A nanostructure scattering measurement configuration optimization method and system based on condition number

Abstract

The present invention discloses a method and system for optimizing the configuration of nanostructure scattering measurements based on condition numbers, belonging to the field of photolithography technology. The method includes: step S1, determining the morphological parameters to be measured of the nanostructure and the fixed parameters in the parameter extraction process of the scattering measurement; step S2, constructing an evaluation function for measuring the fitting error between the measured signal and the theoretical signal in the scattering measurement based on the morphological parameters to be measured and the fixed parameters; step S3, obtaining the error transfer matrix of the morphological parameters to be measured based on the evaluation function; step S4, calculating the condition number of the error transfer matrix of the morphological parameters to be measured under different measurement conditions, and the measurement condition with the smallest condition number is the optimal configuration. The present invention also provides a system for optimizing the configuration of nanostructure scattering measurements based on condition numbers. The present invention provides a new measurement configuration optimization scheme for achieving high measurement precision, accuracy, and good robustness in nanostructure scattering measurements.

Description

Technical Field

[0001] The present invention belongs to the field of photolithography technology, and more particularly, relates to a method and system for optimizing nanostructure scattering measurement configuration based on condition numbers. Background Art

[0002] Nanostructure scattering measurements include those based on optical scattering techniques and those based on small-angle X-ray scattering techniques. The measurement configuration optimization problem is a common problem in nanostructure scattering measurements. Its purpose is to explore the optimal combination of measurement conditions such as wavelength, angle of incidence, azimuth angle, and / or polarization direction to achieve higher measurement accuracy for a given measuring instrument. The uncertainty of nanostructure scattering measurements depends not only on the quality of the measured signal (such as the signal-to-noise ratio), but also on the measurement condition configuration. The accuracy of the measured parameters extracted under different measurement condition configurations often varies. To this end, it is necessary to optimize the measurement condition configuration before parameter extraction to improve the accuracy of the parameter extraction process.

[0003] In the prior art, the configuration optimization method for nanostructure scattering measurement mainly adopts sensitivity analysis, which uses the uncertainty of parameters as the optimization index of scattering measurement. This method is computationally complex and the measurement results are easily disturbed by noise. Summary of the Invention

[0004] In response to the defects and improvement needs of the existing technology, the present invention provides a method and system for optimizing the configuration of nanostructure scattering measurements based on condition numbers, which aims to reduce the complexity of nanostructure scattering measurement configuration optimization while improving the robustness of the measurement results.

[0005] To achieve the above object, according to one aspect of the present invention, a method for optimizing nanostructure measurement configuration based on condition number is provided, comprising:

[0006] Step S1, determining the morphological parameter x to be measured of the nanostructure and the fixed parameter a during the parameter extraction process of the scattering measurement;

[0007] Step S2: constructing an evaluation function for measuring the fitting error between the measured signal and the theoretical signal in the scattering measurement based on the measured topography parameter x and the fixed parameter a;

[0008] Step S3, obtaining an error transfer matrix of the measured topography parameters based on the evaluation function;

[0009] Step S4: calculating the condition number of the error transfer matrix of the measured topography parameter under different measurement conditions. The measurement condition with the smallest condition number is the optimal configuration.

[0010] Furthermore, in step S3, based on the evaluation function, an error transfer matrix of the measured topography parameter is obtained, including:

[0011] Calculating a first-order Taylor expansion of the theoretical signal for the measured morphology parameter x;

[0012] Substituting the first-order Taylor expansion into the evaluation function, the error transfer matrix of the measured morphology parameter is obtained.

[0013] Furthermore, the error transfer matrix of the measured topography parameters is:

[0014]

[0015] Among them, J x is an N×M order Jacobian matrix, and its elements are defined as:

[0016] f(x,a) is the theoretical signal corresponding to the measured signal, x * is the final extracted value of the measured morphological parameter, W is an N×N order diagonal matrix composed of weight factors, Indicates J x The transposed matrix of .

[0017] Furthermore, the measurement conditions with the minimum condition number are obtained by the following optimization method:

[0018]

[0019] Among them, α 1opt , α 2opt ,…,α kopt represents the measurement condition configuration with the minimum condition number, α1, α2, …, α k Represent different measurement conditions, A1, A2, ..., A k They represent the range of variation corresponding to different measurement conditions, Ω is the range of the appearance parameters to be measured, and cond(.) represents the condition number calculation operator.

[0020] Furthermore, a first-order Taylor expansion of the theoretical signal for the measured morphological parameter x is calculated by analytical, numerical modeling or proxy model methods.

[0021] Furthermore, in step S2, the evaluation function is:

[0022]

[0023] Where y=[y1,y2,…,y N ] represents the measurement signal, f(x,a)=[f1(x,a),f1(x,a),…,fN (x, a)] is the theoretical signal corresponding to the measured signal, W is an N×N order diagonal matrix composed of weight factors, x=[x1,x2,...,x M ] T represents the morphological parameters to be measured, a=[a1,a2,...,a L ] T represents the fixed parameter.

[0024] Furthermore, in step S4, the measurement conditions include: measuring the incident angle θ, the azimuth angle Any combination of one or more of wavelength λ and polarization angle Ψ is measured.

[0025] Furthermore, in step S4, the condition number of the error transfer matrix of the measured topography parameter includes: 1-condition number, 2-condition number, ∞-condition number or Frobenius norm condition number.

[0026] According to a second aspect of the present invention, a nanostructure measurement configuration optimization system based on condition number is provided, comprising:

[0027] A parameter determination module is used to determine the measured morphological parameter x of the nanostructure and the fixed parameter a during the parameter extraction process of the scattering measurement;

[0028] An evaluation function construction module is used to construct an evaluation function for measuring the fitting error between the measured signal and the theoretical signal in the scattering measurement based on the measured morphology parameter x and the fixed parameter a;

[0029] An error transfer matrix acquisition module, configured to obtain the error transfer matrix of the topography parameter to be measured based on the evaluation function;

[0030] The optimal configuration calculation module is used to calculate the condition number of the error transfer matrix of the measured morphology parameters under different measurement conditions. The measurement condition with the smallest condition number is the optimal configuration.

[0031] According to a third aspect of the present invention, a computer-readable storage medium is provided, on which a computer program is stored. When the program is executed by a processor, the measurement configuration optimization method as described in any one of the first aspects is implemented.

[0032] In general, the above technical solutions conceived by the present invention can achieve the following beneficial effects:

[0033] (1) The nanostructure measurement configuration optimization method based on the condition number of the present invention constructs an evaluation function for measuring the fitting error between the measurement signal and the theoretical signal in the nanostructure scattering measurement based on the morphological parameters to be measured and the fixed parameters. Based on the constructed evaluation function, the error transfer matrix of the morphological parameters to be measured is obtained. Under different measurement conditions, the smaller the condition number of the error transfer matrix of the morphological parameters to be measured, the smaller the disturbance of the measurement signal on the parameter extraction result. A slight change in the measurement signal will not cause a drastic change in the extraction result of the morphological structure to be measured. The measurement condition with the smallest condition number is the optimal configuration. The measurement result under the optimal configuration is more robust. In addition, the method of the present invention only needs to calculate the error transfer matrix condition number of the morphological parameters to be measured to obtain the optimal configuration condition, and the method is simple.

[0034] At the same time, it can be seen from the simulation results of the present invention that the error distribution of the measured morphological parameters is relatively small, which can improve the precision and accuracy of the nanostructure scattering measurement.

[0035] In summary, the present invention provides a new measurement configuration optimization scheme for achieving high measurement precision, high measurement accuracy and good robustness for nanostructure scattering measurements. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 This is a flow chart of a method for optimizing nanostructure measurement configuration based on condition number proposed in the present invention;

[0037] Figure 2 is a cross-sectional view of a typical one-dimensional grating structure in an embodiment of the present invention;

[0038] Figure 3 is the distribution of condition numbers under different measurement configuration conditions in the embodiment of the present invention;

[0039] Figure 4 is the distribution of the standard deviation of the top line width w1 calculated using Monte Carlo simulation under different measurement configuration conditions in an embodiment of the present invention;

[0040] Figure 5 is the distribution of the standard deviation of the line height h calculated using Monte Carlo simulation under different measurement configuration conditions in an embodiment of the present invention;

[0041] Figure 6 1 is the distribution of the standard deviation of the bottom line width w2 calculated using Monte Carlo simulation under different measurement conditions in an embodiment of the present invention. DETAILED DESCRIPTION

[0042] In order to make the objectives, technical solutions and advantages of the present invention more clearly understood, the present invention is further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely for the purpose of explaining the present invention and are not intended to limit the present invention. In addition, the technical features involved in the various embodiments of the present invention described below may be combined with each other as long as they do not conflict with each other.

[0043] like Figure 1 As shown, the nanostructure measurement configuration optimization method based on condition number provided by the present invention mainly includes the following steps:

[0044] Step S1, determining the morphological parameter x of the nanostructure to be measured and the fixed parameter a during the parameter extraction process of the scattering measurement;

[0045] Step S2: constructing an evaluation function for measuring the fitting error between the measured signal y and the theoretical signal f(x, a) in the nanostructure scattering measurement based on the measured morphology parameter x and the fixed parameter a;

[0046] Step S3: Based on the evaluation function constructed above, an error transfer matrix of the measured topography parameters is obtained;

[0047] Step S4: Calculate the condition number of the error transfer matrix of the measured topography parameters under different measurement conditions. The measurement condition with the smallest condition number is the optimal configuration.

[0048] In this embodiment, in step S1, the morphology parameter x of the nanostructure to be measured is an M-dimensional column vector, x=[x1,x2,...,x M ] T , the fixed parameter a in the process of parameter extraction of scattering measurement is an L-dimensional column vector, a=[a1,a2,...,a L ] T .

[0049] Specifically, in step S1, the nanostructures to be measured include periodic nanostructures and non-periodic nanostructures, and the morphological parameters x to be measured of the periodic nanostructures include line width, period, line height, side wall angle, etc.; the fixed parameters a in the parameter extraction include measurement conditions such as incident angle, azimuth angle, polarization angle or wavelength and some known morphological parameters.

[0050] Specifically, in step S2, based on the measured morphological parameter x and the fixed parameter a, construct χ 2 The evaluation function in the form of is used to measure the fitting error between the measured signal and the theoretical signal;

[0051]

[0052] Where y=[y1,y2,…,y N] represents the measurement signal, f(x,a)=[f1(x,a),f1(x,a),…,f N (x,a)] is the corresponding theoretical signal, and W is the weight factor w n The N×N diagonal matrix is ​​W=diag(1 / σ 2 (y1),1 / σ 2 (y2),...,1 / σ 2 (y N )), σ 2 (.) represents the standard deviation of the measured signal.

[0053] Therefore, the inverse problem in nanostructure measurement can be expressed as:

[0054]

[0055] Among them, x * is the final extracted value of the measured morphology parameter, and Ω is the value range of the measured morphology parameter.

[0056] Specifically, in step S3, based on the evaluation function constructed above, the error transfer matrix of the measured topography parameters is obtained, including:

[0057] Calculate the first-order Taylor expansion of the theoretical signal f(x,a) (i.e., the optical characterization of the nanostructure) corresponding to the measurement signal for the measured morphology parameter x, substitute the first-order Taylor expansion into the evaluation function constructed above to obtain the error transfer matrix of the measured morphology parameter.

[0058] The first-order Taylor expansion of the measured topographic parameter x, corresponding to the theoretical signal f(x,a), is calculated using analytical, numerical, or surrogate model methods. The theoretical signal f(x,a) corresponding to the measured signal can include light intensity, reflectivity, transmittance, ellipsometric parameters, or the Mueller matrix. Analytical or numerical modeling methods include rigorous coupled wave analysis, finite element method, boundary element method, finite time-domain difference method, or scalar diffraction method. Surrogate model methods include polynomial chaos expansion, response surface methodology, kriging method, and artificial neural networks.

[0059] Specifically, it is assumed that the theoretical signal f(x,a) corresponding to the measured signal is at the final extracted measured morphological parameter value x * is smooth near x, so it can be * Do a first-order Taylor expansion nearby:

[0060] f(x,a)=f(x * ,a)+J x ·(xx * ) (3)

[0061] Among them, J x is an N×M order Jacobian matrix, whose elements are defined as:

[0062]

[0063] Let x0 be the expected solution, and let x=x0 in equation (3):

[0064] f(x0,a)=f(x * ,a)+J x ·(x0-x * )=f(x * ,a)+J x Δx (5)

[0065] Where Δx represents the error of the measured topographic parameter x.

[0066] Substituting formula (5) into the evaluation function formula (1) yields:

[0067] χ 2 =[yf(x * ,a)-J x Δx] T W[yf(x * ,a)-J x Δx], (6)

[0068] Take the partial derivative of the measured morphology parameter x on both sides of equation (6) and get * Nearby are:

[0069]

[0070] Further sorting can be obtained:

[0071]

[0072] Among them, considering the above linear equation, It is a coefficient matrix, also called the error transfer matrix of the measured topography parameters, which is generally determined by the measurement configuration conditions and the analytical, numerical modeling or proxy model adopted. Indicates J x is the transposed matrix of , and Δy is the error of the measurement signal.

[0073] If the condition number of the coefficient matrix The smaller it is, the smaller the range of relative error in solving the linear equations will be, and a slight change in the measurement signal will not cause drastic changes in the solution of the equation. That is, under different measurement conditions, the smaller the error transfer matrix condition number of the measured morphological parameters, the smaller the disturbance of the measurement signal on the parameter extraction result will be, and a slight change in the measurement signal will not cause drastic changes in the extraction result of the measured morphological structure. The measurement conditions at this time are the optimal configuration.

[0074] Specifically, in step S4, the different measurement conditions are the measurement conditions to be optimized selected from the nanostructure scattering measurement conditions;

[0075] The measurement conditions to be optimized include: measuring the incident angle θ, azimuth angle Measure any combination of one or more of wavelength λ and polarization angle Ψ, etc.

[0076] Specifically, in step S4, the condition number of the error transfer matrix of the to-be-measured topography parameter includes a 1-condition number, a 2-condition number, an ∞-condition number or a Frobenius norm condition number.

[0077] The condition number of the error transfer matrix of the measured morphology parameters under different measurement conditions is calculated by using optimization methods such as ergodic method, particle swarm optimization, differential evolution algorithm or genetic algorithm.

[0078] In particular, the present invention adopts a "maximum-minimum" optimization process to select the optimal measurement condition configuration:

[0079]

[0080] Among them, α 1opt , α 2opt ,…,α kopt represents the measurement condition configuration with the minimum condition number, α1, α2, …, α k Represent different measurement conditions, A1, A2, ..., A k They represent the range of variation corresponding to different measurement conditions, Ω is the range of the appearance parameters to be measured, and cond(.) represents the condition number calculation operator.

[0081] Take the incident angle and azimuth angle as an example to illustrate the measurement conditions to be optimized:

[0082]

[0083] The optimization process in formula (9) is an optimization process that includes two layers of loops. The inner loop is used to traverse the given range of the measured appearance parameters Ω for each measurement condition configuration to find the corresponding maximum condition number. The outer loop is used to traverse the maximum condition number found by the inner loop within the given range of incidence angle and azimuth angle Θ and Φ Among all the largest condition numbers The combination of incident angle and azimuth angle corresponding to the smallest one among is the optimal measurement condition configuration.

[0084] The principle and working process of the method of the present invention are described in detail below using nanostructure measurement based on ellipsometer as an example.

[0085] (1) In this embodiment, the nanostructure to be measured is a typical one-dimensional grating structure, and its cross-sectional morphology is as follows: Figure 2 As shown, the three topographical structural parameters to be measured are: top line width w1, grating height h and bottom line width w2, that is, x = [w1, h, w2] T The fixed parameters in the parameter extraction process include grating period Λ, incident angle θ, azimuth angle and wavelength λ, i.e.

[0086] (2) Using χ 2 The evaluation function of the form is used to measure the fitting error between the measured signal y and the calculated theoretical signal f(x,a):

[0087]

[0088] In this embodiment, y is the measured Mueller matrix element M meas , f(x,a) is the theoretically calculated Mueller matrix element M calc (x, a), W is an N×N diagonal matrix consisting of the variance of the measurement signal. Therefore, the inverse problem in the nanostructure measurement based on ellipsometer in this embodiment can be expressed as:

[0089]

[0090] (3) Calculate the first-order Taylor expansion of the nanostructure optical characterization quantity f(x,a) with respect to x, substitute it into the evaluation function, and obtain the error transfer matrix of the measured morphological parameters.

[0091] In this embodiment, the rigorous coupled wave analysis method is used to calculate the Mueller matrix element M of the nanostructure. calc (x, a), and when calculating the Jacobian matrix of formula (4), the difference quotient is used instead of the derivative.

[0092] (4) The measurement conditions to be optimized are selected from the nanostructure scattering measurement conditions, and the condition number of the error transfer matrix of the measured morphological parameters under different measurement conditions is calculated. The measurement configuration condition with the smallest condition number is the optimal configuration.

[0093] In this embodiment, the incident angle θ and the azimuth angle are selected to be measured. As the measurement conditions to be optimized, during the optimization process, the measured parameter range Ω: w1∈[30,40]nm, h∈[85,95]nm, w2∈[60,75]nm; the incident angle range Θ:[60°,70°]; and the azimuth angle range Φ:[0°,90°]. In this embodiment, the 2-condition number is used to calculate the condition number of the model error transfer matrix under different measurement conditions:

[0094]

[0095] where σ max is a matrix The largest singular value of σ min is a matrix The smallest singular value of .

[0096] Figure 3 It represents the distribution result of the condition number in the incident angle range Θ: [60°, 70°] and the azimuth angle range Φ: [0°, 90°] using the traversal method. Figure 4-Figure 6 They are the distribution results of the standard deviations of the measured morphological parameters w1, h, w2, etc. calculated by the Monte Carlo method. It can be seen that the distribution range of the measurement configuration with a smaller condition number is basically consistent with the distribution range of the measurement configuration with a smaller standard deviation of the measured parameters calculated by Monte Carlo simulation, which verifies the effectiveness of the method of the present invention.

[0097] The nanostructure measurement configuration optimization method based on the condition number of the present invention constructs an evaluation function for measuring the fitting error between the measurement signal y and the theoretical signal f(x, a) in the nanostructure scattering measurement based on the measured morphological parameter x and the fixed parameter a, obtains the error transfer matrix of the measured morphological parameter based on the constructed evaluation function, and estimates the error of solving the linear equation based on the condition number. Under different measurement conditions, the smaller the condition number of the error transfer matrix of the measured morphological parameter, the smaller the disturbance of the measurement signal on the parameter extraction result, and a slight change in the measurement signal will not cause drastic changes in the extraction result of the measured morphological structure. The measurement condition with the smallest condition number is the optimal configuration, and the measurement result under the optimal configuration is more robust. In addition, the method of the present invention only needs to calculate the condition number of the error transfer matrix of the measured morphological parameter to obtain the optimal configuration condition, and the method is simple.

[0098] The simulation results of the present invention show that the error distribution of the measured topographic parameters is small, which can improve the precision and accuracy of nanostructure scattering measurements. The present invention provides a new measurement configuration optimization scheme for achieving high measurement precision, high measurement accuracy, and good robustness in nanostructure scattering measurements.

[0099] The present invention also provides a nanostructure measurement configuration optimization system based on condition numbers, which is used to implement the nanostructure measurement configuration optimization method based on condition numbers in the above embodiment. The measurement configuration optimization system mainly includes:

[0100] A parameter determination module is used to determine the measured morphological parameter x of the nanostructure and the fixed parameter a during the parameter extraction process of the scattering measurement;

[0101] An evaluation function construction module is used to construct an evaluation function for measuring the fitting error between the measured signal and the theoretical signal in scattering measurement based on the measured morphological parameter x and the fixed parameter a;

[0102] An error transfer matrix acquisition module is used to obtain the error transfer matrix of the measured topography parameters based on the evaluation function;

[0103] The optimal configuration calculation module is used to calculate the condition number of the error transfer matrix of the measured morphological parameters under different measurement conditions. The measurement condition with the smallest condition number is the optimal configuration.

[0104] The specific implementation of each module is detailed in the nanostructure measurement configuration optimization method based on condition number in the above embodiment.

[0105] The present invention also provides a computer-readable storage medium having a computer program stored thereon, which, when executed by a processor, implements the nanostructure measurement configuration optimization method based on condition number as described in the above embodiment.

[0106] It will be easily understood by those skilled in the art that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A method for optimizing nanostructure measurement configuration based on condition number, characterized in that: include: Step S1, determining the morphological parameter x to be measured of the nanostructure and the fixed parameter a during the parameter extraction process of the scattering measurement; Step S2: constructing an evaluation function for measuring the fitting error between the measured signal and the theoretical signal in the scattering measurement based on the measured topography parameter x and the fixed parameter a; Step S3, obtaining an error transfer matrix of the measured topography parameters based on the evaluation function; Step S4, calculating the condition number of the error transfer matrix of the measured topography parameter under different measurement conditions, and the measurement condition with the smallest condition number is the optimal configuration; In step S3, based on the evaluation function, an error transfer matrix of the measured topography parameters is obtained, including: Calculating a first-order Taylor expansion of the theoretical signal for the measured morphology parameter x; Substituting the first-order Taylor expansion into the evaluation function, the error transfer matrix of the measured morphology parameter is obtained.

2. The measurement configuration optimization method according to claim 1, characterized in that: The error transfer matrix of the measured topography parameters is: ； Among them, J x for Jacobian matrix of order, the elements are defined as: , f(x, a) is the theoretical signal corresponding to the measured signal, is the final extracted morphological parameter value to be measured, and W is composed of weight factors A diagonal matrix of order, Indicates J x The transposed matrix of .

3. The measurement configuration optimization method according to claim 2, characterized in that: The measurement conditions with the minimum condition number are obtained through the following optimization method: ； in, Indicates the measurement condition configuration with the minimum condition number, Represent different measurement conditions. They represent the range of variation corresponding to different measurement conditions, Ω is the range of the face parameters to be measured, Represents the condition number calculation operator.

4. The measurement configuration optimization method according to claim 1, wherein: The first-order Taylor expansion of the theoretical signal for the measured morphological parameter x is calculated by analytical, numerical modeling or proxy model methods.

5. The measurement configuration optimization method according to claim 1, characterized in that: In step S2, the evaluation function is: ； Where y = [y1, y2, …, y N ] represents the measurement signal, f(x, a) = [f1(x, a), f1(x, a),…, f N (x, a)] is the theoretical signal corresponding to the measured signal, and W is the weight factor A diagonal matrix of order x = [x1, x2, ..., x M ] T represents the morphological parameters to be measured, a = [a1, a2, ..., a L ] T represents the fixed parameter.

6. The measurement configuration optimization method according to claim 1, characterized in that: In step S4, the measurement conditions include any combination of one or more of the measurement incident angle θ, the azimuth angle φ, the measurement wavelength λ, and the polarization angle Ψ.

7. The measurement configuration optimization method according to claim 1, characterized in that: In step S4, the condition number of the error transfer matrix of the measured topography parameter includes: 1-condition number, 2-condition number, -condition number or Frobenius norm condition number.

8. A nanostructure measurement configuration optimization system based on condition number, characterized in that: include: A parameter determination module is used to determine the measured morphological parameter x of the nanostructure and the fixed parameter a during the parameter extraction process of the scattering measurement; An evaluation function construction module is used to construct an evaluation function for measuring the fitting error between the measured signal and the theoretical signal in the scattering measurement based on the measured morphology parameter x and the fixed parameter a; An error transfer matrix acquisition module is configured to obtain the error transfer matrix of the measured topography parameter based on the evaluation function, specifically comprising: calculating a first-order Taylor expansion of the theoretical signal for the measured topography parameter x; substituting the first-order Taylor expansion into the evaluation function to obtain the error transfer matrix of the measured topography parameter; The optimal configuration calculation module is used to calculate the condition number of the error transfer matrix of the measured morphology parameters under different measurement conditions. The measurement condition with the smallest condition number is the optimal configuration.

9. A computer-readable storage medium having a computer program stored thereon, characterized in that: When the program is executed by a processor, the measurement configuration optimization method according to any one of claims 1 to 7 is implemented.

Images