THE 8th INTERNATIONAL SYMPOSIUM ON ADVANCED TOPICS IN ELECTRICAL ENGINEERING May 23-25, 2013

Bucharest, Romania  

Multiobjective Design of Transverse Flux Induction Heating Device for Semiconductor Processing

Camelia Petrescu1, Lavinia Ferariu2

1“Gheorghe Asachi” Technical University of Iasi, Faculty of Electrical Engineering 2”Gheorghe Asachi” Technical University of Iasi, Faculty of Automatic Control and Computer Engineering

campet@ee.tuiasi.ro, lferaru@ac.tuiasi.ro

Abstract- A multi-objective optimization of an induction device used for heating thin flat semiconductor plates is performed. Two conflicting targets are considered: increase of the uniformity of dissipated power (and thus of the temperature) and the increase of total power absorbed by the semiconductor. Three optimization techniques, mono-objective aggregation, Deb’s algorithm and cluster groups, all implemented using genetic algorithms, are used.

Keywords: induction heating, multiobjective optimization, genetic algorithms

I. INTRODUCTION

Induction heating is a well-established technique used in the processing of flat semiconductor wafers [1-4]. The technology is applied in the fabrication of electronic devices such as photovoltaic cell arrays. High temperatures (up to 2000°C) and short heating times are the main advantages of this heating method. Transverse flux configurations, i.e. a system of coils placed above and/or under the flat heated material, are used in this case. The main demand for a proper processing of the wafer is obtaining a uniform temperature field inside it, this being rather difficult to attain using simple coil arrangements.

A new concept called “zone control”, first described in [3], focuses on the goal of attaining the uniformity of the dissipated power produced by the eddy currents in several subdomains heated by independent coils, instead of monitoring the temperature in the whole domain. The project was further developed in [4]. In both these cases a mono-objective optimization of the heating devices was carried out using evolutionary strategies, considering the design variables to be the currents in the coils and the objective function to regard the uniformity of dissipated power throughout the subdomains.

However, as other researches showed [5], the configuration that renders the highest degree of power density uniformity is one that decreases the total absorbed power, thus leading to a higher heating time.

That is why a multi-objective optimization of the device proposed in [3] is treated in this paper, considering the second target to be the increase of the total power dissipated in the heated region. The objective function that covers the heating uniformity is an amended version of the one used in [3].

The multi-objective optimization is carried out by means of genetic algorithms – a widely recommended alternative in

optimization environments with conflicting targets and a large number of decision variables [6-10].

The paper is organized in five sections. Section II presents the induction heating system, states the magnetic field analysis problem and formulates the objective functions. Section III presents the genetic multiobjective optimization theoretical background and the implementation used. Section IV presents and discusses the obtained results, while section V emphasizes the conclusions of the research.

II. SYSTEM DESCRIPTION AND PROBLEM FORMULATION

An axial cross section of the induction heating device is presented in Fig.1.

The wafer, which is silicon carbide, is heated due to the eddy currents induced by the radio-frequency sinusoidal currents in the exciting coils. A graphite plate is placed between the silicon wafer and the coils. The system of 8 coils, each with two wires and square cross-section, is set on a quartz plate and is connected to high frequency inverters that supply currents with the same frequency, f, and different amplitudes, I1,…, I8.

The governing equation for the magnetic field is:

€

rot µ−1rotA( ) + jωσ − ω2ε0εr( )A = J(e) (1)

where A=Aθeθ is the complex magnetic vector potential having only a tangential component, J(e) is the complex density of the externally applied current, is the angular frequency and σ, ε r, µ are the electric and magnetic constants.

Fig. 1. Induction heating system using “zone control”

978-1-4673-5980-1/13/$31.00 ©2013 IEEE

The magnetic field problem is considered to be uncoupled and quasi-stationary (the material properties are constants, not depending on temperature and on the magnetic field).

The “zone control” technique implies using a system of coils fed by currents of different amplitudes and also using a partition of the domain of interest (the wafer) in a number of subdomains (zones) in which the dissipated power is calculated. The number of zones is usually equal with the number of coils (8 in this case). The mean power dissipated in a subdomain Di is:

€

Pi _mean =PiVi

=

pv (r,z) 2πr dr dzDi∫

2πr dr dzDi∫

, i = 1,8

(2)

where is the power density.

A suitable choice for the objective function that expresses the heating uniformity is:

€

F1 =maxi

Pi _mean( )mini

Pi _mean( ), i = 1,8 . (3)

The decision to use the mean power and not the absolute power, Pi, was taken because the domains have different volumes, although they extend on equal distances in radial direction. During the optimization process F1 is minimized,

F1_best=1. (4) The second objective function is the power dissipated in

the whole domain:

€

F2 = Pii=1

8

∑ = pv2πr dr dzD∫ , D = Di

i=1

8

, (5)

to be maximized. The importance of heating uniformity prevails over the heating time, so that the two objective functions have different weights in optimization.

The two objective functions are in mutual conflict, a solution that increases the uniformity of the dissipated power throughout the subdomains, decreasing the overall absorbed power. The variables in the optimization problem were considered to be the frequency, f, and the amplitudes of the coil currents, normalized with respect to the current amplitude in the innermost coil:

€

Ik I1 , k = 2,8 . (6) Since the objective function surface can be multimodal and

little information is known a priori concerning the optimal solutions, a stochastic search procedure based on genetic algorithms was used in optimization.

III. GENETIC OPTIMIZATION

The design method requests solving a multi-objective optimization (MOO) problem with two conflicting objectives (the minimization of F1 and the maximization of F2). A proper device configuration should be chosen from a large,

highly dimensional search space (i.e. 8 decision variables), without any preliminary knowledge about its landscape.

Briefly explained, the genetic algorithm starts with a random set of potential solutions, uniformly distributed over the whole search space [10]. The population evolves for a predefined number of generations (i.e. iterations). At every iteration new solutions are produced by mating the most valuable individuals from the current population. Then, the population is updated by a competition for survival performed between offspring and existing solutions. At the end of the evolutionary loop, one of the best adapted chromosomes from the last population is declared the result of the algorithm.

For a more intuitive analysis of the search trajectory, float chromosomal encoding is employed. Every chromosome within the population indicates the frequency and the amplitudes of the electric currents passing through the groups of external coils. These amplitudes are encrypted as fractions of the electric current amplitude in the coil of minimum radius. Given the float chromosomal representation, the offspring are produced with the intermediary crossover and the uniform mutation suggested in [10].

In order to solve MOO problems, several customizations should be considered within the standard genetic loop [6-8]. Conflicting objectives lead to an infinite set of optimal points, named the Pareto-optimal set. Any Pareto-optimal solution indicates a potential compromise between the involved objectives that does not allow an improvement in any objective direction, without altering the performances in at least one of the remaining objective directions. With this in mind, the aim of the evolutionary algorithm must be to describe the whole Pareto-optimal set, which translates into finding many diverse solutions placed as close as possible to the Pareto-optimal set, and also distributed as scattered as possible along this front.

Being population-based algorithms, genetic approaches are strongly recommended for MOO [7, 8]. They give the possibility to control the diversity of final solutions, even during a single run. Usually, from the final population, the user selects a solution with higher practical usability – by making use of additional heuristics. Unlike the canonical approaches, the algorithm presented in this paper integrates this additional heuristics within the evolutionary loop. The main advantage lies in guiding the exploration towards the areas of the search space which are suitable for the application. This could allow obtaining appropriate solutions with reduced exploration effort.

There are a lot of genetic MOO approaches suggested in the related literature [6-9]. When dealing with a reduced number of objectives – as claimed by this application, the Pareto methods seem to be the most effective in keeping appropriate diversity within the population, while guiding the search towards the Pareto-optimal set. Basically, the Pareto techniques compute ranking-based fitness values, which are used for filling the recombination pool and/or for insertion. However, unlike the case of mono-objective optimizations, different points in the objective space could be assigned to the

same rank, as the sorting relationship is only partial. Usually, the ranking is done by means of dominance analysis. Within this framework, a solution x is considered to dominate another solution y, if it has better objective values in all the objective directions. Given two arbitrary solutions x and y, one of the following cases could be met: (i) x dominates y; (ii) y dominates x; (iii) x does not dominate y and y does not dominate x. For instance, the ranks for x and y could be assigned as follows: better rank for x in case (i) and for y in case (ii); similar ranks for x and y in case (iii).

A similar idea is exploited in Deb’s algorithm [7] – the most popular in the field. In sequence, several Pareto-fronts of different orders are extracted from the population, all the solutions in a front being assigned with the same rank. The first order Pareto front contains the non-dominated solutions from the set (i.e., the solutions which are not dominated by any other existing solution), the second order Pareto front includes the non-dominated solutions resulted after eliminating the first order Pareto front, and so on. Afterwards, inside each front the ranks are refined by improving the fitness values of solitary solutions. This mechanism is helpful for preserving high diversity within the population – a key issue for the success of Pareto MOO techniques. The efficiency of Deb’s algorithm was empirically proven in many MOO problems, however, this fitness assignment is limited to objectives with the same priority and to a posteriori aggregation between search and decision (i.e. at the end of the search, the decision block picks a useful solution from the resulted set, without any previous cooperation) [7-9].

In better relation to the particularities of the MOO problem, this paper introduces a new special assignment scheme, denoted CFMO (Clustering-based Fitness assignment for Multiobjective Optimization), which is compatible with objectives of different priorities. The priorities are known beforehand at a qualitative level, only. During the evolutionary loop, the algorithm adapts the distance between these priorities in accordance to the content of the population. This adaptation mechanism involves a gradual integration between decision and search, meant to guide the exploration towards the regions preferred by the application.

Basically, CFMO exploits two characteristics of MOO, for a more appropriate fitness computation. Firstly, it is known that the Pareto optimal solutions placed at the ends of the Pareto fronts have no practical use, hence the search around them is progressively avoided. More specifically, this refers to the solutions featuring excellent values for F1, but too small values for F2, or the solutions with large F2 values, but also with too large F1 values. Secondly, the maximization of F2 is less important, so this objective is assigned with a lower priority and/or exploited at a qualitative level, rather than a quantitative one.

This knowledge is incorporated within the CFMO by means of a preliminary grouping. Before ranking, the population is separated in 3 clusters, each one governed by distinct sorting rules. Assuming a set of N individuals

obtained at generation t, , the clusters are

drawn as follows:

€

C1 = {M j ∈ P(t)F1(M j ) ≤ m1,F2 (M j ) ≤ m2}, (7)

€

C2 = {M j ∈ P(t)F1(M j ) ≤ m1,F2 (M j ) > m2}, (8)

€

C3 = P(t) \ (C1 C2 ) (9)

Here, denotes an arbitrary individual from . The borders of the clusters are depicted by controlling the size of each group ( , ): individuals are at the left of on the histogram of F1 and individuals stay at the left of on the histogram of F2. Inside each resulted cluster, the ranks are separately determined, while keeping the ranks from better than those in , and the ranks in

better than those in . As all the individuals belonging to have reasonable F2 values, their ranks are assigned in terms of , only. On the contrary, and can contain solutions with unsatisfactory F2 values. Therefore, inside these clusters the ranks are assigned subject to both objective functions, and . The Pareto-ranks are computed by means of Deb’s algorithm, applied separately in and . Afterwards, the fitness values are linearly assigned according to the resulted ranks.

Given the description above, several important advantages of CFMO can be outlined: i) for the most valuable solutions, the maximization of dissipated power acts at a qualitative level, only; ii) the preliminary clustering combines decision and search, in accordance to the performances of the individuals included in the current population, without the need of rich supplementary initial information; the user can tune the behavior of the CFMO by setting different sizes of clusters; iii) Pareto-ranking is performed in smaller clusters, at lower computational costs; iv) the size of the clusters is directly controlled without the need of other adapting techniques; this enables a pertinent ranking inside each group, without the risk of obtaining unfeasible conclusions due to an ineffective comparison between too few solutions; v) Pareto-ranking is applied for bi-objective optimization only, leading to increased relevance of ranking.

IV. RESULTS AND DISCUSSIONS

The magnetic field problem was solved using the FEM based software COMSOL Multiphysics, namely the AC/DC Azimuthal Induction Currents module. The eddy currents effects were considered in all the conductive regions, wafer, coils and graphite. Table I presents the numerical values of the material properties considered in simulations. The case of heating a wafer disc with a diameter of 300 mm and 1 mm in height, using 8 equally spaced coils with square wire cross section of 1.5 mm2 and having the innermost radius 8 mm and the inner radius of the external coil 148 mm was studied. COMSOL was used to generate the m-file used by the GA.

The performances of CFMO were experimentally investigated in comparison with those provided by two other

design algorithms: O_F1 (which ensures the mono-objective optimization of F1) and D_MO (which addresses the same MOO problem, although it computes the ranks with Deb’s algorithm applied to the whole population, without any preliminary clustering). All these algorithms are implemented by the authors in MATLAB + Comsol.

TABLE I MATERIAL PROPERTIES

Graphite Wafer Quartz Coils Air Conductivity (S/m)

3000 1000 1e-12 5.998·107 0

Permittivity 1 10 4.2 1 1 Relative permeability

1 1 1 1 1

The experiments were carried out for the configurations indicated in Table II. The frequency range is [4e4, 4.5e4].

The selection for recombination is performed by means of stochastic universal sampling. At each generation, N/2 offspring are obtained with intermediary crossover and uniform float-based mutation, applied with the probabilities 0.7 and 0.1, respectively. 30% of the offspring (chosen from the best ones) survive to the next generation. The evolutionary loop is carried out for 20 generations, considering the same initial random population for the trials involving the same number of individuals.

Table III shows the performance of the solutions selected from the final population (columns F1

* and F2*). For CFMO

this is the solution with minimum F1 value, found within C1, for D_MO this is the solution with minimum F1 value from the first order Pareto front and for O_F1 this is the solution with minimum F1 value from the whole population. Table III

also indicates the mean performances achieved within the last population (columns mF1 and mF2). As expected, unlike O_F1 and D_MO, the behavior of CFMO results more diverse, depending on the preset sizes of the clusters. These parameters allow an intuitive preliminary setting of the objective priorities, although at a qualitative level, only. It is important to notice that a too large C1 (#1.4, #1.5) can lead to unsatisfactory results, due to the fact that it induces a premature elimination of the solutions featuring bad F2 values. However, some of these solutions have better chances to provide valuable F1 values, hence the algorithm might focus on F2 improving, even if it evaluates the majority of the individuals in terms of F1. For comparable sizes of C1 (#1.1, #1.2, # 1.3), the size of C2 can have lower impact, if the configuration of clusters still allows selecting numerous parents from C1 and C2. Obviously, ranking the individuals of C2 without Pareto-sorting translates into a bigger selection pressure of the first objective (#1.1 vs #1.6 and #1.2 vs #1.7). As illustrated in Fig. 2 (for #1.2), CFMO is able to preserve the diversity of the population. At the end of the evolutionary loop, its population enables good exploration capabilities by means of diverse and useful solutions. Insight characterization of algorithm behavior is also given by the average performances of the last population, mF1 and mF2. Compared to O_F1, CFMO is also interested in the improvement of F2, so that it provides larger mF2 values. At the same time, CFMO ensures smaller mF1 values than D_MO, being more effective in guiding the search towards the preferred regions.

TABLE II

CONFIGURATIONS USED FOR EXPERIMENTAL TRIALS

1 2 # 1.1 1.2 1.3 1.4 1.5 1.6 1.7 2.1

N 200 200 200 200 200 200 200 400 SR-C2 2 2 2 2 1 1 1 2 l1 0.2*N 0.1*N 0.1*N 0.4*N 0.8*N 0.2*N 0.1*N 0.2*N l2 0.3*N 0.1*N 0.8*N 0.3*N 0.1*N 0.3*N 0.1*N 0.3*N l3 0.5*N 0.8*N 0.1*N 0.3*N 0.1*N 0.5*N 0.8*N 0.5*N

N is the population size. SR-C2 is the ranking strategy employed in C2 cluster: 2 for Pareto-ranking in terms of F1 and F2; 1 for ranking in terms of F1, only. l1, l2 and l3 are the sizes of the clusters formed by CFMO.

TABLE III EXPERIMENTAL RESULTS

CFMO O_F1 D_MO # F1* 103F2

* mF1 103mF2 F1* 103F2

* mF1 103mF2 F1* 103F2

* mF1 103mF2 1.1 1.47

02 4.7607

1.812918

9.164762

1.2 1.3633 5.6315 1.455174 5.984148 1.3 1.4923 5.6483 2.198403 12.436780 1.4 1.9542 11.550 2.010170 11.917712 1.5 1.6728 8.3700 1.734679 8.563564 1.6 1.4106 4.7291 1.724499 7.376872

1

1.7 1.2921 4.1288 1.450022 5.357926

1.1952

4.1854

1.2951

4.8053

1.8583 10.208 2.2578 12.2239

2 2.1 1.2343

4.59887

1.537829

6.637700

1.1910

4.2610

1.4138

5.3045

1.5145 5.3968 2.3388 13.600

F1* and F2

* are the objective values for the selected solution. mF1 and mF2 are the average values of F1 and F2 within the last set.

Because both CFMO and D_MO make use of Pareto-ranking, their results could be acceptable only when working on large enough populations. Therefore, increasing the population size could be beneficial for their behavior (#1.1 vs #2.1 for CFMO and #1 vs #2 for D_MO). However, it should be noticed that there is not a monotonic dependency between N and the performance of the results, thus, a larger N does not guarantee the achievement of better solutions. It is also useful

to remark that D_MO has the tendency to keep inside its population some non-dominant solutions which are unsuitable for the application, thus part of its genetic collection having no practical usability.

Since the values of the eddy currents produced by one coil are smaller at the center of the wafer, the restriction Ik<Ik+1 was imposed in most of the stochastic searches.

A possible solution of interest can be that in case 1.2 (Table III) in which a reasonable (but not perfect) uniformity of the power density throughout the monitored zones is obtained. In this case the values for Ik , k=1,..8, are [1, 0.5360, 0.465, 0.4195, 0.4672, 0.3521, 0.5191, 0.3983]*I1, while the frequency is f=41.624 KHz.

Another solution of interest is 2.1 (Table III), which gives a smaller value for F1, but also a smaller power absorption. In this case the values for Ik , k=1,..8, are [1, 0.6149, 0.3684, 0.4925, 0.4554, 0.4892, 0.3705, 0.4995]*I1 and the frequency is f=43.443 KHz. Fig.3 plots the magnetic flux density lines in this case.

The best value for F1 was 1.19 using O_F1 (which ensures the mono-objective optimization of F1), rendering an absorbed power F2=4.18mW.

It may be also concluded that including the frequency among the design variables does not clearly lead to improved power density uniformities, but leads to a higher complexity of the search space and possibly impairs the convergence of the genetic algorithm to an optimum solution.

Fig. 2. The solutions of the final CFMO population, plotted in the objective space – trial #1.2

r (m)

Fig. 3. Magnetic flux density lines for case 2.1, Table III

z (m)

V. CONCLUSIONS

The study undertaken in this paper reveals that introducing the second objective function in the optimization of the induction heating device (the total absorbed power) makes it very difficult to attain solutions that render an almost uniform temperature distribution in the wafer. If optimization is carried out only with respect to F1, a quasi-uniform power density distribution can be obtained using the zone control technique, as shown by [3, 4] and by the results of this paper. As always, it must be the designer’s choice which solution to adopt.

A solution to further increase the power density uniformity would be to use a larger number of coils and, consequently, monitor a larger number of zones. At the same time, the design variable space could be increased by considering that the coil radii are not fixed (as considered in this study), but can vary within certain limits. However, due to the larger number of design variables, the genetic algorithm can display in this case a slower convergence to optimum values.

Further researches are intended to include the coil radii among the decision variables in the optimization process.

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