Comprehensive Recovery of Point Defect Displacement Field Function in Crystals by Computer X-ray Diffraction Microtomography

Abstract

In the case of the point defect in a crystal, the inverse Radon’s problem in X-ray diffraction microtomography has been solved. As is known, the crystal-lattice defect displacement field function f(r) = h·u(r) determines phases − (±h)-structure factors incorporated into the Takagi–Taupin equations and provides the 2D image patterns by diffracted and transmitted waves propagating through a crystal (h is the diffraction vector and u(r) is the displacement field crystal-lattice-defects vector). Beyond the semi-kinematical approach for obtaining the analytical problem solution, the difference-equations-scheme of the Takagi–Taupin equations that, in turn, yield numerically controlled-accuracy problem solutions has been first applied and tested. Addressing the inverse Radon’s problem solution, the χ²-target function optimization method using the Nelder–Mead algorithm has been employed and tested in an example of recovering the Coulomb-type point defect structure in a crystal Si(111). As has been shown in the cases of the 2D noise-free fractional and integrated image patterns, based on the Takagi–Taupin solutions in the semi-kinematical and difference-scheme approaches, both procedures provide the χ²-target function global minimum, even if the starting-values of the point-defect vector ${\mathcal{P}}_1$ is chosen rather far away from the reference up to 40% in relative units. In the cases of the 2D Poisson-noise image patterns with noise levels up to 5%, the figures-of-merit values of the optimization procedures by the Nelder–Mead algorithm turn out to be high enough; the lucky trials number is 85%; and in contrast, for the statistically denoised 2D image patterns, they reach 0.1%.

1. Introduction

There is nothing supernova if one says the innovative method of computed X-ray diffraction microtomography promotes ongoing tomorrow’s perspectives for the studies of crystal-lattice-defects, pushing one step further by propelling innovations in the semiconductor materials investigation to a more smart level in fabricating new nanoscaled structures with advanced electronic and optical properties. It will occur due to the development and application of relevant high-resolution X-ray diffraction techniques, reciprocal-space mapping (RSM) [1,2,3,4,5], and computer-aided X-ray diffraction microtomography (XRDMT) [5,6,7,8,9,10]. Both gather the 2D image pattern (IP) information recorded within the reciprocal and direct spaces, and these are utilized to decode the 2D IP data in the materials diagnostics. In the concept of a direct method, by decoding the reference 2D IP data, one can recover the crystal-lattice defect structure (see [9,10]). The latter is solving the inverse Radon problem in the XRDMT using the measured and model-simulated 2D IP data to restore the point defect displacement field function in crystals, where the figure-of-merit (FOM) value of recovery depends on the noise level of the 2D Poisson noise IP data. As was shown in [10], the noise- filtering of the 2D IP data with 3–10% noisy levels yields a reduction in the noise level of the order of their values, at least decreasing the convergence FOMs in processing the χ²-target function optimization. One have demonstrated that the 2D IP data denoising effect is effective and can be achieved by statistically averaging several noise-contaminated 2D IP data sets [10].

Further, one uses the term “denoising the 2D IP data” as a sense of digitally noisy filtering without harming the reference signal in the 2D detector pixel. On the contrary, some noisy-filtering methods [11,12,13,14], e.g., the singular-value decomposition-based denoising method [11], the convolutional neural networks [12], bilateral filtering [13], and Hamming’s method [14], lead to uncontrolled digital accuracy of the reference 2D IP data in the post-processing stage.

The present study is a comprehensive exploration of recovering the nanoscale point defect structure in the case of a spherical inclusion incorporated in the middle of the plane-parallel plate of crystal Si(111). An incident X-ray radiation is the monochromatic linear-polarized plane wave with wavelength λ = 0.0709 nm, the X-ray extinction length Λ = 36.287 μm, the diffraction vector is $\mathit{h}=\left[2\overline{2}0\right],$ the Bragg angle θ_B = 10.65°.

The displacement field function $f_{Ctpd}\left(\mathit{r}-{\mathit{r}}_0\right)$ around the spherical inclusion located at point r = r₀ is given by the Coulomb-type function.

$$f_{Ctpd}\left(\mathit{r}-{\mathit{r}}_0\right)=\frac{F}{4\pi }\frac{hx}{{(x^2+y^2+{\left(z-z_0\right)}^2)}^{\nu }+\epsilon }$$

(1)

where $\nu =\frac{3}{2},F=const,\epsilon \to +0$

Hereafter, the thin crystal Si(111) is the plane-parallel plate, and the sample thickness T is assumed to be Λ. Each of the integrated (entire) 2D IP data frames simulated with square lateral dimension $2\Lambda \times 2\Lambda$ contains 182 $\times$ 182 pixels. Accordingly, the linear size of each pixel is about 0.4 μm. In accordance with the characteristics of ESRF-Grenoble synchrotron facilities, the spatial resolution of the CCD hybrid detectors employed at the X-ray diffraction tomography beamlines is about 1 μm, the X-ray flux are about 10³ per pixel, and the exposure time is 0.1 sec. per 2D IP data frame.

The investigation goal is the comprehensive recovery of the reference vector ${\mathcal{P}}^{\left(true\right)}=\{\nu , { z}_0,F\}$ characterizing the displacement field function (1) being retrieved with controlled by FOM-value digital accuracy using the modeling of 2D Poisson-noise IP data frames in both approaches of the semi-kinematical [10] and the difference-scheme [15] for solving the Takagi–Taupin equations in the XRDMT optics.

The schematic of the XRDMT setup is drawn in Figure 1.

It allows bringing the 2D IP data to the frame multimode regime when several 2D IP data frames can be recorded in the sample rotation angle value to be fixed. Such a setup is necessary for improving the single-pixel signal-to-noise ratio by acquiring the 2D IP data frames.

In general, the χ²-target function $\mathcal{F}\left\{{\mathcal{P}}_k\right\}$ is defined as

$$\mathcal{F}\left\{{\mathcal{P}}_k\right\}=\frac{1}{\mathrm{N},\{X,Y\}}\sum _{i=1}^N{\sum }_{\left\{X\left(T\right),Y\left(T\right)\right\}}\frac{{\left(I_{ref}\left[X\left(T\right),Y\left(T\right);{\Phi }_{i,}{\mathcal{P}}^{\left(true\right)}\right]{-I}_{mod}\left[X\left(T\right),Y\left(T\right);{\Phi }_i,\left\{{\mathcal{P}}_k\right\}\right]\right)}^2}{{{(I}_{ref}\left[X\left(T\right),Y\left(T\right);{\Phi }_i{,\mathcal{P}}^{\left(true\right)}\right])}^2}=Min$$

(2)

Here, $I_{ref}\left[X\left(T\right),Y\left(T\right);{\Phi }_i,{\mathcal{P}}^{\left(true\right)}\right]$ is the reference-type 2D noisy IP frame for the sample rotation angle ${\Phi }_i$. $I_{mod}\left[X\left(T\right),Y\left(T\right)|{\Phi }_i,\left\{{\mathcal{P}}_k\right\}\right]$ is the modeling 2D IP frame for ${\Phi }_i$; the current fitting-vectors $\left\{{\mathcal{P}}_k\right\}$={${\nu }_{k, } z_{o,k}, F_k$} of the displacement function $f_{Ctpd}\left(\mathit{r}-{\mathit{r}}_0\right)$, ${\mathit{r}}_0$= (0, 0, z₀) is the defect position in a sample, z₀ = T/2, integer k is the current iteration number, k = 1, 2, 3, …, K, K is the cycle termination number. The function $f_{Ctpd}\left(\mathit{r}-{\mathit{r}}_0\right)$ are determined by the vector ${\mathcal{P}}^{\left(true\right)}=\{1.5, 0.5, 1.8$} in the units of the extinction length Λ/π.

To control fitting the χ²-target function, $\mathcal{F}\left\{{\mathcal{P}}_k\right\}$, one introduces the fitting FOM parameter, ${\mathbf{Ɽ}}_{\mathit{k}}$ defined as

$$\begin{gathered} {\mathrm{Ɽ}}_k={\sum }_i\sum _{\left\{X\left(T\right),Y\left(T\right)\right\}}\left|{I_{ref}\left[X\left(T\right),Y\left(T\right);{{\Phi }_i,\mathcal{P}}^{\left(true\right)}\right]}^{\frac{1}{2}}{{-I}_{mod}\left[X\left(T\right),Y\left(T\right);{\Phi }_i,\left\{{\mathcal{P}}_k\right\}\right]}^{\frac{1}{2}}\right| \\ /{{\Sigma }_i\Sigma }_{\left\{X\left(T\right),Y\left(T\right)\right\}}{I_{ref}\left[X\left(T\right),Y\left(T\right);{{\Phi }_i,\mathcal{P}}^{\left(true\right)}\right]}^{1/2} \end{gathered}$$

(3)

Using the semi-kinematical diffraction optics approach [9] to model the functions $I_{ref}\left[X\left(T\right),Y\left(T\right)|{\Phi }_i,{\mathcal{P}}^{\left(true\right)}\right]$, and $I_{mod }\left[X\left(T\right),Y\left(T\right)|{ \Phi }_i,\left\{{\mathcal{P}}_k\right\}\right]$, one has launched fitting the χ²-target $\mathcal{F}\left\{\mathcal{P}\right\}$ according to the procedure (2), which aims to recover the Coulomb-type point-defect function $f_{Ctpd}\left(\mathit{r}-{\mathit{r}}_0\right)$. The sample Si(111) is the rectangular prism in the coordinates (X, Y, Z) nominated in the units of Λ/π, 0 ≤ Z ≤ π, −π ≤ X(T) ≤ π, −π≤ Y(T) ≤ π, T = $\pi$. Total grid crystal sizes along the dimensionless coordinates (X, Y, Z) are 61 × 61 × 21. The case of the Poisson-noise with levels of 1% and 3% has been taken into consideration. For processing the χ²-target function $\mathcal{F}\left\{{\mathcal{P}}_k\right\}$ for simulating the 2D Poisson-noise IP data frames, one has employed the Poisson random value generator [16].

2. The Recovery Issue in the XRDMT Results

2.1. Semi-Kinematical Diffraction Optics Approach

We begin a computer-aided recovery of the displacement field function (1) using the semi-kinematical diffraction optics approach (cf. [9]). As explained in [4,9], the theoretical approach is valid for describing the 2D frame area directly linked to the nearby vicinity area of the crystal-lattice defect. As was shown in [9], the semi-kinematic approach describes diffraction scattering of X-rays by a Coulomb-type point defect as a purely phase object in a thin crystal. This allows us to provide visualization of the defect in the 2D-IP region, which corresponds to the so-called direct defect contrast [4].

In proceeding with the χ²-target function $\mathcal{F}\left\{{\mathcal{P}}_{\mathit{k}}\right\}$, one applies a joint quasi-Newton–Levenberg–Marquardt–simulated annealing (NLMSA) algorithm (cf. in [9,17,18,19] for details). In this case, the lateral sizes of the calculated 2D IP frames are 61 × 61 grid pixels.

In the previous work [10], the calculations of the 2D IP frames, the Poisson-noise levels of 2% and 4%, and the sample rotation angle range Φ = {−20°, 20°}, have been performed and analyzed. Here, as an illustration, one presents and analyzes the simulated results for the 2D IP frames at the Poisson-noise levels of 1% and 3%.

In Figure 2, there are the 2D IP frames calculated using the semi-kinematical approach of the X-ray diffraction optics theory.

For recovering the displacement field function (1) in accordance with processing the χ²-target function $\mathcal{F}\left\{{\mathcal{P}}_k\right\}$ (2), the 2D Poisson-noise IP frames with noise levels of 1% and 3% have been employed, which were calculated in the semi-kinematical approach of the diffraction optics.

The corresponding results for retrieving the reference vector $\mathcal{P}$^(true) of the point defect displacement function (1) in dependence of the 2D Poisson-noise IP frame quantity are listed in

Table 1. Results of recovering the reference vector $\mathcal{P}$ ^(true). The noise levels of the 2D Poisson-noise frames under consideration are given in the first column. The start vectors $\mathcal{P}$^(start) have been chosen randomly within 40% around ${\mathcal{P}}^{\left(\mathit{t}\mathit{r}\mathit{u}\mathit{e}\right)}.$ The sample rotation angle Φ area is lying from −20° to +20°. Depending on the number of sample rotation angles equidistantly lying in the interval {−20° to +20°}, the 2D Poisson-noise IP frame quantity is pointed out in the second column. The retrieved vectors ${\{\mathcal{P}}_K\}$ are given in the third column.

%-Noise Level	2D IP Frame Quantity	${\mathcal{P}}_{\mathit{K}}$	$\mathcal{F}\left\{{\mathcal{P}}_{\mathit{K}}\right\}$ $\times {10}^5$	${\mathbf{Ɽ}}_{\mathit{K}}$	lucky Trials/Total Trials
1	3	(1.46; 0.50; 1.83)	1.4	0.071	10/12
1	11	(1.48; 0.50; 1.82)	1.2	0.070	10/12
1	23	(1.52; 0.50; 1.81)	1.1	0.066	11/12
1	31	(1.49; 0.50; 1.81)	0.9	0.051	11/12
3	3	(1.55; 0.50; 1.81)	1.7	0.097	9/12
3	11	(1.47; 0.50; 1.81)	1.5	0.092	9/12
3	23	(1.52; 0.50; 1.81)	1.4	0.089	9/12
3	31	(1.49; 0.50; 1.81)	1.1	0.084	10/12

.

Lucky trials are assumed to have FOM values ${\mathbf{Ɽ}}_{\mathit{K}}$ less than 0.1.

Lucky trials, the last column, are assumed to have the FOM values ${Ɽ}_K$ less than 0.1.

Analyzing the datasets in Table 1, one can see a clear tendency when the 2D Poisson-noise IP frame quantity increases, from the number 3 to 31, the corresponding FOM ${\mathrm{Ɽ}}_K$-values decrease, i.e., it does mean the quality of lucky trials is improved. This allows us to conclude that the recovery strategy of further increasing of quantity of the 2D Poisson-noise IP frames would inevitably enhance the digital accuracy of recovering the reference $\mathcal{P}$^(true).

2.2. The Diffraction Optics Rigorous Approach, the Takagi–Taupin Equations in the Finite Differences

In the differential form, the canonic Takagi–Taupin takes the form (cf. [20,21])

$$\begin{array}{c}\widehat{\mathit{S}}\left({\mathit{s}}_0,{\mathit{s}}_{\mathit{h}}\right)\mathit{E}\left({\mathit{s}}_0,{\mathit{s}}_{\mathit{h}}\right)=\widehat{\mathit{M}}\mathit{E}\left({\mathit{s}}_0,{\mathit{s}}_{\mathit{h}}\right),\hfill \\ \widehat{\mathit{S}}=\left[\begin{array}{cc}\frac{\partial }{\partial s_0}& 0\\ 0& \frac{\partial }{\partial s_h}\end{array}\right], \widehat{\mathit{M}}=i\frac{\pi }{\lambda }\left[\begin{array}{cc}{\chi }_0& C{\chi }_{\bar{h}}\mathit{exp}\left[if\left(\mathit{r}\right)\right]\\ C{\chi }_h\mathit{exp}\left[-if\left(\mathit{r}\right)\right]& {\chi }_0\end{array}\right]\hfill \end{array}$$

(4)

where (s₀, s_h) is the oblique-coordinate system, $s_{0,h }=\mp \sin \left({\theta }_B\right)x+\cos \left({\theta }_B\right)z$.

The boundary conditions for the wave field electric vector $\mathit{E}\left(s_0,s_h\right)$ take the form

$$\mathit{E}\left(x,z\right)\left|z=0\right.=\left[\begin{array}{c}\mathit{exp}\left(i\frac{\pi \alpha }{\lambda \mathit{sin}{\theta }_B}x\right)\\ 0\end{array}\right]$$

(5)

Here above, the deviation parameter α = (2k h + h²)/2k² = –sin(2θ_B)Δθ_B is a measure of the angular deviation Δθ_B of an incident X-ray monochromatic plane-wave from the exact Bragg angle θ_B. Coefficient χ₀ is the zero-Fourier- and χ_h,  ${\chi }_{\overline{h}}$ are the h-, –h-Fourier components of electric susceptibility χ(r) for a perfect crystal. In our case, the scalar function f(r) is the point defect displacement field function $f_{Ctpd}\left(\mathbf{r}-{\mathit{r}}_0\right)$ defined by Equation (1).

Recently in [15], a numerical solution of the boundary-valued Cauchy problem (4)–(5) with the point defect displacement field function (1) has been built up for the sample rotation angle Φ = 0° using the Takagi–Taupin equations in the finite differences. The important thing is that the numerical solutions of the boundary-valued Cauchy problem (4)–(5) with a controlled accuracy of 10⁻⁴ have been obtained and digitally controlled by the criterion for thin (non-absorbing) crystal Si(111)

$$\frac{\partial I_0(s_0, s_h)}{\partial s_0}+\frac{\partial I_h(s_0, s_h)}{\partial s_h}=0$$

(6)

where I₀(s₀,s_h) and I_h(s₀,s_h) are the transmitted and diffracted intensities at the current point s₀,s_h.

In the case of using the semi-kinematic theory approach, the Levenberg-Marquardt algorithm was used to solve the problem in combination with the simulated annealing method (qNLMSA) [9]. Such a combination of the above methods allows the program to overcome local minima and increases the robustness of the optimization (advantages), but at the expense of increasing the total number of iterative steps (disadvantage).

A certain limitation for using the qNLMSA method is the number of pixels used in the 2D IP formation, since the size of the arrays (the first derivative Jacobian matrices and the second derivative Hessian matrices) can complicate and/or disrupt the qNLMSA program. In our case, when the image is constructed as 2D frames with sizes of 182 × 182 pixels, other methods that do not use finite-difference derivative matrices should be chosen. In our study, the highly robust Nelder–Mead (N-M) polyhedron method was applied [22].

As an example, the 2D noise-free IP frame calculated along with the numerical solution of the boundary-valued Cauchy problem (4)–(5) is shown in Figure 3, top left in the linear pixel size units; the linear pixel size is 0.4 μm; the total image size is 182 × 182. Appropriately, going back to our work, the 2D Poisson-noise IP frames with a noise level of 5% have been simulated, see Figure 3, top left, using the Poisson random value generator [14] many times, namely: n = {1, 10², 10⁴}, see, e.g., the 2D Poisson-noise IP frame in Figure 3 bottom left for n = 1.

Afterward, the obtained 2D Poisson-noise IP frames were denoised according to the statistical average method [9]. At the final stage, post-processing the χ²-target function $\mathcal{P}$_K, aiming to recover the reference vector $\mathcal{P}$^(true), we employed the Nelder–Mead (N-M) algorithm [22]. The values of the start vector $\mathcal{P}$^(start) were randomly chosen within the 40% range around the reference vector $\mathcal{P}$^(true). The flow equation for applying the (N-M) algorithm is derived in Appendix A, and some details of the computer (N-M) algorithm script are given in Appendix B.

The post-processing results for recovering the reference vector $\mathcal{P}$^(true) of the point defect displacement field function (1) for the sample rotation angle Φ = 0° are listed in

Table 2. Results of the post-processing vector $\mathcal{P}$_K. The values of the start vectors $\mathcal{P}$^(start) were chosen randomly within the 40% range around the true vector ${\mathcal{P}}^{\left(true\right)}$. Lucky trials are assumed to be the FOM ${Ɽ}_{\mathit{K}}$ less than 0.02.

%-Noise	Average Frames, n	$\mathcal{P}$K	$\mathcal{F}\left\{{\mathcal{P}}_{\mathit{K}}\right\}$ $\times {10}^5$	${\begin{array}{c}\mathbf{Ɽ}\end{array}}_{\mathit{K}}$	Lucky Trials/ Total Trials
0	1	{1.50, 0.50, 1.80}	2.21 × 10−7	6.935 × 10−6	15/15
5	1	{1.50, 0.50, 1.80}	1.42 × 10−1	1.408 × 10−2	12/15
5	102	{1.50, 0.50, 1.80}	4.67 × 10−2	2.307 × 10−3	12/15
5	104	{1.50, 0.50, 1.80}	7.25 × 10−4	1.082 × 10−3	13/15

.

As is seen from the dataset in Table 2, there is a stable tendency to decrease the FOM ${Ɽ}_{\mathit{K}}$ with increasing the statistically average 2D IP frame number. Hence, one may claim the statistical average procedure [9] applied to the 2D Poisson-noise IP frames allows us to enhance the accuracy of recovering the point defect displacement field function $f_{Ctpd}\left(\mathbf{r}-{\mathbf{r}}_0\right)$ by improving the signal-to-noise ratio in the computer-aided XRDMT.

3. Discussion

In the present exploration, based on the Poisson-noise statistical properties, there was an aim to exploit them and, if possible, to solve the inverse Radon problem of the XRDMT diagnostics in the case of the Coulomb-type point defect displacement field function $f_{Ctpd}\left(\mathit{r}-{\mathit{r}}_0\right)$. The latter is characterized by the three-parameter reference vector $\mathcal{P}$^(true). For the first time, in the case of the 2D Poisson-noise IP frames-entire area imaging (see Figure 3), higher digital accuracy recovery of the point defect structure $f_{Ctpd}\left(\mathit{r}-{\mathit{r}}_0\right)$ has been achieved by applying the statistical average method [9]. By using the (N-M) algorithm [22], recovering the Coulomb-type point defect structure has been performed.

It is worth emphasizing that, in contrast to using the semi-kinematic theory approach, in manipulating the 2D entire imaging of the Coulomb-type point defect in a crystal Si(111), one has used the robust Nelder–Mead (N-M) polyhedron method, which allows us to avoid the problems of using the qNLMSA algorithm in processing the χ²-target function $\mathcal{F}\left\{{\mathcal{P}}_k\right\}$ optimization. The conducted research allows us to claim that, as large as possible, the number of 2D IP frames recorded with a noise counting level of up to 5% secures the digital-accuracy information and decreases the FOM ${Ɽ}_{\mathit{K}}$ with increasing the 2D IP frames to be statistically averaged (see Table 2).

Here, it is appropriate to emphasize that the key point in this study, which made it possible to improve the digital accuracy of solving the inverse Radon problem in the scope of the XRDMT technology, is the noise-filtering method of the direct statistical averaging of the 2D Poisson-noise entire area imaging frames proposed (see [9] for details).

As to collecting a large number of the 2D IP frames for employing (switching on) the statistical average procedure [9], it is worth keeping in mind that the 2D IP frame registration rate by the 2D CCD hybrid pixel detector has to be high enough. The latter can be altered within a wide range from one kHz to tens of MHz, controlled by the X-ray intensity value per detector pixel. Supposing the exposure X-ray time is about 0.1 sec per one 2D IP frame, one may design one hundred of the 2D IP frames, at least if each pixel catches 10³ photons/sec in the functional pattern part of the 2D IP frames to be collected.

It is interesting that beyond recovering the crystal-lattice defects, the 2D IP frame-by-frame processing allows us to unveil the radiation crystal atomic structure damage by comparing the 2D IP frames one after another and then matching them at the detector registration panel by applying the DATCMP program [23].

4. Conclusions

In this paper, the goal of our study is to unveil a good XRDMT method for purposes of the digital characterization of the crystal-lattice defect structures. The key point of the XRDMT is the fact that it provides a robust, unambiguous procedure to solve a crystal-lattice defect structure with digital accuracy. The test above completes our previous studies in the field of recovering the nanoscale defect structure in crystals.

In contrast to the work [9], the main advance of the present study is the (N-M) algorithm [22] applied for processing the χ²- target function $\mathcal{F}\left\{{\mathcal{P}}_K\right\}$ has allowed us to solve the inverse Radon problem of the XRDMT en Gros in the case of a set of the 2D Poisson-noise entire area imaging frames recorded.

It is essential for future sequent works that this study carried out in this paper shows the way and opens some opportunities to employing the XRDMT technique as an effective digital tool for exploring the crystal-lattice defect structures, such as various clusters, small dislocation loops, quantum wells, and quantum wires.