KR20200025845A - Device of critical dimension based on controlled random search and method implementing thereof - Google Patents

Abstract

The present invention relates to a device for calculating a CD of an optimal model based on a control random search and a method for implementing the same. According to one embodiment of the present invention, the device comprises: a spectrum input unit configured to receive a measurement spectrum measured for each wavelength from a semiconductor device; a library unit configured to store sample data for calculating a test spectrum for comparison with the measurement spectrum; a controller including a simplex algorithm execution unit and a differential evolution (DE) algorithm execution unit, configured to control the library unit to calculate a test spectrum corresponding to sample data calculated from any one of the two algorithm execution units, and configured to repeat a process of calculating sample data for minimizing a difference by comparing the measurement spectrum with the test spectrum; and an output unit configured to output the sample data calculated by the controller and having a minimum difference of the test spectrum.

Description

DEVICE OF CRITICAL DIMENSION BASED ON CONTROLLED RANDOM SEARCH AND METHOD IMPLEMENTING THEREOF}

The present invention relates to an apparatus for calculating a CD of an optimal model based on a control random search and a method of implementing the same.

In the process of fabricating the semiconductor, the pattern of the structures disposed in the semiconductor devices may be measured to confirm physical properties of the semiconductor. In order to confirm the pattern, a critical dimension (CD) of the fine pattern of the structures disposed in the device may be calculated.

A method of modeling a CD parameter of a semiconductor is to compare measurement spectra measured at the semiconductor with test spectra corresponding to the CD parameter modeled for the semiconductor. In addition, a method of estimating the CD parameter of the semiconductor may be used by finding a test spectrum that minimizes the difference between the measurement spectrum and the test spectrum calculated in the comparison process and corresponding CD parameters.

To this end, there is a need for a technique for quickly and accurately estimating semiconductor spectra by calculating CD parameters that can model spectra measured in semiconductors in statistical models.

SUMMARY OF THE INVENTION The present invention has been made to solve the above problems, and an object thereof is to calculate a test spectrum close to the spectrum measured from a semiconductor device and to calculate a corresponding CD parameter.

In addition, an object of the present invention is to generate sample data based on a control random search in order to increase the calculation speed and accuracy in calculating CD parameters.

The object of the present invention is not limited to the above-described object, the structure and manufacturing method of the present invention may have a variety of objects.

An apparatus according to an aspect of the present invention for solving the above technical problem is a spectrum input unit for receiving a measurement spectrum measured for each wavelength from a semiconductor device, a library unit for storing a sample data for calculating a test spectrum for comparison with the measurement spectrum, It includes a simplex algorithm execution unit and a differential evolution algorithm execution unit, and controls the library unit to generate a test spectrum corresponding to the sample data generated by either of these two algorithm execution units, and to measure the measurement spectrum and the test spectrum. And a controller for repeating a process of calculating sample data for minimizing the difference, and an output unit for outputting sample data for which the controller calculates a test spectrum of the minimum difference.

Method for calculating the CD of the optimal model based on the control random search according to an aspect of the present invention comprises the steps of the spectrum input unit receives the measurement spectrum measured for each wavelength from the semiconductor device, the simplex algorithm execution unit and DE (Differential Evolution) A controller comprising an algorithm execution unit calculates sample data from either of these two algorithm execution units, the controller calculates a test spectrum corresponding to the selected sample data from the library, and the controller compares the measurement spectrum with the test spectrum. And repeating the process of calculating sample data for minimizing the difference, and providing the sample data in which the controller calculates a test spectrum of the minimum difference to the output unit and outputting the output unit.

The present invention can calculate the test spectrum close to the spectrum measured from the semiconductor device, calculate the CD parameter corresponding thereto, and accurately and quickly estimate the model CD parameter of the semiconductor device.

In addition, the present invention can increase the calculation speed and accuracy by generating sample data based on the control random search in calculating the CD parameter.

The effects of the present invention are not limited to the above effects, and those skilled in the art can easily derive various effects of the present invention from the configuration of the present invention.

1 is a view showing the configuration of an apparatus for calculating an optimal CD according to an embodiment of the present invention.
2 is a diagram illustrating a CD calculation process according to an embodiment of the present invention.
3 is a diagram illustrating a relationship between a CD parameter and an MSE according to an embodiment of the present invention.
4 is a graph showing local and global optimal solutions.
5 is a diagram illustrating a process of performing a CRS by a controller according to an embodiment of the present invention.
6 is a view showing a detailed configuration of a controller according to an embodiment of the present invention.
7 to 10 are views showing a detailed configuration of a controller according to an embodiment of the present invention.
11 is a diagram illustrating a process of calculating a parameter by a DE algorithm execution unit according to an embodiment of the present invention.
12 is a diagram illustrating a probabilistic selection process of an algorithm according to an embodiment of the present invention.

Certain terms are defined herein for convenience of understanding the invention. Unless otherwise defined herein, scientific and technical terms used herein may have the meanings that are commonly understood by one of ordinary skill in the art.

In addition, the singular forms also include their plural forms, and the plural forms of terms may also include the singular forms thereof, unless the context clearly indicates otherwise.

In order to determine the profile of a semiconductor device, for example, a semiconductor chip, a statistical optimization algorithm is applied to measure a CD (Critical Dimension) value using an optical critical dimension (OCD) device. Can be.

In this specification, a CD parameter of a model that minimizes a difference between a spectrum measured using an OCD device and a spectrum obtained from a rigid coupled wave analysis (RCWA) model is estimated as a CD parameter of an actually measured measurement spectrum. To this end, we will apply the CRS (Controlled Random Search) method in the process of finding the optimized CD parameters from the model.

In the case of applying the above-described CRS, in the present specification, an appropriate CD parameter may be calculated even in a high aspect ratio pattern having a high correlation between parameters (CD values). That is, in the present specification, the CRS may be applied to be most suitable for measurement.

1 is a view showing the configuration of an apparatus for calculating an optimal CD according to an embodiment of the present invention.

The apparatus 100 for calculating a CD parameter outputs a result of computing by using predetermined input information and stored information. Device 100 includes a device capable of executing and implementing a notebook, computer or other algorithm. Alternatively, the device 100 of FIG. 1 includes a computing device that performs only a task of calculating CD parameters. The device 100 may include a software component or a hardware component as detailed configurations.

The configuration of the apparatus consists of a spectrum input unit 120, a library unit 140, an output unit 180 and a controller 160 that controls them and calculates optimal CD parameters. The controller includes one or more of software code for implementing a predetermined optimization algorithm or a hardware chip for implementing the same.

The library unit 140 returns a function value (test spectrum) corresponding to the sample data in calculating sample data (CD values, CD parameter) in the CRS (Controlled Random Search). In addition, the library unit 140 provides the controller 160 with information necessary to calculate the optimal sample data. The test spectrum information corresponding to the sample data is stored in the library unit 140, and the sample data for configuring the library may be fixed or added in the process of newly calculating the sample data. In addition, CD parameters may be added or changed by calculating a spectrum according to RCWA simulation under the control of the controller 160. The output unit 180 outputs the CD parameter of the model that minimizes the difference between the test spectrum calculated by the library unit 140 based on the input spectrum and the CD parameter, that is, the sample data calculated by the controller 160. That is, the controller 160 calculates sample data using a plurality of algorithms, and compares the test spectrum calculated by the library unit with the measurement spectrum and provides sample data for minimizing the difference as a CD parameter.

Using the apparatus 100 of FIG. 1, the measurement spectrum measured for each wavelength from the semiconductor device is input through the spectrum input unit 120. Using the input spectrum, the controller 160 extracts various sample data and receives a spectrum corresponding to each sample data from the library unit 140. The controller 160 calculates sample data, that is, CD parameters of the model, which minimizes the difference between the input spectrum and the spectrum corresponding to the sample data.

In more detail, the controller 160 may calculate a sample data (CD) parameter that minimizes Mean Square Error (MSE). The MSE is a square of the difference between the measured spectrum measured for each wavelength and the spectrum extracted from the library unit 140. It can be calculated as the mean of the sum.

Conventionally, the optimal sample data (CD) were derived using a Gaussian Newton method. This is a progressively stepwise movement using either the first partial differential Jacobian or the second partial differential Hessian in the direction of the gradient vector.

However, the correlation between CD parameters may be high in a pattern having a very high ratio of cross-sectional area to height such as HAR (High Aspect Ratio). In this case, there is a limit in reaching the CD parameter of the global minimum, which must be finally derived from a local minimum solution. That is, in the Gaussian Newton method, proper CD parameter derivation is not possible in a highly correlated structure such as HAR.

In addition, other search or Principal Component Regression (PCC) may be used, but this also has a limitation in obtaining accurate CD parameters in the HAR structure.

Accordingly, in the present specification, an optimal CD parameter is derived by applying CRS (Controlled Random Search).

2 is a diagram illustrating a CD calculation process according to an embodiment of the present invention. First, the measurement device measures the spectrum of the semiconductor device (S1) and provides the measured value to the CD calculation device 100. On the other hand, the library unit 140 of the CD calculation device 100 calculates the spectrum for comparison (S2). S2 extracts a sample stored in the library unit 140 according to an embodiment.

The controller 160 of the CD calculating apparatus 100 compares and analyzes the measured spectrum of S1 and the selected or calculated test spectrum of S2 (S3), and determines whether the difference MSE is smaller than a preset Tolerance. Check (S4). If the difference is greater than the set reference, the CD parameters of the controller 160 are updated again (S5), and the library unit 140 calculates a test spectrum for the updated CD parameters. (S2), the comparative analysis step (S3) is carried out.

On the other hand, if it is less than or equal to the set reference as a result of the check in S4, it is confirmed that the difference between the measurement spectrum and the test spectrum is minimized, and outputs the CD parameters calculated the test spectrum from the output unit 180 (S6).

It looks at the variables used in the process S1 to S6.

The measurement spectrum y measured at S1 is n-dimensional data and is configured as in Equation (1).

[Equation 1]

이며, 이는 p개로 구성되는 파라미터로 수학식 2와 같이 구성된다.And the CD parameters applied to calculate the test spectrum compared to the measurement spectrum

This is a parameter consisting of p number is configured as shown in equation (2).

[Equation 2]

는 2차원 행렬로 구성될 수 있으며 수학식 3과 같이 구성된다. On the other hand, as a fixed input wavelength

May be composed of a two-dimensional matrix and is configured as shown in Equation 3.

[Equation 3]

를 선택하는 것이 필요하다. 아래에서

는 에러값을 의미한다. Based on the above equation, yj can be calculated as shown in equation (4), where the difference between yj and the function f is minimized.

It is necessary to choose. From below

Means an error value.

[Equation 4]

를 샘플 데이터로 구하기 위해 수학식 5와 같이 전개될 수 있다. Equation 4 is a CD parameter that minimizes the sum of squares of the difference between the measured spectrum y and the test spectrum obtained by the model (that is, the test spectrum calculated by the library unit 140 from the sample data) f.

It can be developed as shown in Equation 5 to obtain as sample data.

를 정의한다. 이는 계측 스펙트럼과 테스트 스펙트럼의 각각의 인자들을 비교하여 이들의 차이를 산출하는 것이다. The difference between the measured spectrum y and the test spectrum f

Define. This compares the respective factors of the measurement spectrum and the test spectrum and calculates their difference.

[Equation 5]

The sum of squares of the aforementioned differences is calculated as in Equation 6.

[Equation 6]

)를 산출하기 위해 비교 및 분석을 반복한다. MSE는 파장별로 계측 스펙트럼 y(불변)와, RCWA 모델로 계산된 테스트 스펙트럼 (파라미터 값에 따라 가변)간의 차이 제곱 합의 평균인데, 이 MSE 값을 최소로 하는 파라미터(CD,

) 값을 구할 경우 계측 스펙트럼과 계산된 테스트 스펙트럼이 일치하는 형태를 띄게 된다. That is, the controller 160 may determine a parameter (CD parameter, i.e., minimize the mean square error).

Repeat the comparison and analysis to yield). The MSE is the average of the sum of squared differences between the measurement spectrum y (invariant) for each wavelength and the test spectrum (variable according to the parameter value) calculated by the RCWA model.The parameter that minimizes this MSE value (CD,

), The measurement spectra coincide with the calculated test spectra.

However, when plotting the MSE value according to various values that the parameter CD may have in the HAR pattern, it can be seen that a number of local minimums exist. Therefore, it is necessary to calculate a parameter (CD) value corresponding to the global minimum among these, so that the exact profile of the HAR pattern can be measured.

2, when the spectrum input unit 120 receives a measurement spectrum measured for each wavelength from a semiconductor device (S1), the controller 160 including the simplex algorithm execution unit and the DE (Differential Evolution) algorithm execution unit includes these. The library unit 140 calculates a test spectrum corresponding to the sample data calculated by any one of the two algorithm execution units (S3). The controller 160 repeats the process of calculating sample data for minimizing the difference by comparing the measurement spectrum and the test spectrum (S2 to S5), and finally the controller 160 outputs the sample data for calculating the test spectrum of the minimum difference. The output unit 180 outputs the information to the unit 180 (S6).

3 is a diagram illustrating a relationship between a CD parameter and an MSE according to an embodiment of the present invention. 11 shows the MSE calculated for each parameter. As a result of selecting the CD parameter, the MSE converges from 11a through 11b and 11c to the global optimal solution with the smallest MSE, such as 11d. Parameters can be selected through a heuristic algorithm to produce a CD parameter that converges to the optimal global solution.

12 is the relationship between the spectrum and the wavelength. The dotted line 12a is the spectrum when the MSE is not a global optimal solution. That is, the spectrum calculated using the CD parameter in the library unit 140 is different from the exact spectrum 12b of the actual model. However, when the CD parameter is calculated to reduce the MSE as shown in the graph of 11 and reaches the global optimal solution as shown in 11d, as a result, as shown in 12, the points represented by the multiple circles corresponding to the exact spectrum 12b of the model are shown. By calculating the correct CD parameters.

4 is a graph showing local and global optimal solutions. The local optimal solution yields the minimum MSE when compared with the adjacent left and right parameters in the area indicated by 13. In this state, the parameter at the point indicated by 13 can be determined as the minimum value. The controller 160 of the present invention may apply a heuristic algorithm to calculate a parameter corresponding to the global optimal solution indicated by 14.

That is, the controller 160 of FIG. 1 combines a simplex method having a fast convergence speed in the direction of a solution and a heuristic algorithm called differential evolution suitable for obtaining a global optimal solution.

5 is a diagram illustrating a process of performing a CRS by a controller according to an embodiment of the present invention. The controller 160 may repeat the process of FIG. 5 in order to derive a parameter corresponding to the smallest global best solution using the sample provided by the library unit 140 or newly generated.

를 하나 또는 다수 포함하며, 이들은 수학식 4와 같이 테스트 스펙트럼을 계산하는데 입력될 수 있다. The controller 160 randomly extracts N pieces of sample data in the D-dimensional space to generate a set P (S21). Here, according to an embodiment, N sample data of a D-dimensional space is to calculate a large number of p parameters, as shown in Equation 2. The set P may include some or all of the sample data.

And one or more, which may be input to calculate a test spectrum as shown in Equation 4.

를 탐색한다(S22). 일 실시예로, 계측한 스펙트럼 y와 모델로 얻은 테스트 스펙트럼 f의 차이 또는 차이의 최대값을 가지는 샘플 데이터를

로 한다. 그리고 컨트롤러(160)는 휴리스틱 방법(후술할 심플렉스 알고리즘 또는 DE 알고리즘 등)에 기반하여 새로운 샘플 데이터

를 생성한다(S23).

는 샘플 데이터가 되는 CD 파라미터로

를 일 실시예로 한다. Controller 160 is the sample data in set P

Search for (S22). In one embodiment, the sample data having a difference or maximum value of the difference between the measured spectrum y and the test spectrum f obtained by the model

Shall be. In addition, the controller 160 generates new sample data based on a heuristic method (such as a simplex algorithm or a DE algorithm to be described later).

To generate (S23).

Is the CD parameter that is the sample data.

In one embodiment.

A method of generating new sample data will be described later with reference to FIG. 6.

와 새로운 샘플 데이터

를 함수 f에 입력하여 새로운 샘플

가 더 작은 차이값(MSE, 또는 수학식 5와 같은 차이값)을 산출할 경우

에

를 입력한다(S24). 이로써 새로운 샘플의 선택 과정에서 최대값이 줄어듦으로 인해 전역 최적해에 도달할 가능성을 높인다. 이는 앞서 산출한 값 보다 더 작기에 전역 최적해에 더 가까워진 샘플을 선택하는 것을 의미한다. Afterwards, the new sample and the S22

And new sample data

Into the function f to create a new sample

Yields a smaller difference (MSE, or a difference such as Equation 5)

on

Enter (S24). This reduces the maximum value during the selection of new samples, increasing the likelihood of reaching the global optimal solution. This means selecting samples that are closer to the global optimum because they are smaller than previously calculated.

를 함수 f에 입력한 결과인 테스트 스펙트럼을 산출하여 이를 계측한 스펙트럼 y와의 차이를 좁히는 과정을 S23이 구현한다. S23 means that the maximum value is set to the sample when the function value in which the sample is input is smaller than the function value of the maximum value. Earlier in Equation 2

S23 calculates a test spectrum, which is a result of inputting the function f, to narrow the difference from the measured spectrum y.

를 최대값으로 설정하여 이후 더 작은 스펙트럼을 산출하는 CD 파라미터를 샘플 데이터로 구하는 과정을 반복한다. 이는 이전에 산출된 최소 차이값 보다 작은 테스트 스펙트럼-계측 스펙트럼의 차이가 확인되면, 이 테스트 스펙트럼을 산출한 샘플 데이터인

를

로 하고 S22~S24를 반복하여

에서 산출된 테스트 스펙트럼과 계측 스펙트럼의 차이를 점차 줄여나간다. As a result, when the test spectrum which narrows the difference with the measured spectrum y rather than the value calculated so far is calculated, the corresponding test spectrum is calculated.

Is set to the maximum value, and then the process of obtaining CD parameters as sample data which yields a smaller spectrum is repeated. This means that if the difference between the test spectrum and the measurement spectrum that is smaller than the previously calculated minimum difference value is found,

To

And repeat S22 ~ S24

Gradually reduce the difference between the test spectrum and the measurement spectrum.

The repetition of the calculation may be controlled by using a preset termination condition, and the termination condition may be variously set. In one embodiment, the end condition is a case where the minimum MSE set for calculating the optimal model is reached, and when the condition is reached, the repetition is stopped and the corresponding parameter may be identified as a parameter of the optimal model.

In this case, the controller 160 includes two or more algorithm execution units in S23 to effectively calculate a difference value, that is, a CD parameter for calculating a test spectrum for reducing MSE. The algorithm execution unit includes an algorithm implemented in software code or hardware signal processing to calculate a predetermined output value by applying an input value for a predetermined algorithm.

6 is a view showing a detailed configuration of a controller according to an embodiment of the present invention. The controller 160 processes inputted information and calculates a predetermined result, and may be composed of software code or hardware that calculates the aforementioned result. The controller 160 includes R algorithm execution units 161 to 165.

These algorithm execution units can each calculate a predetermined sample. The controller 160 may select one of the sample data calculated by the algorithm execution units 161 to 165. Alternatively, the sample data related to the test spectrum for calculating the smallest MSE may be selected by performing S24 of FIG. 5 on all of these samples, and the controller 160 may increase the priority of the algorithm execution unit that has calculated the sample data. . The priority may be increased to increase the probability of selecting sample data generated by a specific algorithm execution unit.

The algorithm execution unit that may be included in the controller 160 may be implemented by various algorithms. Examples of function optimization techniques include various algorithms such as the GD (Gradient Descent Method), Newton-Raphson Method, Gauss-Newton Method, and LM (Levenberg-Marquardt Method). Can be. In this specification, the simplex & DE algorithm is applied as an optimal algorithm execution unit for calculating the global optimal solution in the HAR pattern.

The controller 160 may include a simplex algorithm execution unit and a differential evolution algorithm (DE) execution unit. The library 140 described above calculates a test spectrum in response to the sample data calculated by any one of these two algorithm execution units. The controller 160 may repeat the process of calculating the sample data to minimize the difference by comparing the measurement spectrum and the test spectrum. The iterative process may stop the iteration when a minimum difference below a predetermined criterion is identified.

In FIG. 6, the first algorithm execution unit 161 may be a first simplex algorithm execution unit set to the first parameter. According to an embodiment of the present invention, the second algorithm execution unit 162 becomes a second simplex algorithm execution unit set to the second parameter. According to an embodiment of the present invention, the third algorithm execution unit 163 may be a third simplex algorithm execution unit set to the third parameter. Here, the first parameter is a parameter required for executing the simplex algorithm, and the variable elements may be set to different parameters to generate new sample data. Similarly, the second parameter and the third parameter can be set. Of course, the number of simplex algorithm execution units may be configured in one or more ways, but the present invention is not limited thereto.

In an embodiment, the fourth algorithm execution unit 165 may be a differential evolution algorithm (DE) execution unit.

7 to 10 are views showing a detailed configuration of a controller according to an embodiment of the present invention. In FIG. 7, the controller 160 includes a simplex algorithm execution unit 161a implementing the simplex algorithm and a DE algorithm execution unit 165a implementing the DE algorithm. This may be configured as shown in FIG. 7. In the configuration as shown in FIG. 7, the controller 160 may select any one of the plurality of algorithm execution units using a probability proportional to a previous success rate.

That is, the controller 160 may select from various algorithm execution units that implement various heuristic algorithms in the process of searching for new sample data, and select which algorithm execution unit to implement the heuristic methodology with a probability proportional to the success rate. You can decide. The determination method may be variously calculated, such as using a success rate accumulated for a predetermined number of times and using a success rate of the last round. The success rate accumulated for a certain number of times is accumulated in the corresponding number so that the algorithm execution unit having the highest success rate can be selected. This ensures that even if the algorithm execution fails at any one time, the success rate is not set too low, based on previous successful cases. The success rate of the last round may apply the latest success rate. Decision methods based on the two success rates can be mixed or selected depending on the characteristics of the algorithm or the speed or magnitude of the success rate.

를 선택한다(S32). 그리고 이들 선택한 결과를 이용하여 S33과 같이 새로운 샘플 데이터

를 산출한다.An execution method of the simplex algorithm execution unit 161a will be described with reference to FIG. 8. The simplex algorithm execution unit 161a may provide one or more heuristic algorithms based on Nelder-Mead. The simplex algorithm execution unit 161a selects D + 1 samples in P sets (S31). And out of these D + 1 samples

Select (S32). And using these selected results, new sample data like S33

To calculate.

을 산출하는 과정을 제시한다. In S33 in Figure 9

Present the process of calculating.

에서 산출된 테스트 스펙트럼과 계측 스펙트럼의 차이가, 앞서

서 산출된 테스트 스펙트럼과 계측 스펙트럼의 차이 보다 작을 경우,

를

로 대입한다(S34).Afterwards, new sample data

The difference between the test spectrum and the measurement spectrum

Is less than the difference between the test spectrum and the measurement spectrum

To

Substitute in (S34).

9 is a diagram illustrating a process of selecting sample data by a simplex algorithm execution unit according to an embodiment of the present invention. The simplex algorithm execution unit 161a calculates the sample data arranged at the reflection point by using the center point of one or more sample data except for the selected sample data among the sample data selected by the library unit 140. Repeat the sample data calculation process. This process generates CD parameters that minimize the difference between the test spectrum and the measurement spectrum.

는 심플렉스를 정의하는 점들 중 최대값을 가지는

를 지시한다. 그리고

은

를 제외한 나머지 점들의 중심점이다. 이들을 이용하여

을 기준으로 생성한

의 반사점을 새로운 샘플

로 산출한다. 이들의 산출은 수학식 7과 같다.In Figure 9 we indicate samples in the D dimension as possible solutions that exist in the problem space. Meanwhile

Is the maximum of the points defining the simplex

To indicate. And

silver

The center point of the remaining points except for. Using them

Generated based on

The reflection point of the new sample

Calculate These calculations are shown in equation (7).

[Equation 7]

)는

과

간의 반사 거리를 조절해준다. 이전에 산출된 샘플에 따라 반사계수를 설정할 수 있다. 또한, 컨트롤러(160)가 다수의 심플렉스 알고리즘 실행부를 포함할 때, 반사계수를 파라미터로 달리 설정하여 다수의 심플렉스 알고리즘 실행부를 구성할 수 있다. Where reflection coefficient (

)

and

Adjust the reflection distance between the liver. The reflection coefficient can be set according to the sample previously calculated. In addition, when the controller 160 includes a plurality of simplex algorithm execution units, a plurality of simplex algorithm execution units may be configured by differently setting reflection coefficients as parameters.

8 and 9, the simplex algorithm execution unit 161a is applicable when the objective function is a non-linear function and a non-differentiable form. As shown in FIG. 9, the speed of convergence in the direction of the solution is high. On the other hand, there is a disadvantage in that it is easy to fall into the regional optimum solution. Accordingly, the controller 160 may implement the speed and the optimum in calculating the sample parameters by applying the DE algorithm execution unit 165a suitable for calculating the global optimal solution. The simplex algorithm may configure a plurality of execution units by diversifying algorithm setting values such as reflection coefficients. That is, the simplex algorithm execution unit 161a may calculate two or more different samples by executing algorithms configured with two or more different settings.

라 하고 이를 나머지 D개의 샘플들의 중심점을 기준으로 반사한다. 이때 반사란 기준점에 대한 대칭이동을 하는 것을 의미하며, 심플렉스 알고리즘 실행부(161a)는 반사계수를 통해 얼마나 이동할지를 결정한다. 반사된 점(

)을 새로운 샘플 데이터로 한다. 반사계수는 ρ ~ Unif(s,α-s)으로 산출될 수 있다. The simplex algorithm execution unit 161a randomly selects D + 1 samples in the P set when the dimension of x constituting the calculated spectrum is D. The sample with the highest function value (ie, the worst, farthest from the best) among the selected points

This is reflected based on the center point of the remaining D samples. In this case, the reflection means symmetrical movement with respect to the reference point, and the simplex algorithm execution unit 161a determines how much to move through the reflection coefficient. Reflected points (

) As new sample data. The reflection coefficient may be calculated from ρ to Unif (s, α-s).

10 and 11 look at the operation process of the DE algorithm execution unit according to an embodiment of the present invention. The DE algorithm execution unit 165a calculates a better optimal solution by changing some parameters or performing crossover in calculating data in D dimensions. Shows the generation of D-dimensional parameters in each generation.

는 G+1 세대에서의 파라미터로, D개의 차원 중 j번째 차원의 파라미터를 지시한다. 이는 도 11에 보다 상세히 제시된다.

Is a parameter in the G + 1 generation and indicates a parameter of the j-th dimension of the D dimensions. This is shown in more detail in FIG. 11.

로 구성하여 새로운 샘플

에 의해 산출된 테스트 스펙트럼과 계측 스펙트럼 사이의 차이가 이전에 산출된 차이(

에 의해 산출된 테스트 스펙트럼과 계측 스펙트럼의 차이)보다 더 작은 값을 산출할 경우

에

를 입력한다(S48).The DE algorithm execution unit 165a generates a random integer variable l in the D dimension. The probability constant l is a value selected from integers 1 to D. And j starts at 1 and increases by 1 in S45 or S46. When j exceeds the D dimension, the parameters of the generation are calculated, and the calculated parameters (ie, sample data) are replaced with new sample data such as S47.

Organize new samples with

The difference between the test spectrum and the measurement spectrum calculated by

Yields a value smaller than the difference between the test spectrum and the measurement spectrum calculated by

on

Enter (S48).

을 생성한다(S43). 그리고 Uj가 C 보다 작거나 같거나, 혹은 j가 확률 정수 변수 l에 도달한 경우(S44)에 파라미터

를 새로이 뮤테이션(mutation)된 값(

)설정하고 j를 증가시킨다(S45). 반면 S44를 만족시키지 못하는 경우, 파라미터

는 이전 세대의 파라미터 값인

을 입력하고 j를 증가시킨다(S46). On the other hand, if j does not exceed the D dimension (S42), since CD parameters have not been completely calculated yet, the subsequent step is continued to calculate CD parameters. As a result of random random variables

To generate (S43). If Uj is less than or equal to C, or j reaches the random integer variable l (S44)

Is the newly mutated value (

Set and increase j (S45). On the other hand, if it does not satisfy S44, the parameter

Is the parameter value of the previous generation

Input and increase j (S46).

를 산출하는 방식은 수학식 8과 같다.From S45

The method of calculating is as shown in Equation (8).

[Equation 8]

은 집합 P에서 무작위로 선택한 3개의 샘플 데이터이다. F는 변이계수로, P 집합내의 함수의 최대값과 최소값의 차이가 크면 클수록 샘플 데이터를 더 먼 곳에서 생성하도록 하며, 작으면 작을수록 인접한 곳에서 생성하도록 한다. 또한 변이계수에서 너무 가까운 영역에서만 샘플 데이터를 생성하는 것을 방지하기 위해 입력변수로 F_min 값을 추가로 입력변수로 설정할 수 있다. here

Is three sample data randomly selected from the set P. F is a coefficient of variation. The larger the difference between the maximum value and the minimum value of the function in the P set is, the larger the sample data is generated, and the smaller the smaller the value is generated in the adjacent region. In addition, in order to prevent generating sample data only in a region too close to the coefficient of variation, an F_min value may be additionally set as an input variable.

Meanwhile, S46 may be one of crossover methods. That is, setting the parameter value of the previous generation to the parameter value of the new generation.

The DE algorithm execution unit 165a can obtain the global optimal solution when the objective function is a nonlinear function and an indivisible form. It is possible to approach the global optimal solution by intersecting or mutating the D-dimensional parameters from generation to generation. In addition, when parallel computing is applied, computation time can be shortened.

11 is a diagram illustrating a process of calculating a parameter by a DE algorithm execution unit according to an embodiment of the present invention. 51 shows that D-dimensional parameters are calculated in the first generation (Generation 1). The parameters of the first generation may be randomly calculated or selected from predetermined samples. One test spectrum is calculated using the first generation D-dimensional CD parameters, compared with the measurement spectrum, and the next generation CD parameters are generated.

52 shows that D-dimensional CD parameters are calculated in the second generation (Generation 2). Each parameter of the second generation may be calculated by using either the mutation method S45 or the crossover method S46 of Equation 8 by applying the process of FIG. 10. Mutation coefficient F is used in the muting method, and crossover coefficient C is calculated based on the crossover method. This is summarized in Equation (9).

[Equation 9]

)은 이전 세대의 파라미터(

) 또는 이전 세대의 파라미터들에서 산출된 값(

)을 이용하여 산출될 수 있다. The next generation of parameters (

) Is the previous generation of parameters (

) Or the value calculated from the previous generation of parameters (

Can be calculated using

10 and 11, the DE algorithm execution unit 165a calculates a parameter of the (k + 1) th generation in a crossover manner by using the parameters constituting the sample data of the kth generation or the previous generation. Alternatively, the DE algorithm execution unit 165a calculates a parameter of the (k + 1) th generation in a mutated manner by calculating a parameter randomly selected from the previous generation including the library unit 140 or the kth generation. When the entire parameter of the (k + 1) th generation is obtained, this becomes sample data that is a candidate for the CD parameter. The DE algorithm execution unit 165a calculates a test spectrum using the sample data, and compares the result of calculating a difference from the measurement spectrum as shown in Equations 4 and 5 below. If the comparison result is smaller than the previous minimum value, the success value of the DE algorithm increases because the (k + 1) th generation parameter is successful.

12 is a diagram illustrating a probabilistic selection process of an algorithm according to an embodiment of the present invention.

The controller 160 uses both the simplex algorithm execution unit 161a and the DE algorithm execution unit 165a, but can probabilistically select any one of these two execution units. Alternatively, when the simplex algorithm execution unit 161a provides a plurality of algorithms in a plurality of settings, the controller 160 may probabilistically select any one of three or more algorithms.

와 지금까지 확인된

를 비교한 결과 S55와 같이

를 만족하는 경우, 해당 알고리즘은 샘플 데이터를 생성함에 있어서 성공한 것으로 판단한다. Probably one of the heuristic algorithm selected by the simplex algorithm execution unit 161a and the heuristic algorithm selected by the DE algorithm execution unit 165a is selected to generate a new sample existing in the D-dimensional space. And the sample data you created

And confirmed so far

As a result of comparing S55

If satisfies the, the algorithm determines that the sample data is successful in generating sample data.

를 생성함에 있어 선택한 알고리즘(i번째 알고리즘)의 성공 가중치 및 누적 가중치를 갱신하고, 이에 기반하여 각각의 알고리즘의 선정 확률을 갱신한다(S56). 즉, 성공적인 테스트 스펙트럼의 산출에 대응하는 샘플 데이터를 산출한 해당 알고리즘(심플렉스 또는 DE 등)의 성공 가중치를 증가시켜 해당알고리즘의 신뢰도를 높인다. And, the controller 160 is sample data

In generating the S1, the success weight and the cumulative weight of the selected algorithm (i th algorithm) are updated, and the selection probability of each algorithm is updated (S56). That is, the reliability of the algorithm is increased by increasing the success weight of the algorithm (Simplex or DE, etc.) for which sample data corresponding to the successful test spectrum is calculated.

Thereafter, the controller 160 selects an algorithm based on a selection probability of the algorithm and generates the next sample (S57).

Looking at S56 in more detail, the success weight of the i th heuristic algorithm is defined as w_i, the cumulative sum of the success weights of the i th heuristic algorithm is defined as W_i, and the probability that the i th heuristic algorithm is selected as q_i.

F_max and f_min are defined as a maximum f (x) value and a minimum f (x) value in the set P. Based on this, w_ (0, i) is set as an input variable as the initial weight of the i th algorithm.

Also, if δ is an input variable, all values are initialized when any one of the algorithms' selection probabilities becomes smaller than δ.

This initial setting is performed, and S55 to S57 are performed to increase or decrease the success weight of each algorithm. The increased and decreased values are accumulated in the cumulative sum of success weights. When the algorithm is selected in S55, a specific algorithm may be selected by reflecting the success weight and the cumulative weight. In addition, as described above, the success weight can be reset as a whole when the selection probability of any algorithm is smaller than a preset value. However, in this process, the cumulative weight may be maintained without resetting.

Based on this process, the controller 160 updates the selection probability of each algorithm. When the current algorithm is successful, the better samples are obtained based on the maximum and minimum values in the P set, and the greater the success weight is, the more the success weights are summed up to update the probability that the algorithm is selected as the ratio of the weighted sums. do.

The probability of selection of an algorithm that produces a consistently successful sample is updated high. In addition, by calculating the success rate of two or more heuristic algorithms in the process of generating every sample, and selecting the successful algorithm in the next process, the point having the lowest success rate can be effectively removed. Therefore, compared to the existing Nested Partition algorithm, both speed and performance are improved.

As mentioned above, although an embodiment of the present invention has been described, those of ordinary skill in the art may add, change, delete or add elements within the scope not departing from the spirit of the present invention described in the claims. The present invention may be modified and changed in various ways, etc., which will also be included within the scope of the present invention.

100: device
120: spectrum input unit
140: library unit
160: controller
180: output unit

Claims (12)

A spectrum input unit which receives a measurement spectrum measured for each wavelength from the semiconductor device;
A library unit storing sample data for calculating a test spectrum for comparison with the measurement spectrum;
A simplex algorithm execution unit and a DE (Differential Evolution) algorithm execution unit are included, and sample data calculated by one of these two algorithm execution units is selected to calculate a test spectrum, and the measurement spectrum is compared with the test spectrum to make a difference. A controller for repeating a process of calculating sample data to minimize the number of samples; And
And an output unit configured to output sample data in which the controller calculates a test spectrum of a minimum difference.
The method of claim 1,
And the controller is configured to calculate the sample data to minimize the sum of squares of the difference between the measurement spectrum and the test spectrum calculated by the library unit in the sample data.
The method of claim 2,
The controller is a value that maximizes the square of the difference among sample data of the library unit.

Yields,
The controller calculates a test spectrum using sample data calculated by the simplex algorithm execution unit or the DE algorithm execution unit.
If the difference between the test spectrum and the measurement spectrum is less than the minimum difference value that was previously calculated, the controller determines that

And substitute the calculated sample data into.
The method of claim 3,
And the controller increases a success weight of an algorithm execution unit that has calculated sample data corresponding to the test spectrum.
The method of claim 1,
And the simplex algorithm execution unit calculates sample data arranged at a reflection point using a center point of one sample data among the sample data selected by the library unit and at least two other sample data except for the sample data.
The method of claim 1,
The DE algorithm execution unit calculates a parameter of the (k + 1) th generation in a crossover manner by using a parameter constituting sample data of the kth generation or the previous generation, or includes the library unit or the kth generation. A device for calculating a (k + 1) th generation parameter in a mutated manner by calculating a randomly selected parameter from a previous generation.
Receiving, by the spectrum input unit, a measurement spectrum measured for each wavelength from the semiconductor device;
Selecting, by a controller including a simplex algorithm execution unit and a differential evolution (DE) algorithm execution unit, sample data calculated by any one of these two algorithm execution units;
A library unit calculating a test spectrum corresponding to the selected sample data;
Repeating, by the controller, comparing the measurement spectrum with the test spectrum and calculating sample data that minimizes the difference; And
And outputting, by the controller, the output of the output unit by supplying sample data on which the test spectrum of the minimum difference is calculated, to the output unit.
The method of claim 7, wherein
The repeating step is
And calculating, by the controller, the sample data that minimizes the sum of squares of the difference between the measurement spectrum and the test spectrum calculated by the library unit in the sample data, based on a control random search. How to calculate.
The method of claim 8,
The repeating step is
The controller maximizes the square of the difference among sample data of the library unit.

Calculating;
Calculating a test spectrum by the controller using sample data calculated by the simplex algorithm execution unit or the DE algorithm execution unit; And
If the difference between the test spectrum and the measurement spectrum is less than the minimum difference value that was previously calculated, the controller determines that

And substituting the calculated sample data into a CD of an optimal model based on a control random search.
The method of claim 9,
And increasing, by the controller, a success weight of an algorithm execution unit that has calculated sample data corresponding to the test spectrum, based on a control random search.
The method of claim 7, wherein
The simplex algorithm execution unit further comprises the step of calculating the sample data disposed at the reflection point using the center of one of the sample data selected from the library unit and at least two other sample data except the sample data; To calculate the CD of the optimal model based on the control random search.
The method of claim 7, wherein
The DE algorithm execution unit calculates a parameter of the (k + 1) th generation in a crossover manner using the parameters constituting the sample data of the kth generation or the previous generation, or includes the library unit or the kth generation. Calculating a parameter of the (k + 1) th generation in a variance manner by calculating a randomly selected parameter from the previous generation.

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