Control Engineering Practice 8 (2000) 569}579
Implicit linear control law of a close-spaced vapor transport process M. Bonilla Estrada *, V. Rejon, R. Castro-Rodriguez CINVESTAV-IPN, Unidad Zacatenco, Departamento de Control Automa& tico, A.P. 14-740 Me& xico 07000, Mexico CINVESTAV-IPN, Unidad Unidad Me& rida, Departamento de Fn& sica Aplicada, A.P. 73 Cordomex Me& rida, Yucatan 97310, Mexico
Abstract In this paper a linear control scheme for a close-spaced vapor transport process to obtain thin "lm semiconductors is presented. This linear control scheme is composed of two linear control laws. The inner controller is an implicit control law whose aim is to transform the system into a linear time-invariant system without disturbances. The outer controller is a classical PI control law, whose aim is to make the error tend towards zero. 2000 Elsevier Science Ltd. All rights reserved. Keywords: Temperature process; Thin "lms semiconductors; Modeling and control of semiconductor process RTP; CSVT; Implicit systems; PI controllers
1. Introduction Semiconductor thin "lms are widely used in important technological applications like optoelectronic or photovoltaic devices. Numerous processes are available for growing semiconductor layers, e.g., vacuum deposition, hot-wall vacuum evaporation, electron or molecular beam evaporation, chemical vapor deposition, transport from liquid or vapor phase, sputtering, gas}solid reaction, spray pyrolysis, screen printing and electrodeposition. Among these processes, close-spaced vapor transport (CSVT) has been used for several years to deposit epitaxial layers of semiconductor materials. This process is based on heating, to di!erent temperatures, the source of a material to be deposited and a substrate for the epitaxy. Usually, a transporting agent diluted in a carrier gas reacts with the source to form a volatile compound that then migrates to the substrate where the reverse reaction takes place. Control of the doping in the semiconductor materials is important because the usefulness of many electronics devices depends on the accuracy of this control. However, the control of the dopant incorporated into many * Corresponding author. Tel.: #52-5-747-7089; fax: #52-5-7477002/7089. E-mail addresses:
[email protected] (M. Bonilla Estrada),
[email protected] (V. Rejon), romano@kin. cieamer.conacyt.mx (R. Castro-Rodriguez).
semiconductor materials has been di$cult when grown from the vapor (Bajor & Greene, 1983). Fortmann, Fahrenbruch and Bube (1987) and Anthony, Fahrenbruch, Peters and Bube (1985) have investigated dopant incorporation in some II}VI semiconductors such as CdTe using hot-wall vacuum evaporation (HWVE) and CSVT. Also, Hayashi, Susuki and Ema (1988) have reported on the indium incorporation in CdTe by co-evaporation of CdTe and metallic indium. Castro-Rodriguez and Pen a (1993) have reported on a new and novel deposition technique involving the combined use of free evaporation and CSVT (Sosa, Castro & Pen a, 1990) for doping semiconductor thin "lms. They applied this technique to grow indium-doped CdTe "lms with good electrical properties. The main goal of his study was to use the indium source temperature as a pertinent doping control parameter, see Fig. 1. For this purpose, the CSVT technique was performed using a short distance between the source and substrate: typically less than 8 mm. The reaction chamber, consisting of source and substrate separated by a spacer, is held between two graphite blocks. Heating is usually provided by the Joule e!ect, using both graphite blocks as resistances in some cases (Menezes, Fortmann & Casey, 1985). In most cases there is a temperature di!erence of approximately 1003C between the source and substrate. In order to combine the CSVT technique with the free evaporation one, two holes of 5 mm diameter were made in each end of the graphite block for electrical
0967-0661/00/$ - see front matter 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 9 6 7 - 0 6 6 1 ( 9 9 ) 0 0 1 7 5 - 6
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Fig. 1. Close-spaced vapor transport process free evaporation. A: (a) copper wires, (b) thermocouple type K, (c) CSVT-FE process, (d) bell quartz. B: (1) Graphite substrate header, (2) substrate, (3) graphite chamber, (4) CdTe powder, (5) graphite CdTe source header, (6) graphite bell, (7) graphite dopant source header. ¹ , ¹ and ¹ are measured with thermocouple type K.
connections. A threaded hole of 6 mm diameter in the center of the block was for a graphite bell which was screwed into the graphite block. The purpose of the bell is to guide the atoms of the evaporated dopant, introducing them into the CSVT deposition chamber that is in a vacuum of 1;10\ Torr. In the bottom of this graphite bell there is a graphite block or tungsten boat source for dopant evaporation. The graphite substrate heater was similar to the graphite source block but without the center hole. Fig. 1 shows the apparatus. With respect to the temperature control scheme, CastroRodriguez, Rejon, Zapata-Torres, Zapata-Navarro and Pen a (1995) reported a proportional control law. That proportional controller was tuned in a heuristic way and had the disadvantage of being very sensitive to the set point value, and therefore every time the set point was changed the controller had to be re-tuned. For the control of this kind of nonlinear thermal process several approaches have been used. Among them are the following: Breedijk, Edgar and Trachtenberg (1993) described an improved nonlinear dynamic model for the Texas Instrument rapid thermal processing (RTP) with simulations of a nonlinear model predictive controller based on the application of orthogonal collocation to a partial di!erential equation representation of the process. Nonlinear programming was used to compute constrained control moves. Subsequently, Breedijk (1994) developed simulations of a successively linearized quadratic dynamic matrix control (QDMC) strategy and has demonstrated viability for real-time implementation. Schaper, Moslehi, Saraswat and Kailath (1994) have implemented an internal model control (IMC) strategy with gain scheduling on similar RTP systems. Elia (1994) has reported the successful application of optimal multivariable control (using a linear quadratic-Gaussian control-
ler with loop transfer recovery LQG/LTR) to the Applied Materials RTP chamber for both annealing and oxidation. The AM-RTP has eight independently actuated groups of lamps and eight temperature measurements across the width of the chamber. Stuber, Trachtenberg and Edgar (1994) applied a quadratic dynamic matrix control to a Texas Instrument RTP process. In this paper a linear control scheme is used to follow the temperature pro"le in the close-space vapor transport combined with free evaporation (CSVT-FE) process reported by Castro-Rodriguez and Pen a (1993). Since this is an experimental process, a controller which does not need to be tuned again when changing the temperature pro"les or the geometry of the body graphites is needed. This controller is composed of two linear control laws. The inner controller is an implicit control law whose aim is to transform the system in a linear timeinvariant system without disturbances. The outer controller is a classical PI control law, whose aim is to make the temperature errors tend to zero. 2. Models of the CSVT-FE process In this section two models of the process are obtained: a nonlinear low-order model and a linearized one. The "rst model is used in the simulation procedure and the second one is used in the controller synthesis procedure. 2.1. Nominal model A low-order model of the process is "rst obtained. For this, the following "ve assumptions were made: (H1) The physical properties of graphites remain constant during the process.
M. B. Estrada et al. / Control Engineering Practice 8 (2000) 569}579
(H2) The convection heat #ux is neglected. (H3) The temperature and the heat #ux are uniformly distributed in the graphite. (H4) The geometrical body of the graphites is constituted by parallel faces. (H5) The graphites are considered as black bodies. Following the same procedure as Schaper, Moslehi, Saraswat and Kailath (1994) (see also Shearer & Kulakowski, 1990), the following nominal model is obtained by balancing energy (since the graphites are assumed to behave as black bodies, their total emissivity coe$cients is taken as equal to one): m C ¹Q (t)"!pF A (¹ (t)!¹(t)) N ? ? !pF A (¹ (t)!¹ (t)) 1 ! (¹ (t)!¹ (t))#u (t), AS R FI m C ¹Q (t)"!pF A (¹ (t)!¹(t)) ? N ? !F A (¹ (t)!¹ (t)) 1 ! (¹ (t)!¹ (t)) AS R FI #pF A (¹ (t)!¹ (t))#u (t), m C ¹Q (t)"!pF A (¹ (t)!¹(t)) ? N ? 1 ! (¹ (t)!¹ (t)) AS R FI #pF A (¹ (t)!¹ (t))#u (t),
(1)
where m is the mass of the ith graphite, C is the speci"c G N heat of graphite, R G ("¸ /iA G ) is the conduction G AS FI thermal resistances of the ith electrical copper conductor, p is the Stefan}Boltzmann constant, F G and F GH are the geometrical factors, A is the surface area of graphite G? i facing to the exterior environment, and A is the surface GH between graphite i and j. ¹ (t) is the temperature of graphite i, ¹ (t) is the G ? ambient temperature, ¹ G (t) is the copper temperature of AS the electrical conductor of ¸ meters from graphite i and G "nally u (t) is the power input to graphite i. This model G was obtained assuming that ¹ (t)'¹ (t)'¹ (t), since this is the objective. In the Simulations and Experimental Results Section numerical values of the models' parameters are obtained and the model is also validated. 2.2. Implicit model In order to synthesize, in next section, the implicit linear control law, a linear simpli"cation of the nominal model is obtained.
571
For this, "rst linearize (1) around a (variable-time) set point:
¹Q !a (t) ¹Q " a (t) ¹Q 0 u C u # # C u C where
a (t) 0 ¹ !a (t) a (t) ¹ a (t) !a (t) ¹ ¹ ¹ ? # AS C R C R FI FP ¹ ¹ ? # AS , C R C R FI FP ¹ ¹ ? # AS C R C R FI FP
(2)
1 1 1 a (t)" # # , C R C R C R FI FP FP 1 1 , a (t)" , a (t)" C R C R FP FP 1 1 1 1 a (t)" # # # , C R C R C R C R FI FP FP FP 1 1 a (t)" , a (t)" , C R C R FP FP 1 1 1 # # , a (t)" C R C R C R FI FP FP C "m C , R GH G G N FP "(pF GH A (¹(t)#¹(t))(¹ (t)#¹ (t)))\, GH G H G H for R G put j"a. FP Now conjoin all the thermical couplings and disturbances of each graphite block in a simple perturbation term q (t), namely, G ¹Q (t)"a (t)¹ (t)#b u (t)#q (t), i"1, 2, 3, (3) G G G G G G where b "1/C and q (t) are the remainder terms of G G G model (2). In view of the fact that the process works around a "xed bounded temperature band, it is assumed that each a (t) is a time-bounded variable and that each q (t) is G G a bounded disturbance. In the next section an implicit linear control law, for system (3), is proposed that rejects disturbances q (t) and makes the system time-invariant G with a pre"xed dynamics. Namely, the behavior of the closed-loop system will be described by the ordinary di!erential equation: ¹Q (t)"!a ¹ (t)#a v (t), i"1, 2, 3, (4) G G G G G where each coe$cient a is adjusted to obtain a desired G dynamics. That is to say, the implicit linear control law decouples and "xes an invariant-time dynamics for the
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CSVT-TE process. Once, this is achieved by the inner controller, an outer controller, for linear time-invariant systems, is applied in order to set the tracking error to zero, as for example, for a PI controller.
3. Linear control law This section proposes a control scheme that enables the CSVT-FE process to reach the following control objectives: E To track a given temperature pro"le in each graphite block. E To have a steady-state absolute error in temperature less than 13C. E To be able to command the temperature of each graphite block in an independent way, namely to decouple the system. E To have smooth control actions. E Not to have to re-tune the controller each time the set point is changed or when the dimensions and materials of each graphite are changed, since this is an experimental process. To reach such goals the following control scheme composed of two control loops is proposed: Inner control loop: For the inner control loop of each graphite, the following implicit control law (i"1, 2, 3) is used: 1 u " (!a ¹ (t)#a v (t)!u (t)), (5) G b G G G G G G e(u (t)#u (t))"(e (t)#be (t)), (6) G G G G 1 e (t)# e (t) "¹Q (t)#a ¹ (t)!a v (t), (7) G G G G G G q G where a , e, b and q are positive constants. In the next G subsection more details about these constants and the properties of such a controller are given. Outer control law: For the outer control loop of each graphite, the following PI control law (i"1, 2, 3) is used:
v (t)"K e( (t)#K G N G '
R
e( (q) dq, G
e( (t)"¹M (t)!¹ (t), (8) G G G where K and K are the proportional and integration N ' constants and the ¹M (t) are the temperature pro"les to G follow. 3.1. Implicit control law An early version of the implicit control law (5)}(7) was proposed in Bonilla and Malabre (1991), in such a version, the controller was constituted by ideal derivative
actions. In Bonilla and Puga (1996), a proper practical approximation was achieved following the procedure shown in Bonilla, Malabre and Fonseca (1997). The principal properties of the implicit control law (5)}(7) are summarized in the following two theorems (Theorem 1 is proved in the appendix and Theorem 2 is proved in Section 3.1.1). Theorem 1. When the control law (5)}(7) is applied to system (3), the closed-loop behavior is described by the following diwerential equations ( p is the derivative operator d/dt, and i"1, 2, 3): ( p#a )¹ (t)"a v (t)#e (t), G G G G G 1 p# e (t)"e (t), G G q 1 #1 e (t)#(b!1)e (t) (p#1) e p# G G q "e(p#1)g (t), G
(9) (10)
(11)
1 g (t)"f (t)¹ (t)#a (t)e (t)#a v (t)# q (t)# q (t) , G G G G G G G G q G (12)
1 f (t)"a (t)# !a a (t). (13) G G G G q If "a (t)"(R, "a (t)"(R, " q (t)#(1/q )q (t)"(R and G G G G "v (t)"(R for all t*0, then G lim "e (t)#g (0)e\@R""0. G G C
(14)
Theorem 2. There exists e夹'0 such that for all 0(e(e夹, the closed-loop system (9)}(11) is exponentially stable. In view of these theorems, note the following observations: (1) The role of the parameter e is to exponentially stabilize the closed-loop system (see Theorem 2) and to exponentially tend towards the ideal behavior (compare (14) and (9) with (4)). In theory, the smallest value of e obtains the best behavior; but in practice, a very small value of e produces problems in computational integration algorithms. So e should be chosen near to e夹 and lowered bit by bit, without losing the stability of the used integration algorithm until the desired approximation. (2) The role of the parameter b is to determine the ratio of convergence of the control law (see (14)). In other words, the smoothness of the control law is adjusted with b.
M. B. Estrada et al. / Control Engineering Practice 8 (2000) 569}579
(3) The role of the parameter q is only to simplify the stability analysis. (4) The parameters a (i"1, 2, 3) determine the desired G ideal dynamics (see (4)). 3.1.1. Stability analysis The stability of the inner closed-loop system (9)}(11) is studied here. Since the stability of linear systems does not depend on exogenous signals, consider, v (t),0 and G q (t),0 in order to simplify the algebra. G The Lyapunov approach is used. For this, the following three lemmas are needed, proved in the appendix at the end of this paper (in this section the sub-indices i of (9)}(13) are omitted). Lemma 1. The closed-loop behavior (9)}(11) is also well described by the following set of diwerential equations:
1 p# e(t)"e (t), q x (t)"A x(t)#b m(t), mQ (t)"b2 (t)x(t)#d(t, e)m(t), where
(15)
!a
1
(16) (17)
0
!b
(18)
1 b " , (1!b)
,
1 b2 (t)"[ f (t) a(t)], d(t, e)"a(t)!1! , e
(19)
1 a(t)"a(t)#b! . q
(20)
Lemma 2. The following positive real Hermitian matrix is dexned:
a #b
P "
1
,
a #a b#1 b then (I stands for a identity matrix of size 2;2) A2 P #P A "!2a (a #b)I . 1
(21)
P " where
0
0
c
1 A2 (t, e)P #P A (t, e)"! M#N(t) , e
(25)
A b , A (t, e)" b2 (t) d(t, e) M"
(26)
2a (a #b)
0
0
2a (a #b)
!(a #1)
a (a #b)(b!1)!1 b
!(a #1)
a (a #b)(b!1)!1 , b 2c
(27)
N(t)"
0
0
!c f (t)
0
0
!ca(t)
.
(28)
,
1#(a (a #b)(b!1)!1) (a #1) c" # , 4a (a #b)b 4a (a #b)
Furthermore, M is a positive real Hermitian matrix. Proof of Theorem 2. The Lyapunov function is de"ned as (29) C O R e (0)! G G
e\R\H e (j) dj#e\Rg (0)"0, G G
( p#1)
"e(p#1)(a (t)¹ (t)#q (t)). G G G
# (b!1)
Proof of Lemma 1. From (11) and (12)
1 (p#1) e p# #1 e (t)#(b!1)e (t) G G q
1 1 p# 1#e e q
R
viz.,
Thus, adding (A.1) and (A.2), Eqs. (9) and (10) are obtained. (2) Applying the operator e(p#1) to (A.2) and taking into account (6)
e (t)#(b!1) G
e\R\CQ\H e (j) dj ds G
e )lim sup "g (p)" [1!e\>C O C]"0, G 1#e/q C NRW2
E In view that P is positive de"nite and that e'0 and c'0, P will be positive de"nite if and only if (a (a #b)(b!1)!1) (a #1) c' # 4a (a #b)b 4a (a #b) which is the case (24). E From (26), (23), (22) and (19), Eq. (25) is obtained. E Since a (a #b)'0, the real Hermitian matrix M will be positive de"nite if and only if its determinant is positive, namely, a (a #b) '0. b
䊐
References Anthony, T. C., Fahrenbruch, A. L., Peters, M. G., & Bube, R. H. (1985). Electrical properties of CdTe "lms and junctions. Journal of Applied Physics, 57(2), 400}410.
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