* Corresponding author. Tel.: #52-5-747-7089; fax: #52-5-747- 7002/7089. E-mail addresses: mbonilla@enigma.red.cinvestav.mx (M. Bonilla Estrada), vrejon@kin.cieamer.conacyt.mx (V. Rejon), romano@kin. cieamer.conacyt.mx (R. Castro-Rodriguez). 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 photo- voltaic 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 electro- deposition. Among these processes, close-spaced vapor transport (CSVT) has been used for several years to deposit epi- taxial 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. How- ever, the control of the dopant incorporated into many semiconductor materials has been di$cult when grown from the vapor (Bajor & Greene, 1983). Fortmann, Fahrenbruch and Bube (1987) and An- thony, Fahrenbruch, Peters and Bube (1985) have investigated dopant incorporation in some II}VI semiconductors such as CdTe using hot-wall vacuum evaporation (HWVE) and CSVT. Also, Hayashi, Susuki and Ema (1988) have reported on the indium incorpora- tion in CdTe by co-evaporation of CdTe and metallic indium. Castro-Rodriguez and Pen  a (1993) have reported on a new and novel deposition technique involving the com- bined use of free evaporation and CSVT (Sosa, Castro & Pen  a, 1990) for doping semiconductor thin "lms. They applied this technique to grow indium-doped CdTe "lms with good electrical properties. The main goal of his study was to use the indium source temperature as a per- tinent doping control parameter, see Fig. 1. For this purpose, the CSVT technique was performed using a short distance between the source and substrate: typically less than 8 mm. The reaction chamber, consist- ing of source and substrate separated by a spacer, is held between two graphite blocks. Heating is usually provided by the Joule e!ect, using both graphite blocks as resist- ances in some cases (Menezes, Fortmann & Casey, 1985). In most cases there is a temperature di!erence of approx- imately 1003C between the source and substrate. In order to combine the CSVT technique with the free evaporation one, two holes of 5 mm diameter were made in each end of the graphite block for electrical 0967-0661/00/$ - 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