Extended genetic algorithm: application to x-ray ...

INSTITUTE OF PHYSICS PUBLISHING

JOURNAL OF PHYSICS D: APPLIED PHYSICS

J. Phys. D: Appl. Phys. 38 (2005) A235–A238

doi:10.1088/0022-3727/38/10A/046

Extended genetic algorithm: application to x-ray analysis A Ulyanenkov1 and S Sobolewski2 1 2

¨ Bruker AXS, Ostl. Rheinbr¨uckenstr. 50, 76187 Karlsruhe, Germany Byelorussian State University, F. Scariny Av. 4, 220050 Minsk, Republic of Belarus

Received 1 March 2005, in final form 2 March 2005 Published 6 May 2005 Online at stacks.iop.org/JPhysD/38/A235 Abstract The classic scheme of the genetic algorithm is extended to improve the robustness and efficiency of the genetic technique. New genetic operators implemented in this paper are shown to increase the convergence speed and reliability of the optimization process. A complex model function with multiple local extrema and real x-ray reflectivity and diffraction data have been used to prove the modified algorithm. The effectiveness of the new technique is compared with the effectiveness of various optimization methods.

1. Introduction X-ray metrological methods in science and industry have proved to be efficient techniques for sample characterization and growth, process development and control. A large variety of sample structures can be probed by x-rays to examine film thickness, interface roughness, crystallographic lattice strain and distortion, material contamination, etc. Although the measured x-ray intensities can directly be used for evaluation of sample parameters, detailed knowledge about the sample structure can only be obtained using special data interpretation procedures. These procedures usually utilize a trial-anderror technique, which uses the parametrized sample model for simulation of the x-ray scattering process, and then the difference χ 2 between the calculated and measured intensities is minimized with respect to the sample parameters. Thus, an effective and robust optimization algorithm is essential for fast and accurate data interpretation. Moreover, in most experimental set-ups, x-ray measurements provide the magnitude of scattered x-ray intensity, whereas the phase of x-ray waves is lost. Therefore, procedures for a unique reconstruction of the sample physical parameters directly from the measured intensities are not available principally because the loss of x-ray phase information causes ambiguity in the interpretation of results, e.g. several sample models can fit the measured data well. This situation demands that the optimization technique be able to find not only a single global minimum, but a set of most deep minima on the χ 2 hypersurface, every one of which may potentially deliver a real physical solution. Genetic algorithms (GAs) [1], widely used nowadays, seem to be most successful in solving all the above-mentioned problems in x-ray data analysis [2–7]. They 0022-3727/05/SA0235+04$30.00

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combine a stochastic search and strategy aimed feature, that helps to find a global minimum along with other local minima of the cost function χ 2 , which have comparable magnitudes. In this work, we propose a modification of the classic GA (CGA) scheme, which improves the efficiency of the algorithm. These new implementations in standard GA have been verified and the results are compared with those obtained by CGAs. For the tests, the complex model function possessing multiple extrema, and real x-ray diffraction and reflection measurements have been used. In section 2, we describe and qualitatively ground the implemented modifications of the GA. Section 3 presents the test results of the proposed method, called further eXtended GA (XGA) for a model multi-maxima function and comparison with the CGA. The x-ray diffraction and reflection measurements are fitted using both XGA and diverse optimization methods, and the performance of XGA is compared with these optimization techniques [8].

2. Modifications of CGA A detailed explanation of GAs can be found in numerous publications (see, e.g. [1] and citations therein). In this section, only the principle construction of a CGA scheme is described. GAs exploit Darwin’s evolutionary idea and the principle of survival of most adopted individuals in population. The first step in any GA application is the formalization of optimized parameters to unify all the operations with them. The procedure of parameter formalization is called encoding, whereas the reverse operation of obtaining the physical parameters from a binary set is a decoding. After

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the formalization procedure, every point in parameter space represents a unique physical state of the investigated system, and the goal of GA is to explore this space to find the point that gives the best fit, according to the pre-defined fitness criteria. Every complete set of formalized parameters describes the system comprehensively and is considered in GA as individual; a limited number of individuals compose a population, which is evolving on the basis of genetic rules. The primitive GA cycle consists of the following steps. First, the random population of individuals is created (the size of the population is an internal GA parameter). Then, the population begins to evolve by the production of new generations, i.e. creation of successive populations on the basis of the primary one. The principle rules (also called GA operators) used most often for the creation of a new generation of individuals are selection and crossover of parents, and mutation of individuals. The former operator regulates the parents’ selection procedure for the production of offsprings for a new generation. The second rule describes how the parents hand down their features (single parameters) to children. The mutation operator provides irregular changes of offsprings to strengthen the statistical nature of GAs. Applying these basic operators along with optional rules (e.g. elitism, the transfer of the fittest individuals from the current generation to the successive one without changes), after a certain number of generations GA delivers the fittest individual, i.e. the set of parameters, for which the investigated system satisfies the fitness criteria in the best way. The probability to find the best existing set of parameters increases with the number of generations evolved, i.e. with the length of evolution. All operators described above have various implementations and internal parameters, which are used to optimize the GA efficiency for a particular application. The GA, being correctly tuned for a certain problem, demonstrates excellent results as an optimization and search procedure [2–7]. In this work, we propose several modifications of basic GA operators designed to increase the efficiency and reliability of the algorithm. These modifications are as follows: (1) A new basic GA operator is implemented, called the movement operator. Before the selection of parent individuals for mating, the limited number of parents are moved towards the best individual in the population, i.e. the parameters of moved parents are changed towards the parameters of the fittest individual. This operator improves the convergence of GAs by increasing the amount of individuals in the vicinity of the fittest one. (2) A novel effective principle for the formation of new generations is developed. The sequence of formation is the following: first, a limited number of elite individuals is moved to the successive population; then, the population is filled by the individuals from the previous generation and by new offsprings randomly, with the probability depending on the individual’s fitness and the number of children left. This improvement preserves individuals with good fitness and replaces the individuals with rather bad fitness, thus decreasing the loss probability for good solutions. (3) The mutations are forbidden in a certain number of last generations. This modification permits us to use solely GA for an accurate parameter fit. Because of their stochastic nature, conventional GA schemes rarely A236

Figure 1. Flowchart of the modified XGA scheme.

deliver the exact point of the best fitness in parameter space. Usually, gradiental optimization methods have to be further applied in the vicinity of the solution found by GAs to refine the parameters [7]. This modification allows us to precisely localize the final solution that is suitable for data interpretation accuracy by using solely XGA, which reduces the whole optimization time in comparison with the cascade use of the GA and gradiental techniques. Thus, a modified XGA version of the GA has the following principle scheme (flowchart in figure 1): (i) creation of a random population consisting of N (population size) individuals; (ii) evaluation of fitness; (iii) movement of M individuals towards the fittest one; (iv) parents’ selection and production of offsprings; (v) if mutations are permitted in the current generation, m mutations of parameters are carried out; (vi) a new population is created. The evolution proceeds by repeating the cycle in figure 1 either G times (number of generations) or until the requested fitness tolerance is reached.

3. XGA applications To study the effectiveness of the above-mentioned XGA implementations, we first used a smooth two-dimensional analytical function z(x, y) = −[(x − 0.75)2 + (y − 0.625)2 ]× [2 − cos(100(x − 0.75)) − cos(100(y − 0.625))] possessing a single global maximum at (x; y) = (0.75; 0.625) and multiple local maxima (figure 2); the function is defined within the interval [x; y] = [0 . . . 1; 0 . . . 1]. Figure 3 shows a convergence diagram for CGA, CGA with movements and fullfeatured XGA. Figures 3, 5 and 7 show the convergence graphs

Extended genetic algorithm

Figure 2. Model function has numerous local maxima along with a hardly recognizable global maximum.

Figure 4. Measured (◦) and fitted (——) x-ray reflectivity from the Au/Fe3 O4 /MgO sample.

Figure 3. Convergence charts of CGA, partly modified CGA and full-featured XGA recorded for the model function in figure 2.

Figure 5. Convergence of the χ 2 function for various optimization techniques recorded during the fitting of x-ray reflectivity data in figure 4. The iteration scale of SA and SM techniques is reduced to the GA generations scale by adjustment of the computation time.

selected from a large set of fit runs and qualitatively demonstrating the tendency of XGA to quick solution finding. The better fitting of the XGA at the starting point in figures 3 and 5 is irrelevant due to the common stochastic nature of the GAs compared. The general trend of XGA to find a solution fast is reproduced regularly in fits, independently of the start value of χ 2 . We also carried out the tests with a more complicated model function, which is a linear combination of several functions similar to that depicted in figure 2, one with different values of constants. Then, the statistical error of trapping in the wrong maxima has been evaluated for CGA and XGA by performing multiple runs of both algorithms. The error for CGA is found to be 28% against 4% for XGA, and thus XGA demonstrates higher reliability in comparison with the CGA scheme. XGA has also been applied for the fitting of real experimental x-ray reflectivity and high-resolution x-ray diffraction data, the two most commonly used x-ray techniques. The convergence and speed of XGA have been compared with other known optimization methods: simulated

annealing (SA) and the simplex method (SM). Figure 4 shows measured (open dots) and simulated (solid lines) x-ray reflectivities at the x-ray wavelength λ = 0.154 056 nm from a sample consisting of the sequence of thin films Au(55 nm)/Fe3 O4 (120 nm)/MgO(substrate). The fitting has been performed using Parratt’s formalism [9], and fitted parameters were the layer thicknesses tAu and tFe3 O4 and the roughness of the sample surface and interfaces. All methods (CGA, SA, SM, XGA) resulted in acceptable fitness of curves, with slight differences in refined parameters, which is within experimental error. The refined values are tAu  53.8 nm, tFe3 O4  146.3 nm, σsurf  0.78 nm, σAu/Fe3 O4  0.1 nm and σFe3 O4 /MgO  0.3 nm. However, the effectiveness of methods is evidently different: XGA finds the best available solution faster than other methods (figure 5). To adjust the time scales of algorithms, the iteration scales of SA and SM are brought into correspondence with the GA’s generations scale by multiplication of computation time ratios tGA /tSA and tGA /tSM . Figure 5 also demonstrates the advantage of XGA A237

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that gradiental methods have to be used for final refinement in the case of XGA. X-ray diffraction measurements (open dots) from Si1−x Gex /Si (004) Bragg reflection at the x-ray wavelength λ = 0.154 056 nm are shown in figure 6. Two parameters, the thickness of the Si1−x Gex layer and the concentration of germanium, x, have been fitted (solid line). Both SA and XGA methods used for minimization of the χ 2 function, lead to similar values of thickness tSiGe  65.7 nm and concentration x  9.2%. XGA, however, reaches these values after approximately 20 generations, whereas SA requires more than 80 normalized iterations for the same accuracy of χ 2 (figure 7).

4. Conclusions Figure 6. Measured (◦) and fitted (——) (004) Bragg x-ray diffraction from a Si1−x Gex /Si sample.

The XGA is shown to be the most effective and robust optimization technique in comparison with the CGA, SA and SM. Phenomenological tests with complex model functions possessing multiple extrema, as well as with real experimental x-ray data, both reflectivity and diffraction, have shown the advantages of XGA. In view of large computer resources required for the fitting of x-ray data, the robustness and speed of XGA plays an essential role in precise data analysis in x-ray analytical software.

References

Figure 7. Convergence of the χ 2 function for XGA and SA recorded during the fitting of x-ray diffraction data in figure 6. The iteration scale of the SA method is reduced to the GA generations scale by adjustment of the computation time.

due to the third implementation, i.e. the prohibition of mutation in the final stage of evolution helps us to find an accurate global solution. After 100 generations, CGA is still localized in the vicinity of the global minimum of χ 2 , whereas XGA (due to the ban on mutations after the 60th generation) has already found the point, recognized by the SM as an exact global minimum with pre-defined tolerance. Thus, it is not necessary

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