High resolution imaging of superficial mosaicity in single crystals using grazing incidence fast atom diffraction

Actual single crystals always contain defects that induce local distortions to the periodic arrangement. These defects can take several forms: point defects (interstitial atoms, vacancies), linear (dislocations), two-dimensional forms (stacking faults, antiphase boundaries), and three-dimensional forms (amorphous cluster, precipitates) [1–4]. These defects also induce local modifications in the electronic structure, which can affect the physico-chemical properties considerably [5]. Mosaicity is one of the fundamental defects most often present in single crystals. The concept was first introduced to explain the decrease of intensity in x-ray reflectivity studies [6–8]. Mosaic crystals are considered to be formed by a large number of small perfect crystallites slightly disoriented with respect to each other. This is due to an arrangement of dislocations in a pattern separating the crystal into blocks [9, 10]. The spread of misalignment angles of the crystal blocks gives the degree of mosaicity of the material. Generally, mosaicity can be described in terms of two crystallographic parameters: the out-of-plane misorientation called the tilt mosaicity and the in-plane misorientation called the twist mosaicity [11, 12].

In x-ray diffraction, the tilt and twist mosaicities are given by the FWHM of symmetric and asymmetric ω-scan rocking curves respectively [13–16]. However, additional structural imperfections and instrumental broadening also affect the linewidth [17–20], imposing the requirement of measurements for several diffraction orders. Such treatment requires a comparatively high precision and the values determined are often marred by considerable uncertainty. Furthermore, because of the penetration of x-rays in solids, the results are insensitive to the superficial layer, even in grazing geometry where a typical depth around 100 Å is probed. Local microscopies such as STM or AFM are excellent tools for characterizing such crystalline distortions but usually on a comparatively reduced scale. For instance, observation of the twist mosaicity of large domains requires maintaining atomic resolution over large scan distances, which is difficult to achieve with current tools.

In this paper, we present a quantitative characterization of superficial mosaicity, using an exclusively surface sensitive technique: grazing incidence fast atom diffraction (GIFAD). This technique is based on the coherent scattering of helium atoms at energies in the keV range and at grazing incidence angles around 1°. In this geometry, the incident atom energy normal to the surface is below 1 eV, thus precluding penetration below the surface.

All experiments were performed in an ultrahigh vacuum chamber with a base pressure in the 10⁻¹⁰ Torr range. He⁺ ions are extracted from an ion source at 450 eV energy; they are subsequently neutralized by charge exchange in a He cell. Then two variable apertures are used to reduce the beam divergence and the beam size to less than 1 mrad and 100 μm respectively. Under the specific experimental conditions used here, the divergence of the direct beam is 0.3 mrad (0.017°). The beam scattered from the sample, and a small fraction of the direct beam, reach a detector located at L = 865 mm. The latter is of made up of microchannel plates and a phosphor screen; images are recorded by a high resolution CCD camera.

For calibration purposes, the sample is not fully inserted in the beam, allowing a small fraction of the incident beam to hit the detector directly without interacting with the surface. Alternately, the exact beam position can be recorded before or after that of the diffraction pattern with the sample surface in a fully retracted position. A distance dx on the images corresponds to a scattering angle dθ via the small angle formula $\mathrm{d}\theta =\sin \left (\frac{\mathrm{d}x}{L}\right )\approx \frac{\mathrm{d}x}{L}$. The projected lattice parameter a_jkl of a given crystallographic direction 〈jkl〉 is simply given by the distance (angle) between two diffraction peaks $\mathrm{d}\theta =\frac{{k}_{j k l}}{{k}_{0}}$, with ${k}_{j k l}=\frac{2 \pi }{{a}_{j k l}}$ where k₀ is the incident momentum of the helium projectile. The intensity distribution of the diffraction spots (the form factor) reveals the shape of the scattering interaction. In the present case, the interaction responsible for the helium scattering is, to first order, the Pauli repulsion occurring when the helium atom penetrates the surface electronic density. In essence it is the same interaction that pushes away the tip of an AFM. In practice, the two techniques are very much complementary, GIFAD is a reciprocal space technique providing atomic resolution simultaneously on mm² surface (see e.g. [21–23]), whereas AFM or STM allow local information (nonperiodic) such as boundaries to be imaged directly. The commercial NaCl(001) sample, 10 mm × 10 mm in size, was air cleaved and then rapidly introduced into the vacuum chamber; it was further annealed in UHV at 600 °C.

Figure 1 displays diffraction images obtained under an incidence angle close to 0.9° and corresponding to a primary beam aligned along $[\bar {1}1 0]$ and [100], the two main surface channeling directions. It is worth recalling that GIFAD does not allow for exchange of reciprocal lattice vectors along the beam direction. This is due to the fact that the coupling strength is too weak and the associated energy too high [22, 24]. As a result, only one Laue circle intercepting the diffraction spots and the direct beam can be observed. Each image contains two types of information, one related to the macroscopic orientation of the surface and the other connected to the microscopic atomic structure.

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The peak spacing yields reciprocal vectors of 1.57 ± 0.02 Å⁻¹ and 2.22 ± 0.02 Å⁻¹ which correspond well to the projected lattice parameter along the $[\bar {1}1 0]$ and [100] directions of the NaCl(001) surface respectively. These correspond to a lattice parameter of 3.99 ± 0.07 Å and 2.83 ± 0.07 Å in good agreement with the recommended values of 3.98 and 2.82 Å. The geometric description of each image can be summarized with three points: the beam position B, the specular spot position S and the center C of the Laue circle. The line BS gives the direction of the incidence plane which contains the surface normal, while its perpendicular bisector is the projection of the surface plane and contains C. When the beam is perfectly aligned along one of the sample crystallographic axes, C is located on BS.

The intensity of the spots is given by the form factor that, within the hard-wall description [25], is simply the modulus of a Fourier-like transform of the corrugation function. For the simplest corrugation function, a cosine function $z=h\cos \left (\frac{2 \pi }{{a}_{j k l}}y\right )$, the diffracted intensities I_n are given by Bessel functions ${J}_{n},{I}_{n}={J}_{n}^{2}\left (2{k}_{n}\times h\right )$, where z is the elevation on top of the point y,h the corrugation half-amplitude, a_jkl the projected lattice parameter and k_n the wavenumber normal to the surface k_n = k₀sin(θ). Here, the observed intensities are well fitted by such a function, indicating a corrugation half-amplitude h = 0.29 Å, consistent with the value measured with thermal helium scattering [26]. This corrugation amplitude is the elevation difference between the top and bottom of the valley; it is an intrinsic property of the pure NaCl surface and depends only on the projectile normal energy. This corrugation should not be confused with the mean corrugation resulting from defects at the surface and defined as a FWHM in an AFM or STM scan. More importantly, the measured intensity distribution is also extremely sensitive to the angle of incidence (the radius of the Laue circle) and to the angle of misalignment between the beam and the crystallographic axis, so the analysis provides additional evidence of any in-plane misalignment.

Despite the small beam size, the grazing geometry results in a beam length on the sample of several millimeters. At 0.9° incidence, this full length is typically 6 mm, though the actual length is smaller because the sample surface is not fully inserted into the beam; thus a small fraction of the incident beam reaches the detector and provides an angular reference. GIFAD patterns of the same sample but at a different location on the sample surface, recorded under an incidence angle close to 0.5°, are shown in figure 2. At this angle, the projected length of the incident He beam on the surface is nearly twice what it is at 0.9°.

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The image in figure 2(a) was obtained from the $[\bar {1}1 0]$ axial surface channeling; it shows an unexpected superposition of diffraction patterns. One can clearly distinguish a series of diffraction spots, forming well defined Laue circles whose centers are aligned in the specular plane (see schematic representation figure 3(a)). The different diameters are associated with different effective angles of incidence, indicating the presence of different domains within the surface illuminated by the atomic beam. The fact that they appear at different incidence angles indicates that these domains are tilted with a component of the rotation vector located in the surface plane and perpendicular to the beam direction, i.e. parallel to the 〈100〉 direction. Similar patterns are observed when the beam is oriented along the [110] axis. The profile through the line in figure 2(a) is shown in the inset; the multipeak structure is very well described by a regular series of peaks spaced by 0.025° ± 0.005°. We observe similar features for cleaved KBr(001) samples, with a characteristic tilt angle comparable to the value derived for NaCl(001).

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The image in figure 2(b), recorded with the incident beam along the [100] direction, does not contain any sign of tilted domains with rotation vector parallel to the 〈110〉 direction.

The sensitivity of the diffraction pattern to mosaic structures of the surface is illustrated in figure 3. We depict the presence of tilted domains, for the rotation axis normal and parallel to the beam, in figures 3(a) and (b) respectively. These schemes describe well the diffraction images of figures 2(a) and (b); along the $[\bar {1}1 0]$ direction, the fit to the data in figure 2(a) yields α = 0.025°. When the incident beam is parallel to the rotation angle of the tilted domains (see figure 3(b)), Laue circles arising from each domain are simply rotated around the beam position by an amount γ equal to the angle between blocks. Then the separation between subsequent spots is given by γ × D where D is the diameter of the Laue circle. Thus rotation angles below one degree would be difficult to resolve. In fact the quasi-Gaussian broadening, within the Laue circles, of the diffraction spots observed in figure 2 could well originate from such a mosaicity. Assuming that images in figure 1 originate from a single domain and that the extra broadening in figure 2 is due to rotation of the domains around the incident beam direction, an angular distribution with 1.2° FWHM is deduced.

A third situation can be encountered when the atomic beam interacts with twisted mosaic domains; this is shown in figure 3(c). Here the Laue circles arising from the twisted domains are displaced by a distance d_ξ proportional to the twist angle. An example of this type of mosaicity is given in figure 3(d); it was recorded on a cleaved KBr(001) crystal with the beam along the [110] direction.

In the framework of quasi-elastic scattering, which is the case in GIFAD [25], and if we assume that the surface is perfectly flat without defects, the diffracted intensity distributions presented in figures 2–4 would not be allowed according to the energy conservation law. Thus such structured intensity distributions could only be related to topographic features of the surface.

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Features similar to those shown in figure 2(a), with comparable tilt angle values, have been observed using high resolution x-ray diffraction for cleaved NaCl(001) [27] and RbI single crystals [28]. Discrete mosaic distributions have also been reported for thin pseudomorphic ZnSe films on GaAs(001), above a critical thickness of 100 nm [29, 30]. In the latter studies, the tilt angle of the mosaic domains, derived from the positions of satellite spots in SPA-LEED measurements, is found to be 0.2° with respect to the macroscopic plane, which is only slightly larger than the resolution limit in SPA-LEED. With GIFAD, tilt angles as small as 0.01° can be measured; limitations arise only from the divergence of the primary beam. The latter can always be improved, but at the expense of a reduced atom flux that would increase the acquisition time; the data shown here were typically acquired in 10–100 s.

The highly regular distribution of tilt mosaic angles described in figure 2(a) is present on a large fraction of the cleaved NaCl(001) sample surface in the form of patches having a size typically between 100 μm and 1 mm. We also observe, in addition to this patch structure, a strong anisotropy in tilt mosaicity. Along the $[\bar {1}1 0]$ and [110] directions, the diffraction images bear the signature of tilted domains that have a finite component of their in-plane rotation vector normal to these directions. Diffraction images measured along the [100] direction instead show that the domains have no rotation component in this direction. Consequently, we derive that the tilt vector should be parallel to the 〈010〉 direction.

As regards the origin of the observed mosaicity, some speculation can be advanced from comparing samples having different shapes. Using the same sample preparation procedure, we clearly observe that round samples yield more mosaicity than square ones. This could originate from different responses to the cleavage or, alternately, the increased mosaicity could be a response to the mechanical constraint accumulated during the machining of the round samples.

Finally, this quasi-uniform distribution of the tilt mosaic domains within patches appears to be a robust characteristic of some ionic crystals and points to some kind of self-organization of these domains.

To focus on this aspect, the detailed diffraction pattern is not necessarily useful and a so called 'random direction' is more efficient, as shown in figure 4. Here, the probe beam is off the axis with respect to the main crystallographic directions, exactly 7° away from the $[0\bar {1}0]$ direction. The surface corrugation (in terms of the valence electron density), as averaged by the beam, is very flat, so only the zeroth order is observed. As in figure 2(a), a series of equally spaced diffraction spots can be observed, each spot corresponding to specular reflection from a single mosaic domain. A projection along the vertical axis shows two regions, each described in terms of series of peaks equally spaced by 0.027°; however these two regions (marked I and II in figure 4) are phase shifted by a half-period. The typical domain size can be inferred by monitoring the intensity variation of a given Bragg spot while moving the beam in the transverse direction across the sample surface; these measurements indicate a main domain size of a few 100 μm, and this is consistent with the number of illuminated domains at a given incidence angle.

In conclusion, we have demonstrated that grazing incidence fast atom diffraction (GIFAD) can provide a microscopic as well as long range structural analysis of the topmost surface layer. In addition, it can provide a clear signature of surface topographic imperfections. Qualitatively, and quantitatively, we have reported a first experimental characterization of the superficial mosaicity of a NaCl(001) single crystal. Superficial mosaic structures of two types have been observed; (i) twist mosaic structures defined by macroscopic blocks lying parallel to the (100) surface but slightly rotated from one another along the [100] direction, and (ii) tilt mosaic domains with a misalignment angle that is a multiple of an elementary angle, equal to 0.025°.

Because GIFAD is a particularly fast technique than can be operated in situ during thin film growth, it may become a valuable tool for following in real time the formation dynamics of mosaic domains in constrained multilayered systems.

This work was supported by the ANR under contract number ANR-07-BLAN-0160-01.