J Mater Sci: Mater Electron DOI 10.1007/s10854-016-5520-3
Application of a niching genetic algorithm to the optimization of a SiC crystal growth system Xuejiang Chen1 • Juan Su1 • Yuan Li1
Received: 3 June 2016 / Accepted: 8 August 2016 Springer Science+Business Media New York 2016
Abstract It was demonstrated that a niching genetic algorithm (NGA) could be efficient for the optimization of a SiC crystal growth system. And several design parameters of SiC crystal growth system could be optimized at the same time, and high diversity of population was maintained to obtain global optimization solution by NGA. Firstly, the NGA and thermal models were described and applied for SiC crystal growth by physical vapor transport (PVT) method, and the combination method of NGA and thermal models were presented. Then two cases were carried out to demonstrate the automatic optimization of SiC crystal growth system by NGA. One case was a single-objective optimization problem, in which the axial position of coils was optimized to improve the growth rate of crystal. The another was a multi-objective optimization problem, in which the thickness of substrate holder and input current were optimized for uniform temperature distribution along the growth surface for reducing the thermal stresses in growing crystal. Finally, all the optimization results were analyzed.
1 Introduction The PVT technique based on the modified Lely method has been widely used to grow SiC crystal since 1970s [1]. During the growth process, the temperature is important for high quality. But it is difficult to measure temperature
& Xuejiang Chen
[email protected];
[email protected] 1
School of Energy and Power Engineering, Xi’an Jiaotong University, Xi’an 710049, Shaanxi, People’s Republic of China
inside the crucible because the temperature is more than 2000 K. Besides, experimental approach is normally timeconsuming, expensive and ineffective in multi-factor optimization [2]. Therefore, numerical simulation is an effective method used to investigate and optimize the growth process of SiC crystal by PVT. During the SiC crystal growth by PVT, some design parameters, such as parameters of coils, crucible, are very important for the growth progress. Many optimizations were carried out to analyze the influence of design parameters by traditional simulation and experiment methods [3–6]. In these studies, only one design parameter was considered simultaneously in one case, and the feedbacks from optimized value were not considered. Actually, for the optimization of growth system, several design parameters should be considered simultaneously, and the parameters were affected by each other. Furthermore, it was difficult to study several parameters simultaneously, and it was also complicated to analyze their influences on the values to be optimized, especially when the goal function was strongly nonlinear. However, Genetic algorithm (GA) has been proved to be effective for a variety of optimization problems [7], such as the flow-shop scheduling [8], computer aided design [9] and nuclear reactor core reload problems [10]. It was a stochastic optimization method based on the biological concept of natural evolution and genetics introduced by Charles Darwin. GA has obvious advantages in highly non-linear and non-local optimization problems with several parameters compared with other artificial intelligence methods, because the curvature parameters on the objective function are not necessary. In 2004, parameters affecting growth processes of VGFGaAs and Si-Czochralski have been optimized by GA combined with thermal model by Fu¨hner and Jung [11]. In
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their work, although GA with restricted tournament selection operator was applied to reduce the genetic drift, it was still time consuming and could not keep high individual diversity when the problem was complicated and fitness function was highly nonlinear. The niching genetic algorithm (NGA) is an effective method, which can be applied to reduce computation time and maintain high diversity of population. NGA have been applied to optimize the problems of real-world [12]. However, few studies have been reported on optimization of SiC bulkcrystal growth by GA or NGA. Therefore, the main objective of the present study was to optimize the PVT system of SiC crystal growth by NGA. Two optimization cases were carried out for single-objective and multi-objective problems, respectively. And from the analysis of optimization results, the optimization designs on growth system were suggested, and the effectiveness of NGA combined with thermal model of SiC crystal growth was evaluated for the optimization of PVT system. Fig. 1 Pseudo code of clearing procedure
2 The niching genetic algorithm (NGA) The basic idea of NGA is made from an analogy of the nature evolution. There are many subsystems (niches) in an ecosystem, and the subsystems contain different species (subpopulations), and the number of individuals in a subsystem is determined by its resources and ability to take resources (fitness evaluation). In our work, the NGA based on a clearing procedure proposed by Pe´trowski [13] is applied to optimize SiC crystal growth system. Pseudo code of clearing procedure is shown in Fig. 1, where P, r, k and nbwinner are the population array with n individuals, clearing radius, capacity and number of winners of the subpopulation, respectively. Fitness(P[i]) and Distance(P[i], P[j]) represent the fitness value of the i-th individual and the distance between individuals i and j of population P, respectively. Distance(P[i], P[j]) is calculated based on phenotypic sharing theory [7]. For example, distance dij between the individuals xi = [x1,i, x2,i, …, xp,i] and xj = [x1,j, x2,j,…,xp,j] can be calculated as sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p P dij ¼ ðxk;i xk;j Þ2 , where x1,i, x2,i, …, xp,i and x1,j, k¼1
x2,j, …, xp,j are decoded parameters. As shown in Fig. 1, with the clearing mechanism, the individuals which have the first nbwinner fitness are the dominant individuals in a subpopulation. And if the dissimilarity of an individual with the dominants is less than clearing radius r, it belongs to a given subpopulation. The clearing algorithm attributes the whole resource of a niche to nbwinner dominant
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individuals while it resets the fitness of all the other individuals of the same subpopulation to zero. Compared to the traditional GAs, niching procedure based on the clearing mechanism is used after the fitness evaluation and before the selection operation in NGA, and an elitist strategy is applied to improve the performance of NGA before the genetic operator. Therefore, NGA works as follows: Step 1 Initialization. Individuals of initial population are randomly generated and each individual is represented to the bit string with the fixed length by binary coding approach. Step 2 Fitness evaluation. A fitness value is evaluated based on the solution of an individual member in a specific problem. Step 3 Termination judgment. The calculation is terminated as the optimization generation reaches the given generation number. Step 4 Niching procedure. Niching procedure is carried out based on the clearing mechanism here. Step 5 Elitist strategy. The best individual of a population can be memorized and passed to the next generation without any changes. Step 6 Selection. In this procedure, individuals with better fitness value can have more chance to be selected to participate in the next stage. Stochastic ergodic sampling shown in Fig. 2 is applied, in which selected points are chosen at equal interval. If n points are needed to be chosen, the distance between points is 1/n and the
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3 Combination of NGA and thermal model of SiC crystal growth
Fig. 2 Diagram of stochastic ergodic sampling method
first individual is selected in the range of cumulative selection probability [0, 1/n] by number of randomly generated. The selection probability of each individual i N P is Pi ¼ fi =F ¼ fi = fi and the cumulative selection i¼1
probability for the first m individuals is Qm ¼
m P
Pi ,
i¼1
where fi, F and N are fitness value of the individual, total fitness and number of individuals, respectively. Step 7 Crossover. Single-point crossover method is used here, which is shown in Fig. 3. As the figure shown, two individuals are randomly selected as parents after the selection operator. Then one point is also randomly chosen to separate the chromosome of Parent A and B at the same place. The portion of parent A and B are interchanged, then Child A and B are obtained as the offspring. Step 8 Mutation. Mutation is accomplished to describe the unforeseen parts in the search space by randomly changing the individual in the bit string. Single-point mutation is used, such as, 0100101 ? 0000101. Step 9 Return to Step 2.
The SiC crystal growth system studied in this work is shown in Fig. 4, which consists of SiC powder, crucible, insulation shield, pedestal, furnace walls, coils and substrate. For modeling of induction heating, the following major assumptions are made: all of the media are temperature-independent and isotropic, and the displacement current is neglected. And for calculation of heat transfer, it is assumed that radiative transfer is modeled as diffusegray surface radiation. Since the radiative heat transfer is considerably larger than heat transfer by Stefan flow inside the crucible, the effect of gas flow is reasonable to be neglected [15]. Therefore, the governing equations for induction heating and heat transfer are coupled to be solved: o 1 owB o 1 owB ð1Þ þ ¼ lJh or r or or r oz 8 in the coils; < J0 cos xt ð2Þ Jh ¼ : r ow in the conductors; r ot wB ðr; z; tÞ ¼ Cðr; zÞ cos xt þ Sðr; zÞ sin xt
ð3Þ
Since low diversity is the key reason for poor performance and convergence toward local optima in GA [14], phenotype diversity which counts the number of unique fitness values in a population is calculated to make sure a good quality for avoiding premature convergence.
Fig. 3 Diagram of single-point crossover method
Fig. 4 Configuration of SiC crystal growth furnace
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QH ðr; zÞ ¼
rx2 2 ðC þ S2 Þ 2r 2
r ðki rTÞ þ QH ðr; zÞ ¼ 0 Z ~Þ ~ Þ qðx 1 eðx ~; ~ qð~ x ÞdS Kðx x Þ ~Þ ~Þ eðx eðx ~ x 2oV Z 4 ~; ~ ¼ rs T ð~ xÞ Kðx x Þrs T 4 ð~ x ÞdS
ð4Þ ð5Þ
ð6Þ
~ x 2oV
where C(r, z) and S(r, z) are the in-phase and out-of-phase components of the magnetic stream function wB(r, z, t), respectively. l, J0, x, QH(r, z), t, r, ki are the magnetic permeability, charge current density, angular frequency of the electrical current, volumetric heat generation rate, time, electrical conductivity and thermal conductivity for different components, respectively. qð~ xÞ, eð~ xÞ and rs are heat flux, radiation emissivity and Stefan-Bolzmann constant, respectively. ~ x and ~ x are infinitesimal radiative surface * elements on q V, q S is the area of the infinitesimal surface ~; ~ element, and Kðx x Þ is the surface view factor between ~ x and ~ x . For induction heating calculation, the boundary conditions are w = 0 for both r = 0 and (r, z) ? 0. For heat ~Þ ¼ Tcond , transfer simulation, the boundary conditions, Tðr ki ðoT=onÞ ¼ qrad ð~ rÞ and Tcoolingwall ¼ 300 K, are applied. The growth rate is calculated by a one-dimensional mass transfer model [16]. Based on the global thermal field, the equilibrium, stress–strain and strains equations are solved iteratively to predict the thermal stresses in the growing crystal. The equilibrium equations for an axisymmetric thermo-elastic crystal are shown as follows, 1o o r// ¼0 ðrrrr Þ þ ðrrz Þ r oz oz r
ð7Þ
1o o ðrrrz Þ þ ðrzz Þ ¼ 0 r or oz
ð8Þ
where u and v are displacement components in the radial and axial directions, respectively. In order to save time of optimization to perform thousands of evaluations, only models of induction heating, heat and mass transfer are evaluated by NGA. Then the thermal stresses could be calculated. Figure 5 shows the flow chart of NGA used to optimize SiC crystal growth system. NGA is operated from initialization, then the interaction between NGA and thermal model is activated by the fitness evaluation. After values of variable and the range for population n are specified, individuals obtained by decoding are transmitted to thermal model and used to assign the related parameters, such as electric current and parameters of crucible, etc. Then thermal code is executed and simulation results used in fitness function are transmitted to fitness evaluation procedure in NGA. If the calculation does not reach termination criterion, individuals in a new population will be generated based on fitness values by niching procedure, elitist strategy and other genetic algorithm operations.
4 Problem description To optimize the SiC crystal growth system by NGA, two cases with different parameters are carried out to show the validation of our models. Case 1 As a single-objective optimization problem, the design parameter is only the variation of the axial position of coils, Dzcoils and its variation range is 0:06 m Dzcoils 0:06 m. Normally, the growth temper-
where rrr, rzz and r// are normal stresses in the radial, axial and azimuthal directions, respectively, and rrz is the shear stress. The stress–strain relation can be described as: 1 0 10 1 0 c11 c12 c13 0 err brz ðT Tref Þ rrr B r// C B c12 c22 c23 0 CB e// brz ðT Tref Þ C C B CB C B @ rzz A ¼ @ c13 c23 c33 0 A@ ezz bzz ðT Tref Þ A 0 0 0 c44 erz rrz ð9Þ where bij and cij are the thermal expansion coefficient and elastic coefficient of an SiC crystal, respectively. The strains eij can be given by the following formulations: err ¼
ou u ov ou ov ; e// ¼ ; ezz ¼ ; erz ¼ þ or r oz oz or
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ð10Þ
Fig. 5 Flow chart of NGA to optimize SiC crystal growth system
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ature of SiC in PVT furnace is 1900–2400 C [17], so Tc \2673 K is applied as the constraint. The object of optimization is to maximize the average growth rate Vg while the input current is fixed, so the objective function can be written as, Maximize fp ðDzcoils Þ ¼ Vg ð11Þ subject to 0:06 m Dzcoils 0:06 m
ð12Þ
Tc \2673 K
ð13Þ
In this case, the fitness value is the objective function. The parameters for NGA used for three calculations are shown in Table 1. Case 2 In this case, a multi-objective optimization problem is considered, the design parameters are the variation of the thickness of substrate holder Dd, and variation of input current DI. The thickness of the substrate holder d is shown in Fig. 4. Tc \2673 K is also applied as the constraint. Since small temperature gradient along the substrate is necessary to ensure a lower thermal stress in a growing SiC crystal [15]. The objective of optimization is to maintain the temperature variation along the growth surface, DTe-c at 2 K while Tc is kept at 2500 K. So the objective functions can be written as, Minimize fp ðDd; DIÞ ¼ w1 j2500 Tc j þ w2 jDTec 2j
ð14Þ
subject to 0:003 m Dd 0:02 m
ð15Þ
2250 A DI 2250 A
ð16Þ
Tc \2673 K
ð17Þ
where weighting factors w1 and w2 are set as w1 = w2 = 0.5. In this case, the fitness value is the objective function and the parameters of NGA used for three calculations are shown in Table 2.
5 Results and discussion Case 1 Figure 6 shows the best fitness, fp_best with the evolution process in three calculations. As the figure shown, although fp_best develops in different ways with different generations and individuals, the same best fitness
is obtained, and the maximum average growth rate is 460 lm/h, while the axial position variation of coils, Dzcoils is 0.0181 m. Figure 7 shows the growth rate Vg and average growth rate Vg along the substrate surface with the original and optimized axial position of the coils. From the figure, the growth rate of optimized system is higher than that of the original one. Vg of original system is 417.7 lm/ h, which is increased by 10 % by moving up coils for 0.0181 m without increasing power consumption. Balkas pointed out that the growth rate was a linear function of the temperature difference between the substrate surface and the powder, DTp-c, and an exponential function of the reciprocal of minus growth temperature [4]. DTp-c of the optimized growth system is 6 K less than that of the original one, but the growth temperature of the optimized growth system is 18.6 K higher, so increasing the growth temperature is the main reason for the increase of the growth rate. Besides, the results also show that NGA maintains a very good diversity of population, which is from 80 to 98 %. Case 2 The best fitness, fp_best with the evolution process in three calculations is shown in Fig. 8. As shown in the figure, fp_best is converged for different ways in the three experiments, and finally the same minimum value of fp_best reached, 0.09 K, while Dd ¼ 0:0186 m and DI ¼ 4:18 A. If Tc is fixed as 2500 K, for the optimized furnace with thicker substrate holder, the temperature difference along the substrate, DTe-c is decreased from 10.96 to 1.91 K, which is consistent with the result by Chen [6]. The results indicate that the thicker substrate is used, the smaller DTe-c can be obtained. Besides, the population diversity is more than 86 %. In order to show the SiC crystal with high quality obtained in the optimized furnace, the thermal stresses in the growing SiC crystal for the original and optimized furnace are calculated and compared while the crystal grows to the same height, 5.6 mm. The Von Mises stresses rvon in crystal along the center-axis of SiC crystal for the original and optimized furnaces are shown in Fig. 9. The shear stresses rrz along the interface between crystal and crucible for the original and optimized furnace are shown in Fig. 10. As shown in figures, both the Von Mises stresses rvon and shear stresses rrz in the optimized furnace are smaller than that in the original one. It can be found from the results that the radial temperature gradient at the
Table 1 Parameters of NGA for case 1 String length
11
Clearing radius
0.02
Crossover probability
0.7
Mutation probability
0.064
Generations
Individuals
Exp. 1
Exp. 2
Exp. 3
Exp. 1
Exp. 2
Exp. 3
30
30
40
50
30
50
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J Mater Sci: Mater Electron Table 2 Parameters of NGA for case 2 String length
18
Clearing radius
0.015
Crossover probability
0.7
Mutation probability
0.064
Fig. 6 The best fitness, fp_best with the evolution process in three calculations
Fig. 7 The growth rate Vg and average growth rate Vg along the substrate surface in growth system with original and optimized position of the coils
Generations
Individuals
Exp. 1
Exp. 2
Exp. 3
Exp. 1
Exp. 2
Exp. 3
60
40
40
30
40
50
Fig. 8 The best fitness, fp_best with the evolution process in three calculations
Fig. 9 Von Mises stresses rvon along the center-axis of crystal for the original and optimized furnace
same height of the crystal is decreased obviously from 496 to 192 K/m while the axial temperature gradient is almost unchanged. As the non-uniform temperature field and high temperature gradient will cause high thermal stress, the optimized furnace can grow SiC crystal with higher quality.
6 Conclusions The NGA and thermal models were successfully combined to optimize the SiC crystal growth system. Two cases for optimization of the SiC crystal growth system were carried
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Fig. 10 The shear stresses rrz along the interface between crystal and crucible for the original and optimized furnace
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out to obtain a crystal with high quality at higher growth rate by NGA. Results of case 1 showed that the growth rate of the optimized growth system was increased by 10 % without increasing the power consumption by moving up the coils. Results of case 2 showed that the temperature difference along the growth surface could be significantly reduced by making substrate holder thicker, and the SiC crystal with higher quality was obtained for the thicker substrate holder. It can be seen that NGA can be effectively combined with thermal models to optimize SiC crystal growth system.
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