## Main

The Rashba effect1, 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 oscillations15 or the weak antilocalization effect16. Later, Rashba-split bands and their spin polarization were visualized by photoelectron spectroscopy17. The first successful attempts to use the Rashba effect for spin manipulation required low temperatures and found relatively small signals18,19,20, 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 one-dimensional or ballistic devices21,22 or a balance of the Rashba and the Dresselhaus couplings6,23, leading to a persistent spin helix with a momentum-independent spin rotation axis11,12. 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 path-dependent 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 samples21,22,23. Interestingly, the topological protection in spin channels of two-dimensional (2D) topological insulators is probably also limited by fluctuations of the Rashba parameter in combination with electron–electron interactions13 or magnetic impurities14. 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 two-dimensional 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 αR 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 enhancement29,30 dominates at higher B. Thus, the spin splitting at intermediate B can be used to trace the Rashba parameter αR(R) as a function of position R, revealing that αR(R) fluctuates spatially between 0.4 eV Å and 1.6 eV Å. The αR(R) map with a correlation length of 30 nm exhibits a strong correlation with the electrostatic potential of the 2DES V2D(R), as mapped by the spin-averaged LL energy. The observed fluctuations of αR(R) are in agreement with analytic estimates.

The sample is sketched in Fig. 1a. By adsorbing Cs on p-type 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 acceptors28,29,31,32. A strong electric field |E| ≈ 3 × 107 V m−1 within the 2DES results from acceptor doping28, which in combination with the large atomic numbers of In and Sb leads to a large αR. Figure 1c shows the spin-split LLs of this 2DES according to the Bychkov–Rashba model1. 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 αR, while keeping all other parameters identical. Different couplings naturally lead to different nonlinearities, offering an elegant method to locally determine αR. Although αR 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 LL0. For smooth V2D(R) with respect to the magnetic length (cyclotron radius of LL0, : reduced Planck constant, e: electron charge)—that is, the guiding centre dynamics can be described semiclassically—one finds the energies ɛn, 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, ${\sigma }_{{V}_{\text{2D}}}$ < ωc. Here, Fn(r) is the kernel of the nth LL wavefunction (Supplementary Information 1), m is the effective mass, g is the g-factor, ωc = eB/m is the cyclotron frequency, me is the electron mass and ${\sigma }_{{V}_{\text{2D}}}$ is the standard deviation of the histogram of V2D(R). With equations (1) and (2), we can determine αR(R), using ΔESS(B, R) = ɛ0, −(B, R) − ɛ1, +(B, R), if V2D(R), g and m are also known. The error for such an αR(R) determination originates mainly from the neglected higher-order derivatives of V2D(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 previously28 is used to estimate the average Rashba parameter by comparison with the fitting lines. The best agreement is found for  Å, corroborating the earlier results28. Discrepancies between fit and data at higher LLs are due to the neglected nonparabolicity of the conduction band28. The observed strong dip at the Fermi level EF in the experiment is related to the well-known Coulomb gap29,34.

To extract the local Rashba parameter αR(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 EF to maintain the fixed carrier density n and, to a lesser extent, to exchange enhancement29. Reproducible instabilities in the spectroscopy are observed at distinct B (crosses, Supplementary Information 3). Here, the conductance at EF 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 EF again to the Coulomb gap29,34.

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 αR.

Instead, we use the B-nonlinearity of the LL0 spin splitting ΔESS(B, R). Figure 2d shows the measured LDOS at several R and B = 14 T. Double peaks for LL0, and more complicated structures for higher LLs, are found. The complex peak structures appear away from the extrema of V2D(R) owing to the nodal structure of the LL wavefunctions31. The splitting ΔESS(B, R) determined from fitting two Lorentzians to the pair of peaks of LL0 is indicated. It increases for higher average energy—that is, for higher V2D(R). This is opposite to the expectation from the nonparabolicity of g(V2D), which decreases with increasing V2D (ref. 28). Furthermore, a fluctuating peak width is observed, which will be discussed elsewhere.

Figure 2e shows ΔESS(B, R) as deduced from Fig. 2b using Lorentzian fits (inset). Above B = 7 T, ΔESS oscillates, exhibiting maxima at odd filling factors, as expected for exchange enhancement29,30. 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 ΔESS are barely discernible at B < 7 T. Instead a largely linear ΔESS 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 ΔESS0 2.5 meV. Taylor expansion of equation (1) for large B using V2D(R) = constant—that is, V2D, n(R) = V2D, n−1(R) (ref. 33)—reveals that the offset is given by ΔESS0(B → 0) = 4αR2m/2(2 − gm/me), resulting in αR ≈ 0.65 eV Å close to  from Fig. 2a. Figure 2f shows the fitted ΔESS(B, R) for two different potential minima, indicated by crosses in Fig. 3a–d. For the fit, we also consider the V2D, n(R) terms of equation (1) deduced from the potential V2D, n(R) of Fig. 3a (see below). The effective mass is determined from the B-field-dependent slope of the energy splitting between LL0 and LL1 (Supplementary Information 6). The g-factor then results from the relation g(V2D)m(V2D) = g0m0, with g0 = −51 and m0 = 0.0135me at the conduction band minimum, leaving αR as the only fit parameter28. The two resulting αR values differ by a factor of two, indicating strong spatial αR fluctuations.

We can easily rule out that the spatial fluctuation of g is responsible for the differences, as the curve probed at V2D(R) = −121 meV (black) should have a larger g-factor than that probed at V2D(R) = −101 meV (red), in contrast to the experimental observation. Exchange enhancement is also much weaker than the obtained differences in ΔESS.

Having established that αR(R) can be deduced from ΔESS, we map αR(R). A precise determination also requires V2D(R) maps (equation (1)). Therefore, we plot the mean energy of the two LL0 peaks representing V2D(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 V2D(R) maps in Fig. 3b, which are slightly smoother than Fig. 3a. Figure 3c shows the ΔESS(B, R) map at B = 6 T and Fig. 3d shows the resulting αR(R) map according to equation (1). The ΔESS(B, R) and αR(R) maps exhibit similar patterns, but differ in details.

Notably, αR(R) fluctuates between 0.4 eV Å and 1.6 eV Å—that is, by a factor of four. It exhibits a giant , a root-mean-square fluctuation δαR = 0.15 eV Å (Fig. 3e), and a correlation length of ${L}_{{\alpha }_{R}{\alpha }_{R}}$ = 30 nm, being larger than lB, but smaller than ${L}_{{V}_{\text{2D}}{V}_{\text{2D}}}$ (Supplementary Information 5). The latter is due to the stronger relative fluctuations of the E-field of a distribution of Coulomb impurities with respect to the electrostatic potential (Supplementary Information 9). In Methods, we show that δαR and ${L}_{{\alpha }_{R}{\alpha }_{R}}$ are reasonably reproduced by a fitting-parameter-free analytic model using the experimental densities of Cs and bulk acceptors.

It is known that αR(R) depends mostly on the local electric field E(R) perpendicular to the 2DES1, which is not measurable by STS. However, comparing αR(R) with the measured V2D(R), we find a monotonic relation after averaging all αR(R) values belonging to the same V2D (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 previously32, that the potential from the Cs surface donors is spatially rather fixed (δV2D = 1.5 meV) whereas the potential fluctuations are dominated by the more distant acceptors (δV2D = 25 meV). Thus, a stronger electric field (steeper slope of the potential) automatically correlates with a larger V2D(R) (potential folded with the vertical wavefunction of the 2DES). However, the remaining scatter of αR at given V2D () is much larger than the error bar of the αR determination (). This is straightforwardly explained by the remaining scatter in the relation between the simulated V2D(R) and E(R) (Supplementary Information 8). Notice that the spin-splitting map exhibits more spatial detail than the potential map, because Coulomb impurities imply steeper electric fields (1/r2) than potentials (1/r) (Supplementary Information 9).

Finally, we estimate the spin dephasing length lSpin and the spin diffusion length ls from the data of Fig. 3d—that is, we consider only the fluctuations of αR for spin dephasing35 (see Methods). We obtain lSpin ≈ 250 nm, which is slightly larger than the mean free path in this particular system (lMFP ≈ 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 ballistic21,22 or spin-helix6,11,12,23 transistors. For some of these systems, we can estimate lSpin ≈ 23 μm (ref. 21), 60 μm (ref. 23), or 500 μm (ref. 22) and ls ≈ 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 αR(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 αR(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 in-situ transfer into an in-house-built 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 p-doped InSb with acceptor density NA = 1 × 1024 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 2DES28. The density of Cs is chosen to be relatively high, such that the positively charged Cs barely contributes to the potential fluctuations V2D(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 ɛ0, −, ɛ1, +, V2D(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 a1,2, peak widths σ1,2, and peak energies ɛ0, − and ɛ1, + for peak 1 and 2, respectively. Whereas for B > 3.5 T all six parameters are chosen to be free, for lower B, σ1,2 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 EF 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 V2D(R) is, thus, given by the average of ɛ0, − and ɛ1, + determined at the position R with a precession of 1 meV (Supplementary Information 9). The spin splitting is the difference between ɛ0, − and ɛ1, +. Changes in the respective potential and in the spin splitting are not expected on a length scale shorter than lB = 10.5 nm at B = 6 T, hence we smoothed both maps with a Gaussian curve of width lB.

### 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 z-axis 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) |Ψ1(z)|2 as36:

with the maximum position at z0 = 5 nm, corresponding to b = 0.4 nm−1. The local value of the spin–orbit coupling is produced by averaging the z-component of the electric field over the given probability density |Ψ1(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, F2D(R) and F3D(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 white-noise 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 z-component in the form36

where d3r′ integration is taken over the depletion layer (ε0: vacuum dielectric constant). We assume that the dielectric constant is εr = 0.5εr, InSb for positive Cs, because the Cs is surrounded by vacuum in the upper half-space, and εr = εr, 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 〈Ez(r1)Ez(r2)〉 over the disorder using the distribution |Ψ1(z)|2 and the above presented white-noise correlators of the concentrations, one obtains35, 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 material-dependent constant ξ describes the proportionality between the electric field and the Rashba parameter37. The corresponding correlation lengths L2D(3D) are defined as

using the dimensionless range functions F2D(R) and F3D(R) from equations (4) and (5).

This results in

where Dl is the depletion layer depth. Note that because the δ-functions of equation (6) do not have a characteristic nonzero spatial scale, the correlation length of the electric fields produced by the surface charges, L2D, depends solely on the width of |Ψ1(z)|2 (that is, 1/b). However, the bulk dopants are characterized, in addition, by the Dl spatial scale. As a result, L3D includes two length parameters and increases only as the square root with Dl as the far-distant ions produce relatively weak fluctuations of the electric fields. The independence of 〈(δαR,3D)21/2 from the depletion layer depth in the Dlb 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 εr, ξ, and the charge densities.

Substituting the characteristic numbers for the InSb inversion layer used in the experiment: ξ = 526 Å2 (ref. 37), , , εr, InSb = 16.8 for the dielectric constant of InSb, and Dl = 30 nm for the depletion layer width, we obtain δαR : = 〈(δαR,2D)21/2 ≈ 〈(δαR,3D)21/2 ≈ 0.25 eV Å for the variations and L2D ≈ 28 nm, L3D ≈ 11 nm for the correlation lengths, respectively. These numbers, being approximate, show a reasonable agreement with the experiment (δαR = 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 L3D is close to the value of the magnetic length lB.

### 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 × 1016 m−2, we obtain the Fermi wavevector and the corresponding Fermi velocity vF = kF/m ≈ 1 × 106 m s−1, using m = 0.03me. 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 αR(R) parameter causes dephasing35. The spin dephasing length—that is, the length that an electron travels on a random path before losing its spin memory—reads as (m = 0.03me, δαR = 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 lMFP. The mobility of the adsorbate-induced 2DES within an InSb(110) inversion layer has been measured at an acceptor density of to be μ = 1–10 m2 V −1 s−1 (ref. 34) leading to lMFP = kFμ/e = 0.2 − 2 μm. As the acceptor distance within our sample is a factor of ten lower, we estimate lMFP ≈ 20–200 nm. Thus, the spin propagation in our sample is diffusive, with an estimated spin diffusion length ls, 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.