with V and V , while T B keeping them locked at equal strengths. This makes possible a new concept: “stretchable PSHs,” i.e., helical spin patterns with continuously variable pitches P over a wide parameter range. Stretching the PSH, i.e., gate controlling P while staying locked in the PSH regime, provides protection from spin decay at the symmetry point α ¼ β, thus offering an important advantage over other methods. This protection is limited mainly by the cubic Dresselhaus term, which breaks the unidirectionality of the total SO field and causes spin decay at higher electron densities. We quantify the cubic term, and find it to be sufficiently weak so that the extracted spin-diffusion lengths and decay times show a significant enhancement near α ¼ β. Since within the continuous-locking regime quantum transport is diffusive (2D) for charge while ballistic (1D) for spin and thus amenable to coherent spin control, stretchable PSHs could provide the platform for the much heralded long-distance communication ∼8–25 μm between solid-state spin qubits, where the spin diffusion length for α ≠ β is an order of magnitude smaller. DOI: 10.1103/PhysRevX.7.031010 Subject Areas: Condensed Matter Physics, Quantum Information, Spintronics The inextricable coupling between the electron spatial and spin degrees of freedom—the spin-orbit (SO) interaction— underlies many fundamental phenomena such as the spin Hall effects—quantum and anomalous [1]—and plays a crucial role in newly discovered quantum materials hosting dominik.zumbuhl@unibas.ch Permanent address: Department of Physics, Qufu Normal University, Qufu, Shandong, 273165, China. Present address: Naval Research Laboratory, Washington, D.C. 20375, USA. FIG. 1. Stretchable PSHs. Illustration of spin helices at different values of α¼β accessible in the measurements. The position x for Published by the American Physical Society under the terms of one 2π rotation (dashed curve) is changing for the gate-locked the Creative Commons Attribution 4.0 International license. regime α¼β. The gray box highlights how the spin rotation can be Further distribution of this work must maintain attribution to controlled (in situ) at fixed position∼4.8 μmby∼π=2 over the same the author(s) and the published article’s title, journal citation, ˆ ˆ and DOI. range of α¼β.Thex ∥½110 andx ∥½110 axes define the 2Dplane. þ − 2160-3308=17=7(3)=031010(8) 031010-1 Published by the American Physical Society FLORIAN DETTWILER et al. PHYS. REV. X 7, 031010 (2017) Majorana [2] and Weyl fermions [3]. In nanostructures, the injection of spin polarization; see, e.g., Refs. [12,13]. Other SO coupling strength can be varied via gate electrodes [4,5]. spin communication modes can be envisaged with this As recently demonstrated [6], this enables controlled spin setup. The distance is limited mainly by the deviation modulation [7] of charge currents in nonmagnetic (quasibal- from α ¼ β and by the cubic Dresselhaus term, which is listic) spin transistors. small in this range, as we quantify later on, and leads to spin The SO coupling in a GaAs quantum well has two decay with spin-diffusion lengths λ ∼ 8–25 μm over eff dominant contributions: the Rashba [8] and the Dresselhaus which the spin dephases by 1 rad. Note that this type of [9] effects, arising from the breaking of the structural spin manipulation and spin transfer is not possible for a and crystal inversion symmetries, respectively. When the helix with α ≠ β, since λ quickly drops below the helix eff Rashba α and Dresselhaus β SO couplings match at pitch as the SO couplings are deviating from the sym- α ¼ β [10,11], the direction of the combined Rashba- metry point. Dresselhaus field becomes momentum independent, thus The full electrical control of the SO couplings demon- suppressing D’yakonov-Perel spin-flip processes, provided strated in our 9.3-nm-wide quantum well can tune from that the cubic Dresselhaus term be small. The significantly α ¼ β ¼ 5 meV Å to 4 meV Å, thus enabling stretchable enhanced spin lifetime at α ¼ β enables nonballistic spin PSHs with pitches P stretching from 3.5 to 4.4 μm; see transistors and persistent spin helices (PSHs) [10,11]. Fig. 1. Within the shortest spin-diffusion length λ ∼ 8 μm eff However, despite substantial efforts, so far this symmetry for our 9.3-nm well, controlled spin rotations by an angle point has been achieved only at isolated points with finely θ ¼ Qx ¼ 2πx =P can be performed under spin protec- þ þ tuned system parameters [12–14], which is too difficult to tion on any spin sitting at a position x along the stretchable be reliably attained on demand as required for a useful PSH by varying P in the range above. For example, a spin technology. at x ∼ 4.8 μm can be rotated by Δθ ∼ π=2 as P varies in the Stretchable persistent spin helices.—Here, we overcome range above; see gray box shading in Fig. 1. Thus, this outstanding obstacle by (i) using a technique that stretchable helices could provide a platform for long- allows independent control of the SO couplings via a top distance spin communication. gate voltage V and a back gate voltage V while Additional results.—WAL is also used to identify other T B (ii) simultaneously measuring the suppression of weak regimes such as the Dresselhaus regime α ¼ 0 in a more antilocalization (WAL) in an external magnetic field as a symmetrically doped sample. Combined with numerical sensitive probe for matched SO couplings. While gate simulations, we extract the SO couplings α and β, the tuning of the renormalized Dresselhaus coefficient β was bulk Dresselhaus parameter γ, the spin-diffusion lengths, already theoretically described in 1994 [15], we demon- and spin-relaxation times over a wide range of system strate this for the first time here in an experiment, and parameters. We also quantify the detrimental effects employ this tunability to show robust continuous locking of of the third harmonic of the cubic Dresselhaus term, which the Rashba and Dresselhaus couplings at αðV ;V Þ¼ T B mainly limits spin protection. Interestingly, our spin- βðV ;V Þ over a wide range of densities n, i.e., a T B diffusion lengths and spin-relaxation times are significantly “symmetry line” (not a point) in the (V , V ) plane. T B enhanced within the locked α ¼ β range, thus attesting that This allows us to introduce the concept of the “stretchable our proposed setup offers a promising route for spin persistent spin helix,” see Fig. 1, with spin density protection and manipulation. s ∼ sinðQx Þ, s ¼ 0, and s ∼ cosðQx Þ and variable x þ x z þ In what follows, we first explain tuning of the Rashba þ − pitch P ¼ 2π=Q, Q ¼ 4mα=ℏ . The stretchable PSH coupling, then the essential density dependence of the Dresselhaus coupling β that enables the continuous locking makes possible gate control of the spin precession over long distances due to strong protection from spin of the SO fields, how it also leads to spin decay at higher decay by up to 2 orders of magnitude enhanced densities, followed by the relevant weak-localization– spin lifetimes at the symmetry point α ¼ β—without weak-antilocalization (WL-WAL) detection scheme, mea- requiring in-plane electric fields to induce drift [16], surements, and simulations. A full account of our approach, and without relying on micron-width channels to suppress including additional data and details of the model and decay [17]. simulations, is presented in the Appendix and the SM [18]. Long-distance spin communication.—Within the range Controlling the Rashba coupling α.—The Rashba coef- ficient [8] α can be tuned with the wafer and doping profile of the continuously matched-locked SO couplings α ¼ β, quantum transport in the well is diffusive for charge (2D) [12] as well as in situ using gate voltages [4,5] at constant while essentially ballistic (1D) for spins [see Supplemental density n and thus independent of the Dresselhaus term; see Material (SM), Sec. V [18]]. A stretchable PSH could thus below. A change of top gate voltage V can be compen- be used to coherently couple, e.g., spin qubits over sated by an appropriate, opposing change of back gate unprecedented long distances. Figure 1 illustrates how voltage V [see Fig. 2(a)] to keep n fixed [19,20] while spin information can be conveyed between spins via a changing the gate-induced electric field δE in the quantum stretchable PSH. These stretchy waves can be excited upon well, where z⊥2D plane. Another Rashba term due to 031010-2 6.5 5.5 α ≈ β 4.5 3.5 STRETCHABLE PERSISTENT SPIN HELICES IN GAAS … PHYS. REV. X 7, 031010 (2017) 9.3-nm QW, asymmetrically doped the Rashba α term. An additional term with the same (a) (b) 0.4 0 form—the interface Dresselhaus term [24]—could also 1 n =4.5 1 play a role; see SM [18]. 0.2 3 To a very good approximation, the coefficient β ≃ pffiffiffiffiffiffiffiffi -2 γk =4, where the Fermi vector k ≃ 2πn and n is the 5 3 F F 0.0 carrier density of the 2D gas. This neglects the tiny angular 500 μm anisotropy in the Fermi wave vector due to the competition -0.2 -4 between the Rashba and Dresselhaus effects (especially in -2.0 -1.0 0.0 1.0 V [V] GaAs wells). Note that by approximating β ≃ γπn=2, both B 3 the first-harmonic and the third-harmonic parts of the (c) α α n =4.5 w cubic-in-k Dresselhaus term actually become linear in k -6 β α α (see SM for details [18]) and, more importantly, become g +d 6 4 density dependent. We can now group the linear-in-k 3 5 Dresselhaus term β together with the first-harmonic -8 7 contribution β into a single renormalized Dresselhaus term by defining β ¼ β − β . It is this density-dependent 1 3 3 2 1 0 -1 -0.2 0.0 0.2 renormalized coefficient β that can be tuned with a gate δE [V/μm] B [mT] voltage to match the Rashba α coupling continuously. This matching leads to a k-independent spinor (or, equivalently, FIG. 2. Weak localization (WL) as an α ¼ β detector, gate to a k-independent effective SO field), whose direction is control of Rashba α at constant density. (a) Measured charge immune to momentum scattering. In this way we achieve density n (color) versus top gate voltage V and back gate voltage independent, continuous control of the Rashba and V (9.3-nm well). Contours of constant density ð3.5–7.5Þ × 11 −2 Dresselhaus terms by using top gate and back gate voltages. 10 cm are shown. Inset: Optical micrograph of typical Hall This is an unprecedented tunability of the SO terms within bar, with contacts (yellow), gate (center), and mesa (black lines). a single sample. (b) Normalized longitudinal conductivity Δσ=σ ¼½σðB Þ − 0 Z σð0Þ=σð0Þ versus B ⊥2D plane. Curves for gate configurations Spin decay at higher densities.—The strength of the 11 −2 1–7 along constant n ¼ 4.5 × 10 cm are shown (offset third-harmonic contribution of the Dresselhaus term is also vertically), also labeled in (a) and (c). (c) Simulated Rashba α described by the coefficient β . This term, however, is and Dresselhaus β coefficients (see text) against gate-induced detrimental to spin protection as it breaks the angular 11 −2 field change δE , shown for constant n ¼ 4.5 × 10 cm . The symmetry of the other linear SO terms and makes the spinor δE axis—decreasing from left to right—corresponds exactly to k dependent and susceptible to in-plane momentum scat- the V abscissa of (a) for a covarying V , such that n ¼ 4.5 × B T tering, even for matched couplings α ¼ β. As we show, the 11 −2 10 cm constant. Sketches of the well potential at 1, 4, and 6 detrimental effect of the third-harmonic contribution does illustrate the change of α with δE . Note that αðδE ¼ 0Þ ≠ 0 Z Z not prevent our attaining the continuous locking over a since the external E field (see SM [18]) is not zero at δE ¼ 0. relevant wide range of electron densities. Gate-tunable range of the Dresselhaus coupling β.—For donor electric fields [21,22] is negligible in our structures; the narrow quantum wells we use here, β is essentially gate see SM [18]. independent since the wave function spreads over the full 2 2 Linear and cubic Dresselhaus terms in 2D.—Because of width of the well. This also implies hk i ≪ ðπ=WÞ (the the well confinement along the z direction (growth), the infinite well limit), see Fig. 3(d), due to wave function cubic-in-momentum bulk (3D) Dresselhaus SO interaction penetration into the finite barriers. Thus, a change of gives rise to, after the projection into the lowest quantum density by a factor of ∼2.5 changes β =β ¼ πn=h2k i 3 1 z well subband eigenstates, distinct terms that are linear and by the same factor, resulting in a gate-tunable range of cubic in k, the 2D electron wave vector. The linear-in-k 0.08 ≲ β =β ≲ 0.2. In addition, quantum wells of width 3 1 term has a coefficient β ¼ γhk i and turns out practically W ¼ 8, 9.3, 11, and 13 nm were used [12,25], resulting in a 1 z independent of density in the parameter range of interest change of β by roughly a factor of 2. here. The cubic-in-k term, on the other hand, is density Detection scheme for matched SO couplings.—WAL is a dependent and has yet two components with distinct well-established signature of SO coupling in magneto- angular symmetries: (i) the first-harmonic contribution conductance σðB Þ [15,23,26–29] exhibiting a local maxi- proportional to sin ϕ and cos ϕ and (ii) the third-harmonic mum at zero field. In the α ¼β regime, the resulting contribution proportional to sin 3ϕ and cos 3ϕ; here, ϕ is internal SO field is uniaxial, and spin rotations commute the polar angle in 2D between k and the [100] direction and are undone along time-reversal loops. Therefore, WAL (see SM [18]). Interestingly, the first-harmonic contribution is suppressed and the effectively spinless situation display- with coefficient β has the same angular symmetry as both ing weak localization [i.e., σðB Þ exhibiting a local mini- 3 Z the linear-in-k Dresselhaus β term (see Refs. [15,23]) and mum at B ¼ 0] is restored [10,11,14,23]. Away from the 031010-3 V [V] α, β [meV Å] -3 Δσ/σ [x10 ] 0 2.5 6.5 5.5 4.5 3.5 FLORIAN DETTWILER et al. PHYS. REV. X 7, 031010 (2017) (a) 3 (b) 9.3-nm QW small corrections [24,30] even up to room temperature. data fit QW [nm] γ [eV Å ] α α 0.6 8 11.0 ± 1 6 e Also, note that Shubnikov–de Haas oscillations do not 9.3 11.6 ± 1 α α g+d w 11 11.8 ± 1 show any spin-orbit splitting here (see SM [18]) given the 13 10.9 ± 1.6 4 0.4 strength of SO coupling in GaAs, making it clear that locked 2 the quantum corrections in WAL and their suppression at the symmetry point present a very sensitive detector for SO 0.2 coupling. β (= α) Continuous locking α ¼ β.—We proceed to demonstrate β β 1 3 0.0 gate locking of the SO couplings α, β. Figure 2(b) displays σðB Þ of the 9.3-nm well for top gate and back gate -0.2 configurations labeled 1–7, all lying on a contour of constant density; see Fig. 2(a). Along this contour, β is -0.4 0 held fixed since the density is constant (β is essentially 3 4 5 6 -2 -1 0 1 gate independent), while α is changing as the gate voltages 11 -2 V [V] B n [x10 cm ] are modifying the electric field δE perpendicular to the (d) (c) 0.15 quantum well. Across these gate configurations, the con- inf QW calc ductance shows a transition from WAL (configuration 1 0.10 and 2) to WL (4 and 5) back to WAL (7). Selecting the most 0.05 pronounced WL curve allows us to determine the symmetry point α ¼ β. This scheme is repeated for a number of 0.00 8 10 12 14 8 10 12 densities, varying n by a factor of 2, yielding the symmetry W [nm] W [nm] point α ¼ β for each density n [see Fig. 3(a), blue markers], thus defining a symmetry line in the ðV ;V Þ plane. Along T B FIG. 3. Tuning and continuously locking α ¼ β. (a) The this line, β is changing with density, as previously described, markers indicate α ≈ β for four different well widths (asymmetric and α follows β, remaining “continuously” locked at α ¼ β. doping) and various densities (gray contours of constant n, 11 −2 As mentioned earlier, this is a very interesting finding, labeled in units of 10 cm ) in the V and V plane. Error T B as it should allow the creation of persistent spin helices with bars result from the finite number of conductance traces in the ðV ;V Þ space. Theory fits (solid lines) are shown for each well, gate-controllable pitches, as illustrated in Fig. 1. B T with γ as the only fit parameter (inset table, error bars dominated Simulations and fitting of γ.—Self-consistent calcula- by systematic error; see below). The dashed blue line indicates tions combined with the transport data can deliver all SO the slope of constant α ¼ β , neglecting β , which is inconsistent 1 3 parameters. The numerical simulations [31] (see Appendix with the data. (b) Simulation of locked α ¼ β versus density n and SM [18]) can accurately calculate α and hk i. This along solid blue line from (a), showing the various SO contri- leaves only one fit parameter, γ, the bulk Dresselhaus butions (see text). (c) Values of γ from fits for each well width W. coefficient, which can now be extracted from fits to the Red dashed line is the average γ ¼ 11.6  1 eV Å (excluding density dependence of the symmetry point, see solid blue W ¼ 13 nm due to its larger error), gray area is the ∼9% error, line in Fig. 3(a), giving excellent agreement with the data stemming mostly from the systematic uncertainty in the input (blue markers). This procedure can be repeated for a set of parameters of the simulations (see Appendix). (d) hk i as a wafers with varying quantum well width and thus varying function of well width W for realistic (markers) and infinite (blue) potential. β . This shifts the symmetry point α ¼ β, producing nearly parallel lines, as indicated with colors in Fig. 3(a) corre- sponding to the various wafers as labeled. As shown, matched regime, the SO field is not uniaxial, spin rotations locking α ¼ β over a broad range is achieved in all wafers. do not commute, and trajectories in time-reversal loops Since gate voltages can be tuned continuously, any and all interfere destructively upon averaging [26] due to the SO points on the symmetry lines α ¼ β can be reached. Again phases picked up along the loops, thus leading to WAL. performing fits over the density dependence of the sym- Hence, this suppression is a sensitive detector for α ¼β. metry point for each well width, we obtain very good At high β , this detection scheme becomes approximate, giving α ¼β [14]. We note that the WL dip—often used agreement, see Fig. 3(a), and extract γ ¼ 11.6  1 eV Å to determine phase coherence—sensitively depends on the consistently for all wells [Fig. 3(c)]. We emphasize that γ is SO coupling [e.g., curves 3–6 in Fig. 2(b)], even before notoriously difficult to calculate and measure [28,29,32]; WAL appears. Negligence of SO coupling could thus lead the value we report here agrees well with recent studies to spurious or saturating coherence times. At higher [13,32,33]. Obtaining consistent values over wide ranges of temperatures, when quantum coherence is lost, this detec- densities and several wafers with varying well widths tion scheme becomes inoperable, while it is expected that provides a robust method to extract γ. the mechanism for tuning both Rashba and effective Beyond γ, the simulations reveal important information Dresselhaus coefficients continues to function with only about the gate tuning of the SO parameters. The Rashba 031010-4 α ≈ β α ≈ β α ≈ β α ≈ β V [V] γ [eV Å ] 2 -2 <k > [nm ] β [meV Å] α [meV Å] 9 7.5 4.5 STRETCHABLE PERSISTENT SPIN HELICES IN GAAS … PHYS. REV. X 7, 031010 (2017) 11-nm QW, symmetrically doped coefficient is modeled as α ¼ α þ α þ α in the gþd w e (a) (b) 2 1 simulation, with gate and doping term α p n = 6 , quantum well gþd α≈ β 2 0.4 structure term α , and Hartree term α . Along a contour of 0 w e constant density, the simulations show that mainly α and r gþd 0.2 -2 α are modified, while α and β remain constant; see w e t 2 4 -4 Fig. 2(c). The density dependence for locked α ¼ β, on the 0.0 α≈ 0 other hand, shows that while β is nearly constant, β is 1 3 6 -6 -0.2 linearly increasing with n, thus reducing β ¼ β − β ; see 1 3 -8 Fig. 3(b). Hence, to keep α ¼ β locked, α has to be reduced B B Z 2 Z 1 α = 0 8 -0.4 correspondingly. The Hartree term α , however, increases n = 9 -1 -3 -2 -1 0 1 for growing n. Thus, on the α ¼ β line, the other α terms— V [V] q mainly the gate-dependent α —are strongly reduced, gþd -2 (c) maintaining locked α ¼ β, as shown in Fig. 3(b).We α α≈ β n = 6 emphasize that neglecting the density dependence of β 6 3 β -3 and fixing α ¼ β þ const results in a line with slope indicated by the blue dashed line in Fig. 3(a), which is -4 clearly inconsistent with the data. Thus, the density- 0 dependent β enabling gate tunability of the Dresselhaus 4 2 0 -2 -0.1 0.1 δE [V/μm] B [mT] term is crucial here. Z Z Dresselhaus regime.—We now show that α can be tuned FIG. 4. The Dresselhaus and the cubic regime. (a) Locked through β and through zero in a more symmetrically doped regime α ≈ β (black or gray symbols) and Dresselhaus regime wafer, opening the Dresselhaus regime β ≫ α. We intro- α ≈ 0 (red symbols) from the broadest WAL minima (maximal duce the magnetic field B , where the magnetoconduc- SO B ) in the V and V plane for a more symmetrically doped SO T B tance exhibits minima at B ≈−B . These minima Z1 Z2 11-nm well. The solid black line displays the α ¼ β simulation, describe the crossover between WAL and WL, where the while the dashed red line marks the simulated α ¼ 0 contour. Aharonov-Bohm dephasing length and the SO diffusion Open black markers (leftmost V ) are entering the nonlinear gate length are comparable. Beyond the WAL-WL-WAL tran- regime, causing a slight deviation from theory, which assumes sition [Fig. 4(b), upper panel], B is seen to peak and linear gate action. The rightmost V points (gray) are obtained SO B from the minimal B in the presence of WAL. (b) Sequence at decrease again (dashed curve). The gate voltages with SO 11 −2 11 −2 n ¼ 6 × 10 cm (upper panel) and n ¼ 9 × 10 cm (lower maximal B are added to Fig. 4(a) for several densities SO panel), shifted vertically for clarity. Each brown or blue marker in (red markers). We surmise that these points mark α ≈ 0: (a) corresponds to a trace in (b), as labeled by numerals or letters. B signifies the crossover from WL to WAL-like con- SO B is indicated as a guide for the eye by black dashed curves for SO ductance, thus defining an empirical measure for the effects negative B . B increases and peaks (indicating α ¼ 0) before Z SO of SO coupling (larger B , stronger effects). For α ¼ 0, the SO decreasing again (upper panel). Broken spin symmetry regime full effect of β on the conductance becomes apparent (lower panel): WAL is no longer suppressed here due to without cancellation from α, giving a maximal B . SO symmetry breaking from the cubic term at large n. Still, α ≈ β Indeed, the simulated α ¼ 0 curve [dashed red line in can be identified with the narrowest WAL peak. (c) Simulation of 11 −2 Fig. 4(a)] cuts through the experimental points, also α and β along n ¼ 6 × 10 cm . α traverses both β (black reflected in Fig. 4(c) by a good match with the simulated arrow) and for smaller δE also zero (red arrow). α ¼ 0 crossing point (red arrow). Diverging spin-orbit lengths.—For a comparison of vice versa for α ¼ −β. An in-plane rotation of the PSH by a experiment and simulation, we convert the empirical B fixed angle π=2 from α ¼þβ to α ¼ −β was recently SO pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi demonstrated [34]. to a “magnetic length” λ ¼ ℏ=2eB , which we later SO SO Figure 5 shows the theoretical spin diffusion length λ on interpret as a spin-diffusion length, where e> 0 is the eff (see Appendix) and the ballistic λ ,togetherwith the electron charge and the factor of 2 accounts for time- experimental λ , all agreeing remarkably well. Since at reversed pairs of closed trajectories. We also introduce the SO α ¼ β spin transport is ballistic despite charge diffusion, ballistic SO lengths λ ¼ ℏ =ð2m jα  βjÞ. These lengths λ and its diffusive counterpart λ (small β ) are essentially correspond to a spin rotation of 1 rad, as the electrons travel − eff 3 equivalent, as shown in SM [18]. The enhanced λ around along x ˆ and x ˆ , respectively, with spins initially aligned SO þ − α=β ¼ 1 corresponds to an increased spin relaxation perpendicular to the corresponding SO field [e.g., for an time τ ¼ λ =ð2DÞ.Notethat maxðλ ; λ Þ quantifies the electron moving along the x ˆ , its spin should point along SO þ − þ SO x ˆ or z ˆ so spin precession can occur; see SM, Eq. (S20), for deviation from the uniaxial SO field away from α ¼ β,and an expression of the SO field [18]]. For α ¼þβ, thus the extent to which spin rotations are not undone in a closed trajectory due to the non-Abelian nature of spin λ diverges (no precession, indicating that an electron ˆ rotations around noncollinear axes. This leads to WAL, a traveling along x does not precess) while λ is finite, and − þ 031010-5 α = β V [V] α, β [meV Å] -3 -3 x10 Δσ/σ [x10 ] 0 FLORIAN DETTWILER et al. PHYS. REV. X 7, 031010 (2017) semiconductor nanostructures such as quantum wires, quantum dots, and electron spin qubits. Moreover, our work relaxes the stringency (i.e., the “fine-tuning”) of the α ¼ β symmetry condition at a particular singular point (gate) by introducing a “continuous locking” of the SO couplings αðV ;V Þ¼ βðV ;V Þ over a wide range of T B T B voltages, which should enable new experiments exciting persistent spin helices with variable pitches in GaAs wells [12,13], i.e., stretchable PSHs. Further, this concept is also applicable to a range of other III-V semiconductors in various suitable configurations. Another possibility is the generation of a Skyrmion lattice (crossed spin helices) with variable lattice constants, as recently proposed in Ref. [35]. Finally, we stress that within the continuously locked regime of SO couplings we demonstrate in our study, FIG. 5. Experimental and theoretical SO lengths and SO times. pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi SO-coupled quantum transport in our samples shows a very Experimental λ ¼ ℏ=2eB (markers, densities as labeled, SO SO distinctive feature: it is diffusive (2D) for charge while 11 −2 in units of 10 cm ) as a function of the dimensionless ratio ballistic (1D) for spins, thus providing a unique setting for α=β (from SO simulation). The ballistic λ (blue and red dashes) coherent spin control. This ultimately adds a new function- and effective λ (black dashed curve) are only weakly n eff ality to the nonballistic spin transistor of Ref. [10]; i.e., it can dependent (small β ) when plotted against α=β. Thus, curves 11 −2 now be made to operate as the ideal (ballistic) Datta-Das spin for only one density (n ¼ 6 × 10 cm ) are shown. The transistor—but in a realistic 2D diffusive system, with yet experimental uncertainty on λ is captured by the spread given SO by the three slightly different densities. The coherence length controlled spin rotations protected from spin decay. L ≈ 7 μm is added for illustration (obtained from WL curves), setting the visibility of SO effects on the conductance and thus the ACKNOWLEDGMENTS width of the WAL-WL-WAL transition. Inset: Experimental spin We would like to thank A. C. Gossard, D. Loss, D. L. relaxation time τ ¼ λ =ð2DÞ (circles) as a function of α=β for SO SO Maslov, and G. Salis for valuable inputs and stimulating two densities as indicated. Theory curves τ (dashed) now eff discussions. This work was supported by the Swiss include the symmetry-breaking third-harmonic term, preventing divergence at α=β ¼ 1, while λ (main panel) does not. Nanoscience Institute (SNI), NCCR QSIT, Swiss NSF, eff ERC starting grant, EU-FP7 SOLID, and MICROKELVIN, finite B ,and λ ≃ maxðλ ; λ Þ,asobserved(seeFig. 5). SO SO þ − U.S. NSF DMR-1306300 and NSF MRSEC DMR- Unlike the corresponding time scales, the SO lengths are only 1420709 and ONR N00014-15-1-2369, Brazilian grants weakly dependent on density and mobility when plotted FAPESP (SPRINT program), CNPq, PRP/USP (Q- against α=β, allowing a comparison of various densities. NANO), and natural science foundation of China (Grant The third-harmonic contribution of cubic-in-k term No. 11004120). causes spin relaxation even at α ¼ β and becomes visible F. D., J. F., P. J. W., J. C. E., and D. M. Z. designed the at large densities: WAL is present in all traces and through experiments, analyzed the data. and co-wrote the paper. All α ¼ β [Fig. 4(b), lower panel], because the SO field can no authors discussed the results and commented on the longer be made uniaxial, thus breaking spin symmetry and manuscript. S. M. and D. D. A. designed, simulated, and reviving WAL. A partial symmetry restoration is still carried out the molecular beam epitaxy growth of the apparent, where—in contrast to the α ¼ 0 case—a minimal heterostructures. F. D. processed the samples and with B is reached (dashed curves) consistent with α ¼ β [gray SO P. J. W. performed the experiments. J. F. and J. C. E. devel- markers Fig. 4(a) at large n]. We include the cubic β in the oped and carried out the simulations and theoretical work. spin-relaxation time τ (see Appendix), shown in the inset eff of Fig. 5 for two densities, finding good agreement with the APPENDIX: MATERIALS AND METHODS experimental τ ¼ λ =ð2DÞ, where D is the diffusion SO SO 1. GaAs quantum well materials constant. Over the whole locked regime of Fig. 3(b),WAL is absent, and τ is enhanced between 1 and 2 orders of The wells are grown on an n-doped substrate (for details SO magnitude compared to α ¼ 0. Finally, the coherence see SM [18]) and fabricated into Hall bar structures [see length L sets an upper limit for the visibility of SO inset of Fig. 2(a)] using standard photolithographic meth- effects: WAL is suppressed for λ ≫ L , setting the width ods. The 2D gas is contacted by thermally annealed eff φ of the WAL-WL-WAL transition (see SM [18]). GeAu=Pt Ohmic contacts, optimized for a low contact Final remarks and outlook.—This work is laying the resistance while maintaining high back gate tunability (low leakage currents) and avoiding short circuits to the back foundation for a new generation of experiments benefiting from unprecedented command over SO coupling in gate. On one segment of the Hall bar, a Ti=Au top gate with 031010-6 STRETCHABLE PERSISTENT SPIN HELICES IN GAAS … PHYS. REV. X 7, 031010 (2017) dimensions of 300 × 100 μm is deposited. The average time-reversed trajectories: Δφ ¼ 2eAB=ℏ, where A is the gate-induced E-field change in the well is defined as loop area. Here, we take A ¼ λ ¼ 2Dτ as a characteristic SO SO δE ¼ 1=2ðV =d − V =d Þ, with effective distance “diffusion area” probed by our WL or WAL experiment, with Z T T B B d from the well to the top gate or back gate, respectively, τ being the spin-dephasing time, and λ the spin-diffusion T=B SO SO extracted using a capacitor model, consistent with the full length. By taking Δφ ¼ 1 (rad) at B ¼ B , we can extract SO quantum description (see SM [18]). Contours of constant the spin-diffusion length λ and spin-dephasing time τ SO SO pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi density follow δV =d ¼ −δV =d . Deviations from lin- T T B B from the minima of the WAL curves from λ ¼ ℏ=2eB SO SO ear behavior appear at most positive or negative gate −1 and τ ¼ ℏð4eDB Þ , respectively. The factor of 4 here SO SO voltages due to incipient gate leakage and hysteresis. stems from the two time-reversed paths and the diffusion length. 2. Low-temperature electronic measurements The experiments are performed in a dilution refrigerator 5. Effective SO times and lengths with base temperature 20 mK. We use a standard four-wire Theoretically, we determine τ via a spin random walk SO lock-in technique at 133 Hz and 100 nA current bias, process [D’yakonov-Perel (DP)]. The initial electron spin chosen to avoid self-heating while maximizing the signal. in a loop can point (with equal probability) along the s , The density is determined with Hall measurements in the s , and s axes (analogous to x , x , and z, respectively), x z þ − classical regime, whereas Shubnikov–de Haas oscillations þ which have unequal spin-dephasing times τ , τ , are used to exclude occupation of the second subband, DP;s DP;s x x which is the case for all the data we discuss. The WAL and τ . For unpolarized, independent spins, we take DP;s −3 signature is a small correction (10 ) to total conductance. the average τ ¼ðτ þ τ þ τ Þ=3, which leads eff DP;s DP;s DP;s x x z pffiffiffiffiffiffiffiffiffiffiffiffiffi To achieve a satisfactory signal-to-noise ratio, longitudinal to an effective spin-difusion length λ ¼ 2Dτ . eff eff conductivity traces Δσ=σ ¼½σðBÞ − σð0Þ=σð0Þ are mea- Actually, λ is defined from the average variance eff sured at least 10 times and averaged. 2 2 λ ¼ σ¯ ¼ 2Dτ , obtained by averaging the spin- eff eff 2 2 dependent variances σ ¼2Dτ , σ ¼2Dτ , and s DP;s s DP;s x x x x − − þ þ 3. Numerical simulations σ ¼ 2Dτ over the spin directions s , s , and s (this s DP;s x x z z z þ − The simulations calculate the Rashba coefficient α and is equivalent to averaging over the τ’s and not over 1=τ’s). hk i based on the bulk semiconductor band parameters, the In the SM [18], we discuss the spin random walk and well structure, the measured electron densities, and the provide expressions for the DP times including corrections measured gate lever arms. We solve the Schrödinger and due to the cubic β term. Figure 5 shows curves for the Poisson equations self-consistently (“Hartree approxima- spin-dephasing times and lengths presented here. In the tion”), obtain the self-consistent eigenfunctions, and then main panel, the cubic β is neglected in λ since for 3 eff determine α via appropriate expectation values [31]. The 11 −2 n ≤ 7 × 10 cm , WL appears at α ¼ β (small β ). 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