U. Welzel

Stress analysis of polycrystalline thin films and surface regions by X-ray diffraction

The components of the macroscopic mechanical stress tensor of a stressed thin film, coating, multilayer or the region near the surface of a bulk material can in principle be determined by X-ray diffraction. The various analysis methods and measurement strategies, in dependence on specimen and measurement conditions, are summarized and evaluated in this paper. First, different X-ray diffraction geometries (conventional or grazing incidence) are described. Then, the case of macroscopically elastically isotropic, untextured specimens is considered: from the simplest case of a uniaxial state of stress to the most complicated case of a triaxial state of stress. The treatment is organized according to the number of unknowns to be determined (i.e. the state of stress, principal axes known or unknown), the use of one or several values of the rotation angle φ and the tilt angle ψ of the sample, and one or multiple hkl reflections. Next, the focus is on macroscopically elastically anisotropic (e.g. textured) specimens. In this case, the use of diffraction (X-ray) elastic constants is not possible. Instead, diffraction (X-ray) stress factors have to be used. On the basis of examples, it is demonstrated that successful diffraction stress analysis is only possible if an appropriate grain-interaction model is applied.

The “state of the art” of the diffraction analysis of crystallite size and lattice strain.

Abstract This paper addresses both old, but “renovated” methods and new methods for diffraction line-profile analysis. Classical and even extremely simple single-line methods for separating “size” and “strain” broadening effects have merit for characterization of the material imperfectness, but it is generally very difficult to interpret the data obtained in terms of microstructure parameters as used in materials science. Developments of recent years, focusing on distinct anisotropic line-broadening effects, as due to the type, orientation and distribution of dislocations and minute compositional variation, will be touched upon. The most promising development may be the synthesis of line profiles on the basis of a microstructure model and application of the (kinematical) diffraction theory without any further assumption, which contrasts with the other methods. This approach can in principle be applied in single-line and multiple-line variants and also in analyses of the whole diffraction pattern. The advantage is the direct evaluation of microstructure parameters as used in materials science. The challenge is to develop microstructure models which are flexible enough to be applicable in more than one case ...

Local, submicron, strain gradients as the cause of Sn whisker growth.

It has been shown experimentally that local in-plane residual strain gradients occur around the root of spontaneously growing Sn whiskers on the surface of Sn coatings deposited on Cu. The strain distribution has been determined with synchrotron white beam micro Laue diffraction measurements. The observed in-plane residual strain gradients in combination with recently revealed out-of-plane residual strain-depth gradients [M. Sobiech et al., Appl. Phys. Lett. 93, 011906 (2008)] provide the driving forces for whisker growth.

Driving force for Sn whisker growth in the system Cu–Sn

The evolution of residual stress gradients in Sn thin films on Cu substrates upon aging at ambient temperature has been investigated, for specimens which do exhibit and which do not exhibit Sn whisker growth, by performing x-ray diffraction stress measurements at constant penetration depths. Comparison of the measured near-surface stress-depth profiles for both types of specimens, as function of aging time at ambient temperature, showed that a significant negative stress gradient from the surface toward the Sn∕Cu interface is decisive for Sn whisker growth in the system Cu–Sn.

Diffraction stress analysis of macroscopically elastically anisotropic specimens: On the concepts of diffraction elastic constants and stress factors

A unifying, rigorous treatment has been given for the diffraction stress analysis of both macroscopically elastically isotropic specimens and macroscopically elastically anisotropic specimens. The applicability ranges of so-called diffraction elastic constants and diffraction stress factors have been discussed. It has been shown that the diffraction stress factors, originally introduced for the diffraction stress analysis of textured specimens, can be generally used in the diffraction stress analysis of macroscopically elastically anisotropic specimens exhibiting direction-dependent grain interaction, independently of the presence of texture. The grain-interaction tensor has been introduced and investigated for various grain interaction models. The notion surface anisotropy of bulk specimens has been revisited as a special case of direction-dependent grain interaction.

A method for the non-destructive analysis of gradients of mechanical stresses by X-ray diffraction measurements at fixed penetration/information depths

A rigorous measurement strategy for (X-ray) diffraction stress measurements at fixed penetration/information depths has been developed. Thereby errors caused by lack of penetration-depth control in traditional (X-ray) diffraction (sin2ψ) measurements have been annulled. The range of accessible penetration/information depths and experimental aspects have been discussed. As a practical example, the depth gradient of the state of residual stress in a sputter-deposited nickel layer of 2 µm thickness has been investigated by diffraction stress measurements with uncontrolled penetration/information depth and two controlled penetration/information depths corresponding to about one quarter and one tenth of the layer thickness, respectively. The decrease of the planar tensile stress in the direction towards the surface could be well established quantitatively.

Diffraction analysis of internal strain-stress fields in textured, transversely isotropic thin films: Theoretical basis and simulation

Abstract Polycrystalline films produced via physical vapour deposition often possess a microstructure composed of columnar oriented grains. From the mechanical point of view, they cannot be treated as isotropic bulk solids; rather their elastic macroscopic behaviour is only transversely isotropic. In this context, the effect of texture on the elastic response of such films has been investigated for different grain interaction models. In particular, the determination of macroscopic stress by X-ray diffraction methods has been analysed. Traditional grain interaction models such as those due to Voigt and to Reuss have been compared with that proposed by Vook and Witt, compatible with the presence of only transverse isotropy of the body (even in the absence of crystallographic texture). By simulation, it has been demonstrated that, although texture only moderately affects the values of the macroscopic mechanical elastic constants of the transversely isotropic body, it can pronouncedly influence the results from the X-ray diffraction measurements. This effect is shown to depend strongly on the type of grain interaction.

Crystallite size dependence of the coefficient of thermal expansion of metals

The coefficients of thermal expansion (CTEs) of polycrystalline Ni and Cu thin films have been investigated by employing temperature-dependent x-ray diffraction measurements of lattice parameters. Great care has been taken to exclude effects of, in particular, microstructural relaxation and mechanical stresses on the dependences of the lattice parameters on temperature. The CTEs determined in the as-deposited condition, characterized by grain sizes in the range of 25–35nm, are considerably (about 10%) larger than the corresponding literature values of bulk materials. Heat treating the specimens at moderate temperatures induced grain growth and decrease of the crystalline imperfection. After the heat treatment, the CTEs determined for the thin films had reduced considerably and had become equal to (Ni) or approached (Cu) the corresponding literature data for bulk materials.

Use of polycapillary X-ray lenses in the X-ray diffraction measurement of texture

Corrections for instrumental aberrations of X-ray diffraction texture measurements (pole-figure measurements) conducted in quasi-parallel-beam geometry using an X-ray lens have been investigated on the basis of measurements on (texture-free) reference samples. It has been shown that a defocusing correction, which is a major correction in the case of pole figures recorded with divergent-beam geometries, is not necessary when a parallel beam, produced by an X-ray lens, is used. In this case, the major instrumental sources of error stem from the illumination of areas outside the sample surface, i.e. the finite sample size, and the finite area of the detector, both giving rise to a reduction of the recorded signal. Two correction procedures for this reduction, an experimental one and a numerical one, have been tested and are described.

Interdiffusion, phase formation, and stress development in Cu-Pd thin-film diffusion couples: interface thermodynamics and mechanisms

Cu–Pd thin-film diffusion couples (individual layer thicknesses of 50nm) have been prepared by dc-magnetron sputtering on silicon substrates coated with a thin amorphous Si3N4 layer. Stress evolution, microstructural development, and phase formation during interdiffusion have been investigated employing Auger-electron spectroscopy (in combination with sputter-depth profiling), x-ray diffraction, wafer curvature measurements and transmission electron microscopy. Upon annealing at relatively low temperatures (175–250°C) for durations up to 10h, considerable diffusional intermixing occurs. Interdiffusion is accompanied by sequential formation of new phases. First, Cu3Pd forms; subsequently, CuPd forms and grows at the expense of Cu3Pd, which has been interpreted as a consequence of interface thermodynamics. Annealing leads to a slight sharpening of the pre-existing {111}-fiber textures and a little increase in the average grain size. A combination of ex situ (x-ray diffraction) and in situ (wafer curvature) ...