2D Helium Atom Diffraction from a Microscopic Spot

A method for measuring helium atom diffraction with micron-scale spatial resolution is demonstrated in a scanning helium microscope (SHeM) and applied to study a micron-scale spot on the (100) plane of a lithium fluoride (LiF) crystal. The positions of the observed diffraction peaks provide an accurate measurement of the local lattice spacing, while a combination of close-coupled scattering calculations and Monte Carlo ray-tracing simulations reproduce the main variations in diffracted intensity. Subsequently, the diffraction results are used to enhance image contrast by measuring at different points in reciprocal space. The results open up the possibility for using helium microdiffraction to characterize the morphology of delicate or electron-sensitive materials on small scales. These include many fundamentally and technologically important samples which cannot be studied in conventional atom scattering instruments, such as small grain size exfoliated 2D materials, polycrystalline samples, and other surfaces that do not exhibit long-range order.

Two-dimensional systems including surfaces, single atomic layer materials, and surface localized states are all of current interest due to the unique structural, electronic, and mechanical properties that they can exhibit [1][2][3].Examples include the now ubiquitous graphene [4], a multitude of related and layered materials such as hexagonal boron nitride [5] and the transition metal dichalcogenides [6], and topologically protected materials such as bismuth antimonide [7].To study the structure of such materials, techniques are required that are sensitive to the outermost atomic layer.
However, most surface-sensitive techniques, such as low-energy electron diffraction (LEED) and x-ray photoelectron spectroscopy, face a challenge in their penetration depth of up to about 10 nm.As a result, the substrate or bulk material can dominate the data and make it challenging to detect any adsorbed species, including contaminants.Electron and x-ray-based methods can also deposit large amounts of energy and/or charge in the sample, which can compromise sensitive materials.
Thermal energy atom scattering, usually using < 100 meV helium atoms, provides the ultimate surface sensitive technique, thus making it ideal for 2D material characterization.Incident helium atoms interact with and scatter from the outermost electrons in the substrate, typically 2-3 Å above the ion cores of the surface atoms [8].Helium diffraction is ideal for studying structure, and more widely helium scattering is well suited to measuring a range of important material properties [9].However, until now, helium scattering has been limited to samples with a high degree of uniformity over at least a few millimeters.It has been impossible to study small samples, including many 2D materials produced by exfoliation, or materials used in situ within devices.Similar limitations apply to polycrystalline materials, where grain sizes are typically in the micron range, or to other nonuniformities, such as an extended phase change across a surface.
SHeM uses a thermal helium beam that has been focused or collimated to micron scales, which enables microscopic imaging.The angular resolution of the instrument was enhanced, and a new measurement scheme was developed, enabling complete LEED-like diffraction patterns to be obtained.Our measurements were performed on lithium fluoride (LiF), which has been studied extensively using thermal-energy, atom-scattering, and grazing-incidence fast-atom diffraction [16,17].3D potential energy surfaces, are available that have been obtained by comparison with experiment [18][19][20] which allows the modeling of the results to demonstrate consistency with previous work.
Figure 1 illustrates the core geometry of the Cambridge "A-SHeM" that was used in the current Letter.A broad beam is first produced in a free jet atom source, then is collimated by a fine pinhole to form the helium beam with micron-sized lateral extent incident on the sample.The helium atoms scatter from the surface; those scattered at a specific angle θ pass through the detector aperture into a high-efficiency mass spectrometer [21], the signal from which is used to form each pixels' intensity.
To obtain diffraction patterns, both polar and sample azimuthal angles must be varied while accurately maintaining constant incidence conditions at a specific spot on the surface.The SHeM is designed for high-resolution imaging, utilizing three linear nanopositioners [22,23] and a rotation stage (additional details can be found in Supplemental Material [24]) [25] to manipulate the sample, but has a fixed detector aperture, so it is not currently possible to directly move the detector.Instead, as illustrated in Fig. 1, the polar angle can be scanned by varying the z position of the sample along the incident beam direction [26].The motion results in a corresponding change in the detection angle, as shown by the difference between θ 1 and θ 2 in Fig. 1.Variation of the sample azimuthal angle α was achieved by rotating the sample about its normal.Since the center of azimuthal rotation is determined by the sample manipulator, the lateral coordinates of the measurement point needed to be obtained for each azimuthal position.A rotational transformation matrix was used, based on the position of the true center of rotation (COR) for the stage, in the imaging coordinate system.The coordinates of the COR were determined through the acquisition of a series of calibration images at varying sample azimuthal angles.A detailed description of the method calculating the COR can be found in Appendix A which yielded a maximum error in the tracking of 27 μm.We make an important distinction between the imaging resolution, which is dominated by the helium beam spot size [27] (∼10 μm) and the accuracy of the point tracking in the current implementation (estimated as < 27 μm).An alternative point tracking scheme could allow the study of smaller areas, limited by the beam spot size.
Compared to a typical high-resolution SHeM imaging configuration [28], higher angular resolution was required to resolve distinct diffraction peaks.A detector acceptance angle of θ A ≈ 8°was implemented, along with a lateral spot size of R S ≈ 10 μm, to maintain high signal levels [29].Such conditions were implemented using a custom "pinhole plate," that can be manufactured either by conventional machining or 3D printing [30], and employs the geometry shown in Fig. 1 within the wider SHeM instrument.
For elastic scattering at an outgoing angle θ f , the in-plane momentum transfer is given by where θ i ¼ 45°is the incoming beam angle and k i ¼ 2π=λ is the magnitude of the incident helium wave vector.
The intensity can then be measured as it varies with in-plane scattering momentum transfer for each sample azimuth α, by rotating the sample.Note the subtle difference between our approach to measuring the 2D scattering distribution and a typical diffraction measurement with a broad area detector; instead of measuring out-of-plane momenta transfers, we rotate the sample to obtain in-plane scattering at each azimuthal orientation.
2D diffraction scans are plotted by first transforming the collected data from Iðθ f ; α stage Þ into IðΔK; αÞ, using Eq. ( 1) and a shift in the origin of the α axis to match a principle crystallographic direction of the sample.The data is then plotted as a heat map [31] in polar coordinates with the radial axis being the parallel momentum transfer ΔK, the azimuthal axis being the orientation of the crystal α, and the color matching the logarithm of the detected intensity [33].The resulting plots show the data where each colored patch is a data point with specular nominally in the center.
To demonstrate the method, a LiF crystal was cleaved along its (100) plane to form a 2 × 2 × 1 mm sample, which was mounted on a sample stub and installed in the SHeM sample chamber (∼10 −8 mbar).The low reactivity of LiF enabled the sample to be cleaved ex situ, in air, with the subsequent mounting and installation taking a few minutes [26,34,35].Diffraction measurements were collected at the locations marked in the overview SHeM  azimuths, where clear diffraction peaks are seen, highlighting the strength of these spatially resolved diffraction measurements.In contrast, conventional helium atom scattering (HAS) experiments average over a large surface area, and would therefore be degraded by the defects on the sample that are clearly evident in the SHeM micrograph, Fig. 2(d).
Figure 2(c) shows a complete 2D diffraction data set which was taken at the same point.The pattern was translated in 2D by ð5.5; 3.0Þ nm −1 such that ΔK ¼ 0 corresponds to where the principle axes meet.In reciprocal space, the cubic lattice of the fluoride ions dominate the (100) surface, with the diffraction peaks along the [110] direction being particularly weak when compared to the others.By fitting a 2D Gaussian function and a second order polynomial background to each peak independently, a grid of reciprocal lattice points was found.The mean of the reciprocal lattice spacing Ḡ was calculated along with the statistical standard error on the mean.The real-space lattice constant a can then be calculated The value given for the error on a arises from statistical error and from the uncertainty in the magnitude of the incident wave vector in Eq. ( 1), which in turn arises from uncertainty in the temperature of the nozzle and from the finite parallel speed ratio of the helium beam [36].Our measured lattice constant agrees with the accepted literature value to better than one standard deviation [37].We also note the high level of precision and accuracy despite the relatively low angular resolution of our instrument, a result that arises from the large number of diffraction peaks we can observe natively through the full range of azimuthal angles.A more detailed description of the error calculation and of the fitting procedure is provided in Supplemental Material [24].One notable feature of the diffraction data is the weak specular peak, which, although evident in the individual line scans in Fig. 2(b), is barely visible in Fig. 2(c).The effect is due to a combination of: small misalignments between the sample surface, the manipulator, and the rotation axis, the narrow angular width of the specular beam itself and from the intrinsically weak specular intensity relative to that of the other diffraction peaks [16,38].As the sample rotates azimuthally, the specular transitions in and out of the optimal detection condition, resulting in the weak and variable specular signal that is observed.The effect is amplified in the polar diffraction plot, as the data density around the center is highest.Further information and discussion of the sample misalignment is provided in Supplemental Material [24].
Having identified the location of diffraction peaks in reciprocal space, we can return to imaging of the sample either on, or away from, a diffraction peak.These configurations yield different modes of imaging, with some analogy to light and dark field microscopy.Two such images are presented in Fig. 2, with (d) being taken on one of the most intense diffraction peaks, thus enhancing contrast from small scale topographic changes, crystallographic misalignment, and defects.Figure 2(a) was taken away from a diffraction peak so that the contrast is smaller and largely comes from large scale topographic structure.The latter modality is consistent with typical SHeM imaging and produces micrographs similar to the diffuse topographic contrast seen routinely from technological samples [28].
We can now relate our microscale diffraction data to existing helium-surface potentials, thus both demonstrating consistency with large-area helium scattering and explaining the strong variation observed between diffraction peak intensities along h110i and h100i directions.There are several published He-LiF potentials, derived either from ab initio [39] or semiempirical [19,20] methods.Figure 3(a) shows a series of equipotential contours for the interaction potential reported by Celli et al. [19], although all have similar lateral averages and corrugation values.A combination of scattering calculations and ray tracing were used to synthetically generate the diffraction pattern that would be obtained from a surface with that potential in the SHeM instrument.
The close-coupling method [40] was used to determine the probability of helium atoms scattering with a particular surface-parallel momentum transfer, i.e., into a particular outgoing direction.The method involves expanding both the helium wave function and the surface potential as Fourier series, then coupling them through the Schrödinger equation.We used the iterative method of solution implemented by Manolopoulos and Wyatt [41] with up to 158 diffraction channels, both open and closed, and an integration range of z ¼ 1 Å to 17 Å to achieve convergence.Angular distributions were generated at azimuthal increments of 1.25°.In house, Monte Carlo ray-tracing simulations [42,43] were then applied to generate SHeM angle scans, after adaption of the ray-tracing framework to accept numerically supplied scattering probability distributions.The simulation traces straight line paths of atoms as they scatter off the sample and local environment, matching the physical geometry of the instrument precisely.The simulated data was subsequently analyzed in the same way as the experimental measurements.
Figure 3(b) shows the resulting synthetic helium diffraction pattern using the interaction potential reported by Celli et al. [19].The location and intensity of the simulated diffraction peaks are in excellent qualitative agreement with the experimental data in Fig. 2(b), confirming that the intensity variation results from the local helium-surface potential, which in turn is well described by the model potential.Specifically, the low intensities along the h110i directions are replicated in the simulations, suggesting a high degree of out of plane scattering when compared to the h100i directions.There are small quantitative differences, such as with the second order h100i peaks; these differences are slightly increased if the potential from Celli et al. [19] is replaced with the potential from Riley et al. [20].However, the expected energy of the bound states is another important test of the quality of the potentials, with the potential from Riley et al. providing the more accurate prediction.Therefore, there is a deficiency of the current potentials, since neither can simultaneously predict both the bound state energies and the diffraction intensities accurately.Given that the bound state energies are relatively insensitive to the corrugation of the potential [20], while the opposite is true for the diffraction intensities, further work is needed where both of these aspects are simultaneously considered.However, in HAS it is usually more important to identify the presence and positions of peaks, and thus the orientation and symmetry of the surface lattice.Importantly, the ability to obtain microscale scattering data means helium diffraction can be applied to study a range of lattice properties, without the highly restrictive requirement of needing large single crystals.
In summary, we have demonstrated that SHeM can be exploited to measure diffraction from a micron-scale region of surface with only minor instrumental modifications.The resulting 2D diffraction pattern reveals the local lattice type, orientation, and size directly, and further quantitative analysis can be performed to relate the pattern to the underlying helium-surface potential.New imaging modalities have been demonstrated, made possible by recording scattered intensity at a particular condition in reciprocal space, enhancing surface features.Most importantly, the microscale diffraction opens up the possibility of applying established helium scattering methods to a vast new range of samples-the world beyond materials that can be grown into large single crystals.We expect the method to have a major impact on the ability to study small samples, such as 2D materials, where growth or exfoliation processes limit the sample size to the micron range, and polycrystalline materials where large single crystal domains are impractical.In the latter case, the demonstrated methods enable the identification and imaging of the grain structure directly, even in monolayer materials and with weakly bound adsorbate layers.The unique capabilities of helium scattering, such as the ability to measure electron-phonon coupling constants through diffraction [9], coupled with the microscale resolution through SHeM, suggest the technique will be highly profitable for addressing many experimental challenges in condensed matter to come.
A dataset supporting the presented work is published at [44].Appendix A: Point tracking.-Thecalibration of rotational point tracking on a given sample surface can be performed by first taking SHeM images of an identifiable feature.In the current work, images of a corner of the LiF crystal were taken at periodic intervals around 360°of sample rotation.Figure 4 shows these SHeM images, which were plotted in the coordinate system of the sample manipulators.Assuming that the rotational axis is perpendicular to a completely flat sample, we expect the locus of the corner to trace out a circle.The center of rotation was obtained by fitting a circle to the set of points using the least squares method as implemented by Brown [46].Once the center of rotation is known, any arbitrary point on the sample can be tracked by using a standard rotational transformation around the coordinates of the center of rotation.PHYSICAL REVIEW LETTERS 131, 236202 (2023)

FIG. 1 .
FIG.1.Schematic of the SHeM scattering geometry, illustrating the presented measurement mode, where the detection angle θ to the sample surface normal (dashed red lines), is varied by changing the distance z from the pinhole.To keep the same spot on the sample illuminated during the variation of θ, it is also translated in the x direction.The sample azimuthal angle α is varied to obtain a complete 2D reciprocal space measurement.

FIG. 2 .
FIG. 2. (a) Helium micrograph of the cleaved LiF (100) surface, taken with the reciprocal-space detection condition far from any diffraction peak.A suitable measurement point was then chosen in the image, indicated by the yellow dot.A series of detection angle scans were taken on the spot at different azimuthal orientations; two examples are shown in (b), where diffraction peaks are clearly evident.By taking detection angle scans across the full 360°azimuthal range, the in-plane diffraction pattern shown in (c) was built up.All azimuthal directions were measured independently and are not repeated using knowledge of the lattice symmetry.The pattern was translated by ð5.5; 3.0Þ nm −1 so that ΔK ¼ 0 corresponds to where the principle axes meet.The 2D diffraction pattern (c) obtained from the measurements clearly indicates the cubic arrangement of the LiF crystal, where the diffraction peaks observed along h110i have a lower intensity than those along h100i.The first order peaks in these directions consistently have the highest respective intensities.(d) An image taken at the reciprocal-space detection condition corresponding to the strong ð 1 1Þ diffraction peak, as indicated by the white point in panel (c).Because of the strongly varying scattering distribution at the new detection condition, contrast is significantly enhanced in (d) compared to (a), revealing variations in the cleaved surface.

FIG. 3 .
FIG. 3. (a) Contour plot along[100]  showing equipotential lines of the He-LiF potential from[19].(b) 2D diffraction plot generated with the same method as Fig.2(c), but using data calculated using a set of close-coupled equations predicting scattering probabilities, which were subsequently used in a ray-tracing framework to simulate scattered intensities from angular scans of He-LiF.The lower intensities of the diffraction peaks along h100i as compared to h110i, are in agreement with the experimental data presented in Fig.2(c).
The work was supported by EPSRC Grant No. EP/ R008272/1.The authors acknowledge support by the Cambridge Atom Scattering Centre [45] and EPSRC Award No. EP/T00634X/1.The work was performed in part at the Materials and NSW node of the Australian National Fabrication Facility, a company established under the National Collaborative Research Infrastructure Strategy to provide nano and microfabrication facilities for Australia's researchers.S. M. L. acknowledges funding from MathWorks Ltd.B. L. acknowledges financial support from the CSC and the Cambridge Trust.We would also like to acknowledge constructive discussions with Bill Allison and Paul Dastoor.

FIG. 4 .
FIG.4.Positions marked in green of the LiF sample's tracked corner as it was rotated, with the fitted circle plotted in red and its center of rotation marked in blue.The maximum distance between the real positions of the corner and the fitted circle is 27 μm, with the inset demonstrating one of these deviations from the expected trajectory in more detail.