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Abstract
Lateral graded multilayer can realize reflection, collimation and focusing of hard X-ray, and are currently the research frontier and hotspot of synchrotron radiation and high-performance X-ray sources. To reduce the d-spacing error of graded multilayers, a root mean square error optimization method based on double genetic algorithm (DGA-RMSE) is proposed. The theoretical d-spacing distribution is obtained by optical design, and the range is 1.9 ∼ 3.1 nm. The optimized d-spacing distribution is obtained by convolution of particle beam function and continuous monotonic rate distribution line (RDL) which is constructed in the form of a polynomial. The GA is applied to optimize variables from the polynomial twice, and the RMSE of thickness error is optimized and converged to 0.0065 nm. The final thickness error which is measured by the grazing incidence X-ray reflectivity (GIXRR) is consistent with the theoretical calculation. The results show that DGA-RMSE can precisely select polynomial function of RDL, reducing the error in high-precision magnetron sputtering and mask technology.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The X-ray was one of the three major physical discoveries of the 19th century. Its introduction has resulted in rapid advancement and even revolutionary changes in the fields of physics, chemistry, life science, and material science. However, all X-ray sources, including synchrotron radiation sources. X-ray tubes, have some angular divergence or energy bandwidth. As a result, after being emitted from the source, X-ray must be modulated by a series of optical elements to meet the needs of various experiments.
In terms of energy modulation, some X-ray structural analysis experiments (such as high-pressure diffraction, fluorescence analysis, etc.) require X-rays to have appropriate monochromaticity and high flux [1] such as 2-BM of APS [2], ID01 and ID15 of ESRF [3], and most of the high-performance X-ray sources. The total reflection mirror made of a single layer on substrate can achieve high reflectivity and high energy X-ray suppression without changing energy resolution. The multilayer mirror is to deposit different materials periodically and alternately on the substrate so that the periodic thickness and the wavelength of the incident light satisfy the Bragg diffraction law [4]. In terms of divergence angle modulation, the Bragg angle of the multilayer film is 1∼10 times larger than the total reflection angle. As a result, the multilayer mirror has a much shorter mirror length than the total reflection mirror. The multilayer mirror improves the stability of the synchrotron radiation light source's attitude adjustment and increases the integration and portability of the X-ray machine. The graded multilayer can accept a larger divergence angle than the flat multilayer and the total reflection single-layer film so as to achieve higher flux and gain [5]. In addition, gradient multilayer films can obtain a wider range of incident angles or energy, and can be used for EUV optical systems [6] and X-ray supermirrors [7]. Multilayer can be understood as a one-dimensional structural crystal, which can realize the processing of various surface types such as plane, ellipse, and hyperboloid, and can meet the reception of light sources with different divergence angles to achieve reflection, collimation, and focusing [8]. So graded multilayer is widely used in major synchrotron radiation sources and X-ray machines for diffraction, fluorescence, CT, and other related experimental methods. The required period thickness of the multilayer depends on the reflection angle and the energy required by the light source and can be expressed as:
where d is the sum of the thicknesses of the two materials in one period, n is refractive index of material, θ is the grazing incidence angle of X-rays, λ is the wavelength of the incident light, and m is the order of Bragg diffraction.In 2000, Protopopov et al. fabricated a W/C graded multilayer film collimator on a parabolic substrate based on the insertion of specially shaped screens before the substrate [9]. The total output flux at both the Cu Kα1 and Cu Kα2 lines was a factor of 50 larger than that delivered by a conventional Si(111) monochromator [10]. 2011 The Pt/C graded multilayer mirror was processed by Mimura et al. [11], the periodic thickness variation range was 3.3-6.5 nm, and the deposition area was limited by using a narrow rectangular aperture. The scanning speed of the mirror was adjusted such that the convolution of the dwelling time and the deposition profile was equal to the desired thickness [12]. In 2016, Störmer et al. prepared a mirror with a Ru/C graded multilayer. They studied the combination of generator power, argon gas pressure, and the carrier velocity determined the layer thickness in the multilayer stack and achieved picometer-precision in the meter-range [13]. In 2017, the W/B4C graded multilayer mirror, with the central period thickness of 2.58 nm, was prepared by Silva et al. [14]. The MLs were deposited in dynamic mode where the substrate was moved with a controlled speed profile in front of each cathode [15]. In 2019, the diamond synchrotron radiation light source (DLS) Sutter et al. coated a mirror with a mirror length of 1.1 m to produce three kinds of multilayers, respectively. It was Ni/B4C, W/B4C and Pt/B4C multilayer films, the applied photon energy was 40 keV, 76.6 keV and 65.4 keV respectively, the thickness error PV value was less than 1% [16].
Through the above investigations, it can be concluded that the graded multilayers are obtained by optimizing the movement rate of the substrate and the mask with different shape or position. However, there have been many difficulties in substrate motion rate optimization. Moreover, there are few studies on the optimal method of motion rate for differential deposition. To meet this challenge, we have done a detailed study of the optimization method. John J. Grefenstette has proved that genetic algorithms were effective for system optimization problems [17]. Abdullah Konak offered a genetic algorithm developed to solve multi-objective problems that used specialized fitness functions to promote the variety of solutions [18]. Zhang wei Bo et al. reconstructed the genetic algorithm based on the analysis of various constraints in the process route ordering and the convergence of the genetic algorithm, and finally obtained the closest optimal process route [19]. Pietro D. Binda et al. have tested the genetic algorithm on two specific applications and the genetic algorithm has been implemented as a software tool [20] and so on. In this study, we propose a DGA-RMSE model. This model is based on differential deposition theory, to optimize the substrate movement rate and complete the preparation of graded multilayer. Finally, GIXRR was used to measure the thickness error distribution to check the accuracy of the model's theoretical calculation.
Double genetic algorithm is an evolutionary algorithm that combines the advantages of genetic algorithm and evolutionary strategy, hence the name ‘double’. Specifically, the double genetic algorithm uses two populations, breaking the framework of traditional genetic algorithms that rely only on a single population for genetic operations, and introduces two populations for optimization search; different populations are given different control parameters to achieve different search purposes. The two populations are independent of each other and are linked by immigration operators to achieve co-evolution between different populations; the optimal solution is obtained as a comprehensive result of co-evolution of the two populations. The double genetic algorithm can effectively overcome the phenomenon of immature convergence of traditional genetic algorithms.
The specific objects on which the double genetic algorithm acts are the power exponent and the coefficients, respectively.
2. Theoretical calculation
The main factor that determines the optical performance of the graded multilayer is the thickness error of the multilayer in the length direction of the substrate. This thickness error directly affects the shape of the X-ray modulation, such as focal enlargement and collimated beam divergence. If the error is large, the efficiency will be reduced due to the mismatch of Bragg diffraction conditions. Therefore, a DGA-RMSE model is proposed in this paper. Because the fabrication of graded multilayer means that the deposition process of microscopic particles needs to be upgraded to a large-area thickness controllable manner, while maintaining a high level of film quality. The main idea of differential deposition [21] is to calculate the path and speed of each position of the substrate separately, and obtain the thickness according to the particle distribution above the target. The substrate moves back and forth relative to the particle source, and the d-spacing (D) at a given point on the substrate is calculated by integrating along the motion path, and the expression is shown in formula (2) [22]:
By formula (2), a specific thickness gradient along the direction of motion can be obtained by the velocity change of the mirror. At the same time, a suitable mask can be inserted between the source and the substrate to modify the particle beam distribution. It can be known from formula (2) that, the thickness distribution of the graded multilayer mirror should be the convolution of the reciprocal of the mirror motion rate function and the thickness distribution function above the particle source. The positional relationship among motion velocity, graded multilayer d-spacing and particle beam distribution is shown in Fig. 1.
Fig. 1. Positional relationship among motion velocity, graded multilayer d-spacing and particle beam distribution.
The first attempt is the deconvolution operation between the theoretical mirror thickness distribution and the thickness distribution function above the particle source. The rate distribution curve and thickness error curve are then calculated, as shown in Fig. 2(a) and (b). Although the optimized thickness distribution is perfectly consistent with the theoretical curve, but the rate curve has a sharp rise and fall. In terms of acceleration and positioning precision, this is difficult to achieve with mechanical motion control.
Fig. 2. The mirror motion rate curve calculated by deconvolution of theoretical thickness distribution and particle beam function (a) graded multilayer d-spacing and error distribution (b)
Convolution operations can therefore be used to create a continuous and monotonic rate distribution function. The objective function can then be set to gradually improve until convergence while boundary conditions and iterative parameters are determined. The implementation of GA can make it possible to automatically optimize the rate curve and increase the accuracy of the optimization. The objective of optimization is to locate the fitness function within the defined constraints. For the purposes of mirror rate optimization, we assume that the motion rate function's reciprocal takes the shape of a polynomial. Although there is still a thickness error at convergence to the limit, GA gradually approaches the theoretical curve by optimizing the polynomial's coefficients and power exponents. In mechanical motion control, it is simpler to achieve a continuously monotonically changing rate curve.
3. Method and evaluation
Genetic algorithm is an optimization algorithm based on the principle of biological evolution, which searches for optimal solutions by simulating the processes of natural selection, crossover and mutation. In genetic algorithms, there are many parameters to be set, which have a great impact on the performance and search results of the algorithm. The following are common genetic algorithm parameters and their setting methods and effects:
- 1. Population Size: Population size refers to the number of individuals contained in each generation. The setting of population size should be decided according to the complexity of the problem, computational resources and other factors. Generally speaking, the larger the population size, the wider the search space, the better the search effect, but the computation time will also increase accordingly.
- 2. Crossover Probability: Crossover probability refers to the probability that each pair of parent individuals will cross over. The setting of the crossover probability should be decided according to the characteristics of the problem and the search results. In general, the higher the crossover probability, the wider the search space, but it may lead to premature convergence; the lower the crossover probability, the smaller the search space, but it may lead to inefficient search.
- 3. Mutation Probability: Mutation probability refers to the probability of mutation of each gene. The setting of mutation probability should be decided according to the characteristics of the problem and the search results. In general, the higher the mutation probability, the wider the search space, but it may lead to premature convergence; the lower the mutation probability, the smaller the search space, but it may lead to inefficient search.
- 4. Selection strategy (Selection Strategy): selection strategy refers to how to select the next generation of individuals. Common selection strategies are roulette selection, tournament selection, etc. The setting of the selection strategy should be decided according to the characteristics of the problem and the search results. Different selection strategies may lead to different search results and need to be selected according to the actual situation.
- 5. Termination Criteria: Termination Criteria refers to the stopping condition of the genetic algorithm. Common termination conditions include reaching the maximum number of iterations, reaching a certain threshold value of the objective function, etc. The setting of termination criteria should be decided according to the characteristics of the problem and the search results. If the termination condition is not set reasonably, it may lead to the algorithm failing to converge or converging prematurely.
The above are the common genetic algorithm parameters and their setting methods and effects. In practical applications, it is necessary to select the appropriate parameters according to the characteristics of the problem and the experimental results to achieve the optimal search effect.
The continuous monotonic RDL is created as a third-order polynomial (see Eq. (3)), which represents the reciprocal of the motion rate function. Figure 3 illustrates the optimization process for the polynomial coefficients and power exponents.
where a, b, c, and d are the objects to be optimized in this study.Fig. 3. Flowchart of the double GA optimization. Fitness is described in Eq. (4), N1 and N2 are both 10000.
An elite genetic algorithm is introduced in the DGA-RMSE model to enhance the global convergence of the standard genetic algorithm. The purpose of the genetic algorithm in the coating process is to select the best function for the RDL, given the constraints. The fitness function in the genetic algorithm is the RMSE between the optimized d-spacing and the theoretical value of each place in the graded multilayer. The smaller the RMSE is, the better the optimization will be. The RMSE is shown by Eq. (4):
$D_i^\mathrm{^{\prime}}$ and ${D_i}$ represent the optimal and theoretical d-spacing for each graded multilayer point.
The DGA-RMSE flow is shown in Fig. 3, and the process of the algorithm can be summarized as the following steps:
1. initialize power exponential population: randomly generate a certain number of individuals as the initial population.
2. Calculate fitness: According to the requirements of the problem, calculate the fitness value of each individual.
3. Genetic operations: Genetic manipulation consists of three main components: selection, crossover and mutation. Through the three operations of selection, crossover and mutation, genetic algorithms can continuously generate new individuals and gradually optimize the fitness of the population, so as to achieve optimization of the objective function.
4. Immigration operation: the main role is to pass good individuals between different populations. Immigration operations usually take place between multiple populations, each of which has its own set of individuals. In the immigration operation, some good individuals are transferred from one population to another population to increase the diversity and adaptability of the other population. Immigration operation is an important operation in multi-objective genetic algorithm, which can help the algorithm to pass good individuals among multiple populations, increase the diversity and adaptability of the population, and thus improve the performance of the algorithm.
5. Update the population: based on the fitness value, select the best individuals and update the population.
6. Meeting terminating condition: determine whether the algorithm reaches the termination condition, such as reaching the maximum number of iterations or reaching a certain fitness value.
It should be noted that although the double genetic algorithm has certain optimization effects, its operation efficiency is low and it is more sensitive to the parameter settings. In practical applications, it needs to be adjusted and optimized according to specific problems.
4. Results and discussion
4.1 Establishment of source distribution and growth rate
The lateral-graded multilayer studied in this paper is designed at 22 keV by using W and B4C target sputtering alternatively on substrate. The d-spacing range is from 1.9 to 3.3 nm to satisfy Bragg diffraction conditions for incident angles from 15 to 9 mrad. The power of the W target is 30 W, while that of the B4C target is 240 W. The aim is to make the sputtering rates of the two materials approximately equal. A mask with 80 mm wide rectangular holes is placed between the target and the substrate to modulate the particle beam distribution function, and the distance between the mask and the substrate is less than 2 mm. The X-ray diffractometer (XRD) is used to perform the GIXRR test of the d-spacing at different positions of the mirror. The time deposition mode is adopted, that is, the substrate stays above the target, and the periodic multilayer under static conditions is deposited for a fixed deposition time of 100 s. The reflectivity curves of different positions of the substrate are measured by GIXRR, and the peak position of each Bragg diffraction peak can be extracted to calculate the d-spacing distribution at different positions. The same base pressure and coating process parameters are used to prepare periodic multilayers and test the normalized thickness distribution on two consecutive days, respectively, as shown in Fig. 4(a), showing good experimental repeatability. From this thickness distribution, the particle beam distribution matrix f in Eq. (2) can be obtained. In order to calibrate the center growth rate r, two multilayers with a repetition of 10 are prepared under the same process conditions, but in rate mode, that is, the substrates walk over the two targets at a constant moving speed of 10 mm/s, respectively. The tested GIXRR curves are as follows, as shown in Fig. 4(b). The period thicknesses are 2.188 and 2.189 nm, and the difference between the two is only 1 pm. The two test curves are in good agreement, which proves that the experiment has extremely high repeatability.
Fig. 4. Normalized d-spacing distribution above the particle source (a) and GIXRR curve of the multilayer in the center position (b) on two consecutive days respectively show well experimental repeatability.
4.2 Power exponent chosen by DGA-RMSE
In this section, we use the DGA-RMSE method to find the best power exponent variable in Eq. (3). Algorithm hyperparameters: initial population size of 10,000, mutation probability of 0.1, elite retention probability of 0.01, crossover probability of 0.5, and paternal share of 0.4.
Figure 5 shows that the GA converges very quickly in optimizing the power exponent and finds the best combination of power exponents for the coefficients of the RDL equation after 10 iterations. The result of the power exponential variable optimization: the minimum RMSE is 2.2265. This obviously does not satisfy the experimental requirements, so the GA algorithm must be used to optimize the coefficient variables more finely.
4.3 Polynomial coefficients chosen by DGA-RMSE
In this section, in order to obtain more accurate expressions for the RDL function, we again use GA to optimize the coefficient variables of the RDL function (Eq. (3)). The task of optimization is to find the coefficient variables of the RDL function with the smallest RMSE. Hyperparameters for the algorithm include an initial population size of 10,000, a mutation probability of 0.1, a probability of elite retention of 0.01, a crossover probability of 0.5, and a paternal share of 0.4.
Figure 6 clearly shows that after one round of iterations, the RMSE decreases from 2.225 nm in Fig. 5 to 0.00655 nm. The convergence process of optimizing the coefficient variables is significantly slower than that of optimizing the power exponential variables. This is because the optimized coefficients are finding the best combination of coefficients in a very small RMSE range. When the number of iterations reaches 100, the RMSE drops to 0.0065, which is a very good result.
4.4 Test results
The multilayer is deposited on a super-polished silicon substrate using a DC magnetron sputtering machine with a 4-inch target. The base pressure is below 5.0E-5 Pa before deposition, and argon is used as the working gas with a pressure of 0.3 Pa considering the factors of interface roughness. The working gas used in deposition is high-purity Ar gas (99.99%). The substrates used in the experiments are single-sided, polished, 70 mm-long Si (100) substrates with a roughness below 0.3 nm. The GIXRR curves for different positions of the substrate are shown in Fig. 7(a). It can be seen from the change in position of the Bragg peak that the d-spacing gradually decreases along the substrate length. According to the Bragg diffraction peak position, the d-spacing and error distribution at different positions of the substrate can be calculated, as shown in Fig. 7(b). The d-spacing error is 0.0065 nm (RMSE), which is consistent with the theoretical calculation results of the fitness function. This method can be applied to the preparation of graded multilayers with larger mirror size and wider thickness range. On this basis, by optimizing the thickness gradient of each layer, we can further study the preparation methods of conic shape, such as confocal parabola and ellipse, to achieve better X-ray collimation and focusing effect.
Fig. 7. The GIXRR curves (a) and d-spacing distribution (b) of different positions on the substrate.
5. Conclusion
In this study, a new algorithm is proposed to precisely optimize the rate profile function's unknown parameters twice, using an elite GA to optimize a continuous, monotonic rate distribution curve. Considering that the rate distribution of deconvolution operation will make the mechanical control difficult to realize, a DGA-RMSE method for magnetron sputtering and mask technology is developed. According to the experimental findings, the optimized RMSE is 0.0065 nm. The final thickness error measured by GIXRR is consistent with the theoretical calculation. The results show that DGA-RMSE can be a feasible method for high precision selection of the RDL function in magnetron sputtering technology. Numerical analysis and experimental methods on how to reduce the error will be studied further.
Funding
International Partnership Program of Chinese Academy of Sciences (Grant No.113111KYSB20160021); National Natural Science Foundation of China (Grant No.12005250).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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