KR102548410B1 - Device of critical dimension based on controlled random search and method implementing thereof - Google Patents

Abstract

The present invention relates to an apparatus for calculating a CD of an optimal model based on a controlled random search and a method for implementing the same. An apparatus according to an aspect of the present invention includes a spectrum input unit for receiving measurement spectrum measured for each wavelength from a semiconductor device, and a measurement spectrum. A library unit storing sample data for calculating a test spectrum for comparison with a spectrum, a simplex algorithm execution unit, and a DE (Differential Evolution) algorithm execution unit, and corresponding to sample data calculated from either of these two algorithm execution units. A controller that repeats the process of controlling the test spectrum to be calculated in the library unit, comparing the measurement spectrum and the test spectrum to calculate sample data that minimizes the difference, and outputting the sample data from which the controller calculates the test spectrum with the minimum difference. Include an output section.

Description

Apparatus for calculating CD of optimal model based on control random search and method for implementing the same

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

In the process of manufacturing a semiconductor, patterns of structures disposed on semiconductor devices may be measured to check physical characteristics of the semiconductor. In order to confirm the pattern, a critical dimension (CD) of the fine pattern of the structures disposed on the device may be calculated.

As a method of modeling a CD parameter of a semiconductor, a measurement spectrum measured in the semiconductor is compared with a test spectrum corresponding to the CD parameter modeled for the semiconductor. In addition, a method of estimating a CD parameter of a semiconductor through a CD parameter corresponding to the test spectrum that minimizes a difference between the measured spectrum and the test spectrum calculated in the comparison process may be used.

To this end, a technique for quickly and accurately estimating a semiconductor spectrum by calculating a CD parameter capable of modeling a spectrum measured in a semiconductor from a statistical model is required.

SUMMARY OF THE INVENTION An object of the present invention is to calculate a test spectrum close to a spectrum measured from a semiconductor device and to calculate a CD parameter corresponding thereto.

Another object of the present invention is to generate sample data based on controlled random search in order to increase calculation speed and accuracy in calculating CD parameters.

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

An apparatus according to an aspect of the present invention for solving the above-described technical problem includes a spectrum input unit for receiving a measurement spectrum measured for each wavelength from a semiconductor device, a library unit storing sample data for calculating a test spectrum for comparison with the measurement spectrum, It includes a simplex algorithm execution unit and a DE (Differential Evolution) algorithm execution unit, and controls the library unit to calculate a test spectrum corresponding to the sample data calculated by either of these two algorithm execution units, and calculates the measurement spectrum and the test spectrum. It includes a controller that repeats the process of calculating sample data that minimizes the difference by comparison, and an output unit that outputs the sample data from which the controller has calculated the test spectrum of the minimum difference.

According to an aspect of the present invention, a method for calculating a CD of an optimal model based on control random search includes the steps of receiving a measurement spectrum measured for each wavelength from a semiconductor device by a spectrum input unit, a simplex algorithm execution unit and DE (Differential Evolution) A controller including an algorithm execution unit calculates sample data from any one of these two algorithm execution units, a controller calculates a test spectrum corresponding to the selected sample data from a library, and the controller compares the measured spectrum and the test spectrum. and repeating the process of calculating sample data that minimizes the difference, and providing the sample data for which the test spectrum of the minimum difference is calculated by the controller to an output unit, and outputting the sample data to the output unit.

According to the present invention, a model CD parameter of a semiconductor device can be accurately and quickly estimated by calculating a test spectrum close to a spectrum measured from a semiconductor device and calculating a CD parameter corresponding thereto.

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

Certain terms are defined herein for convenience in order to more readily understand the present invention. Unless otherwise defined herein, scientific terms and technical terms used in the specification may have meanings commonly understood by those skilled in the art.

In addition, unless otherwise specified in context, singular terms include their plural forms, and plural terms may also include their singular forms.

Statistical optimization algorithms can be applied when measuring the CD (Critical Dimension) value using OCD (Optical Critical Dimension) equipment to identify the profile of a semiconductor device, for example, a semiconductor chip. can

In this specification, the CD parameter of a model that minimizes the difference between the spectrum measured using OCD equipment and the spectrum obtained from the Rigorous Coupled Wave Analysis (RCWA) model is estimated as the CD parameter of the actually measured spectrum. To this end, the CRS (Controlled Random Search) method is applied in the process of finding the optimized CD parameters from the model.

When the aforementioned CRS is applied, in the present specification, an appropriate CD parameter can be calculated even in a high aspect ratio pattern (HAR pattern) having a high correlation between parameters (CD values). That is, in this specification, CRS can be applied to be most suitable for measurement.

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

The apparatus 100 for calculating the CD parameter outputs a calculated result using predetermined input information and stored information. Device 100 includes a laptop, computer, or other device capable of executing and implementing algorithms. Alternatively, the device 100 of FIG. 1 includes a computing device that only performs an operation of calculating a CD parameter. The device 100 may include software components or hardware components as detailed components.

The configuration of the device is composed of a spectrum input unit 120, a library unit 140, an output unit 180, and a controller 160 that controls them and calculates an optimal CD parameter. The controller includes at least one of a software code implementing a predetermined optimization algorithm or a hardware chip implementing the same.

The library unit 140 serves to return a function value (test spectrum) corresponding to the corresponding sample data in calculating sample data (CD values, CD parameter) in Controlled Random Search (CRS). In addition, the library unit 140 provides the controller 160 with information necessary for calculating optimal sample data. Test spectrum information corresponding to the sample data is stored in the library unit 140, and sample data for constructing the library may be fixed or may be 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 a 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, compares the test spectrum calculated by the library unit with the measured spectrum, and provides sample data minimizing the difference as a CD parameter.

Using the device 100 of FIG. 1 , a 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 returns a spectrum corresponding to each sample data from the library unit 140 . The controller 160 calculates sample data that minimizes the difference between the input spectrum and the spectrum corresponding to the sample data, that is, the CD parameter of the model.

In more detail, the controller 160 may calculate a sample data (CD) parameter that minimizes MSE (Mean Square Error). The MSE is the 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 average of the sum.

Conventionally, optimal sample data (CD) was derived using the Gauss-Newton Method. This is a method of gradually moving as much as a step using 　 first-order partial derivative Jacobian or second-order partial derivative Hessian in the direction of the gradient vector.

However, correlation between CD parameters may be high in a pattern having a structure having a very large ratio of height to cross-sectional area, such as HAR (High Aspect Ratio). In this case, a local minimum occurs, and there is a limit in reaching the CD parameter of the global minimum that is finally to be derived. That is, it is impossible to derive a suitable CD parameter in a structure with high correlation such as HAR using the Gaussian Newton method.

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

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

2 is a diagram showing a CD calculation process according to an embodiment of the present invention. First, the measuring device measures the spectrum of the semiconductor device (S1) and provides the measured value to the CD calculating device 100. Meanwhile, the library unit 140 of the CD calculating device 100 calculates a spectrum for comparison (S2). S2 extracts samples stored in the library unit 140 as an embodiment.

The controller 160 of the CD calculation device 100 compares and analyzes the measured spectrum of S1 and the selected or calculated test spectrum of S2 (S3) to determine whether the difference (MSE) is smaller than a preset standard (Tolerance). Confirm (S4). And, as a result of checking, if the difference is greater than the set standard, the CD parameters of the controller 160 are updated again (S5), and the library unit 140 calculates the test spectrum for the updated CD parameters. (S2), the comparative analysis step (S3) proceeds.

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

Examine the variables used in steps S1 to S6.

The measurement spectrum y measured in S1 is n-dimension data and is configured as in Equation 1.

[Equation 1]

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

, which is configured as in Equation 2 with p parameters.

[Equation 2]

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

Can be composed of a two-dimensional matrix and is composed of Equation 3.

[Equation 3]

를 선택하는 것이 필요하다. 아래에서

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

It is necessary to select 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 in Equation 5 to obtain the sample data.

를 정의한다. 이는 계측 스펙트럼과 테스트 스펙트럼의 각각의 인자들을 비교하여 이들의 차이를 산출하는 것이다. First, the difference between the measured spectrum y and the test spectrum f obtained by the model is

define This is to compare each factor of the measurement spectrum and the test spectrum to calculate the difference between them.

[Equation 5]

Then, the sum of the squares of the above 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 is a parameter (CD parameter, i.e., a parameter that minimizes MSE (Mean Square Error)).

), the comparison and analysis are repeated to calculate MSE is the average of the sum of squares of differences between the measured spectrum y (invariant) for each wavelength and the test spectrum calculated by the RCWA model (variable according to the parameter value). The parameter (CD,

) value, the measurement spectrum and the calculated test spectrum have the same form.

However, when MSE values are plotted according to various values that the parameter (CD) can have in the HAR pattern, it can be seen that there are many local minimums. Therefore, an accurate profile of the HAR pattern can be measured only when a parameter (CD) value corresponding to the global minimum is calculated.

Summarizing the process of FIG. 2, when the spectrum input unit 120 receives the measurement spectrum measured for each wavelength from the semiconductor device (S1), the controller 160 including a simplex algorithm execution unit and a DE (Differential Evolution) algorithm execution unit The library unit 140 calculates a test spectrum corresponding to the sample data calculated by one of the two algorithm execution units (S3). The controller 160 compares the measurement spectrum and the test spectrum to calculate sample data that minimizes the difference (S2 to S5), and finally outputs the sample data for which the controller 160 has calculated the test spectrum with the minimum difference. Provided to the unit 180, the output unit 180 outputs (S6).

3 is a diagram showing the relationship between CD parameters and 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 to the global optimal solution with the minimum MSE, such as 11d through 11a, 11b, and 11c. Parameters may be selected through a heuristic algorithm in order to calculate a CD parameter that converges to an optimal global solution.

12 is the relationship between spectrum and wavelength. The dotted line 12a is the spectrum when the MSE is not the global optimum. That is, the spectrum calculated by using the CD parameters in the library unit 140 is different from the accurate spectrum 12b of the actual model. However, as shown in the graph of Figure 11, if the CD parameters are calculated to reduce the MSE and the global optimum is reached as shown in Figure 11d, as a result, as shown in Figure 12, the points indicated by a number of circles corresponding to the exact spectrum 12b of the model Calculate the exact CD parameters.

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

That is, the controller 160 of FIG. 1 combines a simplex method with a high 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 CRS by a controller according to an embodiment of the present invention. The controller 160 may repeat the process of FIG. 5 to derive a parameter corresponding to a global optimal solution having the smallest MSE using samples provided from 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 generates a set P by randomly extracting N sample data from the D-dimensional space (S21). Here, N sample data in the D-dimensional space is an embodiment in which a plurality of parameters consisting of p are calculated as shown in Equation 2. Set P may include some or all of the sample data.

It includes one or more, and they can be input to calculate the 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 samples data from set P.

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

do it with In addition, the controller 160 performs new sample data based on a heuristic method (such as a simplex algorithm or DE algorithm, which will be described later).

is generated (S23).

is the CD parameter to be the sample data.

as an example.

A method of generating new sample data will be described below 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. After that, a new sample and S22

and new sample data

to the function f to create a new sample

If yields a smaller difference value (MSE, or difference value such as Equation 5)

to

Enter (S24). This increases the probability of reaching the global optimum solution due to the reduction of the maximum value in the process of selecting a new sample. This means selecting a sample that is closer to the global optimal solution because it is smaller than the previously calculated value.

를 함수 f에 입력한 결과인 테스트 스펙트럼을 산출하여 이를 계측한 스펙트럼 y와의 차이를 좁히는 과정을 S23이 구현한다. S23 means that if the function value inputting the sample is smaller than the function value of the maximum value, the corresponding maximum value is set as the sample. In Equation 2 above,

S23 implements the process of calculating the test spectrum, which is the result of inputting to the function f, and narrowing the difference with 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, if a test spectrum that narrows the difference with the measured spectrum y is calculated than the value calculated so far, the test spectrum calculated

is set to the maximum value, and then the process of obtaining the CD parameter that yields a smaller spectrum with sample data is repeated. This means that if a test spectrum-measurement spectrum difference smaller than the previously calculated minimum difference value is confirmed, the sample data that produced this test spectrum

cast

and repeating S22 to S24

The difference between the test spectrum and the measurement spectrum calculated in is gradually reduced.

It is possible to control repetition of the calculation using a preset end condition, and the end condition may be set in various ways. In an embodiment, a case where a minimum MSE set to calculate an optimal model is reached is set as an end condition, and when this condition is reached, repetition is stopped and the corresponding parameter can be checked as a parameter of the optimal model.

Here, the controller 160 includes two or more algorithm execution units in S23 to effectively calculate a CD parameter for calculating a difference value, that is, a test spectrum that reduces MSE. The algorithm execution unit implements the algorithm as software code or hardware signal processing, and calculates a predetermined output value by applying an input value to the predetermined algorithm.

6 is a diagram showing a detailed configuration of a controller according to an embodiment of the present invention. The controller 160 processes the input information and calculates a predetermined result, and may be composed of software codes or hardware that calculates the above results. The controller 160 includes R number of algorithm execution units 161 to 165.

Each of these algorithm execution units may calculate a predetermined sample. Also, the controller 160 may select one of the sample data calculated by the algorithm execution units 161 to 165 . Alternatively, S24 of FIG. 5 is performed on all of these samples to select sample data related to the test spectrum that yields the smallest MSE, and the controller 160 can increase the priority of the algorithm execution unit that calculated the sample data. . In order to increase the selection possibility of sample data calculated by a specific algorithm executing unit, the priority may be increased.

An algorithm execution unit that may be included in the controller 160 may be implemented with various algorithms. As examples of function optimization techniques, various algorithms such as the GD technique (Gradient Descent Method), Newton-Raphson Method, Gauss-Newton Method, and LM technique (Levenberg-Marquardt Method) can be applied. can In this specification, we intend to apply the simplex & DE algorithm 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 (DE) algorithm execution unit. In addition, the above-described library unit 140 calculates a test spectrum corresponding to the sample data calculated by any one of these two algorithm execution units. The controller 160 may repeat a process of calculating sample data to minimize a difference by comparing the measurement spectrum and the test spectrum. The iterative process may stop iterating when a minimum difference below a predetermined criterion is identified.

In FIG. 6, the first algorithm execution unit 161 is a first simplex algorithm execution unit set as a first parameter as an example. In an embodiment, the second algorithm execution unit 162 is a second simplex algorithm execution unit set as a second parameter. An embodiment of the third algorithm execution unit 163 being a third simplex algorithm execution unit set as a third parameter. Here, the first parameter is a parameter necessary for executing the simplex algorithm, and variable elements may be set to different parameters in order to calculate 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 variously configured by one or more, and the present invention is not limited thereto.

In addition, the fourth algorithm execution unit 165 is a DE (Differential Evolution) algorithm execution unit according to an embodiment.

7 to 10 are diagrams showing detailed configurations 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 a simplex algorithm and a DE algorithm execution unit 165a implementing a DE algorithm. In this regard, it may be configured as shown in FIG. 7 . In the configuration shown in FIG. 7 , the controller 160 may select one algorithm execution unit from among a plurality of algorithm execution units using a probability proportional to a previous success rate.

That is, the controller 160 can select from various algorithm execution units that implement various heuristic algorithms in the process of searching for new sample data, and select an algorithm execution unit that implements which heuristic methodology with a probability proportional to the previous success rate. can decide The decision method can be calculated in various ways, such as a method using the success rate accumulated for a certain number of times and a method using the success rate of the previous round. The success rate accumulated for a certain number of times is accumulated at the corresponding number of times, so that an algorithm execution unit with the highest success rate can be selected. This prevents the success rate from being set excessively low based on previous successful cases even if the algorithm execution unit fails at any one time. The latest success rate can be applied to the success rate of the previous round. The two success rate-based decision methods may be mixed or selected depending on the characteristics of the algorithm or the change speed or size of the success rate.

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

를 산출한다.In FIG. 8, an execution method of the simplex algorithm execution unit 161a will be described. 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 from P sets (S31). And among these D+1 samples

is selected (S32). And using these selected results, new sample data such as S33

yields

을 산출하는 과정을 제시한다. In Fig. 9 at S33

presents 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).After that, new sample data

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

If it is smaller than the difference between the test spectrum and the measurement spectrum calculated by

cast

It is substituted into (S34).

9 is a diagram showing 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 sample data disposed at the reflection point by using one sample data among the sample data selected in the library unit 140 and the center point of two or more other sample data excluding the selected sample data. Repeat the sample data calculation process. In this process, a CD parameter that minimizes the difference between the test spectrum and the measured spectrum is generated.

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 FIG. 9, samples in the D dimension are indicated as possible solutions existing in the problem space. Meanwhile

has the maximum value among the points defining the simplex.

instruct and

silver

It is the center point of the remaining points except for . using these

created based on

the reflection point of the new sample

calculated with Their calculation is 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 the reflection coefficient (

)Is

class

Adjusts the distance between reflections. The reflection coefficient can be set according to the previously calculated sample. 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 is non-differentiable, and as shown in FIG. 9, the speed of convergence in the direction of the solution is fast. On the other hand, it has the disadvantage that it is easy to fall into a local optimal solution. Accordingly, the controller 160 may realize speed and optimality in calculating sample parameters by applying the DE algorithm executing unit 165a suitable for calculating a global optimal solution. The simplex algorithm can configure multiple 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 executing unit 161a randomly selects D+1 samples in the P set when D is the dimension of x constituting the calculated spectrum. Among the selected points, the sample with the highest function value (i.e., the worst, far from optimal sample) is selected.

, and reflect it based on the center point of the remaining D samples. At this time, 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. The reflected point (

) as new sample data. The reflection coefficient can be calculated as ρ ~ 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 executor 165a calculates a better optimal solution by changing some parameters or performing crossover in the process of calculating data in D dimensions. It shows the process of generating D-dimensional parameters in every 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 among D dimensions. This is presented 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 1 in the D dimension. The probability integer l is a value selected from among integers from 1 to D. And j starts from 1 and increases by 1 at S45 or S46. When j exceeds the D dimension, the parameters in the generation are calculated, and the calculated parameters (i.e., sample data) are converted to new sample data, such as S47.

A new sample composed of

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

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

to

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 yet been completely calculated, subsequent steps are continued to calculate CD parameters. As a result, a random random variable

is generated (S43). And when Uj is less than or equal to C, or when j reaches the random integer variable l (S44), the parameter

to the newly mutated value (

) and increase j (S45). On the other hand, if S44 is not satisfied, the parameter

is the parameter value of the previous generation

is input and j is increased (S46).

를 산출하는 방식은 수학식 8과 같다.from S45

The method for 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 greater the difference between the maximum and minimum values of the function in the P set, the more distant the sample data is generated, and the smaller the difference, the more adjacent it is generated. In addition, in order to prevent generating sample data only in a region too close to the coefficient of variation, the F_min value can be additionally set as an input variable.

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

The DE algorithm executor 165a may obtain a global optimal solution when the objective function is a non-linear function and is non-differentiable. It is possible to approach the global optimal solution because D-dimensional parameters are intersected by generation or mutation is performed. In addition, when parallel computing is applied, calculation time can be reduced.

11 is a diagram showing a process of calculating parameters by a DE algorithm execution unit according to an embodiment of the present invention. 51 shows that D-dimensional parameters are calculated in Generation 1. The parameters of the first generation can be randomly calculated or selected from a predetermined sample. One test spectrum is calculated using the D-dimensional CD parameters of the first generation, compared with the measurement spectrum, and CD parameters of the next generation 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 either the mutation method (S45) or the crossover method (S46) of Equation 8 by applying the process of FIG. 10 . In the mutation method, the coefficient of variation F is used, and in the crossover method, it is calculated based on the crossover coefficient C. Summarizing this, Equation 9 is obtained.

[Equation 9]

)은 이전 세대의 파라미터(

) 또는 이전 세대의 파라미터들에서 산출된 값(

)을 이용하여 산출될 수 있다. That is, the parameters of the next generation (

) is the parameter of the previous generation (

) or values calculated from the parameters of the previous generation (

) can be calculated using

10 and 11, the DE algorithm executing unit 165a calculates parameters of the (k+1)th generation in a crossover method using parameters constituting sample data of the kth generation or the previous generation. Alternatively, the DE algorithm executing unit 165a calculates parameters randomly selected from the library unit 140 or previous generations including the kth generation, and calculates the parameters of the (k+1)th generation in a variational manner. If all parameters of the (k+1)th generation are obtained, they become sample data that are candidates for CD parameters. Then, the DE algorithm execution unit 165a calculates a test spectrum using the sample data, and compares the result of calculating the difference with the measurement spectrum as shown in Equations 4 and 5. And, as a result of the comparison, if it is smaller than the previous minimum value, the parameter of the (k+1) generation is successful, so the success weight of the DE algorithm increases.

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

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

와 지금까지 확인된

를 비교한 결과 S55와 같이

를 만족하는 경우, 해당 알고리즘은 샘플 데이터를 생성함에 있어서 성공한 것으로 판단한다. From among the heuristic algorithm selected by the simplex algorithm execution unit 161a and the heuristic algorithm selected by the DE algorithm execution unit 165a, one algorithm is probabilistically 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

is satisfied, the corresponding algorithm is determined to be 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. Then, the controller 160 sends the sample data

In generating , the success weight and cumulative weight of the selected algorithm (i-th algorithm) are updated, and based on this, the selection probability of each algorithm is updated (S56). That is, the reliability of the corresponding algorithm is increased by increasing the success weight of the corresponding algorithm (simplex or DE, etc.) that calculated the sample data corresponding to the successful calculation of the test spectrum.

Thereafter, the controller 160 selects an algorithm based on the selection probability of the algorithm to generate the next sample (S57).

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

And, f_max and f_min are defined as the maximum f(x) value and the 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 ith algorithm.

Also, with δ as an input variable, if any of the selection probabilities of each algorithm becomes smaller than δ, all values are initialized.

After initial setting is performed, steps S55 to S57 are performed, and the success weight of each algorithm is increased or decreased. And the increased and decreased values are accumulated in the cumulative sum of success weights. When an 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 may be reset as a whole when the selection probability of an algorithm is smaller than a preset value. However, in this process, the accumulated weight may be maintained without being reset.

Based on this process, the controller 160 updates the selection probability of each algorithm. When the currently applied algorithm succeeds, the more good samples are selected based on the maximum and minimum values in the P set, the larger the success weight is obtained. do.

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

Although one embodiment of the present invention has been described above, those skilled in the art can add, change, delete, or add components within the scope not departing from the spirit of the present invention described in the claims. The present invention can be variously modified and changed by the like, and this will also be said to 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 that 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;
It includes a simplex algorithm execution unit and a DE (Differential Evolution) algorithm execution unit, and a test spectrum is calculated by selecting sample data calculated from either of these two algorithm execution units, and the measurement spectrum and the test spectrum are compared to make a difference. A controller that repeats the process of calculating sample data that minimizes; and
The controller includes an output unit for outputting sample data obtained by calculating a test spectrum of a minimum difference;
The controller calculates the sample data that minimizes the sum of squares of differences between the test spectrum calculated by the library unit in the measurement spectrum and the sample data;
The controller is a value that maximizes the square value of the difference among the sample data of the library unit

Calculate,
The controller calculates a test spectrum using the sample data calculated by the simplex algorithm execution unit or the DE algorithm execution unit;
When the difference between the test spectrum and the measurement spectrum is smaller than the previously calculated minimum difference value, the controller

An apparatus for substituting the calculated sample data into
delete delete According to claim 1,
The controller increases a success weight of an algorithm executing unit that calculates sample data corresponding to the test spectrum.
a spectrum input unit that 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;
It includes a simplex algorithm execution unit and a DE (Differential Evolution) algorithm execution unit, and a test spectrum is calculated by selecting sample data calculated from either of these two algorithm execution units, and the measurement spectrum and the test spectrum are compared to make a difference. A controller that repeats the process of calculating sample data that minimizes; and
The controller includes an output unit for outputting sample data obtained by calculating a test spectrum of a minimum difference;
The simplex algorithm execution unit calculates sample data disposed at a reflection point by using one sample data among sample data selected from the library unit and a center point of two or more other sample data excluding the sample data.
a spectrum input unit that 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 measured spectrum;
It includes a simplex algorithm execution unit and a DE (Differential Evolution) algorithm execution unit, and a test spectrum is calculated by selecting sample data calculated from either of these two algorithm execution units, and the measurement spectrum and the test spectrum are compared to make a difference. A controller that repeats the process of calculating sample data that minimizes; and
The controller includes an output unit for outputting sample data obtained by calculating a test spectrum of a minimum difference;
The DE algorithm execution unit calculates parameters of the (k+1)th generation in a crossover method using parameters constituting sample data of the kth generation or the previous generation, or the library unit or the kth generation including A device that calculates the parameters of the (k+1) generation by a variation method by calculating randomly selected parameters in the previous generation.
receiving a measurement spectrum measured for each wavelength from a semiconductor device by a spectrum input unit;
selecting, by a controller including a simplex algorithm execution unit and a DE (Differential Evolution) algorithm execution unit, sample data calculated by any one of these two algorithm execution units;
calculating, by a library unit, a test spectrum corresponding to the selected sample data;
repeating, by the controller, a process of comparing the measurement spectrum and the test spectrum and calculating sample data minimizing a difference; and
Providing, by the controller, sample data obtained by calculating the minimum difference test spectrum to an output unit, and outputting the sample data to the output unit;
The repeating steps
The controller further comprises calculating the sample data that minimizes the sum of squares of differences between the test spectrum calculated by the library unit in the measurement spectrum and the sample data,
The repeating steps are
A value for which the controller maximizes the square value of the difference among the sample data of the library unit

Calculating;
calculating, by the controller, a test spectrum using sample data calculated by the simplex algorithm execution unit or the DE algorithm execution unit; and
When the difference between the test spectrum and the measurement spectrum is smaller than the previously calculated minimum difference value, the controller

A method for calculating a CD of an optimal model based on control random search, comprising the step of substituting the calculated sample data into .
delete delete According to claim 7,
The method of calculating the CD of the optimal model based on the control random search, further comprising increasing, by the controller, a success weight of an algorithm execution unit that has calculated sample data corresponding to the test spectrum.
receiving a measurement spectrum measured for each wavelength from a semiconductor device by a spectrum input unit;
selecting, by a controller including a simplex algorithm execution unit and a DE (Differential Evolution) algorithm execution unit, sample data calculated by any one of these two algorithm execution units;
calculating, by a library unit, a test spectrum corresponding to the selected sample data;
repeating, by the controller, a process of comparing the measurement spectrum and the test spectrum and calculating sample data minimizing a difference; and
Providing, by the controller, sample data obtained by calculating the minimum difference test spectrum to an output unit, and outputting the sample data to the output unit;
Calculating, by the simplex algorithm execution unit, sample data disposed at a reflection point by using one sample data among sample data selected from the library unit and a center point of two or more other sample data excluding the sample data , a method for calculating the CD of the optimal model based on controlled random search.
receiving a measurement spectrum measured for each wavelength from a semiconductor device by a spectrum input unit;
selecting, by a controller including a simplex algorithm execution unit and a DE (Differential Evolution) algorithm execution unit, sample data calculated by any one of these two algorithm execution units;
calculating, by a library unit, a test spectrum corresponding to the selected sample data;
repeating, by the controller, a process of comparing the measurement spectrum and the test spectrum and calculating sample data minimizing a difference; and
Providing, by the controller, sample data obtained by calculating the minimum difference test spectrum to an output unit, and outputting the sample data to the output unit;
The DE algorithm execution unit calculates parameters of the (k+1)th generation in a crossover method using parameters constituting the sample data of the kth generation or the previous generation, or the library unit or the kth generation including A method for calculating a CD of an optimal model based on a controlled random search, further comprising calculating parameters of a (k+1) generation in a variational manner by calculating randomly selected parameters in a previous generation.

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