Inter-batch control (RtR) is an effective algorithm for semiconductor wafer production process control. However, due to the limitation of measurement means and cost, it is difficult to detect wafer quality data in real time, that is, there is a certain measurement delay, which is usually random and time-varying, and directly affects the performance of batch controller.
Therefore, based on the exponential weighted moving average (EWMA) algorithm, a disturbance estimation method with random measurement delay is proposed in this paper. Based on the analysis of the measurement probability, the perturbation estimation expression including the measurement delay probability is established, and the expected maximization (EM) algorithm is used to estimate the probability of the measurement delay. Then, the static errors which may exist in the system are analyzed, and the corresponding compensation algorithm is given. Finally, the stability of the system is discussed. The effectiveness of the proposed algorithm is verified by simulation examples. Semiconductor manufacturing is the pillar industry of the national economy. Its rapid development has a far-reaching impact on all walks of life. Semiconductor wafer production process includes chemical deposition, photolithography, etching, sputtering and other steps [1], each step is inseparable from quality monitoring and control. In semiconductor wafer production process, run-to-run control (RtR) developed from statistical process control (SPC) and advanced process control (APC) is widely used. It can adjust the disturbance term of RtR in time according to the quality information of the previous batch of wafers, improve the performance of the control system and optimize the quality of the next batch of products. Exponential Weighted Moving Average (EWMA) algorithm [3] is a commonly used algorithm in batch control, but in the face of drift disturbance (such as machine aging), EWMA algorithm is prone to produce deviation, if there is measurement delay, the deviation is more obvious [4-5]. In wafer processing, off-line measurement of wafer quality is the main cause of measurement delay [6]. Because of the measurement delay, the wafer quality information in the current batch can not be used to estimate the disturbance in the next batch in time, which will not only affect the product yield, but also affect the stability of the RtR controller. Some scholars have studied the linear and non-linear systems of the content measurement delay extensively, and proposed the Gauss filtering method [7], the particle filtering method [8], and the method of estimating the unknown constant probability using the maximum likelihood criterion [9]. For RtR system, some scholars have deeply analyzed the relationship between fixed measurement delay and EWMA algorithm performance and system stability [10-12].
Some scholars have proposed the inter-batch control algorithm [13-14] under the framework of output disturbance observer (ODOB). By setting the measurement delay and filter parameters in the filter, the disadvantages of measurement delay to the system can be overcome. Influence. Aiming at the situation that the measurement delay is random and time-varying [15-16], when the measurement delay satisfies the Markov chain condition, based on Markov chain theory, some scholars have deduced the probability of each batch measurement delay [17]. On this basis, the EWMA algorithm is optimized [18], but this method requires that the time delay is independent of the input/output of the system and cannot be based on the system. The actual output adjusts the probability estimate in time. Some scholars also proposed an online estimation method of measurement delay based on Bayesian statistical method, and gave a design method of dEWMA controller based on measurement delay [19]. Based on the analysis of the effect of measurement delay estimation on system performance and the actual output of the system, the random delay probability of the system is estimated by using the expectation maximization (EM) algorithm [20-23]. On this basis, a new disturbance estimation method is proposed to improve the performance of EWMA algorithm. In the formula, u (t) and Y (t) are the inputs and outputs of the t batch, beta is the process gain, alpha is the intercept term, and_ (t) is the time-varying disturbance term. In the formula, T is the set value, B is the estimate of gain beta and_is the discount factor of EWMA. Due to the constraints of production cost and measurement technology, there will always be a certain measurement delay in the actual production process, which will affect the performance of RtR controller, and then affect the quality of wafer. In batch t, the system measurement delay is_(t). Usually the measurement delay is random and time-varying, that is to say, _(t) is not necessarily equal to_(t-1). As shown in Figure 1. Let the wafer quality data measured in batch t be y (t? _ (t), but if the output y (t? _ (t) = y (t? 1? _ (t? 1) of the new system is not measured in time, then y (t? _ (t) = y (t? 1? _ (t? 1) is taken as the wafer quality data in batch t. The measurement delay in the system is assumed as follows. Suppose 1D <=N*, so that t <=N* has_(t)<=D. D is the maximum measurement delay of the system. Where N* is a positive integer set. Define the t batch, the probability of measurement delay_(t)=d is p(d)=P(_(t)=d), D <{1,… D}, then there is, and P D (t) is defined as the estimate of P (d) in the t batch. From the above formula, if we know the estimation of PD (t) of the measurement delay probability in the t batch, we can estimate the perturbation term ^ a (t) of the T batch system, improve the performance of EWMA algorithm, and then improve the quality of wafers to be processed. In the following section, the estimation of measurement delay probability PD (t) is described. The definition denotes the estimation of the measurement delay D in batch t, and the output set of the system whose measurement delay is d according to the estimation of the measurement delay ^ D. The system output y (t) is divided into the set, and N? D (t) represents the quantity of quality data whose measurement delay is d in the set of t batches. EM algorithm includes two steps: E-step and M-step. The optimal solution of the objective function (11) is solved by alternating iterations of these two steps. In the t batch, the initial value f(0)(t)=[~p(0)~u(0)~(0)] and the initial value l=0 of iteration times are defined and iterated successively. .
At the end of the iteration, the number of iterations is the estimated value of the parameters, and the probability estimation vector of the measurement delay of the t-batch system is the probability estimation of the measurement delay of the t-batch system, and the introduction equation (6) estimates the system disturbance. And D takes the position of the largest element for the classification and calculation of equation (8). In addition, let the system produce N batches in total. The ratio of the quantity of quality data to the total number of production batches is defined as the final estimation of the measurement delay probability p (d). For closed-loop systems consisting of equation (1) (6) (4), if the output of the system converges asymptotically to the target value, the system is considered stable. Among them, Zeta=beta/b is the model mismatch coefficient. To meet the requirements of Definition 1, the proof is detailed in Appendix B. The stability conditions are given by Theorem 1. Theorem 1 is stable if the model mismatch coefficient of the system and the parameter_of the EWMA algorithm satisfy the assumption 1, and the system controlled by the perturbation estimator (20) and equation (4) is stable. In order to verify the effectiveness of the proposed algorithm, numerical simulation will be used in this section, and compared with the methods of estimating measurement delay probability based on Markov chain theory, COM-EWMA-RtR method and dEWMA algorithm based on online delay estimation in the literature. The details are as follows. Without losing generality, let the target value of the system be the gain of model matched noise as shown in equation (17). The parameters of IMA (1,1) of noise are as follows: the slope of the first 20 batches of drive disturbance is delta=? 0.25 of the last 80 batches. The maximum measurement delay of the system is set and the average probability of the measurement delay is 1/6. Among them, N is the number of production batches. The parameters of EM-EWMA are set, in which the quality data is taken from the probability estimation of the discount factor measurement delay, thermostatic element and the convergence condition is used to estimate the probability of each measurement delay as shown in Figure 2. The red solid line in each subgraph is the set measurement delay probability p(d),; the blue dotted line is the estimation of measurement delay probability in each batch of this algorithm, and the black dotted line represents the final measurement delay probability estimation of this algorithm. It can be seen from the figure that the probability estimation of measurement delay in each batch varies greatly with the proposed algorithm (EMEWMA), and the final probability estimation of each measurement delay is close to the set value. The result of probability estimation is brought into equation (20), and the disturbance estimation and system output are obtained as shown in figure 3.
Fig. 3 (a) is the output of the system; Fig. 3 (b) is the noise tracking, where the red solid line is the noise and the blue solid line with fork is the noise tracking obtained by Formula 20. The disturbance estimation can track the disturbance in time, so the MSE is smaller. In addition, under different drift disturbances and model mismatch coefficients, the control effect of the proposed algorithm is shown in Table 1.
Case 2-4 is within the stability adequacy condition given by Theorem 1, case 1 and case 5 are outside the upper and lower bounds of the given stability adequacy condition respectively. The system output for the third and fifth cases is shown in Figure 4. Let the system target value, gain, model matching and the IMA (1,1) parameter of noise_(t) be the slope of drive. Let EM-EWMA take quality data, convergence condition and discount factor. The final estimation result of delay probability measured by this algorithm is compared with that of Markov chain theory algorithm (literature [17]) as shown in Fig. 5. Among them, the measured probability of red solid line is p (d); the estimated probability of blue dotted line is Markov chain algorithm (literature [17]), expressed by pMa (d); and the final estimated probability of black dotted line is pEM (d). In the figure, except for the estimation of P (1) and P (4), the Markov chain method (blue dotted line) used in reference [17] is more accurate. The final estimation of the measurement delay probability by the proposed algorithm (black dotted line) is better than that by Markov chain method. When the measurement delay is large, the advantage is more obvious, such as the comparison of the estimation of P (7) and P (8). That is to say, in estimating the probability of measurement delay, especially when the measurement delay is large, the algorithm in this paper has more advantages than the Markov chain method in reference [17]. Using the simulation conditions in reference [18], model matching, the parameters of IMA (1,1) in noise_(t) are taken as the maximum time delay of the system. Reference [18] considers that there may be unmeasured outputs in the system, and assumes that the probability of random measurement delay is that all products are measured, but the measurement delay is larger. Therefore, the probability of unmeasured measurement mentioned in reference [18] is regarded as the measurement delay of 4 to be estimated, and a quality data is obtained. The result of convergence condition is shown in Table 2. For each parameter in Section 4.2, the control results of COM-EWMA-RtR algorithm and this algorithm are shown in Figure 6. Fig. 6 (a) is the control effect of COM-EWMA-RtR algorithm in reference [18]; Fig. 6 (b) is the control effect of this algorithm.
By using the method of reference [19], the rolling window is selected to obtain the control effect of this method and the effect of this algorithm is shown in Figure 8. From Figure 8, it can be seen that the method in reference [19] has large output deviation in batches with inaccurate time delay estimation, especially in time-delay numerical transformation, MSE is relatively large. However, using the proposed algorithm, the weighted mean of the estimation can improve the control effect. Aiming at the random measurement delay in semiconductor wafer production process, EM algorithm is used to estimate the probability of measurement delay, which is integrated into EWMA algorithm. On this basis, the static errors that may be included in the compensated output are analyzed, and the perturbation estimation algorithm that can deal with the random measurement delay is obtained, which improves the performance of the EWMA control algorithm. Secondly, the sufficient conditions for the stability of the system are given. Finally, a simulation example is given to verify the effectiveness of the algorithm.