Temperature sensing and control in multi zone semiconductor thermal processing

TEMPERATURE SENSING AND CONTROL IN MULTI-ZONE SEMICONDUCTOR THERMAL PROCESSING YAN HAN (B.Eng., SJTU) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2009 i Acknowledgements My deepest gratitude is to my advisor, Professor Ho Weng Khuen, for his patience and support throughout my study and research at the National University of Singapore. I have benefited enormously from the constructive advice and critiques that he offered over many of our discussions. I must also extend my gratitude to Professor Ling Keck Voon and Professor Jos´e Romagnoli for the help they rendered to my research. I would also like to thank my friends and colleagues: Dr. Hu Ni, Dr. Wu Xiaodong, Dr. Fu Jun, Dr. Ye Zhen, Dr. Chen Ming, Ms. Wang Yuheng, Mr. Feng Yong, Mr. Shao Lichun, Ms. Lim Li Hong, Mr. Nie Maowen, Mr. Chua Teck Wee, Mr. Ngo Yit Sung, Mr Lee See Chek, Ms. Teh Siew Hong, Mr. Lin Feng, Mr. Tan Kiat An, Mr. Gibson Lee, and many others at the Advanced Control Technology Laboratory (ACT), the Mechatronics & Automation Laboratory and the Control & Simulation Laboratory. I have enjoyed entertaining and inspiring conversations with them and I am thankful for the congenial and conducive working environment to which we have all contributed. ii Finally, my heartfelt thanks to my family for their unfailing support and understanding. They give me purpose, without which I would be a lesser person. iii Contents Acknowledgements Table of Contents Summary iii viii List of Tables xi List of Figures Introduction 1.1 i xiii Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 RTD Bias Estimation . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Control of the Multi-Zone Bake-Plate . . . . . . . . . . . . . . iv 1.2 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 RTD Bias Estimation in Multi-Zone Semiconductor Thermal Processing and Estimator Performance Analysis . . . . . . . . 1.2.2 Multiplexed MPC for Multi-Zone Semiconductor Thermal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 RTD Bias Estimation for Multi-Zone Semiconductor Thermal Processing 13 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.2 Bake-Plate Thermal Modeling . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Bias Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 2.3.1 Least Squares Estimation . . . . . . . . . . . . . . . . . . . . 21 2.3.2 GT-based Estimation . . . . . . . . . . . . . . . . . . . . . . . 23 Analysis of Estimator Performance . . . . . . . . . . . . . . . . . . . 25 2.4.1 Influence Function (IF) . . . . . . . . . . . . . . . . . . . . . . 26 2.4.2 IF of LS Estimator . . . . . . . . . . . . . . . . . . . . . . . . 29 v 2.5 2.4.3 IF of IQR+LS Estimator . . . . . . . . . . . . . . . . . . . . . 29 2.4.4 IF of GT-based Estimator . . . . . . . . . . . . . . . . . . . . 31 2.4.5 Estimation Variance . . . . . . . . . . . . . . . . . . . . . . . 32 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 Appendix 2A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Appendix 2B . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 Appendix 2C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 Appendix 2D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 Appendix 2E . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 Simulation and Experimental Results 43 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2 A Simulation Example . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.3 3.2.1 Problem Setup . . . . . . . . . . . . . . . . . . . . . . . . . . 45 3.2.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 Experimental Verification of Theoretical Results . . . . . . . . . . . . 52 vi 3.4 3.3.1 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3.3 Sample Calculation . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 Appendix 3A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 Multiplexed MPC for Multi-Zone Semiconductor Thermal Processing 75 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 4.2 Bake-Plate Thermal Modeling . . . . . . . . . . . . . . . . . . . . . . 81 4.3 A Review of Multiplexed MPC and Feedforward Control . . . . . . . 83 4.4 4.3.1 Multiplexed MPC . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.3.2 Feedforward Control . . . . . . . . . . . . . . . . . . . . . . . 84 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 4.4.1 Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . . . 85 vii 4.5 4.4.2 Parameter Estimation . . . . . . . . . . . . . . . . . . . . . . 85 4.4.3 Experimental Runs . . . . . . . . . . . . . . . . . . . . . . . . 87 4.4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Appendix 4A . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 Conclusion 97 5.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 5.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 Bibliography 102 Author’s Publications 109 viii Summary The importance of lithography to semiconductor manufacturing is evident. Lithography constitutes about 30% of the cost of manufacturing a chip, and is the key technological enabler for further down-scaling of device dimensions and upgrading of chip performance. Thermal processing is an integral part of lithography. In alignment with the call for ever smaller critical dimension (CD) and due to the fact that the final CD is very sensitive to thermal processing temperature, requirement is increasingly stringent on temperature sensing and control in multi-zone semiconductor thermal processing. Resistance Temperature Detectors (RTD’s) installed in a multi-zone bake-plate typically used for semiconductor thermal processing are subject to measurement bias. Data reconciliation (DR) techniques are extended so that RTD biases can be estimated online from process data. To handle frequently encountered nonnormality in process data, a generalized T distribution (GT) based bias estimator is proposed. Equations are derived which relate variance of a bias estimator to sample size (number of wafers runs per estimation). These equations enable the ix computation of the sample size or the number of wafers needed by the bias estimator to achieve specified variance. With this information, the exact number of wafers can be used for estimation so that bias can be estimated precisely and eliminated as soon as possible to avoid wafer wastage. Alternatively, these equations allow the calculation of the variance of the bias estimator and hence its precision if the number of wafers used is given. The theoretical results on estimator analysis are verified experimentally. In the light of the equations derived, an efficient estimator can be selected. In the presence of outliers that are close to the good data, the equations show that using GT, instead of normal distribution, to characterize process data gives rise to a more efficient estimator than Least Squares (LS) and Interquartile test plus Least Squares (IQR+LS) and therefore enables earlier remedial actions against RTD bias to save semiconductor wafers from sensing-related processing defects. In view of the cost of manufacturing one wafer, a guided choice of an efficient RTD bias estimator and an appropriate sample size for estimation is economically important. To fulfil the stringent requirement on temperature control of a multi-zone bakeplate, Multiplexed Model Predictive Control (MMPC) with feedforward is demonstrated experimentally on a multi-zone bake-plate application. By distributing the control moves over one complete update cycle, MMPC can afford to work with higher sampling rate. It is shown to have the potential to make the bake-plate respond and recover faster than under conventional MPC when disturbance is induced by 95 M −1 i=0 |xk+i+1|k |2q + |∆˜ uk+i|k |2r + F (xk+M |k ) Minimise Jk = wrt ∆˜ uk+i|k , s.t. ∆˜ uk+i|k ∈ Uσ(k+i) , (i = 0, N, 2N, · · · , M − 1) xk+i|k ∈ X, (i = 0, · · · , M − 1) (4.7) (i = 1, · · · , M ) xk+M |k ∈ XI (Kσ(k) ) xk+i+1|k = Axk+i|k + Bσ(k+i) ∆˜ uk+i|k , ∆˜ uk+i|k = ∆˜ uk+i|k−1 , (i = 0, · · · , M − 1) (i = 0, N, 2N, · · · , M − 1) where M = (Mu − 1)N + and Mu is the control horizon, a design parameter which denotes the number of control moves to be optimized per input channel of the original system given by Equation (4.4); F (xk+N ) is a suitably chosen terminal cost, and X and U are compact polyhedral sets containing the origin in their interior. XI (Kσ(k) ) denotes the set in which none of the constraints is active, and which is the maximum positively invariant set [44] for the linear periodic system given by Equation (4.5), when a stabilizing linear periodic feedback controller Kσ(k) is applied. In other words, xk ∈ XI (Kσ(k) ) implies Kσ(k) xk ∈ Uσ(k) and (A + Bσ(k) Kσ(k) )xk ∈ XI (Kσ(k) ). The main idea of MMPC, as captured in the problem formulation (4.7), is to partition the entire system into smaller subsystems, solve the control for each subsystem sequentially, and makes the control update as soon as the solution becomes available. Hence the optimization is carried out with respect to only a subset of the 96 available decision variables, ie, ∆˜ uk+i|k (i = 0, N, 2N, · · · , M −1). This is in contrast to conventional MPC which updates all the control variables simultaneously in one update cycle. Hence, some assumptions must be made about those inputs which have already been planned but which have not yet been executed. MMPC assumes that all such planned decisions are known to the controller, and that they will be executed as planned, ie, ∆˜ uk+i|k = ∆˜ uk+i|k−1 (i = 0, N, · · · , M − 1) 97 Chapter Conclusion 5.1 Summary Temperature in semiconductor thermal processing is an important determinant of CD which quantifies the accuracy of circuit patterns formed over the lithography process. Increasingly stringent requirement on temperature control in semiconductor thermal processing gives rise to the need for a multi-zone bake-plate with unbiased temperature sensing and a computationally efficient temperature control scheme. In this thesis we develop and analyze data reconciliation techniques to efficiently estimate RTD bias from thermal processing data in order for biased RTD’s to be remedied online. We propose and implement MMPC as a control scheme to lessen computational load otherwise incurred by conventional MPC in the multi-zone bakeplate so that potentially superior control performance results without the need for 98 hardware upgrade. In Chapter 2, data reconciliation (DR) is extended so that RTD biases can be estimated online from process data. To handle frequently encountered nonnormality in process data, a generalized T distribution (GT) based bias estimator is proposed. Equations are derived which relate variance of a bias estimator to sample size (number of wafers runs per estimation). These equations enable the computation of the sample size or the number of wafers needed by the bias estimator to achieve specified variance. With this information, the precise number of wafers can be used and wastage can be prevented. Alternatively, these equations allow the calculation of the variance of the bias estimator and hence its precision if the number of wafers used is given. Equations are obtained relevant to the analysis of simple LS, IQR+LS, and the proposed GT-based estimator. In Chapter 3, simulation and experimental examples are given to demonstrate the application of the estimators and estimator analysis developed in Chapter 2. We examine specifically the performance of simple LS, IQR+LS and GT-based bias estimator for the difficult problem where measurement outliers are close to good data such that they cannot be separated easily. The theory is verified experimentally on a multi-zone bake-plate for semiconductor thermal processing. In the light of the equations derived, an efficient estimator can be selected. In the presence of outliers that are close to the good data, the equations show that using GT, instead of normal distribution, to characterize process data gives rise to a more efficient estimator than 99 LS and IQR+LS and therefore enables earlier remedial actions against RTD bias to save semiconductor wafers from sensing-related processing defects. The experiment was performed with 25% of the temperature and power fluctuations coming from distributions with standard deviations times as great as the good data. It is shown that the GT-based estimator with 43 wafer runs achieved the same estimation variance as the IQR+LS estimator with 50 wafer runs. In other words, wafers could be saved from sensing-related processing defects. In Chapter 4, we propose the use of Multiplexed MPC (MMPC) for temperature control in multi-zone thermal processing. MMPC with feedforward is demonstrated experimentally on a multi-zone bake-plate application. It is shown that adding feedforward reduces the effect of disturbance significantly. While most of the temperature drop will be compensated by feedforward control, feedback control is still needed to cope with the errors due, for example, to warped wafers. Depending on the thermal process, the recipe baking time varies and plate temperature should recover fully within the pre-specified period of time. We note from the experiments the potential of MMPC to make plate temperature recover faster than under conventional MPC after disturbance takes place. The sampling rate not being the only factor bearing on closed-loop control performance, MMPC’s key advantage of reduced computational load supports faster sampling and potentially brings about superior control performance. These results are important for the semiconductor wafer baking process, because temperature non-uniformity resulting from poor temperature control performance has adverse impact on the CD of wafers. 100 5.2 Future Work It would be of interest to study the adoption of a Bayesian framework in sensor bias estimation. In what has previously been discussed in this thesis, the probability that an RTD is biased is not factored into calculation. Nor is the probability distribution of the bias of an RTD, since we have assumed that any bias value is equiprobable. For one thing, factoring bias probability may be especially beneficial by bringing about fewer false alarms when bias is not actually existent. For another, factoring bias distribution may bring about further efficiency gains for bias estimators. Bias value tends to be less likely as its magnitude grows so that it maybe modeled to follow a mono-modal distribution centered at zero. The bias estimation problem, with the the statistical properties of bias taken into calculation, is hence to minimize S 2N j=1 i=1 − N ln f (yi (j) − xi − bi e; σi , pi , qi ) + N ln P (ei ) + i=1 ln f (ei ) i=1 with respect to x and e, subject to Ax = 0, where P (e) is the probability an RTD has bias and f (e) is the probability density of an RTD’s bias value. Equations for estimator analysis remain to be worked out, which would be a generalization of the equations we have derived in Chapter 2. As far as the thesis is concerned, sensor bias estimation, or more broadly, data reconciliation (DR), has been executed in a centralized manner (CDR). All measurements from the thermal process are collected, transferred to and reconciled by 101 an estimator. CDR per se cannot handle total breakdown of certain sensors if the quantities these sensors are supposed to measure are not known by the centralized DR as unmeasurable. 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Yan, “Experimental Evaluation of Multiplexed MPC for Semiconductor Manufacturing”, Asian Control Conference, August 2009. K. V. Ling, W. K. Ho, B. Wu, Andreas and H. Yan, “Multiplexed MPC for MultiZone Thermal Processing in Semiconductor Manufacturing”, accepted for publication by IEEE Transactions on Control Systems Technology, 2009. W. K. Ho, H. Yan, K. V. Ling, J. A. Romagnoli and K. V. Ling, “Measurement Bias Detection, Identification and Elimination for Multi-Zone Thermal Processing in Semiconductor Manufacturing”, The 33rd Annual Conference of the IEEE Industrial Electronics Society, November 2007. 110 K. V. Ling, H. Yan, W. K. Ho, J. A. Romagnoli and Y. Joe, “Multi-Zone Thermal System in Semiconductor Manufacturing: Gross Error Treatment”, IFAC Conference on Advanced Process Control for Semiconductor Manufacturing, December 2006. [...]... control of the multi- zone bake-plate Chapter 5 summarizes the work presented in previous chapters and gives a brief outlook on future work 13 Chapter 2 RTD Bias Estimation for Multi- Zone Semiconductor Thermal Processing 2.1 Introduction Thermal processing of semiconductor wafers is common and critical in semiconductor manufacturing [18, 19, 20] Temperature uniformities within a wafer and from wafer... to bring about better temperature control performance, than conventional MPC, by affording faster sampling with its much reduced computational load 1.2 1.2.1 Contributions RTD Bias Estimation in Multi- Zone Semiconductor Thermal Processing and Estimator Performance Analysis As has been discussed in Section 1.1.1, an array of RTD’s are installed in the multizone bake-plate for semiconductor thermal processing. .. step in the lithography sequence is the post-exposure bake step [4, 21] To obtain temperature uniformity, a wafer is heated by multiple independently controlled heating elements simultaneously Each zone is equipped with an RTD for temperature measurement Heating in the presence of RTD bias in such temperature sensitive cases inevitably causes processing defects and reduces wafer yield To maintain temperature. .. independently controlled heating elements is described in [16, 17] Each zone has a resistance temperature detector (RTD) within to provide temperature measurements Heating in the presence of sensor bias in such temperature sensitive processes inevitably causes processing defects and reduces wafer yield To maintain temperature control performance, data reconciliation techniques [22, 23] are proposed in this... to industrial requests with resists that can be manipulated to realize smaller CD, there is increasingly de- 3 manding requirement on the accuracy of temperature control in thermal processing, especially in PEB [7] It has been shown that the application of an advanced thermal processing system in PEB could contribute to a reduction in CD variance by 40% [8] It has also been shown that proper PEB control. .. excellent across-plate temperature uniformity To this end, the bake-plate is typically designed into a multi- zone thermal system A state-of-the-art bake-plate with 49 independently controlled heating elements is proposed in [16, 17], where each zone has an RTD installed within to obtain real-time measurements of zone temperature A typical PEB step begins with a wafer at ambient temperature being transferred... setpoint (between 70°C and 150°C and recipe specific) The wafer is removed immediately after being baked for a pre-specified period of time The multi- zone configuration poses a major challenge to temperature sensing The fact that readings by RTD’s can be biased is a limiting factor on temperature 4 control accuracy Bias should be estimated efficiently online to minimize the effect of inaccurate temperature sensing. .. sensing Another challenge is posed to control engineering: a computationally tractable algorithm is required for real-time temperature control of a multi- zone bake-plate which is then able to reject in a timely manner the disturbance caused by the wafer 1.1.1 RTD Bias Estimation Thermal processing of semiconductor wafers is common and critical in semiconductor manufacturing [18, 19, 20] The most temperature. .. [33] In this case, feedforward control will not be able to eliminate the temperature disturbance completely and feedback control will still be necessary Work on bake-plate temperature control can be found in [34, 35, 36] In [35], PI was used as the feedback controller, while the more sophisticated MPC and LQG controllers were used in [34] and [36] respectively MPC operates by solving a constrained... energy balance and heat transfer in a distributed lumped parameter model An analogy can be made between the thermal system and the electric network shown in Figure 2.2 In parallel with 18 1 i 1 N i N Figure 2.1: A schema of an N zone bake-plate: side view and slant view The bake-plate has radially distributed zones Each zone contains inside itself an RTD for temperature sensing and an individual resistive . sensitive to thermal processing temperature, requirement is in- creasingly stringent on temperature sensing and control in multi- zone semiconductor thermal processing. Resistance Temperature Detectors. TEMPERATURE SENSING AND CONTROL IN MULTI- ZONE SEMICONDUCTOR THERMAL PROCESSING YAN HAN (B.Eng., SJTU) A THESIS. of temperature control in thermal processing, especially in PEB [7]. It has been shown that the application of an advanced thermal processing system in PEB could contribute to a reduction in