MPC Based Feedforward Trajectory for Pulling Speed Tracking Control in the Commercial Czochralski Crystallization Process

Transcription

1 252 Internatonal Khong Journal Lee, Dongk of Control, Lee, Jnguk Automaton, Park, and Moonyong Systems, vol. Lee 3, no. 2, pp , June 25 MPC Based Feedforward Trajectory for Pullng Speed Trackng Control n the Commercal Czochralsk Crystallzaton Process Khong Lee, Dongk Lee, Jnguk Park, and Moonyong Lee* Abstract: In ths work, we propose a smple but effcent method to desgn a target temperature trajectory for pullng speed trackng control of the crystal grower n the Czochralsk crystallzaton process. In the suggested method, the model predctve control strategy s used to ncorporate the complex dynamc effect of the heater temperature on the pullng speed nto the temperature trajectory desgn quanttatvely. The feedforward trajectores desgned by the proposed method were mplemented on 2 mm and 3 mm slcon crystal growers n the commercal Czochralsk process. The applcaton results have demonstrated ts excellent and consstent trackng performance of pullng speed along whole bulk crystal growth. Keywords: Czochralsk crystallzaton process, feedforward trajectory desgn, model predctve control, sem-conductor wafer, slcon crystal control.. INTRODUCTION Czochralsk (CZ) slcon-crystal technology has emerged as an ndspensable technology for semconductor wafer producton and has shown contnuous progress over the years n terms of crystal sze and qualty. Slcon wafers wth dameters up to 2mm are now routnely used, whle producton of 3mm wafers wth ultra hgh qualty s rapdly expandng []. CZ crystal growth s accomplshed by meltng the materal to be crystallzed n an open crucble and then touchng the melted surface wth a seed crystal of the desred materal and crystallographc orentaton. In the CZ crystallzaton process, a sngle slconcrystal s grown from the melt by pullng a seed crystal very slowly whle mantanng a desred thermal envronment n the furnace. The mportant operatonal objectves of the CZ crystal grower are to attan a constant radus, wth a low concentraton of mpurtes and dslocatons, and a unform dopant dstrbuton. To acheve the operaton goal, the classcal PID controllers n a cascade confguraton combned wth emprcally Manuscrpt receved July 3, 24; revsed Aprl 8, 25; accepted May 2, 25. Recommended by Edtoral Board member Young Il Lee under the drecton of Edtor Keum- Shk Hong. Ths work was supported by the Korea Research Foundaton Grant (KRF-2-4-E39). Khong Lee, Dongk Lee, and Moonyong Lee are wth the School of Chemcal Engneerng and Technology, Yeungnam Unversty, Daedong 24-, Kyengsan, Gyeongbuk , Korea (e-mals: khong@yumal.ac.kr, anpols@dreamw. com, mynlee@yu.ac.kr). Jnguk Park s wth LG Sltron Company, Korea (e-mal: pjga@lgsltron.co.kr ). * Correspondng author. derved feedforward trajectores are stll the most wdely utlzed n commercal CZ crystal growers. Although there have been several research efforts to apply state of the art control technques to the control of the CZ crystal growers [2-4], no open report for the real mplementaton of the advanced technques on the commercal growers has been avalable thus far. In the conventonal control scheme of the CZ crystal grower, the crystal dameter s controlled by manpulatng the pullng speed of the crystal. At the same tme, concentraton and dstrbutons of pont defects n the crystal are also manly affected by the crystal growth rate of whch optmal values vary along the crystal growth []. For ths reason, the pullng speed tself should also track a target trajectory, whch s predetermned by experments, precsely along the whole bulk crystal growth. Tght and precse trackng control of the pullng speed s very crucal especally for wafer producton wth larger dameter and hgher qualty specfcaton. Trackng control of the pullng speed can be acheved by adjustng the heater power nput to the crystal grower. However, complex dynamcs wth a large lag tme through the heater, crucble, melt, menscus, and crystal makes t qute dffcult to acheve satsfactory trackng performance wth the feedback control only. For ths reason, a feedforward trajectory for the heater temperature s often added to the feedback control sgnal. In the real feld, however, the feedforward trajectory has been typcally desgned and/or updated merely based on the ntuton and experence of the process engneer, whch often leads to poor and nconsstent trackng performance of the pullng speed and thus poor crystal qualty. In ths work, we present a smple but effcent

2 MPC Based Feedforward Trajectory for Pullng Speed Trackng Control n the Commercal Czochralsk 253 method to desgn the feedforward trajectory of the heater temperature for commercal CZ crystal growers. The proposed method utlzes the model predctve control technque to ncorporate the complex dynamc effect nto the feedforward trajectory desgn, quanttatvely. Through the actual mplementaton on the commercal CZ crystal grower, the proposed method has been found to perform qute satsfactorly. 2. CONVENTIONAL CONTROL STRATEGY OF CZ GROWER Major specfcatons of the slcon sngle crystal produced from the CZ process nclude varous complex physcal and chemcal propertes of crystal such as dslocaton level, mpurty content, and unform radal and axal dopant dstrbuton. However, snce the onlne measurements of these propertes are unavalable, only the crystal dameter and the pullng speed trajectory are normally chosen to be the man control varables to nfer and affect the crystal propertes. Fg. presents the conventonal control scheme used n the commercal CZ crystal growers. Frst of all, control of the crystal dameter s essental not only to mnmze the cuttng waste but also to mprove the qualty of the crystal by reducng fluctuatons n growng condtons. For ths, as shown n Fg., the crystal dameter adjacent to the melt surface s measured by the optcal sensor and s controlled wth the automatc dameter controller (ADC) by manpulatng the pullng speed of the crystal. Fluctuatons of crystal qualtes ncludng dameter durng crystal growth are manly caused by varatons both n crystal pullng speed and thermal condton. The automatc temperature controller (ATC) adjusts the heater temperature to compensate the heater nput aganst thermal dsturbances. The other man control varable s the pullng speed tself. Snce t s closely related to the major crystal propertes, the pullng speed should follow a predetermned target trajectory as closely as possble along Dsp Target Pullng Speed Trajectory Target Dameter Trajectory - - ADC Crystal Dameter AGC Pullng Speed - Target Temperature Trajectory CCD Camera Heater Temperature ATC Heater Input Motor Fg.. Conventonal control scheme of commercal CZ crystal growng process. 2 SCR the whole bulk crystal growng process. Tght trackng control of the pullng speed becomes more essental as larger dameter and hgher qualty are requred. To acheve ths goal, the automatc growng controller (AGC) s used n a cascade confguraton wth the ATC. The AGC receves the pullng speed sgnal and compares wth the pullng speed target to gve the ATC the heater temperature requred. Snce varaton of the target pullng speed s so wde along crystal growth and dynamc effect of the heater nput to the crystal dameter s qute sluggsh and complcated, a target temperature trajectory s often added to the AGC output as a feedforward control sgnal. If the target temperature trajectory s perfectly desgned, the pullng speed would follow ts target value closely enough wth lttle feedback trm from the AGC. 3. APPLICATION OF MPC TECHNIQUE TO FEEDFORWARD TRAJECTORY DESIGN It s well known that the qualty of crystal product manly depends on how accurately the target temperature trajectory s desgned. However, because of ts complex dynamc behavors, t s very dffcult to desgn the feedforward trajectory satsfactorly smply based on ntuton and experence. One of the most proven ways for ncorporatng the complex dynamc effect nto controller desgn s to use the model predctve control (MPC) technque. MPC s by far the most commonly appled advanced control technque for contnuous operaton n the chemcal process ndustry. However, the CZ process s a typcal batch process n whch the operaton proceeds along a tme-varyng reference trajectory over a fnte perod. Recently, there has been extensve research effort to take advantage of the MPC to batch operaton and some promsng advances have been reported [5-7]. In ths study, however, we exclude any attempt to replace the exstng control strategy wth the MPC because of practcal dffcultes n ts actual mplementaton under commercal operaton. Instead, we focused on utlzng the man prncples of MPC to cope wth the quanttatve dynamc effect of heater temperature on the pullng speed nto the desgn of the target temperature trajectory. By dong so, we could mplement the advanced control strategy successfully wth lttle change n the exstng control scheme and algorthm. Now, let us extend the MPC prncples to batch-tobatch feedforward trajectory desgn for the CZ grower. Suppose that the nput sequence over a specfed operaton horzon n the th batch run s T = [ u, u, u2,, un ] u () and the nput sequence results n the output sequence = [, 2, 3,, ] T y y y y y n (2)

3 254 Khong Lee, Dongk Lee, Jnguk Park, and Moonyong Lee whle the setpont trajectory s sp = [ y sp, y sp 2, y sp 3,, y sp ] T n y. (3) In the CZ grower, the actual heater temperature, the actual pullng speed and the target pullng speed correspond to the nput, the output and the setpont of the proposed MPC method, respectvely. Note that any batch run wth the same nput sequence as u would gve the same output sequence as y f no change n dsturbance condtons occurs through batch-to-batch runs. Now consder that a step change s made at the th tme step n the ()th batch run wth no subsequent nput changes,.e., n n = 2 = =. u = u u = = u u whle u u Here, a net ncremental step nput change for the jth tme step n the ()th batch run, u, s defned as j j j j k k= u u u u, j =,..., n. (4) Then the predcted output sequence of the ()th batch run, y ˆ, can be calculated = 2 = 2 2 n = n n j (5) where a j, j=,,n are the step response coeffcents of the output obtaned by makng a unt step nput change to a process. Now suppose the general case where the net ncremental nput changes occur arbtrarly over whole tme steps n the ()th batch run. Then, accordng to the prncple of superposton, the predcted output sequence of the ()th batch run becomes yˆ y a u a u, yˆ y a u a u a u, = 2 = = yˆ y a u a u a u, n = n n n n whch can be re-wrtten n matrx-vector notaton as where ˆ =, (6) y A u y (7) yˆ = y ˆ 2 [, y ˆ T,,, y ˆ ], (8) n T [ u, u,, u n ] u =, (9) a a2 a A =. an an an 2 a () The step response coeffcent matrx A s often called the dynamc matrx n the dynamc matrx control [8]. The desgn objectve of feedforward trajectory for the ()th batch run s to fnd the best u that has the predcted output ncrement yˆ y to approach the current error y sp y as closely as possble n a robust manner. The best soluton for u can be obtaned n the least-squares sense by mnmzng the performance ndex J[ u ] = ˆ T E WEˆ ( u ) W ( u ), () T 2 where ˆ E= A u ( ysp y ) ; W and W 2 are postve-defnte weghtng matrces for penalzng the predcted error and nput movement, respectvely. The net ncremental nput sequence that mnmzes J can be calculated by T T ( 2) u = A WA W A We, (2) where e= ysp y,.e., the error between the setpont trajectory and the actual output of the th run. Fnally, u for the ()th batch run can be calculated usng (4) by j j j k k= u = u u, j =,..., n. (3) 4. IMPLEMENTATION ON THE COMMERCIAL CZ GROWER The desgn of target temperature trajectory requres the step response model between the heater temperature and the pullng speed. However, the dentfcaton test to obtan the requred step response model drectly was practcally dsallowed durng operaton because the control structure needed alteraton. To avod ths problem, we syntheszed the requred model by usng the other step response models that can be obtaned durng operaton easly: a step response of the crystal dameter to the pullng speed and a step response of the crystal dameter to the heater temperature. Crystal bulk growth perod was dvded nto 5 representatve regons based on the dynamc characterstcs and the dentfcaton experments were carred out at each regon. Fgs. 2 and 3 show the examples for the open-loop

4 MPC Based Feedforward Trajectory for Pullng Speed Trackng Control n the Commercal Czochralsk 255 responses of the crystal dameter to the step changes n the pullng speed and n the heater temperature, respectvely. As seen n Fg. 2, the dynamcs between the crystal dameter and the pullng speed can be modeled as an ntegratng plus tme delay process. On the other hand, the dynamcs between the crystal dameter and the heater temperature can be approxmated to a frst order plus tme delay process. Based on the two expermental models, the step response model between the heater temperature and the pullng speed can be derved n context wth the conventonal control scheme shown n Fg. 4. The dynamc models from Fgs. 2 and 3 correspond to G p3 and G p4 n Fg. 4, respectvely. Fg. 5 ndcates the unt step response derved. The dynamc matrx A n () can be obtaned from the derved model straghtforwardly. It s noted that the dynamc behavor of the pullng speed to the heater temperature shows zero gan wth long tme delay and lag. The zero gan ndcates that a double ntegral acton (or at least very strong ntegral acton) s requred to elmnate the trackng off-set n the pullng speed. However, the strong ntegral acton n the AGC s not permtted because of large dead tme and lag tme. Ths confrms why the use of AGC only wth the standard PID algorthm essentally gves poor performance and trackng off-set and the novel desgn of target temperature trajectory as a feedforward sgnal s so mportant. The whole bulk crystal growth regon from mm to 8mm n the ngot length range was consdered as predcton and control horzons. The calculated nput sequences were smoothed by the splne flter aganst excessve wggle n the control acton. The target temperature trajectory for the next batch was updated based on the calculated net ncremental nput change and the actual temperature trajectory of the prevous batch. Target temperature trajectores desgned by the proposed method were mplemented on 2mm commercal CZ crystal growers. Fg. 6 presents the target temperature trajectores by the proposed method and by the old run. The actual temperature trajectores (.e., sum of the AGC output and the target temperature trajectory) are also plotted n Fg. 6. In ths applcaton, W was chosen as the dentty matrx I. The move suppresson matrx W 2 was selected as ki. 2 Crystal Dameter (mm) Actual 8 Model Tme (mn) Fg. 2. Output response of the crystal dameter to.3mm/mn step change n the pullng speed. Fg. 4. Block dagram for dervaton of step response model for the dynamc matrx A: G p, G p2, G p3 denote the transfer functons of heater nputheater temperature, heater temperature-crystal dameter, pullng speed - crystal dameter, respectvely. 2. Crystal Dameter (mm) Model Actual Incremental Pullng Speed (mm/mn) Tme (mn) Fg. 3. Output response of the crystal dameter to step change n the heater temperature Tme (mn) Fg. 5. Derved response of the pullng speed to the unt step change n the heater temperature.

5 256 Khong Lee, Dongk Lee, Jnguk Park, and Moonyong Lee Heater Temperature (C) Target (2nd Run) Actual (2nd Run) Target (Old Run) Actual (Old Run) Ingot Length (%) Target (st Run) Actual (st Run) Fg. 6. Target and actual temperature trajectores n the new and old runs. 75 the actual pullng speed trajectory approaches the target trajectory somewhat more as seen n Fg. 7. To cut back the remanng gap further, addtonal modfcaton of the target trajectory was carred out based on st run data and almost perfect trackng could be acheved as seen n Fg. 7. In the 2nd run, consderng the magntude of the remanng gap, the value of.2 was chosen as the move suppresson factor to force the pullng speed onto the target trajectory more aggressvely. As a model based control technque, performance of the proposed method mght senstvely depend on the model qualty. However, the crude pece-wse lnear model adopted n ths study were found to be enough for satsfactory control performance, whch demonstrated the robustness and practcal usefulness of the method. 5. CONCLUSIONS Pullng Speed (mm/mn) 7 Actual (Old Run) Actual (st Run) Target Trajectory Ingot Length (%) 75 In ths study, we appled the MPC technque to the feedforward trajectory desgn of the commercal CZ crystallzaton process. The method was amed to ncorporate the complex dynamc effect of the heater temperature on the pullng speed nto the desgn of the target temperature trajectory. The proposed method was mplemented on ndustral CZ growers. The applcaton results showed a sgnfcant mprovement n trackng performance and thus crystal qualty. The proposed method s now routnely used n the feedforward trajectory desgn of CZ crystal growers n the LG Sltron Company n Korea. Pullng Speed (mm/mn) Actual (Old Run) Target Trajectory Ingot Length (%) Actual (2nd Run) Fg. 7. Comparson of trackng performance of pullng speed before and after mprovement of target temperature trajectory. Fg. 7 presents the resultng trackng responses of the pullng speed by the correspondng target temperature trajectores n Fg. 6. As can be seen n Fg. 7, n the old run, there was a sgnfcant gap between the target and actual pullng trajectores due to napproprate feedforward acton. Based on the old run data, the target temperature trajectory was modfed n the st run by usng the proposed method. The move suppresson factor k was.7. As a result, REFERENCES [] K. M. Km, Growng mproved slcon crystals for VLSI/ULSI applcatons, Sold State Technology, Pennwell Publcaton Co., Nov [2] M. A. Gevelber and G. Stephanopoulos, Dynamcs and control of the Czochralsk process, Journal of Crystal Growth, vol. 84, no. 4, pp , 987. [3] M. A. Gevelber, M. J. Wargo, and G. Stephanopoulos, Advanced control desgn consderatons for the Czochralsk process, Journal of Crystal Growth, vol. 85, no. -2, pp , 987. [4] R. I. Rvera and W. D. Seder, Model predctve control of the Czochralsk crystallzaton process, Journal of Crystal Growth, vol. 78, no. 4, pp , 997. [5] K. S. Lee, J. H. Lee, I. Chn, and H. J. Lee, A model predctve control technque for batch processes and ts applcaton to temperature trackng control of an expermental batch reactor, AIChE Journal, vol. 45, no., pp , 999.

6 MPC Based Feedforward Trajectory for Pullng Speed Trackng Control n the Commercal Czochralsk 257 [6] K. S. Lee, J. Lee, I. Chn, J. Cho, and J. H. Lee, Control of wafer temperature unformty n rapd thermal processng usng an optmal teratve learnng control technque, Industral & Engneerng Chemstry Research, vol. 4, no. 7, pp , 2. [7] K. S. Lee and J. H. Lee, A generc framework for ntegrated qualty and profle control for ndustral batch processes, Proc. of 6th IFAC Symp. on DYCOPS, pp. 59-7, June 2. [8] C. R. Cutler and B. L. Ramaker, Dynamc matrx control: a computer control algorthm, Prof. of 86th AIChE Natonal Meetng, Aprl 979. Khong Lee receved hs B.S. and M.S. degrees n the School of Chemcal Engneerng and Technology at Yeungnam Unversty n 999 and 2. He s currently workng toward hs Ph.D. degree at the same nsttute. Hs research nterests nclude on-lne process montorng and process control. Jnguk Park receved hs B.S. degree n Electrcal Engneerng from Kyungpook Unversty n 994 and hs M.S. degree n the School of Chemcal Engneerng and Technology at Yeungnam Unversty n 24. He s currently workng at the LG Sltron Company n Korea. Hs research nterests nclude model predctve control, batch process montorng, and statstcal process control. Moonyong Lee receved hs B.S. degree n Chemcal Engneerng from Seoul Natonal Unversty n 982, and hs M.S. and Ph.D. degrees n Chemcal Engneerng from KAIST n 984 and 99. From 984 to 994, he worked n the refnery and petrochemcal plant of SK Company, Korea. Snce 994, he has been wth Yeungnam Unversty where he s currently a Professor n the School of Chemcal Engneerng and Technology. Hs research nterests nclude process control, onlne montorng, and process desgn. Dongk Lee receved hs B.S. and M.S. degrees n the School of Chemcal Engneerng and Technology at Yeungnam Unversty n 999 and 2. Hs research nterests nclude model predctve control and batch control.