- Email: [email protected]
The feedback control of the vertical Bridgman crystal growth process by crucible rotation: two case studies
Computers and Chemical Engineering 29 (2005) 887–896
The feedback control of the vertical Bridgman crystal growth process by crucible rotation: two case studies Paul Sonda, Andrew Yeckel, Jeffrey J. Derby, Prodromos Daoutidis∗ Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave. SE, Minneapolis, MN 55455-0132, USA Available online 19 November 2004
Abstract In this work, we consider the feedback control of flows within the vertical Bridgman crystal growth process. We investigate the use of crucible rotation, via feedback control algorithms, in suppressing oscillatory flows in two prototypical vertical Bridgman crystal growth configurations—a laminar flow regime driven by a time-oscillatory furnace disturbance and a time-varying regime driven by strong buoyant forces. Proportional, proportional–integral, and input–output linearizing controllers are applied to the vertical Bridgman model to attenuate the flow oscillations. Simulation results show that for the first configuration, crucible rotation is an appropriate actuation method for feedback control. In addition, nonlinear control provides superior performance to P and PI control. For the latter case, crucible rotation is less effective, due to its exacerbating effect on the inherent time-dependent flows within this system. © 2004 Elsevier Ltd. All rights reserved. Keywords: Model-based control; Flow control; Crystal growth; Numerical modeling
1. Introduction The vertical Bridgman process is used for the growth of single crystal electronic materials, for use as substrates in optoelectronic and sensing devices. A schematic of the vertical Bridgman process is shown in Fig. 1(a). Polycrystalline material is loaded into a quartz crucible (also known as an ampoule), inside of a high temperature furnace. The material is completely melted, at which time the distribution of temperature in the furnace is adjusted to vary along its length, such that one zone is hotter than the material’s melting point and the other section is cooler. Directional solidification is achieved by slowly translating the crucible towards the cold zone of the furnace. To produce devices of suitable quality, it is necessary to grow low-defect single crystals of homogenous chemical composition. This task is made difficult by the complex coupling of heat, mass, and momentum transport inherent in the ∗
Corresponding author. Tel.: +1 612 625 8818; fax: +1 612 626 7246. E-mail address: [email protected] (P. Daoutidis).
0098-1354/$ – see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2004.09.013
process. One processing issue that continues to challenge the crystal growth community is the occurrence of fine scale variations in composition, known as striations. These composition variations often cause undesirable variations in material and electronic properties. Kim, Witt, & Gatos (1972) used crystal growth experiments to investigate the occurrence of striations in a vertical Bridgman system. It was found that for sufficiently intense flows, the melt exhibited periodic, timevarying behavior. It was then shown conclusively that these time-varying flows were directly related to the occurrence of striations. In previous work (Sonda, Yeckel, Daoutidis, & Derby, 2003, 2004b), we used detailed crystal growth models to numerically simulate the experimental system of Kim, Witt, and Gatos. Our numerical simulations predict that during the early stages of crystal growth, large amplitude, twofrequency oscillations occur within the flow. In later stages of growth, the oscillatory behavior becomes more ordered with decreasing amplitude, eventually reaching a stage in which the flows are steady with time. These simulations are in qualitative agreement with the experimental results of Kim,
888
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
Fig. 1. (a) Schematic of the destabilizing vertical Bridgman system. (b) The finite element mesh used for the simulations.
Witt, and Gatos. Demonstrating the appearance of periodic, time-varying flows was an important first step in our study of crystal striations. In this work, we simulate application of feedback controllers to the system in order to suppress its time-varying behavior. The ultimate goal of this research is to understand how process control can be used to prevent the onset of crystal striations in experimentally grown single crystals. Despite the continuing need for improvement of Bridgman growth processes, there exists minimal work addressing the use of feedback control to improve these processes. Batur and co-workers (Batur, Duval, & Bennett, 1999; Batur, Srinivasan, Duval, Singh, & Golovaty, 1999; Srinivasan, 1994), designed a transparent, multi-heating zone furnace for the growth of lead bromide. The transparent furnace, in coordination with a video camera and real-time imaging software, allowed for the observation of the melt–crystal interface shape. Comparison of the observed interface shape to the desired shape provided an error signal, which was used to adjust the furnace temperature profile. Seidensticker et al. (1999) and Chin and Carlson (1983) used open-loop simulations to investigate the effects of temperature boundary conditions on interface shape and assess their applicability to a control algorithm. Azuma et al. (2003) developed an automatic feedback control system to maintain a constant temperature at the melt–crystal interface during growth of silicon germanium. The position of the interface was automatically detected through in situ monitoring via a charge-coupleddevice camera, and the ampoule translation rate was the manipulated input. Ultrasonic waves were used by Schmachtl, Schievenbusch, Zimmerman, and Grill (1998) to control the shape and velocity of the solid–liquid interface during the Bridgman growth of copper manganese. In related work,
Shiomi and Amberg (2002) applied proportional control to attenuate oscillations arising during growth by the floating zone method, another common method used to grow crystals from the melt. Two sensor/heater pairs were used for measurement and actuation. They found that complete suppression of the oscillations could be achieved for weakly nonlinear flows but was not possible for stronger nonlinear flows. There are many actuation techniques which could feasibly be used for the feedback control of the vertical Bridgman system. Methods that have been shown to perceptibly affect the molten flows include the real-time adjustment of the surrounding thermal environment (Batur, Srinivasan, et al., 1999; Yeckel, Comp`ere, Pandy, & Derby, 2004), the application of vibration (Lan, 2000; Liu, Wolf, Elwell, & Feigelson, 1987), and the application of a magnetic field (Kim, Adornato, & Brown, 1988; Yeckel & Derby, 2004) to the crucible. This work examines the use of crucible rotation as the means for suppressing oscillatory flows in the vertical Bridgman system. Open-loop crucible rotation has been successfully applied to experimental vertical Bridgman systems (Capper & Gosney, 1984) and has been the subject of simulation-based studies (see Yeckel & Derby (2000) and references therein). Proportional, proportional–integral and nonlinear, modelbased controllers are synthesized and implemented within the framework of our crystal growth simulation code (Yeckel & Goodwin, 2003). The controllers are applied to two prototypical vertical Bridgman systems and their performances evaluated.
2. The vertical Bridgman process model Fig. 1(a) presents a schematic of the configuration employed by Kim, Witt, and Gatos (hereafter referred to as KWG). This system, with warm fluid underlying the cooler crystal, is termed destabilizing, since it leads to an inherently unstable melt, with fluid mechanical behavior similar to the classical Rayleigh–B´enard problem (B´enard, 1901; Rayleigh, 1916) of hydrodynamics. We model the vertical Bridgman process using equations for conservation of mass, momentum, and energy, along with appropriate boundary and initial conditions. Our present purposes only require that we study the effect of control on flow, so we do not include conservation of chemical species in the model, although doing so would be necessary for the direct prediction of striations. A key assumption used here is that the system is perfectly axisymmetric, thus all field variables are independent of the azimuthal coordinate. Under these assumptions, the time-dependent energy transport equation is given by Pr m
∂T + v · ∇T − ∇ 2 T = 0 ∂t
(1)
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
in the melt, and Prj
∂T − ∇ 2T = 0 ∂t
assumed a profile which decreases linearly along the z axis, (2)
where the index j designates the material, either the crystal, crucible, or graphite support. T is a dimensionless temperature scaled by the melting temperature, Tmp , t is a dimensionless time scaled by R2 /ν, v(r,z) is the dimensionless velocity field, and ∇ is the dimensionless gradient operator, in which the spatial coordinates are scaled by the crucible radius R. The Prandtl number is defined as Pri ≡ ν/αi , where ν is the kinematic viscosity, and the thermal diffusivity is defined as αi ≡ ki /ρi Cp,i , where ki is the thermal conductivity, ρi the density, and Cp,i is the heat capacity. The velocity field is described by the Navier–Stokes equations for incompressible flow, using the Boussinesq approximation to describe temperature-dependent density variation: 1 ∂v + v · ∇v − ∇ · T − Ra(T − 1)ez = 0, ∂t Pr m
where Tc and Trmh are the minimum and maximum furnace temperatures, respectively, and zr is the furnace reference position. However, in our simulations, we first applied control to a stabilizing vertical Bridgman system, as its fluid dynamical behavior is significantly less nonlinear. The stabilizing configuration is accomplished by inverting the KWG furnace with respect to gravity. To produce time-dependent behavior in this naturally laminar flow, we assume that the furnace experiences a time-sinusoidal external power disturbance, (Trmh − Tc ) Tf (z, t) = Tc + (z − zr ) (1 + sin(wt)), (9) L
T = 1.
(5)
(6)
where κ is the ratio of melt thermal conductivity to solid, κ = km /ks , nsm is the unit vector normal to the melt–crystal interface, pointed towards the melt, St is the Stefan number, St = Hf /Cp,s Tmp , and x˙ is the velocity of the melt–crystal interface. Convective and radiative heat transfer from the furnace to the crucible are represented by a simple flux condition applied at the domain boundary: −∇T |a · naf = Bii (T − Tf (z)) + Rdi (T 4 − Tf4 (z))
(8)
(4)
where P is the dynamic pressure scaled by ρm ναm /R2 , and I is the idemfactor. A thermal flux condition is defined at the melt–crystal interface, (−κ∇T |m + ∇T |s ) · nsm = StPrs (˙x · nsm )
(Trmh − Tc ) (z − zr ), L
(3)
Here the dimensionless velocity is scaled by αm /R, T is the stress tensor, Ra is the Rayleigh number, Ra ≡ gβT R3 Tmp /ναm , where g is the gravitational constant, and βT the thermal expansivity, and ez is the unit vector in the axial direction. The stress tensor is split into the dynamic pressure and the strain rate tensor, = −PI − τ = −PI + (∇v + ∇vT )
Tf (z, t) = Tc +
where w = 0.045/s. The melt–crystal interface is located using the meltingpoint isotherm condition, which demands that the temperature at the interface is equal to the melting temperature of the crystal. In dimensionless terms, this means that:
assuming flow continuity, ∇ · v = 0.
889
(7)
where Tf (z) is the furnace thermal profile, the Biot number, Bii = hR/ki , is a dimensionless heat transfer coefficient, the 3 /k , relates radiative efRadiation number, Rdi = σ i RTmp i fects to conductive effects, is the crucible emissivity, σ is the Stefan–Boltzmann constant, and naf is the unit vector normal to the crucible/furnace interface. In this work, two different furnace thermal profiles were used. To model the KWG experiment (Kim et al., 1972), we
(10)
Thermal conductivity mismatches between the melt, crystal, and crucible along with the release of latent heat during crystallization produce a radial thermal gradient, which results in a curved melt–crystal interface. Accurate representation of this moving boundary requires deformation of the computational mesh. This is accomplished using elliptic mesh equations, as described by de Santos (1991). No-slip boundary conditions are applied at the crucible walls and melt–crystal interface, v = Ωreθ
(11)
where Ω is the crucible rotation rate and eθ is the unit vector in the azimuthal direction. Finally, at the centerline of the system, symmetry conditions are assumed. The governing equations that comprise the model are nonlinear, coupled, partial differential equations. The Galerkin finite element method (GFEM) was employed to spatially discretize equations for energy transport (Eqs. (1) and (2)), momentum transport (Eq. (3)), and mass continuity (Eq. (4)). The finite element mesh used for the simulations is shown in Fig. 1(b). This system of nonlinear DAEs was integrated in time using a second-order implicit trapezoidal method, and the resulting system of nonlinear algebraic equations was solved using Newton’s method. A dimensionless time step of t = 0.01 was shown to provide satisfactory numerical convergence as well as sufficient resolution of the time-varying phenomena within the system. A single factorization of the problem with a discretization of 25592 degrees of freedom required approximately 4.8 s on a mid-priced PC with a processor speed of 2.0 GHz. Additional details of the model and numerical scheme can be found in (Sonda, 2003; Yeckel & Goodwin, 2003).
890
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
3. Controller design A single-input, single-output control system is considered. In an experimental Bridgman system, the choice of the controlled output is a rather difficult decision, since only indirect measurements of events within the crucible are possible. Since our immediate goal is to assess the viability of model-based control for the Bridgman system in the most general sense, we choose an output quantity easily obtained from the numerical model and readily associated with the character of the flows in the melt. Thus, the output is chosen to be the spatial average of the meridional kinetic energy, E, 1 E= (v2 + v2z ) dV. (12) V V r To control flows within the vertical Bridgman system, we derive and implement an input–output linearizing controller. In addition, conventional proportional and proportional– integral controllers are implemented. All states are assumed to be available from the vertical Bridgman process model.
where τ is the response time constant and Esp is the kinetic energy setpoint. The control action was calculated numerically by our crystal growth simulation code (Yeckel & Goodwin, 2003). 3.2. Linear controllers We also implement a proportional controller, Ω(t) = Kc (Esp − E(t))
(16)
where Kc is the controller gain, and a proportional–integral controller, 1 t (17) (Esp − E) dt
Ω(t) = Kc (Esp − E) + τI 0 where τI is the controller integral time constant. Due to the inverse relationship between rotation rate and kinetic energy, the controller gain is negative.
3.1. Nonlinear controller design In this work, we design and implement an input–output linearizing feedforward/state feedback controller (Daoutidis & Kravaris, 1989). A rigorous development of the controller requires the derivation of an ODE representation of the differential-algebraic equation system resulting from the spatial discretization (Kumar & Daoutidis, 1999). Formally, the derivative of the kinetic energy, E, has the form, 2 ∂vr ∂vz vr + vz dV, (13) V V dt dt The derivative of the radial component of the velocity vector, v2θ ∂vr 1 ∂vr ∂vr vr =− − + vz ∂t Prm ∂r r ∂z − (∇ · T)r + Ra(T − 1)ez ,
(14)
is dependent upon the crucible rotation rate, through the centrifugal force term, v2θ /r. This means that this choice of input– output pair produces a system with a relative order of one. Similarly, the derivative of the kinetic energy is dependent on the furnace temperature profile, through the Boussinesq term, Ra(T − 1)ez . Assuming that the controller can measure the disturbance in the temperature field, this results in a system where the relative order of the disturbance is also one. The equal relative orders of the input and disturbance require the use of a static feedforward/state feedback controller (Daoutidis & Kravaris, 1989) to enforce a desired response in the output, E+τ
dE = Esp dt
(15)
Fig. 2. (a) Steady-state effect of steady rotation on spatially-averaged flow speed: stabilizing configuration. (b) Effect of steady rotation (Ω = 3.3 rpm) on oscillatory disturbance.
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
Fig. 3. Proportional control simulation results for Kc = (a) −100; (b) −1000; (c) −2000; (d) −2500: speed vs. time.
Fig. 4. Proportional control simulation results for Kc = (a) −100; (b) −1000; (c) −2000; (d) −2500: Ω vs. time.
891
892
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
4. Simulation results We apply linear and nonlinear controllers to two prototypical crystal growth systems which exhibit time-dependent behavior. The first system considered is the Bridgman process in a stabilizing configuration. Since the underlying melt flow in this system is laminar and steady, a time-periodic state is established by intentionally perturbing the furnace heat transfer with a sinusoidal disturbance. The second system is the destabilizing (KWG) configuration in which an inherently time-periodic flow occurs in the early stages of growth, when the melt height is greater than a critical value. For both systems, the chief control objective is the suppression of time-variations in the flows. 4.1. Control system I: stabilizing furnace with oscillatory disturbance 4.1.1. Effect of steady crucible rotation To begin our analysis, steady-state calculations were conducted. The spatially averaged flow speed was measured as the crucible rotation rate, Ω, was varied from zero to a physically reasonable maximum of 5 rpm. The spatially-averaged flow speed is simply the square root of the kinetic energy, E. The results of this calculation are shown in Fig. 2(a). As seen in the figure, with all other parameters held constant, there is a one-to-one correspondence between rotation rate and flow speed. An increase in rotation rate suppresses the radial and axial components of the velocity field, and consequently the flow speed. This behavior, consistent with the results of Yeckel and Derby (2000), is due to the inhibiting nature of the Coriolis force. Before assessing the effects of the control strategy on this system, we apply steady (i.e. constant w.r.t. time) crucible rotation in transient simulations to see if this is sufficient to suppress the oscillations. Fig. 2(b) shows the melt speed of the stabilizing configuration before and after a steady rotation rate of 3.3 rpm is applied. For t < 0, the stabilizing configuration is simulated without crucible rotation. The temporal variation of the temperature profile clearly influences the buoyancy-driven flows within the melt. Crucible rotation is applied at t = 0; the effect of rotation is seen immediately. Steady rotation suppresses the average meridional flow, yet does not attenuate its time-periodic nature. 4.1.2. Feedback suppression of oscillatory flows The three controllers, defined in Section 3, are applied to the Bridgman system in the thermally stabilizing configuration. To measure controller performance, the spatiallyaveraged flow speed and crucible rotation rate are represented as a function of time and compared to the case with no rotation. For all simulations of the disturbed system, the controller setpoint is 1.02 × 10−4 cm2 /s2 , which is equivalent to a spatially-averaged flow speed of 0.0101 cm/s. This setpoint corresponds to a 15% reduction in the speed relative to the
uncontrolled flow. Since the application of crucible rotation reduces the kinetic energy of the flow, the setpoint must be chosen at a kinetic energy value less than the steady state kinetic energy corresponding to no rotation (Ω = 0). The data in Fig. 2(a) indicate that this setpoint value is physically reasonable. In all simulations, the controller is activated at t = 0. Figs. 3 and 4 show results from P control simulations with controller gains of Kc = −100, −1000, −2000, and −2500. Fig. 3 shows speed versus time and Fig. 4 shows crucible rotation rate versus time. Fig. 3 shows that for a small, negative controller gain, proportional control is ineffective at suppressing oscillations in the speed, and offset is apparent. As the controller gain becomes more negative, the offset decreases and the oscillations are suppressed with increasing effectiveness. For proportional gains less than −2500, the closed-loop system becomes unstable. Fig. 4 shows that at t = 0+ , the crucible rotation rate increases sharply from Ω = 0 to the value calculated by the controller algorithm. As the absolute value of the controller gain increases, the initial rotation rate increases to the point of physical infeasibility. This represents
Fig. 5. PI control simulation results for Kc = −2000 and τI = 1: (a) speed vs. time; (b) Ω vs. time.
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
an actuator limit which will hinder the practical effectiveness of the P controller. Once the initial transient has passed, thereby establishing the average rotation rate corresponding to the controller gain, the closed-loop system starts to rotate in lock-step with the oscillatory disturbance. Note that the system does not change the direction of rotation; instead it accelerates and decelerates while maintaining the same direction. When the flow speed is near a maximum during its oscillation cycle, the rotation rate increases to suppress it. Similarly, when the flow speed is relatively small, the rotation rate decreases. To eliminate offset, a proportional–integral controller is applied to this system. Simulation results for Kc = −2000 and τI = 1 are shown in Fig. 5. Thus, the time constant of the integrator is equal to a dimensionless time unit, as defined in Section 2. As seen in Fig. 5(a), the PI controller produces melt speeds which oscillate with smaller amplitude than the P controller, and zero offset is achieved. Changing the integral time constant τI has little effect on the long time performance of the controller. The rotation schedule for this simulation is shown in Fig. 5(b). In a similar manner to P control, the
Fig. 6. Nonlinear control simulation results for τ = 1: (a) speed vs. time; (b) Ω vs. time.
893
PI algorithm requires that the crucible rotate suddenly and significantly at t = 0+ (see inset). We now consider the case when nonlinear control is applied to the system. Fig. 6 presents closed loop simulation results for time constant τ = 1. As seen in Fig. 6(a), the controller forces a smooth, first-order response to the setpoint, where the oscillations are essentially suppressed. In addition, the nonlinear controller avoids the physically unrealistic actuation spike occurring at t = 0+ in the P and PI control cases. This is shown in Fig. 6(b). System rotation accelerates and decelerates to maintain the process setpoint. 4.2. Control system II: destabilizing furnace with inherently oscillatory flows 4.2.1. Effect of steady crucible rotation Steady crucible rotation was first applied to the KWG system under steady-state conditions. As was seen in the stabilizing case, an increase in crucible rotation produces a decrease in melt speed. This is shown in Fig. 7(a). To understand the
Fig. 7. (a) Steady-state effect of steady rotation on spatially-averaged flow speed: destabilizing configuration. (b) Steady crucible rotation exacerbates naturally occurring oscillations: speed vs. time for Ω = 0 and 3.3 rpm.
894
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
effect of crucible rotation on the inherent instability of the KWG system, we set the melt height to a sufficiently large value to assure time-dependent behavior, and integrated the system in time. The system was allowed to reach a stable, periodic state characterized by a single frequency and constant amplitude. Crucible rotation was applied and the resulting change in the flow was observed. In Fig. 7(b), the spatially-averaged speed is plotted as a function of time for Ω = 0 rpm and Ω = 3.3 rpm under these conditions. Surprisingly, the amplitude of oscillations is greater when rotation is applied. Even though crucible rotation suppresses steady flows in this system (i.e. those below the critical melt height), it also acts to exacerbate the time-varying oscillations in the flow. This seemingly contradictory behavior is attributed to the nonlinear coupling of the intense flows, which produces a destabilizing centrifugal force, thereby promoting the onset of time-periodic flows. These physical arguments are further elucidated in a companion work (Sonda, Yeckel, Daoutidis, & Derby, 2004a). 4.2.2. Feedback suppression of oscillatory flows Through our open-loop simulations, we discovered that crucible rotation complicates the dynamic behavior of the destabilizing Bridgman system. Accordingly, one would surmise that crucible rotation would not provide an effective actuation technique for stabilizing oscillatory flows in the
KWG system. Our simulations support this hypothesis for most cases, however there do exist slightly supercritical flows exhibiting small-amplitude oscillations which are amenable to feedback control. One case of successful control is shown in Fig. 8, where proportional control is applied to the thermally destabilizing KWG system for Ra = 3.6 × 105 and the controller setpoint is set to a speed of 0.478 cm/s. Fig. 8(a) shows the speed as a function of time for simulations of the KWG system under proportional control with Kc = −0.75. The disparity for this controller gain, as compared to the stabilizing case, is due to the significant difference in flow intensity in the two cases (see Eq. (16)). For this parameter value, proportional control is able to suppress the inherent flow oscillations. Indeed, the final value of the speed is within one percent of the setpoint. Fig. 8(b) provides a closer view of the time-variation of speed. The crucible rotation rate is given as a function of time in Fig. 8(c) for the same simulation. The crucible rotates only in one direction, but speeds up and slows down in response to the inherent oscillations in the flow. Interestingly, the crucible rotation rate is quite small, less than 1 rpm. Fig. 8(d) reveals a close-up of the rotation rate, corresponding to the same time window as that in Fig. 8(b). The oscillations in rotation rate follow the flow oscillations closely. For Kc < −1, the proportional controller is ineffective in suppressing flow oscillations. Finally, the application of the nonlinear controller to
Fig. 8. Proportional control of destabilizing (KWG) configuration. (a) Speed vs. time; the P controller requires 12 h to suppress oscillations; (b) close-up view shows temporal variation of speed; (c) rotation rate vs. time; (d) close-up view shows variation of crucible rotation rate in step with the flow oscillations.
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
the KWG model produces an unstable closed-loop system for all values of controller time constant, τ. This result is not surprising, considering the exacerbating influence that crucible rotation has on the inherent flow oscillations in this system.
895
via thermocouple measurements. This point-wise data could be used with state estimation techniques for output feedback control design. This is an active area of our research. Another opportunity for future work is the investigation of additional actuation techniques in closed-loop crystal growth systems.
5. Concluding remarks Acknowledgments Two prototypical vertical Bridgman systems, based on the experimental design of Kim et al. (1972), were used as the model for feedback controller design and simulation. Feedback control successfully attenuated flow oscillations in the stabilizing system but was less successful for the destabilizing system. However, since nearly all vertical Bridgman systems use thermally stabilized configurations for crystal growth, these results are significant. The three controllers exhibited varying degrees of success in suppressing the time-varying disturbance in the stabilizing Bridgman system. Proportional control had an inhibiting effect on the sinusoidal behavior, yet was unable to fully suppress oscillations or obtain zero offset in the output. As expected, proportional–integral control improved the closed-loop performance, suppressing oscillations considerably and providing zero offset. To obtain adequate oscillation suppression, both the P and PI controllers required extremely aggressive controller action at the onset of control. This actuation spike would likely cause practical problems. In contrast to these simple controllers, the input/output linearizing controller provided a smooth transition from the initial state (zero rotation rate) to the final state (non-zero, time-varying rotation rate). In addition, the nonlinear controller provided superior oscillation attenuation as compared to the linear controllers. For the destabilizing configuration, proportional control attenuates the flow oscillations for slightly supercritical melt heights. Thus, the controller, via time-varying crucible rotation, is able to stabilize the previously unstable steady state. Yet our results clearly show that crucible rotation is not an effective actuation technique for the control of the destabilized system. For the case of the stabilizing configuration, the control action is nearly instantaneous; in the KWG system, however, complete suppression requires greater than 1500 rotation cycles. For larger melt heights, larger controller gains, and nonlinear controllers, the closed-loop flows exhibit timevarying oscillations with larger amplitudes than the openloop case. Issues remain with the practical application of control to the vertical Bridgman system. Due to the use of an enclosed crucible, and the need to conduct growth inside a high temperature furnace, it is extremely difficult to measure the states during growth. In this work, we assumed the availability of an accurate volumetric kinetic energy. In practice, this assumption is currently not feasible. However, continued innovations in micromachining and microfluidics could provide this capability in the near future. The data most readily available in crystal growth systems is temperatures within the melt,
This material is based upon work supported by the National Science Foundation under Grant No. 0201486. This work was also supported in part by the Minnesota Supercomputing Institute. PS expresses thanks to the University of Minnesota Graduate School for a Doctoral Dissertation Fellowship.
References Azuma, Y., Usami, N., Ujihara, T., Fujiwara, K., Sazaki, G., Murakami, Y., et al. (2003). Growth of SiGe bulk crystals with uniform composition by utilizing feedback control system of the crystal-melt interface position for precise control of the growth temperature. Journal of Crystal Growth, 250, 298–304. Batur, C., Duval, W., & Bennett, R. (1999). Performance of Bridgman furnace operating under projective control. In Proceedings of the American control conference (pp. 4101–4105). Batur, C., Srinivasan, A., Duval, W., Singh, N., & Golovaty, D. (1999). Online control of solid-liquid interface by state feedback. Journal of Crystal Growth, 205, 395–409. B´enard, H. (1901). Les Tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en r´egime permanent. Annales de Chimie et de Physique, 23, 62–144. Capper, P., & Gosney, J. (1984). Application of the accelerated crucible rotation technique to the Bridgman growth of CdHgTe: simulations and crystal growth. Journal of Crystal Growth, 70, 356–64. Chin, L., & Carlson, F. (1983). Finite element analysis of the control of interface shape in Bridgman crystal growth. Journal of Crystal Growth, 62, 561–567. de Santos, J. (1991). Two-phase cocurrent downflow through constricted passages. Ph.D. thesis, University of Minnesota, available through University Microfilms Inc., http://www.umi.com. Daoutidis, P., & Kravaris, C. (1989). Synthesis of feedforward/state feedback controllers for nonlinear processes. AIChE Journal, 35, 1602–1615. Kim, D., Adornato, P., & Brown, R. (1988). Effect of vertical magnetic field on convection and segregation in vertical Bridgman crystal growth. Journal of Crystal Growth, 89, 339–356. Kim, K., Witt, A., & Gatos, H. (1972). Crystal growth from the melt under destabilizing thermal gradients. Journal of the Electrochemical Society, 119, 1218–1226. Kumar, A., & Daoutidis, P. (1999). Control of nonlinear differential algebraic equation systems: With application to chemical processes. Chapman and Hall/CRC. Lan, C. (2000). Effects of axial vibration on vertical zone-melting processing. International Journal of Heat and Mass Transfer, 43, 1987–1997. Liu, W., Wolf, M., Elwell, D., & Feigelson, R. (1987). Low frequency vibrational stirring: a new method of radial mixing solutions and melts during growth. Journal of Crystal Growth, 82, 589–597. Rayleigh, L. (1916). On convective currents in a horizontal layer of fluid when the higher temperature is on the under side. The Philosopher’s Magazine, 32, 529–546. Schmachtl, M., Schievenbusch, A., Zimmerman, C., & Grill, W. (1998). Crystallization process control during directional solidification in a high-
896
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
temperature-gradient furnace by guided ultrasonic waves and real-time signal evaluation. Ultrasonics, 36, 291–295. Seidensticker, R., Rosch, W., et al. (1999). Active control of interface shape during the crystal growth of lead bromide. Journal of Crystal Growth, 198, 988–994. Shiomi, J., & Amberg, G. (2002). Active control of a global thermocapillary instability. Physics of Fluids, 14, 3039–3045. Sonda, P. (2003). The modeling, simulation, and control of transport phenomena in a thermally destabilized Bridgman crystal growth system. Ph.D. thesis, University of Minnesota, available through University Microfilms Inc., http://www.umi.com. Sonda, P., Yeckel, A., Daoutidis, P., & Derby, J. J. (2003). Improved radial segregation via the destabilizing vertical Bridgman configuration. Journal of Crystal Growth, 260, 263–276. Sonda, P., Yeckel, A., Daoutidis, P., & Derby, J. J. (2004a). Development of model-based control for Bridgman crystal growth. Journal of Crystal Growth, 266, 182–189.
Sonda, P., Yeckel, A., Daoutidis, P., Derby, J. J. (2004b). Hopf bifurcation and solution multiplicity in a model for Bridgman crystal growth. Chemical Engineering Science, in press. Srinivasan, A. (1994). Modeling and control of the solid-liquid interface during crystal growth. Ph.D. thesis, University of Akron. Yeckel, A., Comp`ere, G., Pandy, A., & Derby, J. J. (2004). Three-dimensional imperfections in a model vertical Bridgman growth system for cadmium zinc telluride. Journal of Crystal Growth, 263, 629–644. Yeckel, A., & Derby, J. J. (2000). Effect of accelerated crucible rotation on melt composition in high-pressure vertical Bridgman growth of cadmium zinc telluride. Journal of Crystal Growth, 209, 734–750. Yeckel, A., & Derby, J. J. (2004). Dynamics of three-dimensional convection in microgravity crystal growth: g-jitter with steady magnetic fields. Journal of Crystal Growth, 263, 40–52. Yeckel, A., Goodwin, R. T. (2003). Cats2D (Crystallization and Transport Simulator), User Manual, unpublished (available at http://www.msi. umn.edu/∼yeckel/cats2d.html).
The feedback control of the vertical Bridgman crystal growth process by crucible rotation: two case studies Paul Sonda, Andrew Yeckel, Jeffrey J. Derby, Prodromos Daoutidis∗ Department of Chemical Engineering and Materials Science, University of Minnesota, 421 Washington Ave. SE, Minneapolis, MN 55455-0132, USA Available online 19 November 2004
Abstract In this work, we consider the feedback control of flows within the vertical Bridgman crystal growth process. We investigate the use of crucible rotation, via feedback control algorithms, in suppressing oscillatory flows in two prototypical vertical Bridgman crystal growth configurations—a laminar flow regime driven by a time-oscillatory furnace disturbance and a time-varying regime driven by strong buoyant forces. Proportional, proportional–integral, and input–output linearizing controllers are applied to the vertical Bridgman model to attenuate the flow oscillations. Simulation results show that for the first configuration, crucible rotation is an appropriate actuation method for feedback control. In addition, nonlinear control provides superior performance to P and PI control. For the latter case, crucible rotation is less effective, due to its exacerbating effect on the inherent time-dependent flows within this system. © 2004 Elsevier Ltd. All rights reserved. Keywords: Model-based control; Flow control; Crystal growth; Numerical modeling
1. Introduction The vertical Bridgman process is used for the growth of single crystal electronic materials, for use as substrates in optoelectronic and sensing devices. A schematic of the vertical Bridgman process is shown in Fig. 1(a). Polycrystalline material is loaded into a quartz crucible (also known as an ampoule), inside of a high temperature furnace. The material is completely melted, at which time the distribution of temperature in the furnace is adjusted to vary along its length, such that one zone is hotter than the material’s melting point and the other section is cooler. Directional solidification is achieved by slowly translating the crucible towards the cold zone of the furnace. To produce devices of suitable quality, it is necessary to grow low-defect single crystals of homogenous chemical composition. This task is made difficult by the complex coupling of heat, mass, and momentum transport inherent in the ∗
Corresponding author. Tel.: +1 612 625 8818; fax: +1 612 626 7246. E-mail address: [email protected] (P. Daoutidis).
0098-1354/$ – see front matter © 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.compchemeng.2004.09.013
process. One processing issue that continues to challenge the crystal growth community is the occurrence of fine scale variations in composition, known as striations. These composition variations often cause undesirable variations in material and electronic properties. Kim, Witt, & Gatos (1972) used crystal growth experiments to investigate the occurrence of striations in a vertical Bridgman system. It was found that for sufficiently intense flows, the melt exhibited periodic, timevarying behavior. It was then shown conclusively that these time-varying flows were directly related to the occurrence of striations. In previous work (Sonda, Yeckel, Daoutidis, & Derby, 2003, 2004b), we used detailed crystal growth models to numerically simulate the experimental system of Kim, Witt, and Gatos. Our numerical simulations predict that during the early stages of crystal growth, large amplitude, twofrequency oscillations occur within the flow. In later stages of growth, the oscillatory behavior becomes more ordered with decreasing amplitude, eventually reaching a stage in which the flows are steady with time. These simulations are in qualitative agreement with the experimental results of Kim,
888
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
Fig. 1. (a) Schematic of the destabilizing vertical Bridgman system. (b) The finite element mesh used for the simulations.
Witt, and Gatos. Demonstrating the appearance of periodic, time-varying flows was an important first step in our study of crystal striations. In this work, we simulate application of feedback controllers to the system in order to suppress its time-varying behavior. The ultimate goal of this research is to understand how process control can be used to prevent the onset of crystal striations in experimentally grown single crystals. Despite the continuing need for improvement of Bridgman growth processes, there exists minimal work addressing the use of feedback control to improve these processes. Batur and co-workers (Batur, Duval, & Bennett, 1999; Batur, Srinivasan, Duval, Singh, & Golovaty, 1999; Srinivasan, 1994), designed a transparent, multi-heating zone furnace for the growth of lead bromide. The transparent furnace, in coordination with a video camera and real-time imaging software, allowed for the observation of the melt–crystal interface shape. Comparison of the observed interface shape to the desired shape provided an error signal, which was used to adjust the furnace temperature profile. Seidensticker et al. (1999) and Chin and Carlson (1983) used open-loop simulations to investigate the effects of temperature boundary conditions on interface shape and assess their applicability to a control algorithm. Azuma et al. (2003) developed an automatic feedback control system to maintain a constant temperature at the melt–crystal interface during growth of silicon germanium. The position of the interface was automatically detected through in situ monitoring via a charge-coupleddevice camera, and the ampoule translation rate was the manipulated input. Ultrasonic waves were used by Schmachtl, Schievenbusch, Zimmerman, and Grill (1998) to control the shape and velocity of the solid–liquid interface during the Bridgman growth of copper manganese. In related work,
Shiomi and Amberg (2002) applied proportional control to attenuate oscillations arising during growth by the floating zone method, another common method used to grow crystals from the melt. Two sensor/heater pairs were used for measurement and actuation. They found that complete suppression of the oscillations could be achieved for weakly nonlinear flows but was not possible for stronger nonlinear flows. There are many actuation techniques which could feasibly be used for the feedback control of the vertical Bridgman system. Methods that have been shown to perceptibly affect the molten flows include the real-time adjustment of the surrounding thermal environment (Batur, Srinivasan, et al., 1999; Yeckel, Comp`ere, Pandy, & Derby, 2004), the application of vibration (Lan, 2000; Liu, Wolf, Elwell, & Feigelson, 1987), and the application of a magnetic field (Kim, Adornato, & Brown, 1988; Yeckel & Derby, 2004) to the crucible. This work examines the use of crucible rotation as the means for suppressing oscillatory flows in the vertical Bridgman system. Open-loop crucible rotation has been successfully applied to experimental vertical Bridgman systems (Capper & Gosney, 1984) and has been the subject of simulation-based studies (see Yeckel & Derby (2000) and references therein). Proportional, proportional–integral and nonlinear, modelbased controllers are synthesized and implemented within the framework of our crystal growth simulation code (Yeckel & Goodwin, 2003). The controllers are applied to two prototypical vertical Bridgman systems and their performances evaluated.
2. The vertical Bridgman process model Fig. 1(a) presents a schematic of the configuration employed by Kim, Witt, and Gatos (hereafter referred to as KWG). This system, with warm fluid underlying the cooler crystal, is termed destabilizing, since it leads to an inherently unstable melt, with fluid mechanical behavior similar to the classical Rayleigh–B´enard problem (B´enard, 1901; Rayleigh, 1916) of hydrodynamics. We model the vertical Bridgman process using equations for conservation of mass, momentum, and energy, along with appropriate boundary and initial conditions. Our present purposes only require that we study the effect of control on flow, so we do not include conservation of chemical species in the model, although doing so would be necessary for the direct prediction of striations. A key assumption used here is that the system is perfectly axisymmetric, thus all field variables are independent of the azimuthal coordinate. Under these assumptions, the time-dependent energy transport equation is given by Pr m
∂T + v · ∇T − ∇ 2 T = 0 ∂t
(1)
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
in the melt, and Prj
∂T − ∇ 2T = 0 ∂t
assumed a profile which decreases linearly along the z axis, (2)
where the index j designates the material, either the crystal, crucible, or graphite support. T is a dimensionless temperature scaled by the melting temperature, Tmp , t is a dimensionless time scaled by R2 /ν, v(r,z) is the dimensionless velocity field, and ∇ is the dimensionless gradient operator, in which the spatial coordinates are scaled by the crucible radius R. The Prandtl number is defined as Pri ≡ ν/αi , where ν is the kinematic viscosity, and the thermal diffusivity is defined as αi ≡ ki /ρi Cp,i , where ki is the thermal conductivity, ρi the density, and Cp,i is the heat capacity. The velocity field is described by the Navier–Stokes equations for incompressible flow, using the Boussinesq approximation to describe temperature-dependent density variation: 1 ∂v + v · ∇v − ∇ · T − Ra(T − 1)ez = 0, ∂t Pr m
where Tc and Trmh are the minimum and maximum furnace temperatures, respectively, and zr is the furnace reference position. However, in our simulations, we first applied control to a stabilizing vertical Bridgman system, as its fluid dynamical behavior is significantly less nonlinear. The stabilizing configuration is accomplished by inverting the KWG furnace with respect to gravity. To produce time-dependent behavior in this naturally laminar flow, we assume that the furnace experiences a time-sinusoidal external power disturbance, (Trmh − Tc ) Tf (z, t) = Tc + (z − zr ) (1 + sin(wt)), (9) L
T = 1.
(5)
(6)
where κ is the ratio of melt thermal conductivity to solid, κ = km /ks , nsm is the unit vector normal to the melt–crystal interface, pointed towards the melt, St is the Stefan number, St = Hf /Cp,s Tmp , and x˙ is the velocity of the melt–crystal interface. Convective and radiative heat transfer from the furnace to the crucible are represented by a simple flux condition applied at the domain boundary: −∇T |a · naf = Bii (T − Tf (z)) + Rdi (T 4 − Tf4 (z))
(8)
(4)
where P is the dynamic pressure scaled by ρm ναm /R2 , and I is the idemfactor. A thermal flux condition is defined at the melt–crystal interface, (−κ∇T |m + ∇T |s ) · nsm = StPrs (˙x · nsm )
(Trmh − Tc ) (z − zr ), L
(3)
Here the dimensionless velocity is scaled by αm /R, T is the stress tensor, Ra is the Rayleigh number, Ra ≡ gβT R3 Tmp /ναm , where g is the gravitational constant, and βT the thermal expansivity, and ez is the unit vector in the axial direction. The stress tensor is split into the dynamic pressure and the strain rate tensor, = −PI − τ = −PI + (∇v + ∇vT )
Tf (z, t) = Tc +
where w = 0.045/s. The melt–crystal interface is located using the meltingpoint isotherm condition, which demands that the temperature at the interface is equal to the melting temperature of the crystal. In dimensionless terms, this means that:
assuming flow continuity, ∇ · v = 0.
889
(7)
where Tf (z) is the furnace thermal profile, the Biot number, Bii = hR/ki , is a dimensionless heat transfer coefficient, the 3 /k , relates radiative efRadiation number, Rdi = σ i RTmp i fects to conductive effects, is the crucible emissivity, σ is the Stefan–Boltzmann constant, and naf is the unit vector normal to the crucible/furnace interface. In this work, two different furnace thermal profiles were used. To model the KWG experiment (Kim et al., 1972), we
(10)
Thermal conductivity mismatches between the melt, crystal, and crucible along with the release of latent heat during crystallization produce a radial thermal gradient, which results in a curved melt–crystal interface. Accurate representation of this moving boundary requires deformation of the computational mesh. This is accomplished using elliptic mesh equations, as described by de Santos (1991). No-slip boundary conditions are applied at the crucible walls and melt–crystal interface, v = Ωreθ
(11)
where Ω is the crucible rotation rate and eθ is the unit vector in the azimuthal direction. Finally, at the centerline of the system, symmetry conditions are assumed. The governing equations that comprise the model are nonlinear, coupled, partial differential equations. The Galerkin finite element method (GFEM) was employed to spatially discretize equations for energy transport (Eqs. (1) and (2)), momentum transport (Eq. (3)), and mass continuity (Eq. (4)). The finite element mesh used for the simulations is shown in Fig. 1(b). This system of nonlinear DAEs was integrated in time using a second-order implicit trapezoidal method, and the resulting system of nonlinear algebraic equations was solved using Newton’s method. A dimensionless time step of t = 0.01 was shown to provide satisfactory numerical convergence as well as sufficient resolution of the time-varying phenomena within the system. A single factorization of the problem with a discretization of 25592 degrees of freedom required approximately 4.8 s on a mid-priced PC with a processor speed of 2.0 GHz. Additional details of the model and numerical scheme can be found in (Sonda, 2003; Yeckel & Goodwin, 2003).
890
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
3. Controller design A single-input, single-output control system is considered. In an experimental Bridgman system, the choice of the controlled output is a rather difficult decision, since only indirect measurements of events within the crucible are possible. Since our immediate goal is to assess the viability of model-based control for the Bridgman system in the most general sense, we choose an output quantity easily obtained from the numerical model and readily associated with the character of the flows in the melt. Thus, the output is chosen to be the spatial average of the meridional kinetic energy, E, 1 E= (v2 + v2z ) dV. (12) V V r To control flows within the vertical Bridgman system, we derive and implement an input–output linearizing controller. In addition, conventional proportional and proportional– integral controllers are implemented. All states are assumed to be available from the vertical Bridgman process model.
where τ is the response time constant and Esp is the kinetic energy setpoint. The control action was calculated numerically by our crystal growth simulation code (Yeckel & Goodwin, 2003). 3.2. Linear controllers We also implement a proportional controller, Ω(t) = Kc (Esp − E(t))
(16)
where Kc is the controller gain, and a proportional–integral controller, 1 t (17) (Esp − E) dt
Ω(t) = Kc (Esp − E) + τI 0 where τI is the controller integral time constant. Due to the inverse relationship between rotation rate and kinetic energy, the controller gain is negative.
3.1. Nonlinear controller design In this work, we design and implement an input–output linearizing feedforward/state feedback controller (Daoutidis & Kravaris, 1989). A rigorous development of the controller requires the derivation of an ODE representation of the differential-algebraic equation system resulting from the spatial discretization (Kumar & Daoutidis, 1999). Formally, the derivative of the kinetic energy, E, has the form, 2 ∂vr ∂vz vr + vz dV, (13) V V dt dt The derivative of the radial component of the velocity vector, v2θ ∂vr 1 ∂vr ∂vr vr =− − + vz ∂t Prm ∂r r ∂z − (∇ · T)r + Ra(T − 1)ez ,
(14)
is dependent upon the crucible rotation rate, through the centrifugal force term, v2θ /r. This means that this choice of input– output pair produces a system with a relative order of one. Similarly, the derivative of the kinetic energy is dependent on the furnace temperature profile, through the Boussinesq term, Ra(T − 1)ez . Assuming that the controller can measure the disturbance in the temperature field, this results in a system where the relative order of the disturbance is also one. The equal relative orders of the input and disturbance require the use of a static feedforward/state feedback controller (Daoutidis & Kravaris, 1989) to enforce a desired response in the output, E+τ
dE = Esp dt
(15)
Fig. 2. (a) Steady-state effect of steady rotation on spatially-averaged flow speed: stabilizing configuration. (b) Effect of steady rotation (Ω = 3.3 rpm) on oscillatory disturbance.
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
Fig. 3. Proportional control simulation results for Kc = (a) −100; (b) −1000; (c) −2000; (d) −2500: speed vs. time.
Fig. 4. Proportional control simulation results for Kc = (a) −100; (b) −1000; (c) −2000; (d) −2500: Ω vs. time.
891
892
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
4. Simulation results We apply linear and nonlinear controllers to two prototypical crystal growth systems which exhibit time-dependent behavior. The first system considered is the Bridgman process in a stabilizing configuration. Since the underlying melt flow in this system is laminar and steady, a time-periodic state is established by intentionally perturbing the furnace heat transfer with a sinusoidal disturbance. The second system is the destabilizing (KWG) configuration in which an inherently time-periodic flow occurs in the early stages of growth, when the melt height is greater than a critical value. For both systems, the chief control objective is the suppression of time-variations in the flows. 4.1. Control system I: stabilizing furnace with oscillatory disturbance 4.1.1. Effect of steady crucible rotation To begin our analysis, steady-state calculations were conducted. The spatially averaged flow speed was measured as the crucible rotation rate, Ω, was varied from zero to a physically reasonable maximum of 5 rpm. The spatially-averaged flow speed is simply the square root of the kinetic energy, E. The results of this calculation are shown in Fig. 2(a). As seen in the figure, with all other parameters held constant, there is a one-to-one correspondence between rotation rate and flow speed. An increase in rotation rate suppresses the radial and axial components of the velocity field, and consequently the flow speed. This behavior, consistent with the results of Yeckel and Derby (2000), is due to the inhibiting nature of the Coriolis force. Before assessing the effects of the control strategy on this system, we apply steady (i.e. constant w.r.t. time) crucible rotation in transient simulations to see if this is sufficient to suppress the oscillations. Fig. 2(b) shows the melt speed of the stabilizing configuration before and after a steady rotation rate of 3.3 rpm is applied. For t < 0, the stabilizing configuration is simulated without crucible rotation. The temporal variation of the temperature profile clearly influences the buoyancy-driven flows within the melt. Crucible rotation is applied at t = 0; the effect of rotation is seen immediately. Steady rotation suppresses the average meridional flow, yet does not attenuate its time-periodic nature. 4.1.2. Feedback suppression of oscillatory flows The three controllers, defined in Section 3, are applied to the Bridgman system in the thermally stabilizing configuration. To measure controller performance, the spatiallyaveraged flow speed and crucible rotation rate are represented as a function of time and compared to the case with no rotation. For all simulations of the disturbed system, the controller setpoint is 1.02 × 10−4 cm2 /s2 , which is equivalent to a spatially-averaged flow speed of 0.0101 cm/s. This setpoint corresponds to a 15% reduction in the speed relative to the
uncontrolled flow. Since the application of crucible rotation reduces the kinetic energy of the flow, the setpoint must be chosen at a kinetic energy value less than the steady state kinetic energy corresponding to no rotation (Ω = 0). The data in Fig. 2(a) indicate that this setpoint value is physically reasonable. In all simulations, the controller is activated at t = 0. Figs. 3 and 4 show results from P control simulations with controller gains of Kc = −100, −1000, −2000, and −2500. Fig. 3 shows speed versus time and Fig. 4 shows crucible rotation rate versus time. Fig. 3 shows that for a small, negative controller gain, proportional control is ineffective at suppressing oscillations in the speed, and offset is apparent. As the controller gain becomes more negative, the offset decreases and the oscillations are suppressed with increasing effectiveness. For proportional gains less than −2500, the closed-loop system becomes unstable. Fig. 4 shows that at t = 0+ , the crucible rotation rate increases sharply from Ω = 0 to the value calculated by the controller algorithm. As the absolute value of the controller gain increases, the initial rotation rate increases to the point of physical infeasibility. This represents
Fig. 5. PI control simulation results for Kc = −2000 and τI = 1: (a) speed vs. time; (b) Ω vs. time.
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
an actuator limit which will hinder the practical effectiveness of the P controller. Once the initial transient has passed, thereby establishing the average rotation rate corresponding to the controller gain, the closed-loop system starts to rotate in lock-step with the oscillatory disturbance. Note that the system does not change the direction of rotation; instead it accelerates and decelerates while maintaining the same direction. When the flow speed is near a maximum during its oscillation cycle, the rotation rate increases to suppress it. Similarly, when the flow speed is relatively small, the rotation rate decreases. To eliminate offset, a proportional–integral controller is applied to this system. Simulation results for Kc = −2000 and τI = 1 are shown in Fig. 5. Thus, the time constant of the integrator is equal to a dimensionless time unit, as defined in Section 2. As seen in Fig. 5(a), the PI controller produces melt speeds which oscillate with smaller amplitude than the P controller, and zero offset is achieved. Changing the integral time constant τI has little effect on the long time performance of the controller. The rotation schedule for this simulation is shown in Fig. 5(b). In a similar manner to P control, the
Fig. 6. Nonlinear control simulation results for τ = 1: (a) speed vs. time; (b) Ω vs. time.
893
PI algorithm requires that the crucible rotate suddenly and significantly at t = 0+ (see inset). We now consider the case when nonlinear control is applied to the system. Fig. 6 presents closed loop simulation results for time constant τ = 1. As seen in Fig. 6(a), the controller forces a smooth, first-order response to the setpoint, where the oscillations are essentially suppressed. In addition, the nonlinear controller avoids the physically unrealistic actuation spike occurring at t = 0+ in the P and PI control cases. This is shown in Fig. 6(b). System rotation accelerates and decelerates to maintain the process setpoint. 4.2. Control system II: destabilizing furnace with inherently oscillatory flows 4.2.1. Effect of steady crucible rotation Steady crucible rotation was first applied to the KWG system under steady-state conditions. As was seen in the stabilizing case, an increase in crucible rotation produces a decrease in melt speed. This is shown in Fig. 7(a). To understand the
Fig. 7. (a) Steady-state effect of steady rotation on spatially-averaged flow speed: destabilizing configuration. (b) Steady crucible rotation exacerbates naturally occurring oscillations: speed vs. time for Ω = 0 and 3.3 rpm.
894
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
effect of crucible rotation on the inherent instability of the KWG system, we set the melt height to a sufficiently large value to assure time-dependent behavior, and integrated the system in time. The system was allowed to reach a stable, periodic state characterized by a single frequency and constant amplitude. Crucible rotation was applied and the resulting change in the flow was observed. In Fig. 7(b), the spatially-averaged speed is plotted as a function of time for Ω = 0 rpm and Ω = 3.3 rpm under these conditions. Surprisingly, the amplitude of oscillations is greater when rotation is applied. Even though crucible rotation suppresses steady flows in this system (i.e. those below the critical melt height), it also acts to exacerbate the time-varying oscillations in the flow. This seemingly contradictory behavior is attributed to the nonlinear coupling of the intense flows, which produces a destabilizing centrifugal force, thereby promoting the onset of time-periodic flows. These physical arguments are further elucidated in a companion work (Sonda, Yeckel, Daoutidis, & Derby, 2004a). 4.2.2. Feedback suppression of oscillatory flows Through our open-loop simulations, we discovered that crucible rotation complicates the dynamic behavior of the destabilizing Bridgman system. Accordingly, one would surmise that crucible rotation would not provide an effective actuation technique for stabilizing oscillatory flows in the
KWG system. Our simulations support this hypothesis for most cases, however there do exist slightly supercritical flows exhibiting small-amplitude oscillations which are amenable to feedback control. One case of successful control is shown in Fig. 8, where proportional control is applied to the thermally destabilizing KWG system for Ra = 3.6 × 105 and the controller setpoint is set to a speed of 0.478 cm/s. Fig. 8(a) shows the speed as a function of time for simulations of the KWG system under proportional control with Kc = −0.75. The disparity for this controller gain, as compared to the stabilizing case, is due to the significant difference in flow intensity in the two cases (see Eq. (16)). For this parameter value, proportional control is able to suppress the inherent flow oscillations. Indeed, the final value of the speed is within one percent of the setpoint. Fig. 8(b) provides a closer view of the time-variation of speed. The crucible rotation rate is given as a function of time in Fig. 8(c) for the same simulation. The crucible rotates only in one direction, but speeds up and slows down in response to the inherent oscillations in the flow. Interestingly, the crucible rotation rate is quite small, less than 1 rpm. Fig. 8(d) reveals a close-up of the rotation rate, corresponding to the same time window as that in Fig. 8(b). The oscillations in rotation rate follow the flow oscillations closely. For Kc < −1, the proportional controller is ineffective in suppressing flow oscillations. Finally, the application of the nonlinear controller to
Fig. 8. Proportional control of destabilizing (KWG) configuration. (a) Speed vs. time; the P controller requires 12 h to suppress oscillations; (b) close-up view shows temporal variation of speed; (c) rotation rate vs. time; (d) close-up view shows variation of crucible rotation rate in step with the flow oscillations.
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
the KWG model produces an unstable closed-loop system for all values of controller time constant, τ. This result is not surprising, considering the exacerbating influence that crucible rotation has on the inherent flow oscillations in this system.
895
via thermocouple measurements. This point-wise data could be used with state estimation techniques for output feedback control design. This is an active area of our research. Another opportunity for future work is the investigation of additional actuation techniques in closed-loop crystal growth systems.
5. Concluding remarks Acknowledgments Two prototypical vertical Bridgman systems, based on the experimental design of Kim et al. (1972), were used as the model for feedback controller design and simulation. Feedback control successfully attenuated flow oscillations in the stabilizing system but was less successful for the destabilizing system. However, since nearly all vertical Bridgman systems use thermally stabilized configurations for crystal growth, these results are significant. The three controllers exhibited varying degrees of success in suppressing the time-varying disturbance in the stabilizing Bridgman system. Proportional control had an inhibiting effect on the sinusoidal behavior, yet was unable to fully suppress oscillations or obtain zero offset in the output. As expected, proportional–integral control improved the closed-loop performance, suppressing oscillations considerably and providing zero offset. To obtain adequate oscillation suppression, both the P and PI controllers required extremely aggressive controller action at the onset of control. This actuation spike would likely cause practical problems. In contrast to these simple controllers, the input/output linearizing controller provided a smooth transition from the initial state (zero rotation rate) to the final state (non-zero, time-varying rotation rate). In addition, the nonlinear controller provided superior oscillation attenuation as compared to the linear controllers. For the destabilizing configuration, proportional control attenuates the flow oscillations for slightly supercritical melt heights. Thus, the controller, via time-varying crucible rotation, is able to stabilize the previously unstable steady state. Yet our results clearly show that crucible rotation is not an effective actuation technique for the control of the destabilized system. For the case of the stabilizing configuration, the control action is nearly instantaneous; in the KWG system, however, complete suppression requires greater than 1500 rotation cycles. For larger melt heights, larger controller gains, and nonlinear controllers, the closed-loop flows exhibit timevarying oscillations with larger amplitudes than the openloop case. Issues remain with the practical application of control to the vertical Bridgman system. Due to the use of an enclosed crucible, and the need to conduct growth inside a high temperature furnace, it is extremely difficult to measure the states during growth. In this work, we assumed the availability of an accurate volumetric kinetic energy. In practice, this assumption is currently not feasible. However, continued innovations in micromachining and microfluidics could provide this capability in the near future. The data most readily available in crystal growth systems is temperatures within the melt,
This material is based upon work supported by the National Science Foundation under Grant No. 0201486. This work was also supported in part by the Minnesota Supercomputing Institute. PS expresses thanks to the University of Minnesota Graduate School for a Doctoral Dissertation Fellowship.
References Azuma, Y., Usami, N., Ujihara, T., Fujiwara, K., Sazaki, G., Murakami, Y., et al. (2003). Growth of SiGe bulk crystals with uniform composition by utilizing feedback control system of the crystal-melt interface position for precise control of the growth temperature. Journal of Crystal Growth, 250, 298–304. Batur, C., Duval, W., & Bennett, R. (1999). Performance of Bridgman furnace operating under projective control. In Proceedings of the American control conference (pp. 4101–4105). Batur, C., Srinivasan, A., Duval, W., Singh, N., & Golovaty, D. (1999). Online control of solid-liquid interface by state feedback. Journal of Crystal Growth, 205, 395–409. B´enard, H. (1901). Les Tourbillons cellulaires dans une nappe liquide transportant de la chaleur par convection en r´egime permanent. Annales de Chimie et de Physique, 23, 62–144. Capper, P., & Gosney, J. (1984). Application of the accelerated crucible rotation technique to the Bridgman growth of CdHgTe: simulations and crystal growth. Journal of Crystal Growth, 70, 356–64. Chin, L., & Carlson, F. (1983). Finite element analysis of the control of interface shape in Bridgman crystal growth. Journal of Crystal Growth, 62, 561–567. de Santos, J. (1991). Two-phase cocurrent downflow through constricted passages. Ph.D. thesis, University of Minnesota, available through University Microfilms Inc., http://www.umi.com. Daoutidis, P., & Kravaris, C. (1989). Synthesis of feedforward/state feedback controllers for nonlinear processes. AIChE Journal, 35, 1602–1615. Kim, D., Adornato, P., & Brown, R. (1988). Effect of vertical magnetic field on convection and segregation in vertical Bridgman crystal growth. Journal of Crystal Growth, 89, 339–356. Kim, K., Witt, A., & Gatos, H. (1972). Crystal growth from the melt under destabilizing thermal gradients. Journal of the Electrochemical Society, 119, 1218–1226. Kumar, A., & Daoutidis, P. (1999). Control of nonlinear differential algebraic equation systems: With application to chemical processes. Chapman and Hall/CRC. Lan, C. (2000). Effects of axial vibration on vertical zone-melting processing. International Journal of Heat and Mass Transfer, 43, 1987–1997. Liu, W., Wolf, M., Elwell, D., & Feigelson, R. (1987). Low frequency vibrational stirring: a new method of radial mixing solutions and melts during growth. Journal of Crystal Growth, 82, 589–597. Rayleigh, L. (1916). On convective currents in a horizontal layer of fluid when the higher temperature is on the under side. The Philosopher’s Magazine, 32, 529–546. Schmachtl, M., Schievenbusch, A., Zimmerman, C., & Grill, W. (1998). Crystallization process control during directional solidification in a high-
896
P. Sonda et al. / Computers and Chemical Engineering 29 (2005) 887–896
temperature-gradient furnace by guided ultrasonic waves and real-time signal evaluation. Ultrasonics, 36, 291–295. Seidensticker, R., Rosch, W., et al. (1999). Active control of interface shape during the crystal growth of lead bromide. Journal of Crystal Growth, 198, 988–994. Shiomi, J., & Amberg, G. (2002). Active control of a global thermocapillary instability. Physics of Fluids, 14, 3039–3045. Sonda, P. (2003). The modeling, simulation, and control of transport phenomena in a thermally destabilized Bridgman crystal growth system. Ph.D. thesis, University of Minnesota, available through University Microfilms Inc., http://www.umi.com. Sonda, P., Yeckel, A., Daoutidis, P., & Derby, J. J. (2003). Improved radial segregation via the destabilizing vertical Bridgman configuration. Journal of Crystal Growth, 260, 263–276. Sonda, P., Yeckel, A., Daoutidis, P., & Derby, J. J. (2004a). Development of model-based control for Bridgman crystal growth. Journal of Crystal Growth, 266, 182–189.
Sonda, P., Yeckel, A., Daoutidis, P., Derby, J. J. (2004b). Hopf bifurcation and solution multiplicity in a model for Bridgman crystal growth. Chemical Engineering Science, in press. Srinivasan, A. (1994). Modeling and control of the solid-liquid interface during crystal growth. Ph.D. thesis, University of Akron. Yeckel, A., Comp`ere, G., Pandy, A., & Derby, J. J. (2004). Three-dimensional imperfections in a model vertical Bridgman growth system for cadmium zinc telluride. Journal of Crystal Growth, 263, 629–644. Yeckel, A., & Derby, J. J. (2000). Effect of accelerated crucible rotation on melt composition in high-pressure vertical Bridgman growth of cadmium zinc telluride. Journal of Crystal Growth, 209, 734–750. Yeckel, A., & Derby, J. J. (2004). Dynamics of three-dimensional convection in microgravity crystal growth: g-jitter with steady magnetic fields. Journal of Crystal Growth, 263, 40–52. Yeckel, A., Goodwin, R. T. (2003). Cats2D (Crystallization and Transport Simulator), User Manual, unpublished (available at http://www.msi. umn.edu/∼yeckel/cats2d.html).