X-ray diffraction (XRD) introduction

X-ray diffraction (XRD) is an interferometric phenomenon which results from the coherent superposition of x-ray waves scattered by individual atoms when a material is illuminated with x-ray radiation. Within the ‘kinematical theory’ of diffraction [1] the amplitude of the electric field of the diffracted x-rays is given by the sum:

Equation (1)

where fj is the amplitude scattered by the jth atom in the material, and the scalar product Q⋅r j is the phase difference between each scattered wave, Q and r j being the wave vector transfer (also referred to as the scattering vector) and the coordinate vector of the jth atom, respectively. The amplitude of Q is equal to 4π sinθ/λ, where θ is half the diffraction angle and λ is the x-ray wavelength. The diffracted amplitude is extremely sensitive to, even minute, variations in the atomic coordinates, which explains why XRD has been used since decades to study the structure of crystalline materials, their nano- and microstructure and crystalline defects [14]. However, the quantity measured in an actual XRD experiment is the intensity, i.e. the squared modulus of the amplitude, × E*; the phase of the amplitude is therefore lost in the process, which gives rise to the so-called ‘phase problem’, common to all diffraction techniques, which implies that only coordinate-differences can be obtained via a direct inversion of the intensity. To circumvent this issue, scientists usually rely on modelling and fitting algorithms where a parameterized physical model is iteratively adjusted until a satisfactory agreement with the experimental data is obtained 3 . It should be noted that, for the sake of simplicity, we rely on the kinematical theory of diffraction to present the phase problem, whereas all calculations in the following are performed using the more accurate ‘dynamical theory’ of diffraction. However, as far as the phase problem is concerned, the conclusions drawn using the kinematical theory remain perfectly valid within the dynamical theory.

One important application of XRD is the determination of the spatial distribution of strain in disordered materials, like for instance, those submitted to ion irradiation. Being able to precisely characterize these materials is of critical importance in many fields, for instance to understand strain and defect formation during ion implantation in the semi-conductor industry [5], a family of materials for which XRD is particularly well adapted [6]. This aspect is especially important in this period of time, where increasing the semiconductor supply chain resilience against international crises is now a strategic priority in Europe and the United States of America. For instance, in the USA, the recent CHIPS and Science act 4 will allocate $280 billion in the next 5 years to the semiconductor industry, among which $174 billion will be devoted to investments in Science, Technology, Engineering and Math programs, workforce development, and research & development. A similar, though less ambitious, €43 billion program will be announced in Europe as well 5 . In this context, the development of high precision in-line metrology systems, while maintaining high throughput, is an important challenge to face, and machine learning-based algorithms are here expected to play a crucial role.

The determination of spatial strain profiles in strained materials relies, as mentioned above, on the simulation of experimental XRD data with physical models [714]. These simulation methods are in general based on classic least-squares optimization algorithms which require a good guess of the initial parameters defining the strain profile in order to avoid being trapped in a local minimum of the least-square error hypersurface. Obtaining a physically sound guess may often require several tens of minutes, up to several hours, depending on the complexity of the problem. In any case, it requires a priori knowledge about the system, and the supervision of an expert, in order to set constraints on the fitting parameters during the optimization procedure, so as to avoid the emergence of unphysical solutions. Global search algorithms such as, for instance, simulated annealing [15], may solve the local minimum issue but those are intrinsically slower than usual local minimization techniques.


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