KR20020050015A - Control of wafer temperature uniformity in rapid thermal processing using an optimal iterative learning control technique - Google Patents

Abstract

PURPOSE: A method for controlling the temperature uniformity of a wafer is provided to guarantee the temperature uniformity, by using an optimum input for reducing temperature variations in various positions inside the wafer. CONSTITUTION: A recognizing unit forms a time-varying linear space model from an input/output variable in a previous process. An estimating unit uses the time-varying linear space model made by the recognizing unit and obtains an output error from now on by using a reference output locus, a previous entire input/output variable and a whole input/output variable until the present. A calculating unit makes an object function by using the output error and calculates an input variable locus for minimizing the object function. A process control unit controls a target process according to the input variable locus obtained by the calculating unit.

Description

Control of wafer temperature uniformity in rapid thermal processing using an optimal iterative learning control technique

The present invention relates to an optimal learning control technique for securing the temperature uniformity of a wafer in an RTP apparatus. More specifically, a simple and effective time-varying linear state space model recognition method for an RTP system and an optimal iteration based on the model The present invention relates to an RTP device control method capable of learning control and feedback control.

RTP is a process of processing wafers independently, one by one, using a laser source or halogen lamp arranged in a concentric manner to raise the required temperature in a very short time and then thermal oxidation, nitriding, heat treatment, chemical vapor deposition, etc. The process is designed to be carried out in one device. RTP devices are much smaller than other devices and can be used to reduce the defect rate, making them an economical process, and have advantages such as reduced heat consumption, improved uniformity and cluster compatibility. Despite these outstanding advantages, the widespread application of RTP devices has been delayed because it is possible to raise and maintain the temperature of the entire surface of the wafer uniformly in a short time, and to ensure the uniformity of the temperature distribution to the required level. . With the recent increase in the standard size of wafers from 6 inches to 8 inches, back to 12 inches, the need for proper temperature control is even more urgent.

In order to solve this problem of RTP, many control methods have been studied, but most of the studies have been conducted to improve the existing continuous process control techniques and show limitations in improving control performance. In the early stages of the RTP system control study, a multi-loop PID controller was used, which did not yield satisfactory results due to the strong interaction of the system. Interest has since shifted to model-based multivariate control technologies, including Model Based Predictive Control (MPC), Linear Quadratic Gaussian Control (LQG), and Internal Model Control (IMC). Many studies have used a basic form of iterative learning control, but have not dealt with the spatial distribution of temperature. The performance of the model-based feedback controller is inevitably dependent on the quality of the model. Since RTP is contaminated with chamber walls as the operation is repeated, the characteristics of the RTP system are different. Therefore, the performance of the existing model-based approach is limited unless the model is continuously changed. Also, the traditional integral control method of reducing model error through feedback is not effective in RTP systems.

The present invention provides a method for constructing a time-varying linear state space model for a model-based controller for an RTP system, copes with disturbances generated in real time based on the state space model, and repeated disturbances. Even in the presence of significant model errors, precise tracking control is achieved to provide optimal learning control methods to ensure wafer temperature uniformity.

1 is a block diagram illustrating a temperature uniformity control system of a wafer using an optimal learning control technique according to an exemplary embodiment of the present invention.

2 is a schematic diagram showing an RTP device according to an embodiment of the present invention;

3 is a graph showing a radial temperature distribution of a wafer showing temperature uniformity control of the wafer after 10 runs using the controller of the present invention in the RTP process,

4 is a table comparing and using the controller of the present invention in the RTP process, and the maximum error after 10 runs of the controller using a standard linear secondary Gaussian method,

FIG. 5 is a graph showing the temperature trend of a wafer in the RTP process in a batch that follows a temperature trajectory while minimizing a temperature difference from a set temperature trajectory by learning a process model as the batch progresses using the controller of the present invention. ,

FIG. 6 illustrates the process of learning the process model as the ash progresses using the controller of the present invention in the RTP process, calculating the optimum control output and applying the process to the process so that the temperature of the wafer follows the set temperature trajectory. Graph showing control output.

In order to solve this problem, the present invention uses a recognition means for constructing a time-varying linear state space model for a target process from input / output variable values in a previous process batch, and a reference output trajectory using a time-varying linear state space model configured by the recognition means. Prediction means for predicting an output error in all sections of the current batch based on total input / output variable values of the previous batch, input / output variable values up to the current time of the current batch, and disturbances, output after the current time obtained by the prediction means Optimal learning control for an RTP apparatus including calculation means for making an error function as an objective function, minimizing the objective function, and a process control means for controlling a target process according to the input variable trajectory obtained by the calculation means. Arrange the system.

At this time, the recognition means divides the entire operating temperature range into a finite subrange and finds a linear state space model using a subspace model recognition method at the representative temperature of each subrange, and state space models for different temperature ranges. Converting them to a state space model having the same state basis with respect to the input / output coordinates, and constructing a time-varying linear state space model by interpolating the transformed state space model to have the same state basis according to the time zone temperature relationship of the reference trajectory. Preferably, the process control means repeats the process of controlling the current target process with the first input variable value among the input variable trajectories that minimize the objective function at every sampling time point.

This system uses a first step in constructing a time varying linear state space model for the target process, using the time varying linear state space model, and includes a reference output trajectory, the values of all input and output variables of the previous batch, and the current time of the current batch. And a second step of predicting an output error in the entire section of the current batch based on the statistical characteristics of the disturbance, and minimizing the objective function by using the output error after the current time obtained as a function of the input value in the second step as the objective function. A control method includes a third step of obtaining an input variable trajectory, and a fourth step of controlling a target process according to the input variable trajectory obtained in the third step.

In this case, the second step may include a process of dividing the disturbance into measurement noise and repeated noise, and expressing the expanded state space equation in the form of distinguishing the two noises.

Next, a method of controlling temperature uniformity of a wafer using an optimal learning control technique in a rapid heat treatment apparatus according to an exemplary embodiment of the present invention will be described with reference to the accompanying drawings.

1 is a block diagram showing a temperature uniformity control system of a wafer using an optimal learning control technique according to an embodiment of the present invention, Figure 2 is a schematic diagram showing an RTP apparatus according to an embodiment of the present invention.

As shown in Fig. 1, the control system of the present invention is capable of producing a product having a temperature uniformity by controlling the RTP apparatus 20 as a target process. As shown in Figure 2, the RTP device 20, the raw material wafer is introduced into the RTP device 20 and placed on the rotating shaft support in the chamber, radiates light with a concentric halogen lamp, the temperature of the wafer rises and maintains, A temperature measuring device, such as a pyrometer, measures the temperature of each part of the wafer while performing a control operation with the above-described control system to precisely control the temperature to obtain a uniform product.

At this time, the target process is a discrete time batch process having N sampling points for a limited time interval ([1, ..., N]), with N input variables and Ny controlled variables controlled simultaneously at every sampling time point. It is composed.

When the wafer is inserted into the RTP device once, the same temperature raising and lowering operation is repeated. When the first wafer is inserted and the product is discharged, the first batch, the optimal learning control system is the first in the second batch. The information obtained from the batch and the information generated during the present secondary batch process are used together to perform more precise control to control the temperature of the wafer.

For example, the deviation from the set temperature trajectory is calculated using the output variable y ₁ (t), which is the actual temperature of the wafer for each time in the first batch, and the control variable u ₁ (t) is determined to determine the actual temperature along the temperature set trajectory. To control the power of the heating device, which is the input variable.

In the second batch, input the current time by using quadratic programming consisting of the output variables y ₂ (t) up to the current time point and the input / output variables y ₁ (t) and u ₁ (t) of the _first batch. Compute the variable u ₂ (t).

In the third batch, the output variable before the current point in time and the input / output variables of the batch just before the current batch are used to calculate the input variable at the present time, but the input variable of the second batch contains the information of the first batch. Therefore, the calculation of the input variables of the third batch includes all the information of the past batch.

The present invention utilizes a process model function that formulates the target process, and based on the total I / O variable values of the previous batch and the I / O variables and disturbances up to the current time of the current batch, The predictor 42 predicts the error, and the predictive value of the error is transformed into an objective function to obtain an input variable that minimizes the error. The calculation means 43 satisfies various constraints of the process variable and calculates the objective function. Optimized control input variation to minimize ), And add the previous batch's input to this value to track the best input variable trajectory since the current time ( ) Here the objective function is calculated by quadratic programming. Only the first value of the optimum input variable trajectory is applied to the RTP process (20), and the control is continued after the above process again at the next sampling point. As the batch progresses, the temperature error is continuously reduced to ensure the temperature uniformity of the wafer. It becomes possible.

This explains the specific implementation method.

Consider a discrete time batch iterative process with Nu inputs and Ny outputs, with N sampling points.

In order to apply the proposed control technique, we need a time-varying linear state space model that is recognized along the set point trajectory. In order to obtain a time varying linear state space model, the present invention employs a technique for interpolating time-invariant state space models recognized at two or more operating points. This technique is very useful for RTP systems because it is easy to obtain multiple time-varying models for different temperature ranges. The overall steps are as follows:

Step 1: Divide the full range of operating temperatures into a finite number of subranges. With system recognition at the representative temperature of each subrange, we find a linear state space model using the subspace model recognition method (N4SID).

Step 2: Convert the state space models for different temperature ranges to a state space model with the same state basis for input and output coordinates. Since they have the same state basis, they can be flexibly switched to different state space models without jumping.

Step 3: Construct a time-variable linear state space model by interpolating the transformed models according to the time-to-temperature relationship of the setpoint trajectory.

In step 1, the following statistical state space model is obtained in a balanced form through subspace model recognition method (N4SID).

Where u is the lamp power (Watt), y is the temperature measurement (° K), x is the system state and v is the measurement noise. The partial space recognition method (N4SID) is a partial state space recognition technology that uses the input and output data obtained through the ring-opening experiment to find a state space model with a balanced minimum order. State-space models obtained at different temperature ranges have different state basis for input and output and therefore need to be transformed to have the same state basis. For this reason, a similar transformation of step 2 is necessary and the following is the transformation method.

In step 2, if the current time is t, then, for the current time t, past input and output vectors and future input and output vectors can be defined as follows.

here Is assumed to be large enough for the state order n.

In addition, the future output is represented by a linear combination of the current state and the future input.

here Is the remaining term, and O is the extended observability matrix defined as

Also according to the Kalman filter equation, the current state is represented by a linear combination of past input and past output.

Thus, the future output can be expressed as

When data is obtained through open-loop experiments, future errors are not correlated with other regressors, so the least-squares method , , Can be obtained. For the least squares estimation, assuming that we have s data, we can define the following vector:

Then, Equation 6 may be expressed by the following equation.

Thus, the following equation is derived.

OX can be determined through LP's Singular Value Decomposition, in that the choice of states can be arbitrary under similar transformations, and that the only condition is that the range of OX and LP should be the same. . The following is an outlier decomposition equation.

Since the minimum system order n is equal to the rank of LP, it can be chosen as the number of diagonal terms in S. In addition, the extended observability matrix O and state X can be determined as follows.

The LP matrix, obtained from data sets for different nominal temperatures, has the same basis because it is defined as the actual input and output values. State X, on the other hand, can be defined as a different basis at different nominal temperatures. If the state orders are the same, two nominal temperatures can be expressed as

Wow Has the same basis, so there are similar transformations between the two states:

therefore( Let) be the system matrix recognized at the i-th nominal temperature. then ( ) Is given by

After converting the state space models for the different temperature ranges to the state space model having the same state basis for the input and output coordinates through the above process, in step 3, the models transformed according to the time versus temperature relationship of the setpoint trajectory Interpolate to construct a time-varying linear state space model.

Model system matrices can be recognized for a finite number of temperatures. To construct time-varying models using these models, additional models for several intermediate temperatures must be created. To this end, the present invention applies linear interpolation.

and Is called the neighboring temperature, Let be the basis of the adjusted system matrix. Also To and Assume that the temperature satisfies the following equation.

The frequency gain from the input to the state The parallel system is Appears. Temperature, interpolating these two gains linearly The system matrix in can be obtained by knowing the relation of the following two frequency gains.

The above equation is valid in the selected frequency range and the interpolation matrix can be determined by the least-squares method.

By multiplying before and after the equation, the above equation is summarized as follows.

here, Wow Represents a real matrix. The above equation is summarized as follows.

Is directly obtained by the first and second equations. Also, Can be obtained by the least square method using the third and fourth equations.

Therefore, through the above process, the time-varying linear state space model can be obtained through interpolation.

Next, a process of performing optimal iterative learning control using the time-varying linear state space model obtained above will be described.

In a batch process or a repetitive operation process, the input trajectory for the currently running ash is corrected using information from past ashes as well as information measured in the current ash. Optimal iterative learning control refers to an input correction algorithm that optimally modifies the input trajectory using the measurements of past batch operations.

In order for the controller to use the error information in the previous batch, it is necessary to associate the information on the disturbance in the previous batch with the information on the disturbance in the next batch in a time-varying state space equation. The first step to this is to introduce a reasonable assumption about the nature of disturbance and to build a model for the transition from one batch to the next based on that assumption. In the following, Indicates an ash index.

Measurement noise in time-varying state-space equation Is a sequence that is independent of the time and is uniformly distributed, but in different batches Shows an important correlation. In fact, for the ash index k In addition to the random distribution shows the continuous and drift characteristics. Such measurement noise ( ) Shows dynamic characteristics ( ), And showing drifting characteristics ) Can be expressed as

That is, here Wow Is a random sequence of independent mean zeros. I.e. bar Is the error term that continues to occur in all future batches, Is the missing error term.

Now, the time-varying state space expression Wow With one part being pushed into, It can be divided into another part that is promoted by.

Of course, the measurement variable Satisfies

Also, When the set trajectory of is expressed as r (t), the error is expressed by the following equation.

To convert the above equation into a standard form so that the external noise is an independent sequence for t and k, we can derive the following equation by taking the difference between the two equations between the subsequent batches.

In addition, by equation (21) Can be expressed as follows.

To include the term, first Define as

Then, the following equation is obtained from the above equation.

Now, as follows Define.

According to the definition above, bar The t + 1th element of Same as Now, in state space 21, 24 If you include, you get the following expanded state-space expression:

Where H (t) is Matrix to extract the t + 1th element of. Therefore, the following equation is satisfied.

The transition from one batch to the next is represented by the following equation.

This method describes how to calculate input variables using this state-space equation.

The lamp power, an input variable to ensure temperature uniformity of the wafer, suppresses the movement of the input variable if possible, and minimizes the tracking error of the entire batch of controlled variables as follows. Is determined to minimize.

Here, M, Q, and R represent weighting matrices, and their selection can tune the performance of the system. The objective function above can also be configured to include constraints on input variables and controlled variables. The optimal solution of the standard form can be found, which is estimated by the Kalman filter equation and the linear quadratic regulator (LQR). Given by the combination of.

Here, the estimated state value is obtained as follows.

This is the standard optimal solution every sampling time, i.e. every time a measurement is made. Only the first term value of is applied to the process. Thus, as the batch is repeated, an optimum input is applied to the process that reduces the temperature deviations of the various points in the wafer, thereby ensuring temperature uniformity of the wafer.

Figure 3 is a graph showing the radial temperature distribution of the wafer after 10 runs using the controller of the present invention in an RTP process, Figure 4 is a controller using the present invention and also using a standard linear secondary Gaussian method. This is a table showing the maximum error in the radial direction after 10 cycles of operation compared with the controller.

Referring to FIG. 3, it can be seen that the temperature variation in the radial direction of the wafer is not evenly distributed, and FIG. 4 shows that the controller using the present invention shows much better performance than the controller using the standard linear secondary Gaussian method. can see.

FIG. 5 is a graph illustrating a temperature trend of a wafer showing a process of optimally learning a process model as the batch progresses in an RTP process and minimizing errors by following a set output trajectory. In the RTP process, as the ash progresses, the process model is learned to calculate the power supplied to the heating lamp, which is the optimum control output, to show the temperature change of the optimum wafer of FIG. .

Referring to FIG. 5, the second batch process trajectory is considerably deviated from the reference trajectory, but as the batch is repeated, the second batch process trajectory is corrected closer to the reference setting trajectory.

The present invention provides a system for ensuring temperature uniformity of wafers by using optimal learning control, which is one of the advanced control techniques for the RTP process. The system provided provides a method for constructing a time-varying process model, and the optimum learning control function feeds back process variables in batches to perform more precise control as the ash is repeated despite significant model errors. It has a structure to ensure temperature uniformity. It also has the ability to respond to various disturbances. In addition, the provided control system is configured to satisfy various constraints of the process variable and to optimally satisfy the performance index of the set objective function.

Claims (7)

Recognition means for constructing a time-varying linear state space model for the target process from input / output variable values in the previous process batch, Prediction means for calculating an output error after a current time using a time-varying linear state-space model configured by the recognition means, and obtaining an output error after the current time from a reference output trajectory, all input / output variable values of the previous batch, and all input / output variable values up to the current time of the current batch; Calculation means for making an output error after the current time obtained by the prediction means into an objective function and obtaining an input variable trajectory that minimizes the objective function; Process control means for controlling the target process according to the input variable trajectory obtained by the calculation means Optimal learning control system for RTP device comprising a. In claim 1, The recognition means Dividing the entire operating temperature range into finite subranges and finding a linear state space model using subspace model recognition at the representative temperature of each subrange, Converting the state space models for different temperature ranges into a state space model having the same state basis for input and output coordinates, And a step of constructing a time-varying linear state space model by interpolating the transformed state space model to have the same state basis according to the time zone temperature relationship of the reference trajectory. In claim 2, And the process control means repeats the process of controlling the current target process with the first input variable value among the input variable trajectories that minimize the objective function at every sampling time point. A first step in constructing a time varying linear state space model for the target process, The output error of the current batch section is calculated by using the time-varying linear state space model and based on the reference output trajectory, the total input / output variable value of the previous batch, the input / output variable up to the current time of the current batch, and the statistical characteristics of the disturbance. Step 2, A third step of obtaining an input variable trajectory for minimizing the objective function by using an output error after a current time obtained as a function of an input value as the objective function in the second step, A fourth step of controlling the target process according to the input variable trajectory obtained in the third step Optimal learning control method of the RTP device comprising a. In claim 4, In the second step, the disturbance is divided into measurement noise and repetitive noise, and the extended learning of a state space equation in the form of dividing the two noises comprises the step of optimal learning control method of the RTP device. In claim 4, The first step is Dividing the entire operating temperature range into finite subranges and finding a linear state space model using subspace model recognition at the representative temperature of each subrange, Converting the state space models for different temperature ranges into a state space model having the same state basis for input and output coordinates, And interpolating the transformed state space model to have the same state basis according to the time-zone temperature relationship of the reference trajectory to construct a time-varying linear state space model. In claim 4, In the fourth step, the optimal learning control method for the RTP device to repeat the process of controlling the current target process with the first input variable value of the input variable trajectory that minimizes the objective function at every sampling time point.