Numerical Backstepping for Diameter Control of Silicon Ingots in the Czochralski Process

Transcription

1 5st IEEE Conference on Decision and Control December 0-3, 202. Maui, Hawaii, USA Numerical Backstepping for Diameter Control of Silicon Ingots in the Czochralski Process Parsa Rahmanpour and Morten Hovd Abstract The Czochralski crystallization process is an important process for the production of monocrystalline silicon for semiconductor and solar cell applications. We use a design procedure motivated by backstepping for the control of crystal diameter - since the system model is not of a form for which backstepping is directly applicable. Similarly to backstepping, the design procedure is iterative, and allows for application of problem-specific engineering understanding at each stage of the procedure. the designed controller to the process model, whereas section 5 contains discussion and conclusions. II. THE CZOCHRALSKI CRYSTALLIZATION PROCESS AND A MODEL FOR THE CRYSTAL RADIUS A. Czochralski Process Description A sketch of the Czochralski process is shown in Fig.. I. INTRODUCTION One of the few techniques where the crystal lateral surface shaping is performed without any contact with the container walls, is the Czochralski (Cz) method. The idea is to heat silicon contained in a crucible up to its melting point, while dipping a seed crystal (of same material) into the melt and then pulling it up slowly. At the meniscus, i.e. the interface between the melt and the crystal, crystallization will take place. The crystallization process can be described mathematically using the conservation of mass and the heat balance around the meniscus. Jan Czochralski discovered this method in the early 900 s, but it has only found wide practical application during the last decades because of the development of semiconductor engineering and the solar industries. After production of the silicon ingot, it is cut into thin wafers, and each wafer cut into a specific shape (typically quadratic) depending on the final application. The crystal diameter must therefore be sufficiently large for the final application, but too large crystal diameter will mean that expensive material is wasted. Naturally, the production rate is also of great importance for the Czochralski process. However, since the production rate correlates with the amount of defects in the crystal structure, a straight forward maximization of the production rate is not desired. This paper addresses the control of the crystal diameter only, with the crystal pulling speed as the control input. It is implicitly assumed that the production rate is controlled by using the heaters to control the metal bath temperature. Section 2 will give a brief introduction to the Czochralski process and present a simplified process model, section 3 will give a brief introduction to backstepping and present the control design method used for the Czochralski diameter control problem. Section 4 presents the results of applying This work was supported by Prediktor AS and the Research Council of Norway. Parsa Rahmanpour and Morten Hovd are with the Engineering Cybernetics Department, Norwegian University of Science and Technology, N-749 Trondheim, Norway parsa.rahmanpour,morten.hovd@itk.ntnu.no Fig.. An illustration of the main parts of the Czochralski crystallization process (This Figure is licensed under a Creative-Commons BY-NC-SA license). The solid silicon put in a crucible. Electrical heaters are used both to melt the silicon, and to maintain an appropriate temperature trajectory throughout the crystallization process. A small crystal seed is put in contact with the molten silicon, and the crystal is produced by slowly pulling the seed out of the melt. Initially, the crystal diameter is increased quite quickly, whereas for most of the duration of the process it is desirable to keep the crystal diameter constant. A sketch of the region around the meniscus is shown in Fig. 2. During the crystallization, the heat of fusion (a.k.a. latent heat) is released at the crystal/meniscus interface. The crystal extends from the molten metal bath into the colder environment. This will set up a temperature gradient, with corresponding heat transfer from the bath, through the crystal and to the environment. The heat transfer rate through the crystal from the interface to the environment must therefore balance the sum of the heat transfer from the bulk of the /2/$ IEEE 703

2 Fig. 2. A sketch of the crystal and its contact with the molten metal(this Figure is licensed under a Creative-Commons BY-NC-SA license). molten metal to the interface and the rate of release of latent heat at the interface. The heat transfer from the molten metal bath to the interface will be affected by meniscus geometry. The meniscus geometry will also affect the crystal growth angle, and thereby also the crystal diameter. The crystal diameter will in turn affect the heat transfer from the interface to the environment. Therefore, in order to pull an optimal mono-crystal silicon, it is essential to ensure the appropriate shape of the melt meniscus []. B. A Crystal Formation Model for the Czochralski Process The basic phenomena that need to be covered by a model for the Czochralski process are the capillary problem and the thermal conditions. From the theory of hydrostatics, the equilibrium shape of the liquid surface is described by the Laplace capillary equation [2]: σ LV R + σ LV R 2 + gρ L z = const. () where σ LV is the liquid surface tension coefficient at the three-phase boundary, ρ L shows the difference between the densities of the ambient fluid (melt) and liquid shaping the meniscus, and R and R 2 are the principal radii of curvature of the meniscus. With definition of the capillary constant a = (2σ LV /gρ L ) /2, we get a dimensionless coordinate z/a = z where the z-axis is directed vertically upwards. The liquid surface meniscus for cylindrical or tubular crystals is obtained by rotating the profile curve around this axis. The shape of the free liquid surface is given by solution of the Laplace-Young equation [2]: z r + z ( + z 2 ) ± 2(d z)( + z 2 ) 3/2 = 0, d = (2) P a 2σ LV Here z and r represent the vertical and radial coordinates, while a is the Laplace constant and P is a constant depending on the constant defined in (). Furthermore, the prime denotes differentiation with respect to r, the plus sign in the equation corresponds to the meniscus part with z < 0 and the minus sign is for the case with z > 0. For the thermal part of the model, much research has been devoted to calculation of the temperature field in the crystal-melt system. However, because of the variety of growth configurations and the presence of a great number of elements that must be considered while studying the thermal conductivity problems, a complete mathematical description of heat transfer throughout the process is extremely difficult. Analytical solutions are usually achieved by applying many simplifications [2]. Having this perspective in mind, Carslaw [3] derived the following equation by reducing the Navier-Stokes equation in order to describe the temperature distribution of the crystal-melt system in a one-dimensional approximation: T i k i t = d2 T i dz 2 v p dt i λ i dz µ i F (T i T amb ) (3) k i Here i = L, S (liquid and solid body), T i denotes the temperature, λ i is the thermal diffusivity coefficient, z is the vertical coordinate, µ i denotes the coefficient of heatexchange with the environment, F defines the crystal (meniscus) cross-section perimeter-to-its area ratio, T amb is the ambient temperature, and k i is the thermal conductivity coefficient. According to Tatarchenko [2], the temperature variable in (3) should be interpreted as the temperature averaged over the cross-section of the crystal/meniscus, with real isotherms replaced by flat ones. An accurate model for the Czochralski process will necessarily result in a highly complex model involving coupled PDE s [2]. Such simulation models do exist, but they are typically based on assuming quasi-stationarity conditions and are anyway too complex for controller design. Following [4], we use instead the simple model defined below: ṙ = v g (r, h, t) tan(θ) ḣ = u v g (r, h) where the input u = v p (pulling velocity), θ defines a nearly constant growth angle, and v g is the growth velocity normal to the interface along with the pulling direction and defined as: v g (r, h) = (4) [ 2H k s H r (T M T amb ) (5) k ] l h (T B T M ) Here H is the latent heat of fusion per unit volume, k s,l are the thermal conductivities respectively in crystal and melt, T B is the temperature at the base of the meniscus, T M is the melting temperature (at the lower surface of the cylindrical crystal) and T amb represents the ambient temperature at the top of the crystal. H = (4ɛ s σ/k s )Tamb 3 is the linearized heat transfer coefficient, ɛ s being the emissivity of the crystal and Stefan s constant is denoted by σ [5]. The 704

3 growth angle θ is small and nearly constant, but cannot be measured during normal operation. Instead we replace it with its approximation defined in (6) []. sin(θ) ( h a )2 ( + a r 2 ) θ (6) Here a, as above, represents the Laplace constant. In normal operation, the angle θ will stay within a few degrees from zero. Thus, we can use the approximation tan(θ) sin(θ) θ for θ 0. In order to make the model easy-to-follow, some new variables are introduced to replace the temperature relations in v g (r, h, t): 2H ζ = H k s(t M T amb ) (7) and η = k l H (T B T M ) (8) This results in the model h 2 η a h 2 r + ηh a 2 ṙ = ζ r η h ζ a r ( a + r ) + 2 ḣ = u ( ζ r η h ) (9) Although the model consists of two states only, it is still relatively complex. One should of course keep in mind that it is only an approximation of the true process, and therefore can only be expected to be accurate within a limited operational range. Clearly, the model only makes sense for positive r and h. It is also important to keep the growth rate v g positive, both for productivity and because melting and regrowing the crystal will ruin the crystal structure. III. CONTROLLER DESIGN The controller design is motivated by backstepping, and we will therefore give a brief introduction to this controller design methods. Readers in need for more information about backstepping are referred to [6] and [7]. A. Backstepping Design We will illustrate backstepping design with a two-state system, but note that systems with a higher number of states can be acommodated by backstepping multiple steps. Consider the system ẋ = f(x ) + g(x )x 2 (0) ẋ 2 = u () The functions f and g are assumed to be smooth. We assume further that a Lyapunov function V (x ) is known for (0) together with a corresponding stabilizing state feedback x 2 = φ(x ) such that V [f(x ) + g(x )φ(x )] W (x ) x (2) where W (x ) is a positive definite function. Adding and subtracting g(x )φ(x ) in (0) we obtain ẋ = f(x ) + g(x )φ(x ) + g(x )(x 2 φ(x )) (3) Performing the change of variables z = x 2 φ(x ) we obtain the system ẋ = f(x ) + g(x )φ(x ) + g(x )z (4) ż 2 = u + φ (5) where φ = φ (f(x ) + g(x )x 2 ) = φ ẋ. Observing that (4) is asymptotically stable when z is zero, we choose an overall Lyapunov function V (x, z) = V (x ) + V 2 (z) = V (x ) + 2 z2. The time derivative of V (x, z) can now be found to be V = V [f(x ) + g(x )φ(x )] (6) + V g(x )z + zż W (x ) + V g(x )z + zż Choosing the input u such that ż = kz V g(x ) for some k > 0 therefore ensures V W (x ) kx 2 2 and hence the stability of the system. The corresponding state feedback is given by u = φ V g(x ) kz (7) We note for later reference that g(x ) = [ẋ (x, x 2 ) ẋ (x, φ(x ))] /z. B. Controller Design for the Czochralski Crystallization Process Attempting to apply the backstepping methodology for the control of the radius r in Czochralski crystallization, we find that (9) is not of the same form as (4-5), and that backstepping in its basic form therefore does not apply. However, numerically it is fairly straight forward to determine a desired value for h (denoted h d ), such that r is given some desired behavior. That is, we choose V (r) = 2 (r r 0) 2, where r 0 is the reference for r. We then find that V = (r r 0 )ṙ (8) We want to achieve ṙ = c (r r 0 ), and considering (9) we therefore define the function F (r, h) = ζ η r h ζ ( a r a + r 2 ) (9) η + a h 2 r + ηh a 2 + c (r r 0 ) The value of h d is therefore the value of h that solves F (r, h) = 0 for the present value of r. We observe that finding h d involves finding the a root of a cubic function. While procedures for doing this exists, it is essential to consistently pick the correct root. For the case at hand, engineering understanding of the problem can be used. Having thus found h d, we define as above z = h h d, and find that ż = ḣ ḣd = u ( ζ r η h ) ḣd (20) h 2 705

4 An expression for ḣd is therefore required. This is available from (9) - not only do we wish to find h d such that F (r, h d ) = 0, we also need to determine ḣd such that F = 0. Expressed mathematically, this means that d dt F (r, h d ) = r F r,h d ṙ + F r,hd ḣ d = 0 (2) h d which is straight forward to solve for ḣd. Choosing V 2 (z) = 2 z2, we therefore find that V (r, z) = d dt (V (r)+v 2 (z)) < 0 if ż = c 2 z (r r 0 )ψ(r, h) (22) for some c 2 > 0 and We note that ψ(r, h) = [ṙ(r, h) ṙ(r, h d (r))] /(h h d ) (23) ṙ lim ψ(r, h) = h d h h r,h=h d (r) (24) and ψ(r, h) is therefore a well defined function also as h d h provided ṙ is smooth at (r, h d ). This results in the input u = c 2 z + ( ζ r η h ) + ḣd (r r 0 )ψ(r, h). (25) Clearly, the term involving ψ(r, h) corresponds to the term V g(x ) for the conventional backstepping. IV. SIMULATION RESULTS The control design described above has been simulated on the plant model. In the simulation, a change in the reference for the crystal diameter from r = 5cm to r = 4cm is made at time t = 2000s. The response in crystal radius is shown in Fig. 3. It can be seen that the crystal radius tracks the reference well (and yes, the time scale is right - this is a slow process). Crystal radius (m) Fig Crystal radius Radius reference 0.38 Response in crystal radius to a setpoint change from 5cm to 4cm. The corresponding responses in meniscus height and the crystal pulling speed (the manipulated input) are shown in Figs. 4 and 5, respectively. It is important to ensure that the control adheres to operational constraints. One such operational constraint is that the crystal growth rate must always be kept positive. Meniscus height (m) 6.2 x Fig. 4. Response in meniscus height to a crystal radius setpoint change from 5cm to 4cm. Crystal pulling speed (m/s) 3.5 x Fig. 5. Crystal pulling speed (manipulated input) to a crystal radius setpoint change from 5cm to 4cm. Attempting to increase the crystal radius from 5cm to 6cm, the controller reduces the meniscus height so much that the growth rate according to (5) goes to zero, and crystal growth stops. To avoid this problem, we constrain h d to h d > 4e 3m. This means that we no longer select the h d which solves F (r, h) = 0 in (9), since more important concerns take priority. The response of the crystal radius with the applied constraint in h d is shown in Fig. 6. The corresponding responses in meniscus height and the crystal pulling speed (the manipulated input) are shown in Figs. 7 and 8, respectively. It is clear that the controller quickly hits the constraint in h d. This results in restricting the rate with which the crystal radius is allowed to increase. When the constraint is no longer active, the normal control of crystal radius takes over. In Fig. 8, a spike in the manipulated input can be observed as the constraint in h d becomes inactive. Such spikes are often considered undesirable, but as can be seen from Figs. 6 and 8, the dynamics of this process are so slow that the spike is of no significance. An alternative could be to filter away the spike in the input using a low pass filter (which can be made very fast compared to the dynamics of the process). V. DISCUSSION AND CONCLUSIONS A non-linear controller is designed for crystal diameter control in the Czochralski crystallization process. The con- 706

5 x Crystal radius (m) Crystal radius Radius reference 0.48 Crystal pulling speed (m/s) Fig. 6. Response in crystal radius to a setpoint change from 5cm to 6cm, with constraint in h d. Fig. 8. Crystal pulling speed (manipulated input) to a crystal radius setpoint change from 5cm to 6cm, with constraint in h d. Meniscus height (m) 5.5 x Fig. 7. Response in meniscus height to a crystal radius setpoint change from 5cm to 6cm, with constraint in h d. troller design is inspired by backstepping, but because the process model is not of the required form for backstepping, a numerical (rather than analytical) approach is used to calculate the manipulated input value. The controller needs to find the correct root of F in (9). Clearly, some care is needed to ensure that the correct root is found and selected - if the root of F is unique and easily found, conventional backstepping can typically be used. Engineering understanding of the system to be controlled is of great help in finding the correct root of F, and knowing in what range of h to search for this root. In a manner reminiscent of conventional backstepping [6], the approach applied here can easily be modified to avoid cancelling beneficial/stabilizing terms - simply do not include such terms in F in (9). Uncertain or complicated terms may be replaced by overapproximations in (9). The second simulation case illustrates how engineering understanding can be used to avoid operational problems by constraining intermediate variables in the controller calculations. For the control of the Czochralski crystallization process, there is a lot of further work to be done. The model should be extended to include the electrical heaters and the temperature of the molten metal, to allow including crystal growth rate control. Improved modelling of heat transfer in the crystal will allow for better radius and growth rate control. State and parameter estimation will need to be addressed. The Czochralski crystallization process is a batch process, and run-to-run control is therefore another interesting and relevant area of research. ACKNOWLEDGEMENT Helpful discussions with professor Ole Morten Aamo are gratefully acknowledged. REFERENCES [] T. Duffar, Crystal Growth Processes Based on Capillarity: czochralski, fl oating zone, shaping and techniques. Chippenham, Wiltshire, GB: John Wiley &Sons, Ltd., Publication, 200. [2] Y. Tatarchenko, Shaped Crystal Growth. Dordrecht, The Netherlands: Kluwer Academic Publishers, 993. [3] H. Carslaw and J. Jaeger, Conduction of heat in solids, 959. [4] D. Hurle, G. Joyce, M. Ghassempoory, A. Crowley, and E. Stern, The dynamics of czochralski growth, Journal of Crystal Growth, vol. 00, pp. 25, 989. [5] D. T. Hurle, Crystal Pulling from the Melt. Berlin, Germany: Springer- Verlag, 993. [6] M. Krstic, I. Kanellakopoulos, and P. Kokotovic, Nonlinear and Adaptive Control Design. John Wiley &Sons, 995. [7] H. K. Khalil, Nonlinear Systems, third edition. Upper Saddle River, New Jersey 07458: Prentice Hall,