Abstract
As the Rashba effect is an electrically tunable spin–orbit interaction^{1}, it could form the basis for a multitude of applications^{2,3,4}, such as spin filters^{3}, spin transistors^{5,6} and quantum computing using Majorana states in nanowires^{7,8}. Moreover, this interaction can determine the spin dephasing^{9} and antilocalization phenomena in two dimensions^{10}. However, the real space pattern of the Rashba parameter, which critically influences spin transistors using the spinhelix state^{6,11,12} and the otherwise forbidden electron backscattering in topologically protected channels^{13,14}, is difficult to probe. Here, we map this pattern down to nanometre length scales by measuring the spin splitting of the lowest Landau level using scanning tunnelling spectroscopy. We reveal strong fluctuations correlated with the local electrostatic potential for an InSb inversion layer with a large Rashba coefficient (∼1 eV Å). This type of Rashba field mapping enables a more comprehensive understanding of its fluctuations, which might be decisive towards robust semiconductorbased spintronic devices.
Main
The Rashba effect^{1}, which lifts spin degeneracy by breaking inversion symmetry at surfaces and interfaces, was first probed in transport using the beating pattern in Shubnikov–de Haas oscillations^{15} or the weak antilocalization effect^{16}. Later, Rashbasplit bands and their spin polarization were visualized by photoelectron spectroscopy^{17}. The first successful attempts to use the Rashba effect for spin manipulation required low temperatures and found relatively small signals^{18,19,20}, probably due to a D’yakonov–Perel’ (DP)type spin relaxation where each scattering event randomly changes the direction of the Rashba field, which is orthogonal to the momentum. Options to overcome this limit are to use more onedimensional or ballistic devices^{21,22} or a balance of the Rashba and the Dresselhaus couplings^{6,23}, leading to a persistent spin helix with a momentumindependent spin rotation axis^{11,12}. For such cases, where the DP mechanism is suppressed, other dephasing mechanisms limit device functionalities. An obvious candidate is the spatial fluctuation of the Rashba parameter which induces pathdependent spin rotation even on ballistic trajectories. Estimates of the dephasing rates due to this mechanism (see Methods and Supplementary Information 10) reveal that it could explain the remaining dephasing in optimized samples^{21,22,23}. Interestingly, the topological protection in spin channels of twodimensional (2D) topological insulators is probably also limited by fluctuations of the Rashba parameter in combination with electron–electron interactions^{13} or magnetic impurities^{14}. However, experimental evidence of the Rashba field fluctuations has not been published so far.
A natural method to investigate electronic disorder is scanning tunnelling spectroscopy (STS)^{24,25}, which has already revealed fingerprints of the Rashba effect in twodimensional electron systems (2DES)^{26,27,28}, but so far without probing its spatial pattern.
Here, we use an InSb 2DES, produced by Cs adsorbates on the (110) surface (see Methods), as a model system to probe the Rashba parameter α_{R} in real space. STS in a perpendicular magnetic field B reveals a nonlinear spin splitting of the Landau levels (LL), which fits to the Rashba model at intermediate B = 3 − 7 T, whereas exchange enhancement^{29,30} dominates at higher B. Thus, the spin splitting at intermediate B can be used to trace the Rashba parameter α_{R}(R) as a function of position R, revealing that α_{R}(R) fluctuates spatially between 0.4 eV Å and 1.6 eV Å. The α_{R}(R) map with a correlation length of 30 nm exhibits a strong correlation with the electrostatic potential of the 2DES V_{2D}(R), as mapped by the spinaveraged LL energy. The observed fluctuations of α_{R}(R) are in agreement with analytic estimates.
The sample is sketched in Fig. 1a. By adsorbing Cs on ptype InSb(110), the valence and conduction bands are bent down towards the surface, forming an inversion layer with one occupied subband (Fig. 1b)^{28}. The Cs coverage (1.8% of a monolayer) is low enough to barely disturb the mapping of the 2DES by STS, but large enough such that the disorder is dominated by the more loosely spaced bulk acceptors^{28,29,31,32}. A strong electric field E ≈ 3 × 10^{7} V m^{−1} within the 2DES results from acceptor doping^{28}, which in combination with the large atomic numbers of In and Sb leads to a large α_{R}. Figure 1c shows the spinsplit LLs of this 2DES according to the Bychkov–Rashba model^{1}. One recognizes crossing points of opposite spin levels (dashed ellipses) and a nonlinearity of the spin splitting at low B. Figure 1d shows this splitting for different α_{R}, while keeping all other parameters identical. Different couplings naturally lead to different nonlinearities, offering an elegant method to locally determine α_{R}. Although α_{R} is a strictly local parameter, the measured spin splitting is related to wavefunctions, such that the spatial resolution of the method is limited to approximately the cyclotron radius, being smallest for the lowest Landau level LL_{0}. For smooth V_{2D}(R) with respect to the magnetic length (cyclotron radius of LL_{0}, ℏ: reduced Planck constant, e: electron charge)—that is, the guiding centre dynamics can be described semiclassically—one finds the energies ɛ_{n, λ}(B, R) for different LLs n and spin labels λ = +, − (ref. 33):
provided that Landau level mixing by disorder can be neglected—that is, {\sigma}_{{V}_{\text{2D}}} < ℏω_{c}. Here, F_{n}(r) is the kernel of the nth LL wavefunction (Supplementary Information 1), m^{∗} is the effective mass, g is the gfactor, ω_{c} = eB/m^{∗} is the cyclotron frequency, m_{e} is the electron mass and {\sigma}_{{V}_{\text{2D}}} is the standard deviation of the histogram of V_{2D}(R). With equations (1) and (2), we can determine α_{R}(R), using ΔE_{SS}(B, R) = ɛ_{0, −}(B, R) − ɛ_{1, +}(B, R), if V_{2D}(R), g and m^{∗} are also known. The error for such an α_{R}(R) determination originates mainly from the neglected higherorder derivatives of V_{2D}(R) within equation (1), and is on average 5% for the probed 2DES (Supplementary Information 2 and 6).
Figure 2a shows the density of states (DOS)—that is, the spatially averaged local density of states (LDOS)—of the 2DES at B = 7 T. The characteristic beating pattern of the LLs found previously^{28} is used to estimate the average Rashba parameter by comparison with the fitting lines. The best agreement is found for Å, corroborating the earlier results^{28}. Discrepancies between fit and data at higher LLs are due to the neglected nonparabolicity of the conduction band^{28}. The observed strong dip at the Fermi level E_{F} in the experiment is related to the wellknown Coulomb gap^{29,34}.
To extract the local Rashba parameter α_{R}(R), we record local LL fans. Figure 2b shows the measured LDOS of a single spatial point at different energies and B. LLs and spin levels of two subbands are discernible as marked. The individual levels collectively undulate with B, which we ascribe to the undulation of all LLs with respect to E_{F} to maintain the fixed carrier density n and, to a lesser extent, to exchange enhancement^{29}. Reproducible instabilities in the spectroscopy are observed at distinct B (crosses, Supplementary Information 3). Here, the conductance at E_{F} drops below ∼3 pS—that is, an insulating sample area close to integer filling factors prohibits current flow at these values of B. We ascribe the slight suppression of LDOS around E_{F} again to the Coulomb gap^{29,34}.
Multiple crossings of levels are present, for example, in the boxes marked I–III enlarged in Fig. 2c. The dashed lines (guides to the eye) reveal that the marked levels cross away from B = 0 T, such that they cannot belong to simple Landau and spin energies, both being linear in B and crossing at B = 0 T. A natural way to explain the crossings is the Rashba effect and, indeed, some of the crossings appear at values of B similar to those in the calculations (Fig. 1c). Discrepancies, most obvious at lower B, are attributed to the local confinement within the potential minimum, where the data are recorded. This complication hampers the use of the crossings for an accurate determination of α_{R}.
Instead, we use the Bnonlinearity of the LL_{0} spin splitting ΔE_{SS}(B, R). Figure 2d shows the measured LDOS at several R and B = 14 T. Double peaks for LL_{0}, and more complicated structures for higher LLs, are found. The complex peak structures appear away from the extrema of V_{2D}(R) owing to the nodal structure of the LL wavefunctions^{31}. The splitting ΔE_{SS}(B, R) determined from fitting two Lorentzians to the pair of peaks of LL_{0} is indicated. It increases for higher average energy—that is, for higher V_{2D}(R). This is opposite to the expectation from the nonparabolicity of g(V_{2D}), which decreases with increasing V_{2D} (ref. 28). Furthermore, a fluctuating peak width is observed, which will be discussed elsewhere.
Figure 2e shows ΔE_{SS}(B, R) as deduced from Fig. 2b using Lorentzian fits (inset). Above B = 7 T, ΔE_{SS} oscillates, exhibiting maxima at odd filling factors, as expected for exchange enhancement^{29,30}. Because the exchange interaction depends exponentially on the overlap of the wavefunctions, which roughly scales with , it decays rapidly at lower fields, being below 1 meV for B < 6 T (ref. 29). Accordingly, oscillations of ΔE_{SS} are barely discernible at B < 7 T. Instead a largely linear ΔE_{SS} is observed at 3–7 T, decaying more rapidly at lower fields, as in the curves of Fig. 1d. Extrapolating the linear part to B = 0 T (dashed line) reveals an offset of ΔE_{SS}^{0} ∼ 2.5 meV. Taylor expansion of equation (1) for large B using V_{2D}(R) = constant—that is, V_{2D, n}(R) = V_{2D, n−1}(R) (ref. 33)—reveals that the offset is given by ΔE_{SS}^{0}(B → 0) = 4α_{R}^{2}m^{∗}/ℏ^{2}(2 − gm^{∗}/m_{e}), resulting in α_{R} ≈ 0.65 eV Å close to from Fig. 2a. Figure 2f shows the fitted ΔE_{SS}(B, R) for two different potential minima, indicated by crosses in Fig. 3a–d. For the fit, we also consider the V_{2D, n}(R) terms of equation (1) deduced from the potential V_{2D, n}(R) of Fig. 3a (see below). The effective mass is determined from the Bfielddependent slope of the energy splitting between LL_{0} and LL_{1} (Supplementary Information 6). The gfactor then results from the relation g(V_{2D})m^{∗}(V_{2D}) = g_{0}m_{0}^{∗}, with g_{0} = −51 and m_{0}^{∗} = 0.0135m_{e} at the conduction band minimum, leaving α_{R} as the only fit parameter^{28}. The two resulting α_{R} values differ by a factor of two, indicating strong spatial α_{R} fluctuations.
We can easily rule out that the spatial fluctuation of g is responsible for the differences, as the curve probed at V_{2D}(R) = −121 meV (black) should have a larger gfactor than that probed at V_{2D}(R) = −101 meV (red), in contrast to the experimental observation. Exchange enhancement is also much weaker than the obtained differences in ΔE_{SS}.
Having established that α_{R}(R) can be deduced from ΔE_{SS}, we map α_{R}(R). A precise determination also requires V_{2D}(R) maps (equation (1)). Therefore, we plot the mean energy of the two LL_{0} peaks representing V_{2D}(R) (Fig. 3a). The resulting potential fluctuates by about ±10 meV, with a correlation length {L}_{{V}_{\text{2D}}{V}_{\text{2D}}} = 50 nm. Convolving with the LL wavefunction kernel (equation (2)) leads to the V_{2D}(R) maps in Fig. 3b, which are slightly smoother than Fig. 3a. Figure 3c shows the ΔE_{SS}(B, R) map at B = 6 T and Fig. 3d shows the resulting α_{R}(R) map according to equation (1). The ΔE_{SS}(B, R) and α_{R}(R) maps exhibit similar patterns, but differ in details.
Notably, α_{R}(R) fluctuates between 0.4 eV Å and 1.6 eV Å—that is, by a factor of four. It exhibits a giant , a rootmeansquare fluctuation δα_{R} = 0.15 eV Å (Fig. 3e), and a correlation length of {L}_{{\alpha}_{R}{\alpha}_{R}} = 30 nm, being larger than l_{B}, but smaller than {L}_{{V}_{\text{2D}}{V}_{\text{2D}}} (Supplementary Information 5). The latter is due to the stronger relative fluctuations of the Efield of a distribution of Coulomb impurities with respect to the electrostatic potential (Supplementary Information 9). In Methods, we show that δα_{R} and {L}_{{\alpha}_{R}{\alpha}_{R}} are reasonably reproduced by a fittingparameterfree analytic model using the experimental densities of Cs and bulk acceptors.
It is known that α_{R}(R) depends mostly on the local electric field E(R) perpendicular to the 2DES^{1}, which is not measurable by STS. However, comparing α_{R}(R) with the measured V_{2D}(R), we find a monotonic relation after averaging all α_{R}(R) values belonging to the same V_{2D} (red circles in Fig. 3f). This can be rationalized by numerically calculating the electric field and the potential from randomly distributed Cs surface donors and bulk acceptors (Supplementary Information 7). It turns out, as found previously^{32}, that the potential from the Cs surface donors is spatially rather fixed (δV_{2D} = 1.5 meV) whereas the potential fluctuations are dominated by the more distant acceptors (δV_{2D} = 25 meV). Thus, a stronger electric field (steeper slope of the potential) automatically correlates with a larger V_{2D}(R) (potential folded with the vertical wavefunction of the 2DES). However, the remaining scatter of α_{R} at given V_{2D} () is much larger than the error bar of the α_{R} determination (). This is straightforwardly explained by the remaining scatter in the relation between the simulated V_{2D}(R) and E(R) (Supplementary Information 8). Notice that the spinsplitting map exhibits more spatial detail than the potential map, because Coulomb impurities imply steeper electric fields (∝1/r^{2}) than potentials ∝(1/r) (Supplementary Information 9).
Finally, we estimate the spin dephasing length l_{Spin} and the spin diffusion length l_{s} from the data of Fig. 3d—that is, we consider only the fluctuations of α_{R} for spin dephasing^{35} (see Methods). We obtain l_{Spin} ≈ 250 nm, which is slightly larger than the mean free path in this particular system (l_{MFP} ≈ 20–200 nm), such that , being similar to the spin precession length (80 nm). This type of spin dephasing becomes decisive, for example, if the DP mechanism is avoided, as in ballistic^{21,22} or spinhelix^{6,11,12,23} transistors. For some of these systems, we can estimate l_{Spin} ≈ 23 μm (ref. 21), 60 μm (ref. 23), or 500 μm (ref. 22) and l_{s} ≈ 4 μm (refs 21,23) or 20 μm (ref. 22), which is compatible with the remaining signs of dephasing there (Supplementary Information 10). In particular, whereas DP and Elliott–Yafet dephasing disappear in ballistic channels, the fluctuations of α_{R}(R) still cause dephasing as long as the conducting channel is wider than {L}_{{\alpha}_{R}{\alpha}_{R}}, being, for example, 80 nm (ref. 21) or 450 nm (ref. 22) in typical devices. Consequently, a detailed understanding of α_{R}(R) fluctuations, as uniquely provided by our method, becomes crucial for these prospective devices, in particular when it comes to assessing reliability issues in spin transistor networks.
Methods
Preparation of clean InSb(110) surface.
InSb single crystals were glued with a conductive epoxy onto a molybdenum sample holder. A 1 mm deep notch was cut into the crystal to support cleaving along the (110) surface. A small screw was glued on top of the crystal. Inside an ultrahigh vacuum (UHV) chamber, at a base pressure of 10^{−10} mbar, the crystal was cleaved at the notch by pushing the screw towards the chamber wall. After insitu transfer into an inhousebuilt STM within one hour and direct cooling to 9 K, 4 K, 1.5 K, and 400 mK, respectively, atomically clean and flat terraces with widths of several μm were found.
2DES inversion layer.
The cleaved surface of the pdoped InSb with acceptor density N_{A} = 1 × 10^{24} m^{−3} was transferred in UHV into a sample stage held at T = 30 K, and Cs was evaporated from a Cs dispenser onto the surface. The Cs dispenser, operated at 470 °C, contains caesium chromate. After three evaporation cycles of 180 s, the surface coverage is 1.8% of a monolayer of Cs, as determined by counting the Cs atoms. One monolayer is defined as one Cs atom per InSb unit cell (that is, there is one Cs atom for approximately 60 InSb unit cells or the Cs atoms are on average about 4 nm apart). These Cs atoms are surface donors inducing the 2DES^{28}. The density of Cs is chosen to be relatively high, such that the positively charged Cs barely contributes to the potential fluctuations V_{2D}(R) (ref. 32). Estimates in Supplementary Section 7 reveal that Cs contributes ∼1.5 meV to the potential fluctuations of the 2DES, whereas the bulk acceptors contribute ∼25 meV. During the whole preparation, the pressure did not exceed 1.6 × 10^{−9} mbar. After the evaporation process, the sample was immediately transferred into the STM and cooled down.
Peak fitting for determination of ɛ_{0, −}, ɛ_{1, +}, V_{2D}(R), and spin splitting.
To discriminate the two spin levels of the lowest Landau level, we fit a double Lorentzian peak to the LDOS curves according to
with amplitudes a_{1,2}, peak widths σ_{1,2}, and peak energies ɛ_{0, −} and ɛ_{1, +} for peak 1 and 2, respectively. Whereas for B > 3.5 T all six parameters are chosen to be free, for lower B, σ_{1,2} were both fixed to 5.6 meV to deal with the less pronounced spin splitting. This is justified, because the distance of the two levels to E_{F} barely changes, leading to similar lifetimes and, thus, similar peak widths.
To first order, the LLs probe the electrostatic potential with a resolution approximately equal to the cyclotron radius (Supplementary Information 9). The electrostatic potential V_{2D}(R) is, thus, given by the average of ɛ_{0, −} and ɛ_{1, +} determined at the position R with a precession of 1 meV (Supplementary Information 9). The spin splitting is the difference between ɛ_{0, −} and ɛ_{1, +}. Changes in the respective potential and in the spin splitting are not expected on a length scale shorter than l_{B} = 10.5 nm at B = 6 T, hence we smoothed both maps with a Gaussian curve of width l_{B}.
Expected fluctuations of the Rashba coupling.
Here, we present analytical estimates of the random Rashba coupling and corresponding correlation lengths based on the approach developed in ref. 35, to compare with the experimental results. The model has no fitting parameters—that is, all inputs are known either from the experiment or from independent calculations and modelling. We use the experimentally determined densities of surface Cs atoms and bulk acceptors, as well as the electron density distribution along the zaxis as input parameters. Within our model, fluctuations in the spin–orbit coupling appear as a result of random electric fields of the negatively charged acceptors in the inversion layer and the positively charged Cs ions at the surface. For the random charge density we consider a random 2D distribution for positive charges and a random 3D distribution for negative charges.
We take the electron probability density distribution (see Fig. 1b) Ψ_{1}(z)^{2} as^{36}:
with the maximum position at z_{0} = 5 nm, corresponding to b = 0.4 nm^{−1}. The local value of the spin–orbit coupling is produced by averaging the zcomponent of the electric field over the given probability density Ψ_{1}(z)^{2}.
As a result of having two sources of the electric field fluctuations, we obtain two 2D correlation functions of the Rashba parameters:
Here, F_{2D}(R) and F_{3D}(R) are the corresponding range functions. They can be calculated straightforwardly using lengthy integral expressions for corresponding correlators. The subscript 2D is used for the contribution of the surface Cs donors with the mean density and the subscript 3D is used for that of the bulk acceptors with the mean density . We assume an uncorrelated whitenoise distribution of dopants, both on the surface and in the bulk, described by:
where r is a 3D coordinate. These densities of charged particles produce, at a point r, a random electric field with an unscreened zcomponent in the form^{36}
where d^{3}r′ integration is taken over the depletion layer (ε_{0}: vacuum dielectric constant). We assume that the dielectric constant is ε_{r} = 0.5ε_{r, InSb} for positive Cs, because the Cs is surrounded by vacuum in the upper halfspace, and ε_{r} = ε_{r, InSb} for negative acceptor charges, located in the bulk. Details of the coordinate dependence of the dielectric constant do not have a considerable effect on our results.
By averaging the products 〈E_{z}(r_{1})E_{z}(r_{2})〉 over the disorder using the distribution Ψ_{1}(z)^{2} and the above presented whitenoise correlators of the concentrations, one obtains^{35}, after a lengthy but straightforward calculation, the correlation functions of the random contribution to the Rashba parameters. The variations of the spin–orbit coupling have the form
where the materialdependent constant ξ describes the proportionality between the electric field and the Rashba parameter^{37}. The corresponding correlation lengths L_{2D(3D)} are defined as
using the dimensionless range functions F_{2D}(R) and F_{3D}(R) from equations (4) and (5).
This results in
where D_{l} is the depletion layer depth. Note that because the δfunctions of equation (6) do not have a characteristic nonzero spatial scale, the correlation length of the electric fields produced by the surface charges, L_{2D}, depends solely on the width of Ψ_{1}(z)^{2} (that is, 1/b). However, the bulk dopants are characterized, in addition, by the D_{l} spatial scale. As a result, L_{3D} includes two length parameters and increases only as the square root with D_{l} as the fardistant ions produce relatively weak fluctuations of the electric fields. The independence of 〈(δα_{R,3D})^{2}〉^{1/2} from the depletion layer depth in the D_{l}b ≫ 1 limit is shown by equation (8). It is interesting to mention that both correlation lengths depend solely on the system geometry and not on the material and sample parameters, such as ε_{r}, ξ, and the charge densities.
Substituting the characteristic numbers for the InSb inversion layer used in the experiment: ξ = 526 Å^{2} (ref. 37), , , ε_{r, InSb} = 16.8 for the dielectric constant of InSb, and D_{l} = 30 nm for the depletion layer width, we obtain δα_{R} : = 〈(δα_{R,2D})^{2}〉^{1/2} ≈ 〈(δα_{R,3D})^{2}〉^{1/2} ≈ 0.25 eV Å for the variations and L_{2D} ≈ 28 nm, L_{3D} ≈ 11 nm for the correlation lengths, respectively. These numbers, being approximate, show a reasonable agreement with the experiment (δα_{R} = 0.15 eV Å, {L}_{{\alpha}_{R}{\alpha}_{R}} = 30 nm) and allow us to attribute the experimental results to the fluctuations in the electric fields produced by randomness in the distribution of dopants. Notice that the variation in the calculated Rashba parameter due to the acceptors would be smeared in the experiment because the calculated L_{3D} is close to the value of the magnetic length l_{B}.
Spin dephasing by random Rashba coupling.
Here we present the characteristic values of the length scales related to spin propagation in our experiment. The conventional D’yakonov–Perel’ (DP) mechanism describes spin relaxation in a uniform Rashba field due to random spin precession directions between the scattering events. The Elliott–Yafet (EY) mechanism describes spin relaxation due to the spin flip at the scattering events. In contrast, the spin relaxation by the random Rashba field does not require any scattering. Thus, the corresponding spin relaxation rate is simply proportional to the correlation length of the Rashba disorder.
Taking the electron concentration n ≈ 1 × 10^{16} m^{−2}, we obtain the Fermi wavevector and the corresponding Fermi velocity v_{F} = ℏk_{F}/m^{∗} ≈ 1 × 10^{6} m s^{−1}, using m^{∗} = 0.03m_{e}. Spin dephasing is given by the deviation from the regular precession due to the average Rashba effect within the random Rashba field. It has been calculated for the case in which DP and EY mechanisms are absent and only the disorder in the α_{R}(R) parameter causes dephasing^{35}. The spin dephasing length—that is, the length that an electron travels on a random path before losing its spin memory—reads as (m^{∗} = 0.03m_{e}, δα_{R} = 0.15 eV Å, {L}_{{\alpha}_{R}{\alpha}_{R}} = 30 nm):
It is larger than the spin precession length . Most likely, it is also larger than the mean free path in our 2DES l_{MFP}. The mobility of the adsorbateinduced 2DES within an InSb(110) inversion layer has been measured at an acceptor density of to be μ = 1–10 m^{2} V ^{−1} s^{−1} (ref. 34) leading to l_{MFP} = ℏk_{F}μ/e = 0.2 − 2 μm. As the acceptor distance within our sample is a factor of ten lower, we estimate l_{MFP} ≈ 20–200 nm. Thus, the spin propagation in our sample is diffusive, with an estimated spin diffusion length l_{s}, which is the effective propagation distance of electrons within a sample before dephasing:
Data availability.
The data that supports the plots within this paper and other findings of this study are available from the corresponding author on request.
Change history
07 September 2016
The original version of this Letter mistakenly neglected to acknowledge the support of the National Science Center in Poland to Eugene Ya. Sherman (research project no. DEC2012/06/M/ST3/00042). The Acknowledgements have been updated accordingly in all versions of the Letter.
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Acknowledgements
We appreciate helpful discussions with D. HernangómezPérez, S. Florens, T. Champel, S. Becker, A. Georgi, C. Saunus, N. M. Freitag, M. M. Glazov, V. K. Dugaev and R. Winkler. Financial support by the German Science foundation via MO 858/112 and INST 222/7761 is gratefully acknowledged. J.U. acknowledges financial support via the Alexander von HumboldtStiftung. The work of E.Y.S. was supported by the University of the Basque Country UPV/EHU under program UFI 11/55, Spanish Ministry of Economics and Competitiveness (grant FIS201236673C0301), and ‘Grupos Consolidados UPV/EHU’ program of the Basque Country Government (grant IT47210). The work of E.Y.S. was also supported by the National Science Center in Poland as a research project No. DEC2012/06/M/ST3/00042.
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J.R.B. prepared the samples, conducted the experiments with the help of M.P. and M.L., evaluated the data with the help of M.L., and wrote the first version of the manuscript together with M.M.; E.Y.S. supported the evaluation of the data and provided the analytic theory to determine Rashba disorder, spin dephasing and spin diffusion length; J.U. provided the exact diagonalization results; M.M. and E.Y.S. devised the overall idea of the experiment. All authors contributed to the interpretation of the data and revising the manuscript.
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Bindel, J., Pezzotta, M., Ulrich, J. et al. Probing variations of the Rashba spin–orbit coupling at the nanometre scale. Nature Phys 12, 920–925 (2016). https://doi.org/10.1038/nphys3774
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DOI: https://doi.org/10.1038/nphys3774
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