Nanoscale tensile strain in perfect silicon crystals studied by high-resolution X-ray diffraction Free

There are several ways to change distances between atoms in crystals. Mechanical strain,^1,2 temperature, and chemical composition^3,4 are examples; each method has its advantages and disadvantages depending on its purpose. Perfect crystals are key components of X-ray optics. Bragg’s equation, λ = 2d sin θ, gives the diffraction angle θ for scattering at wavelength λ from lattice planes with spacing d. In structural analysis, λ and θ are the parameters normally varied to maximize the experimental performance while the d spacing, determining the Bragg diffraction planes, is considered constant. High-resolution X-ray diffraction (HRXRD) techniques can be used to detect strain and defects in “perfect crystals” and to measure atomic distances. The term “perfect single crystal” (PSC) is used here to describe single crystals with an extended defect concentration below 10⁴/cm²; the intrinsic X-ray rocking curve (RC) FWHM of these crystals does not differ from that calculated with the dynamical theory.^5,6

A PSC with a lattice deformation controllable by a uniaxial tensile stress^7,8 could be used in X-ray optics as a tunable monochromator. With the appropriate sample geometry, it should be possible to obtain a controllable strain field producing a range of d spacing appropriates for a given application. For example, an X-ray monochromator with an appropriate value of the d spacing for one set of Bragg planes could be used to select a single wavelength from a divergent polychromatic beam without a need to change the incident angle. Another application of such crystals would be to have a crystal with the d spacing slightly different from its natural value, e.g., to tune energy around θ Bragg at 90°.⁹ 

The main difficulty here is spurious external and internal sources of strain that perturb the intended gradient or uniform field. The main source of spurious or unintended strain is normally the grip mechanism holding the sample. Those effects can be attenuated by contact-free laser-based excitation,¹⁰ by using an electromagnetic stress, or by gluing the sample (with proper alignment) onto the stressing device.¹¹ Extended structural defects,¹² dopant concentrations, surface finish, and notches or corners¹³ are also important spurious sources.

Precise stress-strain measurements for structurally well-characterized perfect silicon crystals are difficult to find in literature. HRXRD can render such measurements possible on the nanoscopic (i.e., angstrom-length) scale of Bragg plane spacing. Stress-strain curves⁶ from such measurements may not agree between measurements made, for instance, with microscopic and macroscopic samples of the same material as a result of the size effect^14–18 (see below).

At the nanoscopic or atomic scale, the crystal homogeneity is broken and possesses both orthotropy and inhomogeneity, conditions under which anisotropic elasticity is used to describe macroscopic mechanical properties of crystals.¹⁹ In this case, the tensorial description must be used to describe its mechanical properties. Because the silicon crystal cell is cubic, only 3 components of the tensor must be determined. The total sample length variation was not measured, and consequently, it is not possible to assure that it is proportional to the corresponding lattice variation.

It is difficult to find measurements of Δd/d in the literature for small values of induced strain in silicon under tensile stress. Most recent papers on the mechanical properties of silicon single crystals are motivated by the needs of the electronic industry, such as for the failure strain point.^20,21 In addition to needed information on the samples’ precise crystallographic orientation, the lack of structural characterization of the silicon crystal that is used in each experiment is a fundamental fault of many such papers. The temperature, the concentrations of extended defects,¹² dopants,²² oxygen,²³ the surface finish quality, the precision of the applied force, the sample dimensions, and the strain field uniformity in the measured region are examples of specifications required to compare experimental and theoretical results²⁴ or to compare different experiments. The strain rate of the test procedure is an important point,^24,23 often forgotten. Experimental stress-strain results of mechanical properties of materials are usually made on the traditional universal testing machine (UTM),⁸ typically with large samples requiring large forces on the sample-holding grips, introducing spurious strains that compromise the results.

It is well known that the size effect gives different experimental measurements of stress-strain results for silicon whiskers^25,26 and bulk samples.²⁸ The dependence of stress-strain measurements on the size scale is an important point in comparing such measurements to theoretical predictions^20,29,30 made by ab initio calculations^31–33 such as molecular dynamics.²¹ Recent studies of structural plasticity under tensile load, specifically on metallic³⁴ nanocrystals by TEM and silicon nanocrystals by X-ray diffraction,³⁵ show the importance of surfaces in the mechanical behavior of such materials. The area by volume ratio of a sample is higher for smaller samples, and the presence of a surface is a kind of defect or strain on the first few layers (see Fig. 6).²⁷ 

An understanding of the relation between the macroscopic and the nanoscopic (lattice) strain response to external uniaxial stress in PSCs could validate theoretical approaches that extrapolate to macroscopic samples, using the properties of a unit cell having variable atomic bond lengths and angles³⁶ under external stress.

In principle, an ideal tensorial model should be valid for both the unit cell of a crystal and for a macroscopic ideal perfect crystal. The relation between what Segmüller and Neyer³⁷ called internal strain (lattice) and external strain (macroscopic sample) should be clarified. Cousins³⁸ and Cousins et al.³⁹ determined, theoretically and experimentally, the internal elastic constants of silicon. The results of the present work, the variation of atomic plane distances with external tensile strain, will not be used here to determine Young’s modulus of silicon with high precision because these results may not represent the sample’s macroscopic mechanical behavior. Furthermore, the results presented here give nonlinear stress-strain curves, suggesting a nonlinear approach to elasticity theory is needed in this case. This nonlinearity may be intrinsic to this kind of experiment and result from problems such as with the bonding polymer or poor sample alignment. Assuming that the underlying stress-strain relation is actually linear, the calculated Δd/d is nonetheless of the same order of magnitude as calculated with tensorial elasticity theory.

When the macroscopic description is used in this work, it is assumed that the sample length variation in each direction is proportional to the corresponding lattice variation for the applied force, since what is measured by X-ray diffraction here is the lattice parameter variation.

In this report, the design, construction, and testing of a mechanism to apply uniaxial tensile strain to a sample that is illuminated by an X-ray beam are described. Such a device is mounted on the second axis of a double-axis X-ray diffractometer, and the variation of the Bragg condition, that is, the distance between atomic planes, in the quasistatic condition is determined by rocking curve measurements. The goal is to obtain a controllable strain field in a PSC to allow its use in X-ray optics and in precise measurements of the crystal’s mechanical properties.

The design of the strain device required a small size and low mass to allow it to be attached to the second axis of our double crystal diffractometer and to be fabricated in hard steel. The tensile force is applied to the sample by using a calibrated spring that is pulled by a differential screw especially made for the apparatus, with a reproducibility of 5.825 µm per turn of the slow screw that pulls the spring, and a total range of 10.0 mm. With a calibrated spring (k = 28.1 N/mm), a maximum force of 280.4 N may be applied.Figure 1 shows a schematic drawing and Fig. 2 shows the apparatus photo. The center of the sample is vertically positioned on the center of the diffractometer’s second axis. This device is similar to the apparatus described by D’Amour et al.,⁷ designed to be used on a four-circle diffractometer, but is different from the Nanox stress rig⁴⁰ that was designed for polycrystalline samples and imaging. The strain device described here is to be used for single crystals and to be attached to a high-resolution double-axis diffractometer with 0.06 arc sec per step on each axis. Experiments of mechanical properties of materials, usually made on the traditional universal testing machine (UTM), normally with large samples, when strong forces are necessary on the grips that hold the sample, may introduce spurious strains that compromise the results. In measurements normally made with UTM equipment (see Ref. 8) the initial values of the stress-strain graphic are disregarded as a consequence of the way this equipment is operated. The initial force is used to eliminate the mechanical backlash proper of its construction (UTM) and to align the sample. For silicon samples, stress values below one GPa are not considered. The present apparatus can be used to measure forces as small as 0.17 N and tensions below 260 MPa, that is, the results presented here should not be compared with strain measurements made with values above 1 GPa.

The sample design was a dog-bone shape (Fig. 3) to avoid corners and notches that are sources of spurious strain (Fig. 9). The silicon samples were made by Finnlitho Ltd. (Finland) from a well characterized [100] and 200 μm-thick silicon wafer. The samples were cut by laser ablation with edge uniformity (notches and corners) better than 20 µm. The small holes in one sample were introduced to avoid nonuniform tension from the grips reaching the central regions where the measurements are made. Both samples were tested for measurements but with equal results because the mounting is the dominant source of spurious strain and not the grips. For these reasons, only samples without the small holes were used on the measurements. For the Bragg-Bragg (BB) case, the (400) symmetric diffraction Bragg plane was used with a calculated convoluted Darwin width (FWHM) of 4.8 arc sec and an absorption depth of 19.8 µm for the CuKα radiation was used, calculated with the XOP code.⁴¹ For the Bragg-Laue (BL) case, the plane (040) was used with the second crystal, the calculated FWHM Darwin width was 4.3 arc sec, and the absorption depth (Laue case) was 28.6 µm (Figs. 4 and 5).

The sample was carefully bonded at both ends onto the stress device with polymer adhesive. The increase of the RC FWHM with the applied tensile strain was the main criterion used to evaluate the quality of the strained PSC as an X-ray optics device.

Simulation with finite element modeling (FEM) was carried out to clarify the influence of spurious strain sources on the uniformity of the strain field, especially in the sample’s central region. Figure 6 shows the simulation when a uniform tensile force is applied on the sample ends. Spurious sources of strain-like torsion (Fig. 7), out-of-plane misalignment (Fig. 8), or possible sample cutting irregularities [see Fig. 9(a)], such as small notches [Fig. 9(b)] or protrusions [Fig. 9(c)] on the edge of the sample, show that in the surface region near those irregularities, the tensile strain deformation is nonlinear and that the tensile strain field can be uniform only in the absence of such defects. The simulations also show that the sample’s narrow neck should be as long as possible to minimize strain gradients in the central region from the grips.

A conventional CuKα (8.047 keV) X-ray source was used. The measured increase in the convoluted Darwin FWHM with tensile strain was used to evaluate the quality of the illuminated region (see Fig. 11). The Bragg case first crystal was mounted on a small translator that is itself mounted to allow adjustment of the tilt angle. This set is fixed on the diffractometer’s first axis.

In situ quasistatic measurements were done with double-crystal X-ray diffraction in nondispersive BB and BL modes. In the BB case, what is measured is the variation of the atomic distance within the first microns of depth penetrated by the X-ray beam; this variation with strain should be negative, that is, the transverse width (Z direction) should reduce with the applied longitudinal force (Poisson effect). In the BL case, because the diffracted beam passes through the whole sample thickness, the atomic distances within the sample bulk are measured and the variation is positive perpendicular to the diffracting crystal planes (Y direction) and parallel to the external strain direction.

To test the reliability of the diffractometer’s second axis stepping motor, which drives the angular Δθ scale (scan scale), a reference crystal was attached on the fixed support near the sample; it possesses the same diffracting planes and was oriented nearly parallel to the sample. The incident beam illuminated both crystals (Fig. 10): the unstrained reference crystal (only one side bonded) and the minimum tensile-strained crystal (both sides bonded). The angular distance between the diffraction peaks (Fig. 11) indicates the nonparallelism between the reference crystal and the sample. After several tests, it was concluded that it was reliable to use the number of motor steps for the RC angular scan scale. The set-up’s thermomechanical instability during the measurement time, 30 min for one complete strain-stress cycle of data collection, was less than 2.0 arc sec.

Measured RCs for the BB and the BL configurations (Fig. 12) provide typical examples of such measurements, where Δθ is the change in the angle between the two RCs resulting from application of stress ΔF.

From elasticity theory for macroscopic samples, for the Bragg scattering geometry, the calculated size variation Δdz in the direction normal to the sample blade’s face and normal to its length is −2.96 µm for an applied tension of 100 MPa; the C11 value of 165.6 GPa for Young’s modulus from the recent literature^42,43 was used. For the Laue geometry, the variation Δdy along the blade length, also for 100 MPa applied tension, is calculated to be +47.4 µm. Because of the small expected value of this displacement, no attempt was made to measure it simply and with reasonable precision.

The X-ray beam divergence was around 10⁻⁴ rad, the beam size on the sample was 2 H × 5 V mm², and the temperature was kept at 23.5 ± 0.5 C. For each applied force, a RC is measured, and the relative angular peak position (indicating a change in internal d spacing) becomes one point on a stress-strain graph (Fig. 12).

In all determined stress-strain graphs (e.g., Fig. 12), the dependence of Δd/d on T is nonlinear. It is not possible to distinguish if the relation is quadratic or exponential, in part as a result of the maximum force allowed with the apparatus. Such nonlinearity may be real in the sense that it represents the variation of the atomic distance with the applied force in this strain range [see Eqs. (1) and (2)] or it may be an experimental anomaly because the measurements were done in the region of small tension with high sensitivity (of the order of Δd/d ∼ 10⁻⁵) and below 280 MPa; these are not typical conditions in measurements that show linear dependence. This means that a Young’s modulus cannot be obtained from such results because of the nonlinearity. More results are necessary to clarify the nonlinear dependence of the stress-strain curve for a perfect single crystal at low stress.

A tensile strain device to be attached to a double-axis diffractometer was developed and tested with an applied force up to 280 N. It was used to vary atomic distances in perfect silicon crystals, and the results show that the increase of the Darwin width and the drop in the peak intensity with the applied force are small enough to allow such a stressed crystal to be used for X-ray optics, with Δd/d up to at least 10⁻³. This device could be used to tune a silicon monochromator from CuKα1 to CuKα2 (ΔE = 20 eV), for instance, while keeping a constant incident angle. The main difficulty of the experimental method is to avoid spurious strain introduced by the grips or by the mounting. Because of the nonlinearity in the stress-strain curves, a precision experimental determination of the Young’s modulus of a PSC using this method was not possible. The parameters that influence this kind of measurement should be evaluated in terms of the mechanical structural properties of perfect crystals.

From the nonlinearities seen in Fig. 12, classical elasticity theory may not be appropriate to explain the elastic behavior of PSCs in the tensile strain range explored here.

The tension modification of atomic distances with the proposed PSC device is not meant to substitute for classical monochromators but to be used in special cases where its unique properties could compensate for a small loss of crystalline perfection.

The author gratefully acknowledges CNPq/PQ for research fellowship Grant No. 309614/2013-9. The author is grateful to Wilmar Teixeira for the mechanical project execution of the strain device, Petri Karvinen (Finnlitho Ltd., Finland) for cutting the Si samples, Guilherme Suguinoshita for the FEM results, Herica Selzelin Batista for excellent graphic/editorial work, and Irineu Mazzaro for fruitful discussions.