JP6820902B2 - Plasma processing equipment, data processing equipment and data processing method - Google Patents

Description

The present invention relates to a data processing device and a data processing method for a series of numerical data of a target system (device, analytical data, etc.).

Various models have been proposed as techniques for data smoothing and data prediction of a series of numerical data in the target system. In addition, for the purpose of grasping the state of the target system, in order to detect the extreme value (maximum value, minimum value) or inflection point that is the change point of the data, the first derivative, second derivative, etc. of the data Differentiation processing is often used. In particular, in the case of time-series data such as measurement data including noise, etc., the change point in the target system is detected by performing data smoothing processing and differential processing with high accuracy, and control of the target system is performed. Is an important technical issue.

Conventional data smoothing and data prediction techniques include a curve fitting method, a moving average method, and the like as described in Non-Patent Document 1. As a curve fitting method, there is a polynomial fitting method (Savitzky-Golay method) as described in Patent Document 1. Further, as a digital filter, there is a Butterworth type low-pass filter or the like. Further, the moving average method includes an exponential smoothing method and the like as described in Non-Patent Document 2.

Non-Patent Document 2 discloses a one-step exponential smoothing method (one smoothing parameter) and a two-step exponential smoothing method (two smoothing parameters), and is related to economic relations such as supply and demand forecasting. It is used in the field of. Further, Non-Patent Document 3 discloses a response type one-stage exponential smoothing method in which a smoothing coefficient changes according to input data (particularly sequential input data). Cases in which the smoothing coefficient changes according to | relative error / absolute error | and cases in which the smoothing coefficient changes according to the logistic function are described. In general, there are many cases where the smoothing coefficient shown in Eqs. (1) to (4) changes according to | relative error / absolute error |.

Data smoothing: S1 _{t + 1} = α1 _t Y1 _t + (1-α1 _t ) S1 _t Equation (1)
Smoothing coefficient: α1 _t = ｜ δα _t / Δα _t ｜ Equation (2)
Relative _{error: δα t = A1 (Y1 t} - S1 t) + (1 - A1) δα t-1 Equation (3)
Absolute error: Δα _t = A1 | Y1 _t --S1 _t | + (1 --A1) Δα _t-1 Equation (4)
Here, the input data is, for example, time series data Y1 _t : t = 1, 2, ..., And the predicted value of smoothing of the output data one period ahead is S _{t + 1} . A1 is an arbitrary constant. Although each symbol is different from Non-Patent Document 3, it is described so as to be the same as the symbol in the following examples as much as possible.

The finite difference method is often used in the conventional data differentiation processing technology. Further, as described in Patent Document 1 and Non-Patent Document 4, the polynomial adaptation method (Savitzky-Golay method) may be used. Usually, the polynomial adaptation method (Savitzky-Golay method) uses a plurality of consecutive data and often derives the differential processing result at the center point of the period of the continuous data. However, Non-Patent Document 4 discloses a method of performing data smoothing processing and first-order differential processing by a polynomial matching method (Savitzky-Golay method) at any point of the data to be used.

Patent Document 2 is an example of controlling a target system by performing data smoothing processing on time-series data such as measurement data including noise, detecting a change point of the data by primary differential processing, and secondary differential processing. Disclosed in "A method of smoothing spectral intensity signal data from plasma emission in a plasma etching processing apparatus by moving average processing and determining an etching end point based on a primary differential value and a secondary differential value". Further, Non-Patent Document 3 discloses a method of estimating the optimum smoothing parameter by minimizing the sum of the errors of the predicted values in the first period in the one-step exponential smoothing method.

Japanese Unexamined Patent Publication No. 2000-228397 Japanese Unexamined Patent Publication No. 61-53728

"The Behind the Scenes of Data Processing" by Kazuhiro Takahashi, Journal of Surface Analysis Vol7.No 1 (2000) p68-p77. "Introduction to Time Series Models" by A.C. Harvey, translated by Naoto Kunitomo / Taku Yamamoto, (1985), p173, University of Tokyo Press, (Original: TIME SERIES MODELS by A.C. Harvey) James W Taylor, Journal of Forecasting, 2004, (23), pp385-394. Peter A. Gorry, Anal. Chem. 1990, (62), pp570-573.

In the case of data smoothing processing that removes noise from data in which noise is added to the true signal (signal) such as measurement data, increasing the S / N ratio (signal / noise ratio) is one of the technical issues. Is. Even in the case of data differentiation processing, increasing the S / N ratio (signal / noise ratio) is one of the technical issues.

For example, if the data smoothing processing is not sufficient when the data differentiation processing is performed by the difference method as described in Patent Document 1, the output result of the first-order differentiation processing contains a lot of noise and is smooth (smooth) data. Instead, the S / N ratio (signal / noise ratio) becomes smaller. When the data differentiation processing is performed again by the difference method using the above data, the output result of the second-order differentiation processing further increases noise, is not smooth data, and further reduces the S / N ratio. That is, it is a technical problem that it is difficult to increase the S / N ratio (signal / noise ratio, hereinafter referred to as S / N ratio) as the data first-order differentiation processing and the data higher-order differentiation processing are performed rather than the data smoothing processing. It becomes.

Data processing includes a method of performing processing after acquiring all data (offline processing) and a method of performing processing while acquiring data (online processing or sequential data processing). In the case of online processing, the data processing result is often used in real time, for example, for controlling a process processing device. Therefore, one of the technical problems is that the delay time of the data processing result (the amount of time lag from the true processing result) is small.

If the difference interval is increased by the data differentiation process by the difference method, the output result of the first derivative process and the output result of the second derivative process will be smooth data, and the S / N ratio will increase (however, in the difference interval). Has an optimum value), and in the case of online processing of time-series data, there is a problem that the data delay time due to data differentiation processing increases. Further, in a low-pass filter or the like, when the cutoff frequency is reduced, the output data of the first-order differential processing and the output data of the second-order differential processing become smooth data, and the S / N ratio increases (however, the cutoff). There is an optimum value for the frequency), and there is a problem that the data delay time due to the data differentiation processing increases as described above.

Further, when data differentiation processing is performed by the polynomial adaptation method (Savitzky-Golay method), it is common to require a plurality of consecutive data and derive a differential value at the time at the center of the data period. Therefore, in the case of sequential data processing, there is a problem that a time delay occurs in principle by at least the time difference from the time of the latest data to the data at the center. When the differential value is derived when the latest data is acquired by the polynomial matching method (Savitzky-Golay method) described in Non-Patent Document 4, the differential value at the data center point is derived by the ordinary polynomial matching method (Savitzky-Golay method). The data delay time is reduced as compared with the case where the above is derived, but on the other hand, the S / N ratio is reduced.

Therefore, as the number of data used increases, the S / N ratio increases and improves. However, on the other hand, there is a problem that the responsiveness to a sudden change deteriorates due to the use of a large number of data, and the responsiveness is limited depending on the number of data used. Specifically, there is a problem that the high frequency component contained in the true signal is missing in the signal after data processing.

The Savitzky-Golay method fits a plurality of data into a quadratic curve or a cubic curve. Therefore, if the number of data is increased, the fitting accuracy is improved. However, the reason why the above-mentioned high-frequency component is missing is that it cannot cope with changes of quadratic or cubic curves or more depending on the period of the number of data.

Further, there is a method disclosed in Patent Document 2 as a method for determining the end point of plasma etching in a plasma etching processing apparatus, but the following 2 is mainly used for the method for determining the end point of plasma etching disclosed in Patent Document 2. There are two challenges.

The current semiconductor device has a high step structure such as a FinField Effect Transistor (FinFET) structure due to high performance and high integration. Further, in normal plasma etching, microloading occurs, which is a difference in etching performance between the sparse portion pattern and the dense portion pattern. Further, the film to be etched may be non-uniform in the wafer surface.

From these facts, for example, the time series data of the spectral intensity signal data from the plasma emission used for determining the end point of the plasma etching may change in two stages. In this way, when the time series data of the spectral intensity signal data from plasma emission changes in two stages and the end point of plasma etching is determined by the first or second change, the first change and the second change are short. Since it occurs in a period, the data processing cannot follow the two changes and the end point of the plasma etching cannot be detected. Here, the time point at which the time series data of the spectral intensity signal data from the plasma emission changes is set as the end point of the plasma etching.

Further, for example, even in plasma etching for a short time with respect to the delay time of the first-order differential value or the second-order differential value, the data processing for calculating the first-order differential value or the second-order differential value is the spectral intensity signal data from the plasma emission. Cannot follow changes in time series data. That is, there is a problem that the responsiveness of detecting the end point of plasma etching is insufficient.

Next, the mask patterns for plasma etching are roughly classified into groove patterns and hole patterns. Further, usually, the aperture ratio of the wafer of the hole pattern is smaller than the aperture ratio of the wafer of the groove pattern, and the aperture ratio may be less than 1%. Further, as the aperture ratio becomes smaller, the change in the spectral intensity of plasma emission becomes smaller.

Therefore, for example, in the case of a wafer having an aperture ratio of less than 1%, the change in the time series data of the spectral intensity signal data from the plasma emission is too small, and it becomes difficult to detect the end point of the plasma etching. In other words, there is a problem that it cannot cope with a low S / N ratio. Here, the aperture ratio is the ratio of the area to be etched to the area of the entire wafer.

Non-Patent Document 3 discloses a responsive one-step exponential smoothing method in which the smoothing coefficient changes according to input data (particularly sequential input data). Cases in which the smoothing coefficient changes according to | relative error / absolute error | and cases in which the smoothing coefficient changes according to the logistic function are described. In both of the above two cases, the smoothing coefficient is a data response type, but since the base is a one-stage exponential smoothing method, there is a problem that it is difficult to achieve both an S / N ratio and data responsiveness.

In addition, in the case of using the logistic function, it is presumed that the main reason is that the output suitable for the movable range of the smoothing coefficient is simply 0 to 1, and the relationship between the smoothing coefficient and the error is fully considered. I wasn't. Therefore, there is a problem that the S / N ratio performance of the data smoothing process is not always sufficient. Furthermore, the logistic function has a plurality of parameters, and there is a problem that it is difficult to set the parameters and it is not easy to use.

As described above, in the data smoothing process and the data differentiation process, there is a trade-off relationship between the improvement of the S / N ratio performance and the improvement of the data response performance (reduction of the delay time). In the data smoothing processing method and the data differentiation processing method, it is necessary to optimize the parameters of each data processing method according to the input data to be processed.

Conventionally, data processing is performed by sequentially changing data processing parameters, and a comprehensive bird's-eye view of each data smoothing processing waveform, data processing waveform such as data differential processing waveform, and numerical data such as S / N ratio and delay time. And found the optimum parameters. However, in the above method, it takes a long time to find the optimum parameter, and there is a problem that knowledge and experience of data processing are required to shorten the time to find the optimum parameter.

Further, Non-Patent Document 3 discloses a method of estimating the optimum smoothing parameter by minimizing the sum of the errors of the predicted values in the first period in the one-step exponential smoothing method. However, in this method, since the smoothness of the curve of the predicted value is not taken into consideration, the noise is large and the S / N ratio is not good in the first-order differential processing and the second-order differential processing for detecting the change point. There was a challenge.

Therefore, the present invention has a high S / N ratio and a reduction in data delay time in a plasma processing device including a data processing device for processing data, a data processing device for processing data, and a data processing method. And provide a plasma processing device, a data processing device, and a data processing method that can achieve both improvement in data responsiveness. Further, the present invention provides a plasma processing apparatus, a data processing apparatus, and a data processing method capable of easily and automatically finding the optimum parameters for data smoothing processing and data differentiation processing of input data.

The present invention is a data processing method for processing data using a responsive two-stage exponential smoothing method in which the smoothing parameter is changed according to an error between the input data and the predicted value of the smoothed data. It is characterized in that the responsiveness coefficient of the smoothing parameter is derived by using a probability density function (normal distribution type function, Gaussian distribution type function) with the error from the predicted value of the converted data as a parameter.

Further, the present invention is a data processing method for processing data by using a response type two-stage exponential smoothing method in which a smoothing parameter is changed according to an error between an input data and a predicted value of the smoothed data. It is characterized by using an exponential load transfer average of the error between and the predicted value of the smoothed data, and using a responsive addition coefficient in which the coefficient of the exponential load transfer average changes according to the slope of the smoothed data. ..

Further, according to the present invention, in a data processing method for processing data by using the two-stage exponential smoothing method, the result of performing data smoothing processing by the backward Savitzky-Golay method using a plurality of continuous input data is obtained in 2. The result of the first-order differential processing by the backward Savitzky-Golay method using the predicted values of the smoothing of a plurality of consecutive data used as the input data of the data smoothing formula of the step exponential smoothing method is 2 It is characterized in that it is used as input data of the slope of the predicted value of data smoothing in the formula of the slope of the smoothing data of the step exponential smoothing.

Further, the present invention measures the light emission data at the time of plasma processing of the sample in the plasma processing method for plasma processing the sample, and responds to the error between the measured light emission data and the predicted value of the smoothed light emission data. A probability density function (normal distribution type function, Gaussian distribution type function) that uses the response-type two-stage exponential smoothing method that changes the smoothing parameter and uses the error between the input data and the predicted value of data smoothing as a parameter. To derive the responsiveness coefficient of the smoothing parameter, or to average the error between the input data and the predicted value of smoothing of the data by exponential load transfer average, and the coefficient of the exponential load transfer average is the gradient of the smoothing data. The result of data smoothing by the backward Savitzky-Golay method using a responsive addition coefficient that changes according to the predicted value or using a plurality of continuous input data is obtained by the two-stage exponential smoothing method. The result of first-order differential processing by the receding Savitzky-Golay method using the input data of the data smoothing formula and the predicted values of the smoothing of a plurality of consecutive data is used for two-stage exponential smoothing. First-order differential processing obtained by utilizing the two-stage exponential smoothing method, which uses at least one of the smoothing data slope equations as input data for the slope of the predicted value of data smoothing. In the result or the second-order differential processing result, a change in the plasma processing state of the sample or the plasma processing state of the sample is detected based on at least one processing result, and the plasma processing of the sample is performed according to the detection result. It is characterized by controlling.

Further, in the present invention, there is an error between the data input / output device that receives the data to be processed, the storage device that stores the data received by the data input / output device data, and the input data and the predicted value of smoothing of the input data. A probability density function (normal distribution type function, Gaussian distribution type) that uses a response-type two-stage exponential smoothing method that changes the smoothing parameter according to the error and uses the error between the input data and the predicted value of data smoothing as a parameter. By (function), the responsiveness coefficient of the smoothing parameter is derived, or the error between the input data and the predicted value of data smoothing is exponentially loaded with moving average, and the coefficient of the exponentially loaded moving average is the smoothing of the data. Two-step exponential smoothing is the result of data smoothing by the backward Savitzky-Golay method using a response-type addition coefficient that changes according to the predicted value of the slope of, or using a plurality of continuous input data. The result of first-order differential processing by the backward Savitzky-Golay method using the input data of the smoothing formula of the data of the conversion method and the predicted values of the smoothing of a plurality of consecutive data is used as a two-stage index. Data processing for storing the data processing program of the two-stage exponential smoothing method using at least one of the smoothing data slope equations used as input data for the slope of the predicted value of data smoothing. It is a data processing device including a program storage device and a data calculation processing device that performs data processing based on the data processing program.

Further, the present invention changes the smoothing parameter according to an error between the processing chamber to be controlled, the measuring device that acquires the data related to the processing chamber, and the input data and the predicted value of smoothing of the input data. In addition to using the response-type two-stage exponential smoothing method, the smoothing parameter is determined by the probability density function (normal distribution type function, Gaussian distribution type function) with the error between the input data and the predicted value of data smoothing as a parameter. Derivation of the responsiveness coefficient or exponential load transfer averaging of the error between the input data and the predicted value of data smoothing, and the coefficient of the exponential load moving average depends on the predicted value of the slope of data smoothing. The result of data smoothing by the backward Savitzky-Golay method using a variable, responsive addition coefficient, or using multiple consecutive input data, is used to smooth the data by the two-stage exponential smoothing method. The result of the first-order differential processing by the receding Savitzky-Golay method using the input data of the equation and the predicted values of smoothing of a plurality of consecutive data is the inclination of the smoothing data of the two-stage exponential smoothing. A data calculation processing device that performs data processing based on the data processing program of the two-stage exponential smoothing method using at least one of the input data of the gradient of the predicted value of data smoothing in the formula of It is provided with a control device that detects a change in the state of the processing chamber or the state of the processing chamber by using the data processing result obtained by the data calculation processing device and controls the processing chamber according to the detection result. It is a processing device characterized by this.

Further, the present invention comprises a processing chamber in which a sample placed on a sample table is subjected to plasma processing, a measuring device for acquiring light emission data during plasma processing of the sample, and data and data acquired by the measuring device. A probability density function (a probability density function that uses a response-type two-stage exponential smoothing method that changes the smoothing parameter according to the error from the predicted smoothing value and uses the error between the input data and the predicted smoothing value as a parameter. Derivation of the responsiveness coefficient of the smoothing parameter by (normal distribution type function, Gaussian distribution type function), or exponential load transfer averaging of the error between the input data and the predicted value of the smoothed data, and the exponential load transfer Data smoothing is performed by the backward Savitzky-Golay method using a responsive addition coefficient in which the average coefficient changes according to the predicted value of the slope of the smoothed data, or using a plurality of consecutive input data. The result is used as the input data of the data smoothing formula of the two-stage exponential smoothing method, and further, using the predicted values of the smoothing of a plurality of consecutive data, the receding Savitzky-Golay method is used for the first-order differentiation. The two-stage exponential smoothing method, wherein the processed result is used as input data for the inclination of the predicted value of data smoothing in the expression for the slope of the smoothing data of the two-stage exponential smoothing. A change in the plasma processing state of the sample or the plasma processing state of the sample is detected by using the data processing device that performs data processing based on the data processing program and the data processing result obtained by the data calculation processing device. The processing device is provided with a control device that controls the processing chamber according to the detection result.

Further, the present invention is a data processing method in which a plurality of discrete, continuous, numerical data are used as input data, data smoothing processing or data differentiation processing is performed, and the data processing method has at least one parameter, and the input data and data smoothing processing are performed. The evaluation function is the value obtained by multiplying the average square error of the result and the square average value of the second-order differential of the data smoothing processing result by the coefficient of an arbitrary numerical value, and the evaluation function is minimized. This is a data processing method characterized by deriving the parameters of the data processing by indexing the fact.

According to the present invention, in a plasma processing device provided with a data processing device for processing data, a data processing device for processing data, and a data processing method, the S / N ratio of the data processing result is improved and the data processing result is improved. It is possible to achieve both improved responsiveness (reduction of delay time of data processing results). In addition, the optimum parameters of the data processing method can be easily and automatically found in a short time. As a result, it is possible to provide a processing device having good usability, including a data processing device, a data processing method, and a control device for controlling a processing room.

It is a figure which shows the whole structure of the data processing apparatus 1 which concerns on Example 1. FIG. It is a figure which shows the flow of the data processing which concerns on Example 1. FIG. It is a figure which shows the definition of the deviation amount of the predicted value from the mean value in random number data. It is a figure which shows the relationship between the deviation amount / standard deviation σ, and the response coefficient F _t in the data of FIG. 3A. It is a figure which shows the change example of the slope of the signal output which becomes the input of data processing. It is a vertical sectional view of the magnetic field microwave plasma etching apparatus which concerns on Example 1. FIG. It is a flow chart of typical data processing for etching end point detection. The data processing result when the response type two-stage exponential smoothing method (type IV) of equations (33) to (42) is used in the data processing apparatus 1 according to the first embodiment using the data processing flow of FIG. is there. This is the data processing result when N = 0 in the response type two-stage exponential smoothing method (type I) of equations (5) to (14). This is the data processing result when N = 1 in the response type two-stage exponential smoothing method (type I) of equations (5) to (14). This is the data processing result when N = 5 in the response type two-stage exponential smoothing method (type I) of equations (5) to (14). The input data (Y1 _t ) and the data smoothing processing result (S1 _t ) when one parameter of the type I responsive two-stage exponential smoothing method (N = 0) is sequentially changed are shown in (A). It is a figure which consists of (B) which shows the 2nd order differential processing result (B2 _t ). In the case of FIG. 11, it is a figure which showed the parameter dependence of E which is a mean square error, D which is the root mean square of the second order differential, and W which is an evaluation function. (A) showing the input data (Y1 _t ) and the data smoothing processing result (S1 _t ) when the parameter of the position marked with × automatically derived by using the evaluation function W in FIG. 12 is used, and the second order. It is a figure which consists of (B) which shows the differential processing result (B2 _t ).

Hereinafter, examples of the present invention will be described with reference to the drawings.

First, the data processing apparatus according to the first embodiment of the present invention will be described with reference to FIGS. 1 to 10. Here, an example in which the present invention is applied to etching end point detection by plasma spectroscopy for the purpose of high-precision etching processing in a magnetic field microwave plasma etching apparatus will be described.

FIG. 1 shows a configuration diagram of the data processing device 1 of the first embodiment. In this embodiment, the data processing device 1 is composed of a data input / output device 2, a data storage device 3 which is a storage device, a data processing program storage device 4, and a data calculation processing device 5, and data is transferred to each other. Is connected so that

Further, in addition to the above, a data display device (not shown) is provided as needed. The data processing device 1 can input / output data to and from the target system 6 (device, analysis data, etc.). As a result, the target system 6 is controlled with high accuracy. In the case of this embodiment, the target system 6 is a microwave plasma processing apparatus. Further, the data processing device 1 may be used alone, and can be used for data analysis and the like.

The data input / output device 2 can input / output processing data, parameters of a data processing program, and the like. The data input / output device 2 receives data to be processed collectively or sequentially from the target system 6 or the like. The data received by the data input / output device 2 is stored in the data storage device 3 such as RAM, and the data smoothing process is performed by the data calculation processing device 5 according to the data processing program stored in the data processing program storage device 4 such as RAM. Data differentiation processing is performed. After the data is calculated by the data calculation processing device 5, the data smoothing processing result data and the data differentiation processing result data are output to the target system 6 or the like by the data input / output device 2 and used for controlling the target system 6.

FIG. 2 shows an overall flow diagram of the data processing method stored in the data processing program. Receives data to be processed collectively or sequentially, and data is input. The input data was time series data Y1 _t : t = 1, 2, ... Next, the initial value is derived by the method described later. Next, the first two-stage exponential smoothing process is performed to obtain the predicted value S1 _t for smoothing the data of the first output and the predicted value B1 _t for the slope of the smoothed data.

In this case, the treatment was carried out by the type I responsive two-stage exponential smoothing processing method shown in the following equations (5) to (14).

Data smoothing: S1 _t = α1 _t Y1 _t + (1-α1 _t ) (S1 _t-1 + B1 _t-1 ) Equation (5)
Slope of smoothed data: B1 _t = γ1 _t (S1 _t --S1 _{t -1} ) + (1 -γ1 _t ) B1 _t-1 Equation (6)
Smoothing coefficient: α1 _t = (K _α -L _α ) F _α + L _α equation (7)
Response _{factor: F αt = (| δα t} / Δα t | + φ) N (8)
Relative _{error: δα t = A1 (Y1 t} - S1 t) + (1 - A1) δα t-1 Equation (9)
Absolute error: Δα _t = A1 | Y1 _t --S1 _t | + (1 --A1) Δα _t-1 + φ equation (10)
Smoothing coefficient: γ1 _t = (K _γ -L _γ ) F _γ + L _γ equation (11)
Response _{factor: F γt = (| δγ t} / Δγ t | + φ) N (12)
Relative _{error: δγ t = A2 {(S1} t - S1 t-1) - B1 t} + (1 - A2) δγ t-1 Equation (13)
Absolute _{error: Δγ t = A2 | (S1} t - S1 t-1) - B1 t | + (1 - A2) Δγ t-1 + φ Equation (14)
Here, the input data is, for example, time series data Y1 _t : t = 1, 2, ..., And the predicted value S1 _t for smoothing the output data and the predicted value B1 _{t for} the slope of the smoothed data. Can be obtained by sequential data processing. In addition, K _α , L _α , K _γ , L _γ , N, A1, A2, and φ are arbitrary constants. However, 1> K _α > L _α > 0, 1> K _γ > L _γ > 0, 1>A1> 0, 1>A2> 0. φ is the absolute error [Delta] [alpha] _t and [Delta] [gamma] _t, or response factors F _{[alpha] t} and F _{[gamma] t} is in order to avoid a zero value. In addition, a very small value is selected as φ so that the influence on the normal calculation is very small.

When N = 0 in Eqs. (8) and (12), the response coefficients F _α = 1 and F _γ = 1, and the smoothing coefficients α _t = K _α and γ _t = K _γ , which are constant values. .. Therefore, when N = 0, the usual two-stage exponential smoothing method is used. When N = 1, each smoothing coefficient is a response-type two-stage exponential smoothing method in which the range of K _{α to} L _α and K _{γ to} L _γ is changed in proportion to each relative error / absolute error. Similarly, when N = 5, each smoothing coefficient changes in the range of K _{α to} L _α and K _{γ to} L _γ in proportion to the fifth power of each relative error / absolute error. It becomes a conversion method.

The responsive two-stage exponential smoothing method improves the data smoothing performance by reducing each smoothing coefficient when each relative error / absolute error is small. On the other hand, when each relative error / absolute error is large, each smoothing coefficient is increased to improve the data response performance. The data smoothing performance and the data response performance are in a trade-off relationship, and the balance between the two changes with the above N value. Therefore, the above N value is selected according to the characteristics of the input data.

Next, the above-mentioned method of deriving the initial value will be described. In general, the initial value of the predicted value S1 for data smoothing and the initial value of the predicted value B1 for the slope of the smoothed data in the two-stage exponential smoothing process are derived by, for example, the following method. The initial value of the predicted value S1 for data smoothing is S1 ₁ = input data Y1 ₁ (method A1), or S1 ₁ = average value of the initial N input data ({Y1 ₁ + Y1 ₂ + ... +) Derived by Y1 _N } / N) etc. (method A2).

The initial value of the predicted value B1 of the slope of the smoothed data is B1 ₁ = Y1 ₂ -Y1 ₁ (method B1), B2 ₁ = {(Y1 ₂ -Y1 ₁ ) + (Y1 ₄ -Y1 ₃ )} / Derived by 2nd magnitude (method B2). In general, the two-stage exponential smoothing process has a problem that an error is large immediately after the start of data processing. One of the causes is that the initial value by the above-mentioned conventional derivation method has a large error between the predicted value of the true initial smoothing of the data and the predicted value of the slope of the true initial smoothed data. It was.

Here, a polynomial approximation formula is derived by the least squares method using the initial data Y1 _t (t = 1, 2, ..., N) after the start of inputting N desired data. We also used the initial 10 time-series data at equal-time intervals. From the polynomial approximation formula derived above, the predicted value S1 ₀ for smoothing the data, which is virtual data immediately before the input data t = 0, and the predicted value B1 ₀ for the slope of the smoothed data are derived.

Here, a linear linear equation is used as the polynomial approximation equation, and the predicted value S1 ₀ for smoothing the data and the predicted value B1 _{0 for} the slope of the smoothed data are Eqs. (15) and (16), respectively. Derived by.
Predicted data smoothing value S1 ₀ = {330 Y1 ₁ + 275 Y1 ₂ + 220 Y1 ₃ + 165 Y1 ₄
+ 110 Y1 ₅ + 55 Y1 ₆ + 0 Y1 ₇ --55 Y1 ₈ --110 Y1 ₉
--165 Y1 ₁₀ } / 825 Equation (15)
Predicted slope of smoothed data B1 ₀ = {-45 Y1 ₁ --35 Y1 ₂ --25 Y1 ₃ --15 Y1 ₄
―― 5 Y1 ₅ + 5 Y1 ₆ + 15 Y1 ₇ + 25 Y1 ₈ + 35 Y1 ₉
+ 45 Y1 ₁₀ } / 825 Equation (16)

Further, in the second two-stage exponential smoothing process shown in FIG. 2, the initial value S2 ₁ of the predicted value of data smoothing and the initial value B2 ₁ of the predicted value of the slope of the smoothed data are respectively. , S2 ₁ = S1 ₁ and B2 ₁ = 0 are set. The above-mentioned initial value setting method has an effect that the initial input data after the start of data input also has a small error and can perform data smoothing processing and data differentiation processing with high accuracy.

Here, the predicted value S1 ₀ for smoothing the data, which is virtual data immediately before the input data t = 0, and the predicted value B1 ₀ for the slope of the smoothed data are derived from the polynomial approximation formula, but t = 1 etc. , Each predicted value by the polynomial approximation formula of arbitrary time may be used as each initial value. However, in this case, it is disadvantageous for data smoothing processing, data differentiation processing, etc. in a shorter time step than when virtual data with t = 0 is used.

As described above, the first two-stage exponential smoothing process is performed to obtain the predicted value S1 _t for smoothing the data of the first output and the predicted value B1 _t for the slope of the smoothed data. Next, the predicted value B1 _t of the slope of the smoothed data of the first output is used as the second input data Y2 _t , and the second two-stage exponential smoothing process is performed by the following equations (17) and (18). As a result, the predicted value S2 _t for smoothing the data of the second output and the predicted value B2 _t for the slope of the smoothed data are obtained.

Data smoothing: S2 _t = α2 Y2 _t + (1-α2) (S2 _t-1 + B2 _t-1 ) Equation (17)
Slope of smoothed data: B2 _t = γ2 (S2 _t --S2 _t-1 ) + (1 -γ2) B2 _t-1 equation (18)
Next, the data smoothing processing result data S1 _t , the data first-order differential processing result data S2 _t , and the data second-order differential processing result data B2 _t are output as batch or sequential data. Here, the smoothing parameter α2 for data smoothing and the smoothing parameter γ2 for the slope of the smoothed data in the second two-stage exponential smoothing are set to arbitrary constants in advance. However, 0 <α2 <1 and 0 <γ2 <1. Since the predicted value B1 _t of the slope of the smoothed data of the first output also corresponds to the first-order differential processing result, this may be used, but since the data results vary widely, the second two-stage index A data smoothing process is performed by a smoothing process.

In the data smoothing processing result, the data first-order differential processing result, the data second-order differential processing result, etc. by the above-mentioned type I response type two-stage exponential smoothing method and the two-stage exponential smoothing processing performed twice, S / We were able to greatly improve both the increase in the N ratio and the reduction in the delay time. However, when the change point of the target system is detected by the first-order differential data or the second-order differential data and the target device is controlled closer to real time and with higher accuracy, the time delay can be further shortened. The control accuracy of the target device can be improved by improving the data processing performance such as improving the responsiveness and achieving both a high S / N ratio.

Therefore, in the first two-stage exponential smoothing process of FIG. 2, in the case of type II of this embodiment, data processing was performed using the following equations (19) to (24).
Data smoothing: S1 _t = α1 _t Y1 _t + (1-α1 _t ) (S1 _t-1 + B1 _t-1 ) Equation (19)
Slope of smoothed data: B1 _t = α1 _t (S1 _t --S1 _{t -1} ) + (1 -α1 _t ) B1 _t-1 Equation (20)
Smoothing coefficient: α1 _t = (K --L) F _t + L equation (21)
Response coefficient: F _t = 1 --Exp [-δα _t ² / (2σ _t ² )] Equation (22)
Relative _{error: δα t = A1 (Y t} - S1 t) + (1 - A1) δα t-1 Equation (23)
Prediction error variance: σ _t ² = A1 (Y _t --S1 _t ) ² + (1 --A1) σ _t-1 ² equation (24)
Here, K, L, and A1 are arbitrary constants. However, 1>K>L> 0 and 1>A1> 0. Further, the initial value of σ _t was calculated from the standard deviation of the error between the result of polynomial approximation using the data of several initial points and the input data in the input data in the same manner as the above-mentioned initial value derivation method. The response coefficient F _t is obtained by subtracting the probability density function (normal distribution type function, Gaussian distribution type function) from 1. In general, the error often has a normal distribution. Therefore, the response coefficient F _t using the probability density function that expresses the normal distribution is used. As a result, a smoothing coefficient suitable for minimizing the error can be set according to the data change.

As shown in FIG. 3A, for example, in random number data following a normal distribution with an average of 0 and a standard deviation of σ = 1, the value at which the predicted value is most likely to occur and the amount of deviation from the average value are defined. FIG. 3B shows the relationship between the deviation amount / standard deviation σ and the response coefficient F _t . Here, N = 0, N = 1, and N = 5 are the response coefficients in the type I response type two-stage exponential smoothing method shown in equations (5) to (14). 1-PDF is the response coefficient in the response type two-stage exponential smoothing method using the type II probability density function shown in equations (19) to (24).

Here, PDF shows a probability density function (Probability Density Function). 1-PDF shows characteristics of response coefficient similar to N = 5, but 1-PDF has a large response coefficient and deviation when the deviation amount / standard deviation σ <about 1 compared to N = 5. When the quantity / standard deviation σ> about 1, the response coefficient is small. Due to this response characteristic, there is an effect that data smoothing performance (S / N ratio) and data responsiveness (reduction of delay time) can be compatible, and data smoothing processing and data differentiation processing can be performed.

In the above-mentioned first two-stage exponential smoothing process, in the case of Type III of this example, data processing was performed using the following equations (25) to (32).
Data smoothing: S1 _t = α1 _t Y1 _t + (1-α1 _t ) (S1 _t-1 + B1 _t-1 ) Equation (25)
Slope of smoothed data: B1 _t = α1 _t (S1 _t --S1 _{t -1} ) + (1 -α1 _t ) B1 _t-1 Equation (26)
Smoothing coefficient: α1 _t = (K --L) F _t + L equation (27)
Response coefficient: F _t = 1 --Exp [-δα _t ² / (2σ _t ² )] Equation (28)
Relative _{error: δα t = A1 t (Y1} t - S1 t) + (1 - A1 t) δα t-1 Equation (29)
Prediction error variance: σ _t ² = A1 _t (Y1 _t --S1 _t ) ² + (1 --A1 _t ) σ _t-1 ² equation (30)
Response type addition coefficient: A1 _t = MAX (A1, β _t A1 _max ) Equation (31)
Slope coefficient: β _t = 1 --Exp [-B _t ² / (2 NN C) ² ] Equation (32)
Here, A1 _max is the upper limit of the addition coefficient (0 <A1 _max <1), and NN is a sensitivity coefficient and is an arbitrary constant. Further, C is a slope calculated from the result of the first-order approximation formula using the data of several initial points in the input data as in the above-mentioned initial value derivation method. Type III is a type II with a response type addition coefficient introduced, and the formulas (25) to (30) of the type III are almost the same as the formulas (19) to (24) of the type II. FIG. 4 shows an example of a change in the slope of the input signal output. In this case, the slope increases with reference to the initial slope C in the calculation period of the initial value, and the slope changes to the same level as the initial slope again.

In this embodiment, the addition coefficient is changed according to the change in inclination. From equations (29) and (30), the addition coefficient A1 _t corresponds to the coefficient of exponential load averaging. Therefore, the coefficient of the exponential load averaging process is changed according to the initial slope. In other words, when the slope B1 _t of the smoothed data becomes larger than the initial slope C, the response type addition coefficient A1 _t is made smaller so that the ratio of the coefficients of the latest data is effectively increased. Improve responsiveness. For the response-type addition coefficient A1 _t , the slope coefficient β _t is first calculated by Eq. (32).

The slope coefficient β _t is calculated using the probability density function (normal distribution function, Gaussian function) in the same manner as the response coefficient F _t described above. For the response-type addition coefficient A1 _t , the set value A1 is normally used according to equation (31), and when the slope of the smoothed data becomes large and β _t A1 _max becomes larger than the set value A1, this value is added. Used as a coefficient. However, the upper limit value A1 _max can be set in order to prevent the responsiveness from becoming excessively improved.

If the slope of the data smoothed by the introduction of the responsive addition coefficient A1 _t is the same as the initial slope, the addition coefficient is kept at the set value. When the slope of the smoothed data, which indicates a change in the data state, becomes large, the weight of the latest data increases by increasing the response-type addition coefficient A1 _t , which further improves the responsiveness of data processing. It has the effect of improving.

In the above-mentioned first two-stage exponential smoothing process, in the case of Type IV of this example, data processing was performed using the following equations (33) to (42).

Data smoothing: S1 _t = α1 _t SGB0D _t + (1-α1 _t ) (S1 _t-1 + B1 _t-1 ) Equation (33)
Slope of smoothed data: B1 _t = α1 _t SGB1D _t + (1 -α1 _t ) B1 _t-1 Equation (34)
SGB0D _t = (83Y1 _t + 54Y1 _t-1 + 30Y1 _t-2 + 11Y1 _t-3 --3Y1 _t-4 --12Y1 _t-5 -16Y1 _t-6
―― 15Y1 _t-7 ―― 9Y1 _t-8 + 2Y1 _t-9 + 18Y1 _t-10 ) / 143 Equation (35)
SGB1D _t = (945S1 _t + 456S1 _t-1 + 67S1 _t-2 --222S1 _t-3 --411S1 _t-4 --500S1 _t-5
--489S1 _t-6 --378S1 _t-7 --167S1 _t-8 + 144S1 _t-9 + 555S1 _t-10 ) / 4290
Equation (36)
Smoothing coefficient: α1 _t = (K --L) F _t + L equation (37)
Response coefficient: F _t = 1 --Exp [-δα _t ² / (2σ _t ² )] Equation (38)
Relative _{error: δα t = A1 t (Y} t - S1 t) + (1 - A1 t) δα t-1 Equation (39)
Prediction error variance: σ _t ² = A1 _t (Y _t --S1 _t ) ² + (1 --A1 _t ) σ _t-1 ² equation (40)
Response type addition coefficient: A1 _t = MAX (A1, β _t A1 _max ) Equation (41)
Slope coefficient: β _t = 1 --Exp [-B _t ² / (2 NN C) ² ] Equation (42)

Here, SGB0D _t uses the input data Y1 _{t to} Y1 _t-10 of 11 consecutive terms by the backward Savitzky-Golay method described in Non-Patent Document 3, and data smoothing at the time of the latest data Y1 _t. This is the processing result. Similarly SGB1D _t is the backward type Savitzky-Golay method, the time of the most recent predicted values S1 _t of the smoothed data with using the predictive value S1 _t ~S1 _t-10 the smoothing data of consecutive 11 Section It is the result of the first-order differential processing in. However, Non-Patent Document 3 seems to have some errors and the like, and has been corrected and used.

Equations (33), (34), and (37) to (42) are substantially the same as the above-mentioned Type III equations (25) to (32). The meanings of equations (5) and (6), which are the basic parts of the two-stage exponential smoothing method and the type I responsive two-stage exponential smoothing method, are the following equations (43) and (44), respectively. ..
Data smoothing: S1 _t = α1 _t (input data) _t + (1-α1 _t ) (S1 _t-1 + B1 _t-1 )
Equation (43) Slope of smoothed data: B1 _t = γ1 _t (Slope of predicted value of data smoothing) _t + (1 -γ1 _t ) B1 _t-1
Equation (44)
In Type IV, the result of data smoothing processing using the backward Savitzky-Golay method is used as preprocessing instead of the input data of Eq. (43). In addition, instead of the slope of the predicted value of the smoothed data in Eq. (44), the result of first-order differential processing using the backward Savitzky-Golay method is used as preprocessing. Usually, the polynomial adaptation method (Savitzky-Golay method) uses a plurality of consecutive data and often derives the data processing result at the center point of the period of the continuous data. Here, this is referred to as the central Savitzky-Golay method. On the other hand, the case of deriving the data processing result at the time of the latest data is called the backward Savitzky-Golay method.

In the case of the central Savitzky-Golay method, since the data processing result at the central point of continuous data is derived, it is inevitable that a delay time will occur in the data processing. In the Savitzky-Golay method, the S / N ratio improves as the number of data used increases, but in the central type, the delay time increases with the number of data used. In the case of the backward Savitzky-Golay method, since the data processing result at the time of the latest data is derived, there is almost no delay in data processing. However, the receding type has a lower S / N ratio than the central type.

As a result of the examination, as a result of comparing the case of the central type with 5 data and the case of the backward type with 11 data, it was found that almost the same S / N ratio can be obtained. Since the Savitzky-Golay method is a polynomial approximation to a quadratic or cubic curve using multiple data, if the number of data increases, a change of the third order or higher occurs during the period of the number of data used. Can not. That is, as the number of data used increases, there is a risk that high frequency components will be lost.

In view of the above, in Type IV, the data smoothing process of 11 data of the backward Savitzky-Golay method data is used for the input data of the formula (33), and the slope of the predicted value of the smoothing of the data of the formula (34) is set back. Type Savitzky-Golay method Data of 11 data First-order differential processing values were used. Therefore, according to this embodiment, there is an effect that the data delay time is not increased, the frequency characteristic of the data processing is suppressed to a small extent, and the S / N ratio of the data smoothing processing and the data differentiation processing is improved. In Type IV, 11 pieces of backward Savitzky-Golay method data were used, but other data numbers may be used depending on the required data processing performance, or data may be used at some point between the central type and the backward type. The Savitzky-Golay method for processing may be used.

Furthermore, the result of data smoothing processing of the input data as preprocessing other than the Savitzky-Golay method is used as the input data of Eq. (43), and the result of first-order differential processing of the predicted value of data smoothing is obtained by Eq. (44). It may be used for the slope of the predicted value of smoothing of the data of.

FIG. 5 shows a vertical cross-sectional view of the magnetic field microwave plasma etching apparatus according to the first embodiment of the present invention. In this embodiment, the magnetic field microwave plasma etching apparatus corresponds to the target system 6 (device, analysis data, etc.) in FIG. The exhaust opening / closing valve 12 is opened inside the processing chamber 11 partitioned by the container 7, the discharge pipe 8, the quartz plate 9, and the quartz window 10, and the pressure is reduced by the vacuum exhaust device 13. The etching gas passes through the gas pipe 14 via a mass flow controller (not shown), passes between the quartz plate 9 and the quartz shower plate 15, and is introduced into the processing chamber 11 through the gas holes of the quartz shower plate 15. .. The etching gas introduced into the processing chamber 11 adjusts the pressure in the processing chamber 11 to a desired pressure by the exhaust speed variable valve 16.

Further, the processing chamber 11 is in a magnetic field region generated by the coils 17 and 18 and the yoke 19. Microwaves oscillated from the magnetron 20 in this case with a frequency of 2.45 GHz pass through an isolator (not shown), a power monitor (not shown), and a matching unit 21 to the rectangular waveguide 22.
It propagates in the rectangular TE10 mode, and propagates in the circular waveguide 24 via the circular rectangular converter 23 in the circular TE11 mode. After that, the microwave is introduced into the cavity resonator 25, passes through the quartz plate 9 and the quartz shower plate 15, and is incident on the processing chamber 11. In the processing chamber 11, a magnetic field region having a magnetic flux density of 875 Gauss that causes electron cyclotron resonance with the 2.45 GHz microwave introduced is perpendicular to the central axis of the processing chamber 11 and the microwave introduction direction, and in the processing chamber. It is formed on the entire surface in the cross-sectional direction with respect to the central axis of 11.

The wafer 27 mounted on the wafer mounting electrode 26, which is a sample table, is etched by the plasma mainly generated by the interaction between the 2.45 GHz microwave and the 875 Gauss magnetic field. Further, in order to control the etching shape of the wafer 27 as a sample, a high frequency power supply 28 is connected to the wafer mounting electrode 26 via a matching device (not shown), and a high frequency voltage can be applied. There is. Further, a chiller unit (not shown) is connected to the wafer mounting electrode 26, and the temperature of the wafer 27 can be controlled.

The processing chamber 11, the wafer 27, and the wafer mounting electrode 26 are arranged coaxially with each other. Further, the gas hole region of the quartz shower plate 15 into which the etching gas is introduced, the exhaust opening / closing valve 12 which is a vacuum exhaust portion, the exhaust speed variable valve 16, and the vacuum exhaust device 13 are also arranged coaxially with the processing chamber 11. .. Therefore, the gas flow on the wafer 27 is coaxially symmetric. Since the coils 17, 18 and yoke 19 that generate the magnetic field are also arranged coaxially with respect to the processing chamber 11, the magnetic field profile in the processing chamber 11 and the electron cyclotron resonance region of the magnetic flux 875Gauss are formed coaxially with the processing chamber 11. Will be done. Further, since the circular waveguide 24 and the cavity resonator 25 are also arranged coaxially with the processing chamber 11, the microwave introduced into the processing chamber 11 is also introduced coaxially with the processing chamber 11.

Since the magnetic field is generated coaxially with the processing chamber 11 and the microwave is also introduced coaxially with the processing chamber 11, the plasma formed by the interaction between the magnetic field and the microwave is generated coaxially with the processing chamber 11. Will be done. As a result, the electrons and ions in the plasma are transported coaxially with the wafer 27. Further, since the flow of the etching gas is also coaxial with the processing chamber 11, radicals generated by plasma and reaction products obtained by etching the wafer 27 are introduced and exhausted coaxially with the wafer 27. Therefore, the etching process processing performance such as the etching rate, the material selection ratio, and the etching shape can be uniformly etched on the wafer surface.

The light emitted from the plasma generated in the processing chamber 11 and from the side of the processing chamber 11 is introduced into the spectroscope 30 through the quartz window 10 and the optical fiber 29, and is output as wavelength-dependent time-series data of the light intensity. Will be done. Further, the light emitted from the plasma from above the processing chamber 11 passes through the quartz shower plate 15, the quartz plate 9, the cavity resonator 25, the circular waveguide 24, the circular rectangular converter 23, and the optical fiber 31 to the spectroscope 32. It is introduced and output as wavelength-dependent time-series data of light intensity.

Etching gas and etching reaction products from the wafer 27 are introduced into the processing chamber 11, and these are dissociated by the interaction between the microwave and the magnetic field to generate a plasma. Therefore, the light emitted from the plasma generated in the processing chamber 11 includes information on the atoms, molecules, radicals, and the reactants constituting the etching gas and the etching reaction product.

For example, in a typical poly-Si etching of a Si substrate in which a poly-Si film and a SiO ₂ film are arranged below a pattern mask, it is required to perform Poly-Si etching with a high selectivity with the underlying SiO 2. A halogen-based gas is used as the etching gas, and the etching reaction product contains Si and halogen, which are materials to be etched. Since the etching reaction product is re-dissociated by the plasma, the light intensity of the emission of 288 nm wavelength due to Si from the plasma is monitored by the spectroscope 30 or the spectroscope 32.

In this case, when the etching of the poly-Si film is completed and the underlying SiO2 appears, the etching rate of the underlying SiO2 is small, so the plasma emission intensity at a wavelength of 288 nm due to Si sharply decreases and finally reaches a constant value. Get closer. The change in plasma emission is monitored to detect the end point of the etching process.

The emission of plasma from the side of the processing chamber 11 includes information on the etching gas and the etching reaction product, but the emission of plasma from above the processing chamber 11 includes the plasma light of the wafer 27 in addition to the above information. Since interference occurs in the film structure and the step structure, information on the film structure and the step structure of the wafer 27 is also included. By analyzing the emission data of this plasma, it is possible to monitor the film thickness, etching depth, etc. during etching. In this embodiment, for the sake of simplicity, plasma emission data from the side of the processing chamber 11 was used for the etching end point monitor. A typical data processing flow for detecting the etching end point is shown in FIG. The input data Y1 _t was created by the evaluation function of Eq. (45) simulating the change in plasma emission intensity during etching.

Y1 _t = H / [1 ＋ exp {-A (t --T)}] ＋ C t ＋ D ＋ F (R -0.5) Equation (45)
Here, H, A, T, C, D, and F are arbitrary constants, and R is a random number from 0 to 1. If the above evaluation function is used, the analytical true value of the data smoothing process, the primary differential processing, and the secondary differential processing is known. Therefore, in various data processing methods, the absolute error from the true value and the data processing can be used. Data processing performance such as the accompanying delay time and S / N ratio (signal / noise ratio) can be compared and evaluated.

As shown in FIG. 6, a typical data processing flow for the etching end point is shown in FIG. 6 after performing data smoothing processing on the input data waveform as shown in FIG. 6 (A) as shown in FIG. 6 (B). The first-order differential processing is performed as shown in 6 (C), and the second-order differential processing is performed as shown in FIG. 6 (D). For input data containing a lot of noise, the change point becomes clear by the data smoothing process. This change point is detected as a peak value point (time) in the first derivative processing and as a zero crossing point (time) in the second derivative processing. The etching end point is determined based on the zero crossing point (time), the etching apparatus is controlled, and high-precision etching processing is performed.

The change point can be determined more clearly and more easily by the peak of the first derivative processing and the zero cross of the second derivative process, but the absolute value of the signal intensity becomes smaller and smaller. Therefore, data processing with a high S / N ratio is important. In particular, in the case of etching a mask pattern having a small aperture ratio and a small area to be etched, the change in plasma emission intensity before and after the etching end point is small, so further data processing with a high S / N ratio is required.

Generally, in data smoothing processing and data differentiation processing, the higher the S / N ratio, the longer the delay time and the larger the absolute error from the true value. That is, the S / N ratio, the delay time, and the absolute error are in a trade-off relationship, and data smoothing processing and data differentiation processing that simultaneously satisfy the S / N ratio, the delay time, and the absolute error are required.

In this embodiment, the data processing of FIG. 6 is performed using the data smoothing process of FIG. 2 and the data differentiation processing flow. Further, the input data of FIG. 6 corresponds to the output data from the spectroscope 30 that monitors the plasma emission during etching in the magnetic field microwave plasma etching apparatus of FIG. As shown in FIG. 5, a system control device 33 (including a data input / output device, a data processing device, a data display device, etc.) that controls a magnetic field microwave etching device as a system and a data processing device 1 of the present embodiment are provided. Has been done. The data processing device 1 may be incorporated as a part of the system control device 33.

The output data from the spectroscope 30 and the spectroscope 32 is transmitted to the data processing device 1. The data smoothing processing result, the data first-order differentiation result, and the data second-order differentiation result are transmitted to the system control device 33, which is a control device. The system control device 33 determines the etching end point based on the data smoothing processing result, the data first-order differentiation result, and the data second-order differentiation result, and controls the magnetic field microwave etching device as a system. In FIG. 5, the system control device 33 is shown to be connected to the magnetron 20 and the high frequency power supply 28 in order to mainly control the plasma generation in the etching end point determination. Further, although not shown in FIG. 5, the system control device 33 is also connected to other devices constituting the system.

FIG. 7 shows a case where the data processing apparatus 1 of the embodiment of the present invention uses the data processing flow of FIG. 2 and uses the responsive two-stage exponential smoothing method of type IV of equations (33) to (42). It is the data processing result of. FIG. 8 shows the data processing results when N = 0 in the type I responsive two-stage exponential smoothing method of equations (5) to (14). Further, FIG. 9 is a data processing result when N = 1 as in FIG. 8. FIG. 10 is the same as in FIG. 8 and shows the data processing result when N = 5.

In FIGS. 7 to 10, (A) shows the input data (Y1 _t ) and the data smoothing processing result (S1 _t ), (B) shows the first-order differential processing result (B1 _t ), and (C). ) Indicates the first-order differential smoothing processing result (S2 _t ), and (D) indicates the second-order differential processing result (B2 _t ). Further, the same input data in FIGS. 7 to 10 was used.

Comparing FIGS. 7 to 10, first, in the data smoothing process of (A), FIG. 7 (A), which is an example of Type IV of the present invention, is of Type I, FIGS. 8 (A) and 9 (A). ), It can be seen that the data smoothing processing result follows the change of the input data as compared with FIG. 10A, has less overshoot and the like, has better responsiveness, and has a smaller error. Next, in the first-order differential smoothing process of (C), FIG. 7 (C), which is an example of Type IV of the present invention, shows Type I, FIGS. 8 (C), 9 (C), and 10 (C). ), The peak time of the first derivative is earlier, and the half width of the waveform is smaller. Therefore, it can be seen that the delay time of data processing is small and the responsiveness is good.

In the second derivative processing of (D), FIG. 7 (D), which is an example of Type IV of the present invention, is more than that of Type I, FIGS. 8 (D), 9 (D), and 10 (D). The zero cross time of the second derivative is fast, and the overall width of the second derivative waveform is also small. Furthermore, the waveform is smooth and the signal strength of the second derivative is also large. Therefore, it can be seen that the delay time of data processing is small, the responsiveness is good, and the S / N ratio is also good. When compared numerically, FIG. 7 (D), which is an example of type IV of the present invention, FIG. 8 (D) of type I (N = 0), FIG. 9 (D) of type I (N = 1), type. The S / N ratios of the respective second derivative waveforms in the order of FIG. 10 (D) of I (N = 5) are 1862, 68, 621, and 957.

The delay time is 3.9 seconds earlier than that of FIG. 8 (D) of type I (N = 0) in FIG. 7 (D), which is an embodiment of the present invention, in the zero cross time of the second-order differential waveform. The delay time is shortened by 1.6 seconds earlier than FIG. 9 (D) of = 1) and 0.9 seconds earlier than FIG. 10 (D) of type I (N = 5). Therefore, in the embodiment of the present invention, both the delay time and the S / N ratio are improved as compared with the type I, and there is an effect of achieving both a reduction in the delay time (improvement in responsiveness) and an improvement in the S / N ratio. ..

Therefore, it can be seen that the peak point (time) of the first derivative, which is the reference for determining the etching end point, and the zero cross point (time) of the second derivative can be clearly detected. Therefore, according to the embodiment of the present invention, there are effects that the absolute value error is small, the S / N ratio is high, the delay time is short, and the data smoothing process and the data differential process can be sequentially processed in real time.

Further, in the (B) first-order differential processing, FIG. 7 (B), which is an example of the type IV of the present invention, is from FIGS. 8 (B), 9 (B), and 10 (B) of the type I. The waveform is smooth. This is the effect of pretreatment by the backward Savitzky-Golay method. Therefore, in the processing flow of FIG. 2, since the first output data (S1 _t ) and (B1 _t ), which are the second input data, have smooth waveforms, the smoothing of the second two-stage exponential smoothing is performed. The conversion factor can be increased. As a result, the delay time between the first-order differential smoothing result (S2 _t ) and the second-order differential processing result (B2 _t ), which are the second outputs, can be shortened and the responsiveness can be improved.

Further, when only the data smoothing processing result and the data first-order differential processing result are required, according to this embodiment, the first two-stage exponential smoothing processing is performed by the effect of the preprocessing by the backward Savitzky-Golay method. Since a smooth waveform of the first derivative can be obtained, the first output data (S1 _t ) and (B1 _t ) may be used without performing the second two-stage exponential smoothing process. In this case, there is an effect that the data processing program becomes simple and the data processing speed is improved.

Furthermore, the case where the receding Savitzky-Golay method is applied to the response-type two-stage exponential smoothing method of type I shown in equations (5) to (14) is defined as type V, and the following equations (46) to (57) are used. Shown.
Data smoothing: S1 _t = α1 _t SGB0D _t + (1-α1 _t ) (S1 _t-1 + B1 _t-1 ) Equation (46)
Slope of smoothed data: B1 _t = γ1 _t SGB1D _t + (1 -γ1 _t ) B1 _t-1 Equation (47)
SGB0D _t = (83Y1 _t + 54Y1 _t-1 + 30Y1 _t-2 + 11Y1 _t-3 --3Y1 _t-4 --12Y1 _t-5 -16Y1 _t-6
―― 15Y1 _t-7 ―― 9Y1 _t-8 + 2Y1 _t-9 + 18Y1 _t-10 ) / 143 Equation (48)
SGB1D _t = (945S1 _t + 456S1 _t-1 + 67S1 _t-2 --222S1 _t-3 --411S1 _t-4 --500S1 _t-5
--489S1 _t-6 --378S1 _t-7 --167S1 _t-8 + 144S1 _t-9 + 555S1 _t-10 ) / 4290
Equation (49)
Smoothing coefficient: α1 _t = (K _α -L _α ) F _α + L _α equation (50)
Response _{factor: F αt = (| δα t} / Δα t | + φ) N Equation (51)
Relative _{error: δα t = A1 (Y1 t} - S1 t) + (1 - A1) δα t-1 Equation (52)
Absolute error: Δα _t = A1 | Y1 _t --S1 _t | + (1 --A1) Δα _t-1 + φ equation (53)
Smoothing coefficient: γ1 _t = (K _γ -L _γ ) F _γ + L _γ equation (54)
Response _{factor: F γt = (| δγ t} / Δγ t | + φ) N Equation (55)
Relative _{error: δγ t = A2 {(S1} t - S1 t-1) - B1 t} + (1 - A2) δγ t-1 Equation (56)
Absolute _{error: Δγ t = A2 | (S1} t - S1 t-1) - B1 t | + (1 - A2) Δγ t-1 + φ Equation (57)
In the case of type V, the responsiveness is inferior to that of type IV, but there is an effect that the data processing program becomes simple. The required level and complexity method may be selected according to the required data processing performance.

Furthermore, the case where the receding Savitzky-Golay method is applied to the type II responsive two-stage exponential smoothing method shown in equations (19) to (24) is defined as type VI, and the following equations (58) to (65) are used. Shown.
Data smoothing: S1 _t = α1 _t SGB0D _t + (1-α1 _t ) (S1 _t-1 + B1 _t-1 ) Equation (58)
Slope of smoothed data: B1 _t = α1 _t SGB1D _t + (1 -α1 _t ) B1 _t-1 Equation (59)
SGB0D _t = (83Y1 _t + 54Y1 _t-1 + 30Y1 _t-2 + 11Y1 _t-3 --3Y1 _t-4 --12Y1 _t-5 -16Y1 _t-6
―― 15Y1 _t-7 ―― 9Y1 _t-8 + 2Y1 _t-9 + 18Y1 _t-10 ) / 143 Equation (60)
SGB1D _t = (945S1 _t + 456S1 _t-1 + 67S1 _t-2 --222S1 _t-3 --411S1 _t-4 --500S1 _t-5
--489S1 _t-6 --378S1 _t-7 --167S1 _t-8 + 144S1 _t-9 + 555S1 _t-10 ) / 4290
Equation (61)
Smoothing coefficient: α1 _t = (K --L) F _t + L equation (62)
Response coefficient: F _t = 1 --Exp [-δα _t ² / (2σ _t ² )] Equation (63)
Relative _{error: δα t = A1 (Y1 t} - S1 t) + (1 - A1) δα t-1 Equation (64)
Prediction error variance: σ _t ² = A1 (Y1 _t --S1 _t ) ² + (1 --A1) σ _t-1 ² equation (65)
Type VI is also inferior to Type IV in responsiveness, but has the effect of simplifying the data processing program. When N = 0 in type V, the receding Savitzky-Golay method is applied to the two-stage exponential smoothing method with a fixed smoothing coefficient, which is the simplest. The required level and complexity method may be selected according to the required data processing performance.

Type II is a type I responsive two-stage exponential smoothing method in which a data responsive smoothing coefficient using a probability density function is introduced. Type III is a type II with a data response type addition coefficient introduced. In addition, Type IV is Type III with the introduction of pretreatment by the backward Savitzky-Golay method. Similar to type V, the data response type smoothing coefficient and the data response type addition coefficient using the probability density function, which are the improvement factors, may be individually applied to the type I response type two-stage exponential smoothing method. .. Further, the data response type smoothing coefficient using the probability density function, the data response type addition coefficient, and the backward Savitzky-Golay method may be appropriately combined and applied to the type I response type two-stage exponential smoothing method. .. As described above, the required level and complexity method may be selected according to the required data processing performance.

As described above, in the data smoothing process and the data differentiation process, the improvement of the S / N ratio performance and the improvement of the data response performance (reduction of the delay time) are in a trade-off relationship. Therefore, the above-mentioned type I responsive two-stage exponential smoothing method, or the data smoothing processing method and data differentiation processing method for type II, type III, type IV, type V, type VI, etc. described in this embodiment. Then, it is necessary to optimize the parameters of each data processing method according to the input data to be processed.

Conventionally, data processing parameters are sequentially changed, data processing is performed, and numerical data such as data smoothing waveforms, data differential waveforms, and S / N ratios and delay times are comprehensively viewed to find the optimum parameters. Was there. However, in the above method, it takes a long time to find the optimum parameter, and there is a problem that knowledge and experience of data processing are required to shorten the time to find the optimum parameter.

Further, Non-Patent Document 3 discloses a method of estimating the optimum smoothing parameter by minimizing the sum of the errors of the predicted values in the first period in the one-step exponential smoothing method. However, since this method does not consider the smoothness of the curve of the predicted value, there is a problem that the noise is large and the S / N ratio is not good in the first-order differential processing and the second-order differential processing for detecting the change point. there were.

Therefore, a method that can easily and automatically find the optimum parameters for data smoothing processing and data differentiation processing of input data is shown below.

First, the evaluation function W of the following equation (66) is used.

Evaluation function W = mean square error (E) + coefficient λ x root mean square (D) equation (66)
Here, E, which is the mean square error, is the mean square error of the input data and the data smoothing processing result, and evaluates the goodness of fit of the data smoothing processing result with respect to the input data. D, which is the root mean square of the second derivative, evaluates the smoothness of the curve of the data smoothing processing waveform. The coefficient λ is an arbitrary numerical value and adjusts the ratio of the importance of the goodness-of-fit evaluation and the smoothness evaluation of the curve to the above input data.

The greater the goodness of fit of the data smoothing processing result with respect to the input data, the less overshoot and the like, the better the responsiveness of the data processing, and the smaller the delay time of the data processing. Further, the smoother the data smoothing processing waveform is, the smoother the data first-order differential processing waveform and the data second-order differential processing waveform are. As a result, the S / N ratios of the data first-order differential processing and the data second-order differentiation processing are improved. The smoothness of the curve is evaluated by the root mean square D of the second derivative of the data smoothing processed waveform. Here, the second derivative was calculated by the difference method.

The root mean square D of the second derivative becomes smaller as it approaches a straight line. Therefore, if not only the root mean square D of the second-order differential is small, but also the mean square error E is a small value, and the goodness of fit of the data smoothing processing result with respect to the input data is good, the S / N ratio is compatible. Is good, and the responsiveness of data processing is good (delay time is small), which is the optimum data processing. The evaluation function W used in this embodiment is shown in the following equation (67).

W = Σ (Y1 _t --S1 _t ) ² / N + λ × Σ {(S1 _{t + 1} --2 × S1 _t + S1 _{t + 1} ) / ΔT ² } ² / N
Equation (67)
Here, N is the number of data, and ΔT is the sampling time (time interval) of the input data. The optimum parameters for the data smoothing process and the data differential process are derived by using a gradient method such as the steepest descent method so that the evaluation function W becomes the minimum value. In order to show the characteristics of the evaluation function W in a two-dimensional graph, the case of a simple type I responsive two-stage exponential smoothing method will be described with reference to FIGS. 11 to 13. However, the input data is similar to the input data used in FIGS. 7 to 10, but different data is used.

FIG. 11 shows (A) and the second order showing the input data (Y1 _t ) and the data smoothing processing result (S1 _t ) when one parameter of the type I responsive two-stage exponential smoothing method is sequentially changed. It is a figure which consists of (B) which shows the differential processing result (B2 _t ). The numerical values are larger in the order of parameters 1, 2, 3, and 4. In the case of the parameter 4 having the largest numerical value, the input data waveform and the data smoothing waveform almost match, and it can be seen that the goodness of fit is good. However, it can be seen that the second derivative waveform has a fast zero cross time and good responsiveness, but has a lot of noise and the S / N ratio is not good. On the other hand, in the case of the parameter 1 having the smallest numerical value, it can be seen that the data smoothing waveform has an overshoot with respect to the input waveform, the goodness of fit is not good, and the responsiveness is not good. Further, it can be seen that the second derivative waveform has a small noise, a small signal strength, a slow zero cross time, and a large delay time. In the case of FIG. 11, from the second derivative processing waveform of (B), it can be determined that the S / N ratio and the delay time are compatible with each other in the case of parameter 2.

FIG. 12 is a diagram showing the parameter dependence of the mean square error E, the root mean square D of the second derivative, and the evaluation function W in the case of FIG. Further, the positions of the parameters corresponding to FIG. 11 are indicated by arrows. Further, the x mark indicates the position of the parameter whose minimum value is the evaluation function W automatically derived by using the gradient method such as the steepest descent method and the position of the numerical value of the evaluation function W. FIG. 13 shows the input data (Y1 _t ) and the data smoothing processing result (S1 _t ) when the parameter of the position marked with × automatically derived using the evaluation function W in FIG. 12 is used (A). It is a figure including (B) which shows the 2nd order differential processing result (B2 _t ).

As shown in FIG. 12, it can be seen that the mean square error E decreases as the numerical value of the parameter increases, and the root mean square D of the second derivative increases. As a result, it can be seen that the evaluation function W has a minimum value. As a result, it can be seen that by finding the minimum value of the evaluation function W, it is possible to find a parameter that has both an S / N ratio and responsiveness. The parameter dependence of the characteristics of the qualitative data processing waveform shown in FIG. 11 is quantitatively shown in FIG.

The optimum values of the parameters marked with x in FIG. 12 are substantially the same as those of parameter 2. Further, it can be seen that the data smoothing processed waveform and the data second-order differential processing waveform shown in FIG. 13 are substantially the same as the data smoothing processed waveform and the data second-order differential processing waveform of the parameter 2 of FIG.

In this embodiment, for convenience of explanation, a case where one parameter is changed has been described, but a plurality of parameters can be derived by using a gradient method such as the steepest descent method. In this case, it is necessary to consider the presence or absence of the local minimum value (optimal value) and the initial value of the parameter and the search range. Further, in the embodiment, for convenience of explanation, the case of the simple type I response type two-stage exponential smoothing method has been described, but the above-mentioned type II, type III, type IV, type V, and type VI data processing has been described. Also in the method, or other data smoothing processing such as 1-step exponential smoothing method (exponentially weighted moving average: EWMA), responsive 1-step exponential smoothing method, low-pass filter, Kalman filter, or data differential processing by difference method, etc. Similarly, by minimizing the evaluation function W, it is possible to easily find the optimum parameter that achieves both the S / N ratio and the responsiveness.

Further, in this embodiment, the root mean square D of the second derivative of Eqs. (66) and (67) uses the second derivative value of the data smoothing processing result, but the first derivative processing result or the second derivative processing result. You may use the root mean square of the second derivative of. In this case, the smoothness of the curve of the first-order differential waveform and the smoothness of the curve of the second-order differential waveform are directly evaluated. In this case, since the absolute value of the numerical value of the first-order differential processing result and the second-order differential processing result becomes smaller in order as compared with the data smoothing processing result, the coefficients λ of the equations (66) and (67) are increased. Etc. need to be adjusted.

In the response-type one-stage exponential smoothing method shown in equations (1) to (4) described in Non-Patent Document 3, the input data Y1 _t at the current time t and the data smoothing at the current time t The predicted value S _t + 1 for data smoothing at the time t + 1 one period ahead is derived using the predicted value S _{1 t} . On the other hand, in the example of type I~ type VI, using the prediction value S1 _t-1 data smoothing the input data Y1 _t and one period before the time t-1 of the current at time t at the current time t The predicted value S1 _t of data smoothing is derived. The former can be called "one-term forecast" and the latter can be called "current estimation". Even in the case of the examples of Type I to Type VI, it may be changed to "1st period forecast" by conversion such as S1 _t-1 → S1 _t , S1 _t → S1 _{t + 1} .

However, in general, "current estimation" has better accuracy of data smoothing prediction value than "one period ahead prediction". Further, Non-Patent Document 3 discloses a method of estimating the optimum smoothing parameter by minimizing the sum of the errors of the one-term advance prediction values in the one-stage exponential smoothing method of "one-term advance prediction". Has been done.

In the case of "current estimation", if the smoothing coefficient is 1, Y1 _t = S1 _t , and the error or average squared error is zero. Therefore, it is estimated as the optimum smoothing parameter using the steepest descent method or the like. Can not do it. However, as shown in the example, even in the "current estimation", the optimum smoothing parameter can be derived by using the evaluation function W of the equation (66) considering the mean square error and the smoothness of the curve. it can.

According to this embodiment, there is an effect that the optimum parameters for data smoothing processing and data differentiation processing of input data can be automatically found in a short time without relying on the knowledge and experience of data processing. This has the effect of providing a processing device that is easy for general users to use, that is, has good usability, and includes a data processing device, a data processing method, and a control device that controls a processing room.

As described above, the present invention is particularly effective for a short-time etching process and detection of the end point of etching accompanied by a change in a short time. With the increasing integration and miniaturization of semiconductor devices, the number of steps for etching a multilayer thin film is increasing in semiconductor etching, and it is possible to detect the etching end point in a short-time etching process and an etching step that involves a short-time change. It has become important.

This short-time process and short-time change process correspondence can detect clear first-order differential waveforms and second-order differential waveforms by improving the overall performance of both the improvement of S / N ratio and the improvement of responsiveness according to the present invention. Will be. As a result, the etching end point can be determined with high accuracy, and by controlling the process in the plasma processing chamber based on this, the semiconductor wafer can be finely processed with stable performance and with high accuracy.

In the above-described embodiment, the case where etching is performed with high accuracy by applying to the detection of the end point of etching in the microwave plasma etching apparatus has been described in detail, but other plasma generation methods (induction coupling type, parallel plate type, etc.) The state of the apparatus by applying the data processing apparatus and the data processing method of the present invention to the etching apparatus and the film forming apparatus of the above, or the processing apparatus and other apparatus in other fields by inputting the numerical data obtained from the apparatus and the like. Can be monitored and the change point of the state can be detected with high accuracy. Therefore, there is an effect that the target device can be controlled with high accuracy. Further, the same action and effect are obtained in the control of other devices.

Further, in the economic and financial fields such as demand supply forecasting, there is an effect that data can be analyzed with high accuracy by applying the data processing apparatus and data processing method of the present invention.

Further, according to the present invention, in the sequential data processing, the data processing and the data differential processing are performed with a high S / N ratio and a small data delay, and the initial period of data processing is also reliable. High data processing can be performed.

Further, the present invention can obtain a data smoothing value, a first-order differential value, a high S / N ratio with a second-order differential value, a short delay time, or even at the initial stage of data processing with high reliability in sequence in real time. Further, the present invention can control the target system with high accuracy by using the data smoothing value, the first derivative value, and the second derivative value.

The present invention is not limited to the above-described examples, and includes various modifications. For example, each of the above-described examples has been described in detail in order to explain the present invention in an easy-to-understand manner, and is not necessarily limited to those having all the described configurations. It is also possible to replace a part of the configuration of one embodiment with the configuration of another embodiment, and it is also possible to add the configuration of another embodiment to the configuration of one embodiment. Further, it is possible to add / delete / replace a part of the configuration of each embodiment with another configuration.

Further, in Examples 1 and 2, "smoothing" and "smoothing" are mixed, but "smoothing" and "smoothing" in Examples 1 and 2 are used as synonyms. ing.

1 ... Data processing device, 2 ... Data input / output device, 3 ... Data storage device, 4 ... Data processing program storage device, 5 ... Data calculation processing device, 6 ... Target system, 7 ... Container, 8 ... Discharge Tube, 9 ... Quartz plate, 10 ... Quartz window, 11 ... Processing chamber, 12 ... Exhaust opening / closing valve, 13 ... Vacuum exhaust device, 14 ... Gas piping, 15 ... Quartz shower plate, 16 ... Variable exhaust speed valve, 17 ... Coil, 18 ... Coil, 19 ... Yoke, 20 ... Magnetron, 21 ... Matcher, 22 ... Rectangular waveguide, 23 ... Circular transducer, 24 ... Circular waveguide, 25 ... Cavity resonator, 26 ... Wafer mounted Placement electrode, 27 ... wafer, 28 ... high frequency power supply, 29 ... optical fiber, 30 ... spectroscope, 31 ... optical fiber, 32 ... spectroscope, 33 ... system control device

Claims (2)

In a data processing method for smoothing data related to plasma emission in a plasma etching apparatus using an exponential smoothing method.
The mean square error between the plurality of data and the plurality of predicted values corresponding to the plurality of data calculated by performing smoothing processing on the plurality of data by the exponential smoothing method, and the plurality of predicted values. Calculate the smoothing parameter of the exponential smoothing method that minimizes the sum of the squared mean value of the quadratic differential value of and the product of the coefficient.
A data processing method characterized in that smoothing processing is performed on the data using the calculated smoothing parameters. The data processing method according to claim 1.
Instead of the squared mean value of the second derivative values of the plurality of predicted values, the squared mean value of the second derivative value of either the first derivative value or the second derivative value of the plurality of predicted values is used. Data processing method.

Images