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[B. W. Grobecker](https://orcid.org/0009-0007-0344-0764), [A. V. Poshakinskiy](https://orcid.org/0000-0003-4514-1061), [S. Anghel](https://orcid.org/0000-0003-4517-4314), [T. Mano](https://orcid.org/0000-0002-6955-260X), [G. Yusa](https://orcid.org/0000-0003-3053-7629), [M. Betz](https://orcid.org/0000-0002-5676-3432)

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[Enhancing spin diffusion in GaAs quantum wells: The role of electron density and channel width](https://mdr.nims.go.jp/datasets/dac91e04-1834-46a1-bc34-c9d0b81d50e9)

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Enhancing spin diffusion in GaAs quantum wells: The role of electron density and channel widthViewOnlineExportCitationRESEARCH ARTICLE |  MAY 08 2025Enhancing spin diffusion in GaAs quantum wells: The role ofelectron density and channel widthB. W. Grobecker  ; A. V. Poshakinskiy  ; S. Anghel   ; T. Mano  ; G. Yusa  ; M. Betz J. Appl. Phys. 137, 183902 (2025)https://doi.org/10.1063/5.0257898Articles You May Be Interested InSpin helices in GaAs quantum wells: Interplay of electron density, spin diffusion, and spin lifetimeJ. Appl. Phys. (August 2022)Long-lived nanosecond spin coherence in high-mobility 2DEGs confined in double and triple quantumwellsJ. Appl. Phys. (June 2016)Dependence of the 0.5 × (2e2/h) conductance plateau on the aspect ratio of InAs quantum point contactswith in-plane side gatesJ. Appl. Phys. (February 2017) 12 May 2025 05:42:05https://pubs.aip.org/aip/jap/article/137/18/183902/3346573/Enhancing-spin-diffusion-in-GaAs-quantum-wells-Thehttps://pubs.aip.org/aip/jap/article/137/18/183902/3346573/Enhancing-spin-diffusion-in-GaAs-quantum-wells-The?pdfCoverIconEvent=citejavascript:;https://orcid.org/0009-0007-0344-0764javascript:;https://orcid.org/0000-0003-4514-1061javascript:;https://orcid.org/0000-0003-4517-4314javascript:;https://orcid.org/0000-0002-6955-260Xjavascript:;https://orcid.org/0000-0003-3053-7629javascript:;https://orcid.org/0000-0002-5676-3432https://crossmark.crossref.org/dialog/?doi=10.1063/5.0257898&domain=pdf&date_stamp=2025-05-08https://doi.org/10.1063/5.0257898https://pubs.aip.org/aip/jap/article/132/5/054301/2837275/Spin-helices-in-GaAs-quantum-wells-Interplay-ofhttps://pubs.aip.org/aip/jap/article/119/21/215701/374006/Long-lived-nanosecond-spin-coherence-in-highhttps://pubs.aip.org/aip/jap/article/121/8/083901/144535/Dependence-of-the-0-5-2e2-h-conductance-plateau-onhttps://e-11492.adzerk.net/r?e=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&s=3920gUfC7hmp0bwX-CUg8C4p2okEnhancing spin diffusion in GaAs quantum wells:The role of electron density and channel widthCite as: J. Appl. Phys. 137, 183902 (2025); doi: 10.1063/5.0257898View Online Export Citation CrossMarkSubmitted: 13 January 2025 · Accepted: 19 April 2025 ·Published Online: 8 May 2025B. W. Grobecker,1 A. V. Poshakinskiy,2 S. Anghel,1,a) T. Mano,3 G. Yusa,4 and M. Betz1AFFILIATIONS1Experimentelle Physik 2, Technische Universität Dortmund, Otto-Hahn-Straße 4a, D-44227 Dortmund, Germany2ICFO—Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, 08860 Castelldefels, Spain3National Institute for Materials Science, Tsukuba, Ibaraki 305-0047, Japan4Department of Physics, Tohoku University, Sendai 980-8578, Japana)Author to whom correspondence should be addressed: sergiu.anghel@tu-dortmund.deABSTRACTThis study explores the relationship between spin diffusion, spin lifetime, electron density, and lateral spatial confinement in two-dimensionalelectron gases hosted in GaAs quantum wells. Using time-resolved magneto-optical Kerr effect microscopy, we analyze how Hall-bar channelwidth and back-gate voltage modulation influence spin dynamics. The results reveal that the spin diffusion coefficient increases with reducedchannel widths, a trend further amplified at lower electron concentrations achieved via back-gate voltages, where it increases up to 150% for thenarrowest channels. The theoretical model developed in this work suggests that the spin diffusion coefficient is spatially inhomogeneous acrossthe channel cross section. Near the channel edges, where the electron density is reduced, the spin diffusion coefficient is enhanced due toweaker electron–electron scattering. As a result, narrower channels, which contain a relatively larger proportion of these low-density edgeregions, exhibit overall faster spin diffusion. Our results underscore the importance of tuning electron density and spatial geometry to optimizespin transport and coherence, providing valuable design considerations for spintronic devices where efficient spin manipulation is crucial.© 2025 Author(s). All article content, except where otherwise noted, is licensed under a Creative Commons Attribution (CC BY) license(https://creativecommons.org/licenses/by/4.0/). https://doi.org/10.1063/5.0257898I. INTRODUCTIONIn low-dimensional non-centrosymmetric semiconductor struc-tures, in particular in two-dimensional electron gases (2DEGs)within quantum wells (QWs), the spin–orbit (SO) interactiongoverns a wide range of spin-related phenomena.1–3 Among these isthe persistent spin helix (PSH),4,5 a helical spin texture thatemerges when the Rashba6 and Dresselhaus7 SO couplings areequal in magnitude. This condition restores SU(2) spin rotationsymmetry,8 suppresses spin relaxation due to the Dyakonov–Perelmechanism, and creates robust spin coherence with a spatiallystriped spin texture.4 The PSH state has been experimentally demon-strated through techniques like transient spin grating spectroscopy5,9and Kerr-rotation microscopy,4,10–28 showcasing its long-lived natureand potential for spintronic applications.The PSH arises due to the unidirectional momentum-dependenteffective magnetic field BSO(k), which protects the spin state fromdephasing caused by spin-independent scattering. This protection,coupled with the ability to tune Rashba and Dresselhaus interactionsthrough quantum well design, gate voltage modulation,20,29 or opticaldoping,19,21 allows for precise control of PSH dynamics. Studies havedemonstrated that factors such as the spin diffusion coefficient,15cubic Dresselhaus terms,30,31 and carrier heating16 influence PSHlifetimes and diffusion properties. Adjustments in carrier density,excitation energy, and applied electric fields also impact thetransition from ballistic to diffusive motion,11 modulating thespin transport efficiency. Recent advances link PSH dynamicsto structured light and topological concepts10 by using a spatiallystructured light such as vector vortex beam that enables spatiallyvariant polarization states, offering applications in optical com-munication, metrology, and spintronic devices. Future devicescould harness the synergy of long-lived PSH spin states andspatial spin modes for applications in information processing,quantum computing, and advanced optics. However, ensuringrobust spin transport and coherence remains challenging, particu-larly due to quantum well design constraints that affect electronmobility, spin diffusion, and PSH lifetime (τs).Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 137, 183902 (2025); doi: 10.1063/5.0257898 137, 183902-1© Author(s) 2025 12 May 2025 05:42:05https://doi.org/10.1063/5.0257898https://doi.org/10.1063/5.0257898https://pubs.aip.org/action/showCitFormats?type=show&doi=10.1063/5.0257898http://crossmark.crossref.org/dialog/?doi=10.1063/5.0257898&domain=pdf&date_stamp=2025-05-08https://orcid.org/0009-0007-0344-0764https://orcid.org/0000-0003-4514-1061https://orcid.org/0000-0003-4517-4314https://orcid.org/0000-0002-6955-260Xhttps://orcid.org/0000-0003-3053-7629https://orcid.org/0000-0002-5676-3432mailto:sergiu.anghel@tu-dortmund.dehttps://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/https://doi.org/10.1063/5.0257898https://pubs.aip.org/aip/japA possible way to further increase the lifetime of spin texturesis to consider quantum wells with lateral potential that confineselectron gas to quasi-1D channels of a few-micrometer width.Such channels are still much wider than the electron-mean-freepath and preserve the diffusive character of transport. However,they can be narrower than the spin–orbit precession length, leadingto the suppression of the Dyakonov–Perel mechanism spin relaxationmechanism.32–36 It was theoretically predicted that in this regime,the spin lifetime should increase with a power law 1/d2, where d isthe channel width.32 The experimental studies also evidenced thespin lifetime enhancement,24,25,37 though limited by the effect ofk-cubic spin–orbit interaction.24 Importantly, in all studies the spindiffusion coefficient (Ds) has been either assumed or experimentallyshown to be independent of the channel width.The current paper seeks to advance knowledge in this areaby investigating how the channel width of the Hall-bar, combinedwith back-gate voltage modulation, affects the spin lifetime andspin diffusion coefficient. Using time-resolved magneto-opticalKerr effect (TR-MOKE) microscopy, we find that Ds increases signifi-cantly as the width of the Hall-bar channels decreases. This increaseis further enhanced when the density of the two-dimensional electrongas is decreased by the back-gate voltage. Since the spin relaxationrate in Dyakonov–Perel mechanics is proportional to Ds, the increasein the latter in the narrow channels tries to speed up the spin relaxa-tion, competing with the described effect of the slowing down of thespin relaxation.II. EXPERIMENTAL DETAILSThe sample under investigation is a modulation-doped(001)-oriented 15 nm GaAs quantum well, grown by molecularbeam epitaxy and sandwiched between Al0.33Ga0.67As barriers( for more details, see Fig. S1 in the supplementary material).The QW is patterned in a series of five channels with differentwidths as shown in Fig. 1(d) with AuGeNi ohmic contacts.The channels are oriented along [1�10] crystallographic direction,i.e., the direction of the spatial spin precession when the PSHconditions are met. Two Si δ-doping layers are placed above theQW providing a resident electron concentration n in the QW thatcan be modified by the back-gate voltage UBG.38 Using magneto-transport measurements, we verify that both n and electron mobil-ity μ have a linear dependence on the back-gate voltage in therange of 1:5 V , UBG , �2:5 V (see Fig. S2 in the supplementarymaterial). To create robust electron spins, the sample resides in acompact cold-finger cryostat, ensuring a lattice temperature of3:5 K for all performed measurements.The time-resolved magneto-optical Kerr effect (TR-MOKE)microscopy measurements are performed using pulses with a tempo-ral width of �35 fs derived from a 60 MHz mode-locked Ti:sapphireoscillator. Subsequently, they are split into pump and probe paths,which are spectrally tuned independently by grating-based pulseshapers.22 The resulting pulses have a bandwidth of �0:5 nmand allow for a transform-limited temporal resolution of �1 ps.The probe pulses are linearly polarized, while the pump pulses aremodulated between left (σþ) and right (σ�) circular polarizationby an electro-optical modulator (EOM). Both probe and pumppulses are collinearly focused on the sample surface through a50× microscope objective. The full width at half maximum (FWHM)diameter of pump and probe pulses are w0 ¼ 3+ 0:1 μm and1+ 0:1 μm, respectively. The reflected pump light is filtered outwith a monochromator, and the Kerr-rotation of the reflected probepulse is measured using balanced photodiodes connected to alock-in amplifier referenced to the EOM frequency. The delaytime t between the pump and probe pulses is adjusted by amechanical delay stage with tmax ¼ 1:8 ns. The spatial overlap ofthe pump with the fixed and centered probe is adjusted througha lateral translation of the input lens of a beam-expanding tele-scope in the pump path.39,40 In our setup, we scan the positionof the pump beam while keeping the probe beam fixed. As aresult, the obtained spatial maps can be considered “inversemaps,” as they depict spin polarization at a fixed detection pointwhile varying the location where the spin polarization is initiallygenerated. The pump and probe photon energies are chosenbased on the spectral response of the 2DEG (see, e.g., Ref. 16).All measurements are performed with the pump photonenergy set to Ep ¼ 1:57 eV, which is 40 meV above the bandgapenergy (1.53 eV), and a peak power density of Ip ¼ 4:7 MW/cm2.Since the energy separation between the first and second electronlevels in the QW is about 52 meV, the pump excites electrons tothe first sublevel only. The probe photon energy is tuned toEpr ¼ 1:53 eV with a pulse peak irradiance of Ipr ¼ 2:36 MW/cm2.III. RESULTS AND DISCUSSIONFirst, the sample is investigated at UBG ¼ �1:55 V, corre-sponding to an electron concentration of n ¼ 3:25� 1010 cm�2.This aimed to ensure the longest possible spin lifetime underthe given experimental conditions (note that this voltage doesnot correspond to the PSH condition, as shown in Ref. 15).FIG. 1. Schematics of the channel mask with five different channel widths ofthe investigated GaAs QW sample (the yellow and cyan lines indicate the mesaand ohmic contacts, respectively) (d) and examples of 2D spatial “inversemaps” of the spin polarization Sz at a delay time of t ¼ 570 ps for the channelswidths of (a) 20, (b) 8, and (c) 4 μm respectively.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 137, 183902 (2025); doi: 10.1063/5.0257898 137, 183902-2© Author(s) 2025 12 May 2025 05:42:05https://doi.org/10.60893/figshare.jap.c.7782500https://doi.org/10.60893/figshare.jap.c.7782500https://doi.org/10.60893/figshare.jap.c.7782500https://pubs.aip.org/aip/japFigures 1(a)–1(c) show the spatial evolution examples of the mea-sured spin polarization Sz within three selected Hall-bar channelsof different widths. Here, the time delay between the initial pumppulse and the scanning probe pulse was fixed to t ¼ 570 ps. Thismakes sure that the helical spin pattern has enough time to estab-lish itself. The dimension of each respective channel is highlightedby a dotted line. The existence of some signal outside of thoseborders corresponds to the relatively big pump diameter of3+ 0:1 μm. Especially for the narrowest channels, this lies withinthe order of magnitude of channel width d. Therefore, by using aprobe-stationary approach, spin polarization is induced by the remain-ing overlap of the pump spot and embedded channel, although thecenter of the pump pulse is already off.To investigate the interplay of the channel dimensions and theelectron density in manners of spin dynamics, it is essential to mapthe latter in both space and time. Given spatial constraints imposedby the channel widths, the spatial resolution is limited to the y axis(1D case), which aligns along the crystallographic axis [1�10].Figure 2(a) showcases an example of spatiotemporal measurementfor the 8 μm channel and UBG ¼ �1:55 V. The broadeningand decay of the initial Gaussian shaped spin polarization Sz(t, y)is caused by diffusion and relaxation mechanisms, respectively.To visualize the spin distribution amplitude and broadening in spaceand time, five examples of spatial cuts are shown superimposed withthe original signal. At any given delay time t, the spatial spin distri-bution is fitted with the phenomenological equation, which repre-sents the product of a Gaussian function and a cosine function,capturing both the diffusive broadening and the oscillatory nature ofthe spin precession (for more details, see Ref. 4)Sz(y) ¼ A � e4ln(2) (y�yG )2w2y cos2π(y � yc)λSO� �, (1)where A(t) is the amplitude of the spin polarization,wy(t) is the FWHM of the Gaussian envelope, andλSO(t) ¼ λ0wy(t)2/(wy(t)2 � w20) is the momentary spin precessionlength, whereas λ0,y ¼ π�h2/(m�jα þ βj) is the precession length ofthe long-lived spin helix. A more comprehensive discussion of λ0dependence on the back-gate voltage, including its theoretical back-ground and experimental observations, can be found in Ref. 15.Due to the significant spatial restrictions imposed by the investigatedHall-bar channels, especially when it comes to the narrow channels,we assume that spatial dynamics mostly occur only in one dimen-sion. Therefore, the temporal evolution of the total number of spinscan be quantified by A � wy(t). This expression accounts for all ini-tially excited spins, not only those that are oriented along z at themoment of detection and contribute to measured Sz but also thosespins, which, due to the spatial precession, lie in the QW plane. Thistotal spin volume, whose time dependence is shown in Fig. 2(b),allows us to retrieve the spin lifetime τs by fitting it with a single-exponential decay. Furthermore, the time-dependent square of theGaussian envelope FWHM, illustrated in Fig. 2(c), provides access tothe spin diffusion coefficient Ds via a linear regression ofw2y(t) ¼ w20 þ 16 ln (2)Dst: (2)When fitting the time dependencies of the spin volume andFWHM, we restrict the data to delay times * 400 ps to avoid tran-sient effects that occur at shorter times. These include the faster decayof spin polarization due to its initial spread across the channel24,25and the formation of the spin-helix pattern.41 The same procedure ofanalyzing the experimental data is applied to all the subsequent mea-surements done for each channel seen in Fig. 1(d).This paper aims to investigate the dependence of the spin dif-fusion coefficient, spin lifetime, and electron concentration n onthe channel width d and various back-gate voltages. Specifically, itfocuses on the relative changes in these parameters compared totheir behavior in the absence of spatial restrictions imposed bychannel width. To this end, measurements were conducted for dif-ferent channel widths while varying the back-gate voltage UBGbetween þ0:5 V and �2:0 V. Figure 3 presents the experimentalresults as ratios relative to the corresponding parameter valuesobtained for d ¼ 20 μm (the absolute values are given in Table S1in the supplementary material), a channel width that does notimpose significant lateral confinement. These reference values aredenoted by a subscripted zero (D0, τ0). The first striking result isthat Ds/D0 ratios increase drastically with limiting the spatialdynamics in one dimension: the smaller the d, the higher the Ds.Additionally, when different UBG were applied, it can be observedthat this effect is further enhanced by decreasing the UBG, i.e., byreducing n. Hence, the range of amplification varies from only∼25% (for a UBG ¼ 0:5 V) up to ∼150% (UBG ¼ �2:0 V) at thenarrowest channel width of 4 μm. It gives the impression that thenarrower the channel, the greater the influence of the electronFIG. 2. (a) Spatiotemporal mapping of the induced spin polarization distributionSz(t, y) of the 8 μm channel and UBG ¼ �1:55 V. For better visualization fivearbitrary spatial scans are highlighted. For any given delay time t the spatialdependence gets fitted to Eq. (1). Two of the resulting fit parameters (A, wy) areused to display (b) the spin volume A � wy and (c) the square of FWHM. The zaxis is restricted to arbitrary units, and only the time axis is shared by all plotsthroughout the figure.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 137, 183902 (2025); doi: 10.1063/5.0257898 137, 183902-3© Author(s) 2025 12 May 2025 05:42:05https://doi.org/10.60893/figshare.jap.c.7782500https://pubs.aip.org/aip/japconcentration on the acceleration of diffusion in one dimension.In Fig. 3(b), the ratios of τs/τ0 are visualized. Here, the curve ofUBG ¼ �1:55 V stands out as it shows a rather voltage-independentratio of 1:0 with a small deviation at d ¼ 10 μm. This can beattributed to the maximum spin lifetime, which was observed atthis specific voltage during the characterization of the sample.The remaining voltages tend to allow a small rise in the spin life-time by reducing the channel width. The parameters obtained forhigh electron concentrations (UBG ¼ 0:5 V) show large errors,since the diffusion is slower and the initial spread across thechannel and the formation of the spin-helix pattern takes longer.Finally, the ratio of spin diffusion length Ls/L0 (whereL ¼ ffiffiffiffiffiffiffiDτsp) is displayed in Fig. 3(c). For low n (UBG ¼ �2:0 V)and small d (4 μm), the ratio gets almost doubled, in comparisonwith having no lateral restriction. Due to the rather stable lifetimeat UBG ¼ �1:55 V, here the extension of Ls is not as pronounced.Still, it is enhanced by 30%, due to the sudden increase of Ds atd ¼ 4 μm. The ratios for remaining voltages lie in between,although the spike for the smallest channel is suppressed. It emergesas a rather steady increase by decreasing d.To explain the above seen relation between UBG, Ds, andchannel width d, we develop a theoretical description of spin dif-fusion of the 2D electrons inside the channel �d/2 , x , d/2,formed by the external potential U(x) and described by a spatiallyinhomogeneous spin diffusion coefficient Ds(x) [see Fig. 4(a)]. Wesuppose that the channel has a symmetric profile, so U(x) ¼ U(�x)and Ds(x) ¼ Ds(�x).The diffusion equation for the spin distribution functionS(x, y, t), accounting for the spin–orbit interaction, reads@S@t¼Xα¼x,yΛα � @@rα� �Ds Λα � @@rα� 1T@U@rα� �S, (3)where tensor Λα describes the spin rotation in the spin–orbit field,Λx,yS ¼ 2m�(β+ α)ey,x � S/�h2. We also suppose that the electrongas is non-degenerate and described by the temperature TFIG. 3. Results from analyzing the interplay of the channel width d and back-gate voltage UBG in manners of spin dynamic parameters: (a) spin diffusioncoefficient ratio Ds/D0 and the respective fits to Eq. (12), (b) spin lifetime ratioτs/τ0, and (c) spin diffusion length ratio Ls/L0 L ¼ ffiffiffiffiffiffiffiDτsp� �, where D0, L0 andτ0 are the corresponding parameters obtained for the widest channeld ¼ 20 μm. The lines in (b) and (c) serve as guides to the eye.FIG. 4. (a) Spatial profile of the external potential U(x), which induces a spa-tially inhomogeneous spin diffusion coefficient Ds(x). (b) Extracted fit parametera of Eq. (15) in dependence on the back-gate voltage UBG.Journal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 137, 183902 (2025); doi: 10.1063/5.0257898 137, 183902-4© Author(s) 2025 12 May 2025 05:42:05https://pubs.aip.org/aip/jap(measured in the energy units), as it was shown in a previousstudy.15 The spin diffusion equation (3) should be accompanied bythe boundary condition,Λx � @@x� 1T@U@x� �S(r)����x¼+d/2¼ 0, (4)which reflects the conservation of the spin at the channel edge.We note that, due to the presence of the external potential U(x),the differential operator on the right-hand side of the spin diffusionequation (3) is not symmetric. Consequently, spin diffusion isgenerally not reciprocal, meaning that interchanging the pumpand probe positions can lead to changes in the detected signal.However, in our study of spin relaxation and diffusion along thechannel, we specifically positioned both the pump and probe alongthe central line of the channel (x ¼ 0). In this configuration, reci-procity is restored, ensuring consistent measurements.We search for the eigen solutions of the diffusion equationthat have the form,Sk(x, y, t) ¼ e�U(x)/TþikyþΛxx�ΓktPk(x): (5)Here, the spin polarization degree Pk(x) should be obtainedby solving the Sturm–Liouville problem,� e�U(x)/T ddxeU(x)/TDs(x)dPk(x)dxþ kþ e�ΛxxiΛyeΛxx� �2Ds(x)Pk(x) ¼ ΓkPk(x): (6)Since there is no analytical solution for this equation in thegeneral case, we use the perturbation theory, assuming kΛxwk � 1and following the approach of Ref. 32. First, we describe unper-turbed solutions, i.e., for Λx ¼ 0, which corresponds to the PSHregime. The solutions with the lowest decay rate have a homoge-neous distribution of spin polarization across the channel,Pk(x) ¼ p(σ), where p(σ) is an eigenvector of Λy and satisfiesΛyp(σ) ¼ ik0p(σ) with k0 ¼ σm(α þ β), σ ¼ 0, +1, and jp(σ)j2 ¼ 1.The optically accessible modes, which have a nonzero spin compo-nent along the z axis, correspond to σ ¼ +1. Note that for k ¼ k0,the unperturbed decay rate of the modes vanishes Γk0 ¼ 0, which isthe signature of the PSH state. There are other solutions of theunperturbed problem, which are described by the eigenfunctionsPk(x) that have n � 1 zeros and fast decay rates Γk � Ds/d2.To consider the deviation from the PSH regime, we rewriteEq. (6) as� e�U(x)/T ddxeU(x)/TDs(x)dPk(x)dxþ (k0 þ iΛy)2 þ V(x)�  Ds(x)Pk(x) ¼ ΓPk(x), (7)whereV(x) ¼ (k0 þ iΛy), (k0 þ e�ΛxxiΛyeΛxx � iΛy) �þ k0 þ e�ΛxxiΛyeΛxx � iΛy� �2, (8)k0 ¼ k� k0, and {A, B} ¼ ABþ BA. We expand V(x) up to termsquadratic in k0 and Λxx and getV(x) ¼ k0 � ix[Λx , Λy]� �2þ (k0 þ iΛy), k0 � ix[Λx , Λy]� ix22[Λx , [Λx , Λy]]� �� :(9)Next, we determine the effect of the perturbation V(x) on theunperturbed solution p(σ). Note that for the considered mode(k0 þ iΛy)p(σ) ¼ 0, so only the first term in Eq. (9) contributes.The parity of the problem also constrains the mixing of the modes.Up to the order /d2, the mixing of the p(σ) with different σ doesnot occur. The mixing of the p(σ) mode with the fast-decayingmodes of the odd parity leads to the contributions to the decayrate of the former of the order /d4. Therefore, in the order /d2,the decay of the eigen solution p(σ) is determined by averaging thefirst line of Eq. (9), which yields Γk0 þ k0 ¼ �Γþ �Dsk02, with theeffective spin diffusion coefficient,�Ds ¼ÐDs(x)e�U(x)/TdxÐe�U(x)/Tdx, (10)and the spin relaxation rate is�Γ ¼ 16m�4(1� σ2/2)(α2 � β2)2�h8�Ðx2Ds(x)e�U(x)/TdxÐe�U(x)/Tdx: (11)Finally, we suppose the spin diffusion coefficient is limited byelectron–electron collisions, thus inversely proportional to the elec-tron density Ds(x)/ eU(x)/T .15 We neglect here the possible effecton Ds(x) of the electron scattering by impurities, phonons, orchannel edge imperfections. Then, for the optically accessiblemodes with σ ¼ +1, the above expressions for �Ds and �Γ arethen simplified to�Ds ¼ Ds(0)dd � a, (12)�Γ ¼ �Ds2m�4(α2 � β2)2d23�h8, (13)where d is the channel width, U(0) ¼ 0 is assumed, and the param-eter a isa ¼ð(1� e�U(x)/T )dx (14)quantifies the effective width of the channel regions where theexternal potential deviates from zero, U(x)/T *1.The simplest model of the U(x) profile is illustrated inFig. 4(a). In this model, the potential is assumed to be flat[U(x) ¼ 0] in the central region of the channel, which has a widthd0, while it increases near the channel edges. As a result, thecentral region exhibits high electron density, leading to slower spinJournal ofApplied PhysicsARTICLE pubs.aip.org/aip/japJ. Appl. Phys. 137, 183902 (2025); doi: 10.1063/5.0257898 137, 183902-5© Author(s) 2025 12 May 2025 05:42:05https://pubs.aip.org/aip/japdiffusion. In contrast, the lower electron density near the channeledges causes faster spin diffusion in those areas. Then, the value ofa [Eq. (14)] is determined by the channel border only and can beassumed to be independent of the channel width d, provided thatd . d0.We use Eq. (12) to fit the experimental curves in Fig. 3(a)with the fitted curves superimposed on the same figure. From thesefits, we extract the fit parameter a for different back-gate voltages.The dependence of a on UBG is shown in Fig. 4(b). With the increasein UBG, the density of electrons grows, and they start to screen thechannel profile potential U(x), which becomes flatter. This leads to adecrease in the effective width of the channel edge regions whereU(x)/T *1, and the decrease of parameter a [see Eq. (14)].The change in the spin lifetime τs ¼ 1/�Γ with the channelwidth d, shown in Fig. 3(b), shall be explained by the competition ofthe factor d2 in Eq. (13) and the decrease in �Ds. To eliminate theeffect of �Ds, we plot in Fig. 3(c) the spin diffusion lengthLs ¼ (�Ds/�Γ)1/2, which, according to Eq. (13), should be described byLs ¼ffiffiffiffiffiffiffi3/2p�h4m�2jα2 � β2jd : (15)However, note that while the behavior of the diffusion coeffi-cient �Ds is well described by Eq. (12) in a wide range of channelwidth d, the applicability of Eqs. (13) and (15) is limited to thesmall values of d only. Accordingly, only the initial part of theexperimental data in Fig. 3(c) follows the 1/d trend, while for largerd, the spin diffusion length tends to a constant value, which corre-sponds to the 2D diffusion in the absence of spatial restrictionsimposed by the channel widths.IV. CONCLUSIONSThis study offers a detailed examination of spin dynamics inquasi-1D channels within GaAs quantum wells, emphasizing therelationship between the spin diffusion coefficient, spin lifetime,and electron density under varying geometrical and electronicconditions. Using TR-MOKE microscopy, the findings highlightthe significant impact of lateral confinement and back-gate voltagemodulation on spin transport. Specifically, the spin diffusion coeffi-cient increases notably with narrower Hall-bar channel widths, reach-ing up to a 150% enhancement in the narrowest channels, an effectfurther amplified at reduced electron concentrations controlled viaback-gate voltages.A theoretical framework corroborates these experimental obser-vations, demonstrating how spatial variations in the electron densityinfluence spin diffusion and relaxation. The model suggests two dis-tinct regions within the channels: a central region, where the electrondensity is high and spin diffusion is slow, and areas near the channeledges, where the lower electron density results in faster spin diffu-sion. These spatially varying densities significantly shape spin trans-port behavior.The findings underscore the potential for optimizing spincoherence and transport efficiency through careful tuning of electrondensity and spatially constrained geometries. This has profoundimplications for advancing spintronic devices. Future research couldfocus on refining these strategies, including innovations in materialdesign and external modulation techniques, to harness these insightsfor real-world applications in spin-based technology.SUPPLEMENTARY MATERIALSee the supplementary material for additional details on thewafer structure, magneto-transport measurements, 2D spatiotem-poral data, and the justification for using a single-exponential fit todescribe the time evolution of the spin volume signal.ACKNOWLEDGMENTSThe authors thank T. Takayanagi and Y. Takahashi for theirfruitful discussion. This work was supported by a Grant-in-Aid forScientific Research (Grant Nos. 19H05603, 21H05182, 21H05188,and 24H00399) from the Ministry of Education, Culture, Sports,Science, and Technology (MEXT), Japan. A.V.P. acknowledgesthe funding from the postdoctoral fellowship Beatriu de Pinós(2023 BP 00136), Government of Spain, under Severo Ochoa GrantCEX2019-000910-S (MCIN/AEI/10.13039/501100011033), Generalitatde Catalunya (CERCA program), Fundació Cellex, and FundacióMir-Puig.AUTHOR DECLARATIONSConflict of InterestThe authors have no conflicts to disclose.Author ContributionsB. W. Grobecker: Conceptualization (equal); Data curation (equal);Formal analysis (equal); Investigation (equal); Methodology (equal);Software (equal); Writing – original draft (equal); Writing – review &editing (equal). A. V. Poshakinskiy: Conceptualization (equal);Data curation (equal); Formal analysis (equal); Investigation(equal); Software (equal); Validation (equal); Writing – originaldraft (equal). S. 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