CN117743732A - Theoretical spectrum acquisition method, electronic device and medium - Google Patents
Theoretical spectrum acquisition method, electronic device and medium Download PDFInfo
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Abstract
The invention discloses a method for acquiring a theoretical spectrum, which comprises the steps of firstly carrying out Fourier series expansion on different optical characteristic parameters of a sample, substituting the optical characteristic parameters into a Maxwell equation set, then calculating reflection and transmission coefficients of each wavelength point through an operation component of a multi-core central processing unit, wherein the operation component comprises a plurality of parallel operation modules, each operation module calculates the reflection and transmission coefficients of one wavelength point at a time, and finally determining the theoretical spectrum of the sample according to the reflection and transmission coefficients of all the wavelength points. According to the method, the algorithm flow of the strict coupled wave analysis is optimized from the physical meaning level, serial logic is changed into parallel calculation, so that a plurality of wavelength intervals can be solved simultaneously, the calculation efficiency is effectively improved, and the calculation time is shortened.
Description
Technical Field
The present invention relates to the field of optical technologies, and in particular, to a method, an electronic device, and a medium for obtaining a theoretical spectrum.
Background
Chip yield is an important index in the semiconductor industry, and accurately obtaining the critical optical dimension (OCD) of a sample to be detected can effectively ensure chip yield. The field distribution and propagation characteristics of light in a particular structure can be obtained by solving maxwell's equations describing the basic laws of physics of electromagnetic wave propagation.
There are various solutions to the common maxwell's equations, such as Finite Difference Time Domain (FDTD), finite Element Method (FEM), and Rigorous Coupled Wave Analysis (RCWA). The strict coupled wave analysis method is high in accuracy, strong in expandability, physical and visual, and suitable for periodic electromagnetic fields, so that the method is widely applied to solution of Maxwell's equations. However, when the strict coupled wave analysis method is adopted, because a plurality of coupled wave equations need to be solved, the calculation complexity is high and the solving cost is high, so that in the detection of some multi-layer large-scale structures, longer calculation time is required, and the overall efficiency of the final software product is affected by the overlong calculation time.
There are currently some existing solutions to the time cost problem of rigorous coupled wave analysis. The more common method is to use a Graphics Processing Unit (GPU), and shorten the running time of the whole system by accelerating a large number of matrix operation parts in a strictly coupled wave analysis model, wherein the matrix operation generally comprises matrix multiplication, eigenvalue solution, inverse matrix solution and the like. However, the GPU has a large difference from the CPU, and needs to be developed by adopting a specific programming language CUDA, which increases the development cost of the program. Meanwhile, the GPU is adopted to carry out data transmission, synchronization, communication and the like of heterogeneous computation, and the inherent problems of data transmission, synchronization and communication of heterogeneous computation also increase the potential risks of codes. In addition, the operation by using the GPU only focuses on a matrix computing part, the algorithm is optimized by a pure mathematical computing layer, and the parallelism of a physical layer is not involved, namely, a large number of complicated logic judgment and frequent data reading and writing of the model are not optimized.
Disclosure of Invention
Aiming at part or all of the problems in the prior art, the first aspect of the invention provides a method for acquiring theoretical spectra, which is based on the idea of parallel calculation, and based on a strict coupled wave analysis method, a multi-core Central Processing Unit (CPU) is used for improving and optimizing the numerical solving process of a Maxwell equation set, so that the calculation time of an optical model is shortened, and the calculation efficiency is improved. The method comprises the following steps:
performing Fourier series expansion on different optical characteristic parameters of the sample, and substituting the different optical characteristic parameters into a Maxwell equation set;
on a multi-core central processing unit, calculating the reflection and transmission coefficients of each wavelength point through an operation assembly, wherein the operation assembly comprises a plurality of parallel operation modules, and one operation module calculates the reflection and transmission coefficients of one wavelength point at a time; and
and determining the theoretical spectrum of the sample according to the reflection and transmission coefficients of all the wavelength points.
Further, the optical characteristic parameters of the sample include geometry, boundary conditions, and characteristics of the incident light.
Further, the geometry, boundary conditions include the size, shape, number of layers, and surface periodicity of the sample.
Further, the characteristic parameters of the incident light include wavelength, angle and polarization state.
Further, the number of terms of the fourier series expansion is determined according to the required calculation accuracy and/or the expected calculation duration.
Further, the number of the operation modules is greater than or equal to the number of the wavelength points, and each operation module calculates simultaneously.
Further, the number M of the operation modules is smaller than the number N of the wavelength points, each operation module calculates the reflection and transmission coefficients of ⌊ N/M ⌋ wavelength points, and one or more operation modules which preferably finish the calculation calculate the reflection and transmission coefficients of the remaining N-M ⌊ N/M ⌋ wavelength points.
Further, the calculation of the reflection and transmission coefficients for each wavelength point includes:
solving maxwell's equation set, calculating the electromagnetic field distribution of reflection and transmission to obtain a scattering matrix; and
and obtaining the reflection coefficient and the transmission coefficient through matrix operation.
Further, the operation component realizes parallel computation based on openMP.
Based on the method of acquiring a theoretical spectrum as described above, a second aspect of the invention provides an electronic device of acquiring a theoretical spectrum, comprising a memory and a processor, wherein the memory is configured to store a computer program which, when run by the processor, performs the method of acquiring a theoretical spectrum as described above.
The third aspect of the invention also provides a computer readable storage medium storing a computer program which, when run on a processor, performs a method of acquiring a theoretical spectrum as described above.
According to the method, the electronic device and the medium for acquiring the theoretical spectrum, the Maxwell equation set in the strict coupled wave analysis method is solved by adopting the multithread parallel computing method, and specifically, optical parameters of a plurality of wavelength points are synchronously computed through a plurality of parallel threads, so that the algorithm flow of the strict coupled wave analysis is optimized from the aspect of physical significance, serial logic is changed into parallel computing, a plurality of wavelength intervals can be solved simultaneously, the computing efficiency is effectively improved, and the computing time is shortened. In addition, the method can be realized by adopting a multi-core CPU without additionally configuring a display card and a graphic processing unit, so that on one hand, the risk brought by heterogeneous computation can be avoided, and meanwhile, the cost can be saved.
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To further clarify the above and other advantages and features of embodiments of the present invention, a more particular description of embodiments of the invention will be rendered by reference to the appended drawings. It is appreciated that these drawings depict only typical embodiments of the invention and are therefore not to be considered limiting of its scope. In the drawings, for clarity, the same or corresponding parts will be designated by the same or similar reference numerals.
FIG. 1 is a schematic flow diagram of a prior art rigorous coupled wave analysis method;
FIG. 2 shows a flow diagram of a method of acquiring a theoretical spectrum in accordance with one embodiment of the invention; and
fig. 3 shows a schematic diagram of a method of acquiring a theoretical spectrum in comparison to the calculation time required by the prior art according to an embodiment of the present invention.
Detailed Description
In the following description, the present invention is described with reference to various embodiments. One skilled in the relevant art will recognize, however, that the embodiments may be practiced without one or more of the specific details, or with other alternative and/or additional methods or components. In other instances, well-known structures or operations are not shown or described in detail to avoid obscuring aspects of the invention. Similarly, for purposes of explanation, specific numbers and configurations are set forth in order to provide a thorough understanding of embodiments of the present invention. However, the invention is not limited to these specific details.
Reference throughout this specification to "one embodiment" or "the embodiment" means that a particular feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the present invention. The appearances of the phrase "in one embodiment" in various places in the specification are not necessarily all referring to the same embodiment.
It should be noted that the embodiments of the present invention describe the steps of the method in a specific order, however, this is merely for the purpose of illustrating the specific embodiments, and not for limiting the order of the steps. In contrast, in different embodiments of the present invention, the sequence of each step may be adjusted according to the adjustment of the actual requirement.
Rigorous coupled wave analysis is a method for calculating the electromagnetic behavior of periodic optical structures. The electromagnetic behavior of the periodic optical structure is calculated by adopting a strict coupled wave analysis method, related parameters are initialized firstly, the range of the wavelength to be calculated is determined, N is divided equally, then the reflection coefficient of each wavelength is calculated in sequence, the calculation result of the corresponding wavelength band is stored until all the wavelengths are traversed, and finally the theoretical spectrum is obtained. FIG. 1 is a flow chart of a prior art method of tightly coupled wave analysis, as shown in FIG. 1, which is performed using a single core single thread, and thus must be cycled through all wavelength points within a defined wavelength range. Assuming that the time length required for calculating a certain wavelength point is t, and that the defined wavelength range includes N wavelength points, the time cost of serial calculation increases linearly with the wavelength range, that is, n×t, and the calculation efficiency is low. In order to improve the calculation efficiency, the invention introduces the concept of parallel calculation, after parameter initialization, the defined total wavelength range is divided equally, namely, one total task is divided into a plurality of subtasks, the subtasks are executed simultaneously on different calculation units by starting a proper multi-thread calculation tool, each calculation unit is responsible for solving one wavelength point at the same time and does not interfere with each other, and after all the subtasks are calculated and completed, the calculation results of all the subtasks are collected to obtain a theoretical spectrum.
The technical scheme of the invention is further described below with reference to the accompanying drawings of the embodiments.
Fig. 2 shows a flow diagram of a method of acquiring a theoretical spectrum according to an embodiment of the invention. As shown in fig. 2, a method for obtaining a theoretical spectrum includes:
first, in step 201, parameters are initialized. The parameter initialization is a preparation step necessary before intensive calculation, and in the embodiment of the invention, fourier series expansion is mainly performed on different optical characteristic parameters of a sample, and the parameters are substituted into maxwell's equations. In one embodiment of the invention, the optical characteristic parameters include geometry, boundary conditions, constraints of the sample to be detected, and determining the characteristics of the incident light. The accuracy of the subsequent calculation result is determined by the accuracy of the light characteristic parameters, and the accuracy is the basis of spectrum calculation. Wherein the geometric shape and boundary conditions relate to the size, shape, layer number, surface periodicity and the like of the sample to be detected, and the characteristics of the incident light include physical parameters such as wavelength, angle, polarization state and the like, and the physical parameters can influence the frequency, the incident direction, the polarization mode and the like of the incident light. The fourier series expansion is used to convert a complex dielectric function into a linear combination of a series of sine and cosine functions, and the dielectric function is decomposed into a finite number of frequency components by discretization, so that subsequent computation becomes feasible. It can be appreciated that the number of terms of the fourier series expansion has a relation with the calculation accuracy, the more the number of terms, the higher the calculation accuracy, but the calculation amount can also be increased significantly, based on which, in one embodiment of the present invention, the number of terms of the fourier series expansion is determined based on the calculation accuracy and/or the expected calculation duration, so as to satisfy the balance requirement between the calculation accuracy and the calculation cost;
next, at step 202, tasks are divided. As described above, in the embodiment of the present invention, the theoretical spectrum is obtained by the rigorous coupled-wave analysis method, based on which, before starting the calculation, the calculation task is split into N tasks according to the wavelength range to be calculated, that is, the wavelength range is equally divided into N wavelength bands, and each sub-task performs the calculation of one wavelength band, specifically, the transmission and reflection coefficients of one wavelength point;
next, in step 203, the computation is performed in parallel. On a multi-core central processing unit, the reflection and transmission coefficients of each wavelength point are calculated through an operation assembly, the operation assembly comprises a plurality of parallel operation modules, the operation modules synchronously operate, and one operation module executes a subtask, namely, the reflection and transmission coefficients of one wavelength point are calculated. In one embodiment of the invention, at each wavelength point, the following calculations are performed: solving maxwell's equation set, calculating the reflected and transmitted electromagnetic field distribution to obtain scattering matrix, and obtaining the reflection coefficient Rs and the transmission coefficient Rp through proper matrix operation. In one embodiment of the invention, the arithmetic component implements parallel computation based on openMP. The openMP is one of the most widely applied parallel tools, has high calling speed and simple grammar, and can be compatible with Windows systems and Linux operating systems. Before parallel operation is started, an openMP environment needs to be configured, and the configuration of the openMP environment is realized by adopting common technology in the field and is not described herein. And after the openMP environment is configured, parallel computing can be started and multithreading can be started according to openMP grammar rules. In step 203, the number of threads that are turned on should comprehensively consider the total number of threads of the CPU used for code running and the number of total wavelength points contained in the wavelength range. In one embodiment of the present invention, the number of the operation modules, that is, the maximum number of threads that can be opened is greater than or equal to the number of the wavelength points, and in this embodiment, each operation module may perform calculation at the same time, and if the time required for calculating a certain wavelength point is t, the whole calculation process may be completed only by the time t in an ideal state. In yet another embodiment of the present invention, the number of the operation modules, that is, the maximum number of threads that can be opened is smaller than the number of the wavelength points, all the operation modules are opened, the calculation tasks are first distributed evenly, and then the operation modules that preferentially finish the tasks execute the redundant subtasks. Specifically, each operation module calculates the reflection and transmission coefficients of ⌊ N/M ⌋ wavelength points, and one or more operation modules which preferably complete the calculation calculate the reflection and transmission coefficients of the remaining N-M ⌊ N/M ⌋ wavelength points. Assuming that the existing computing task of 101 wavelength points, the computing component comprises 32 computing modules, namely, the maximum number of CPU threads is 32, all 32 threads are started first, all threads sequentially bear the computing tasks of 3 wavelength points, and then a total of 32×3=96 tasks are executed, and the remaining 5 tasks are automatically distributed to the computing modules which finish the previous 3 tasks first. In addition, in the process of realizing parallel computing, attention is paid to data synchronization and sharing among threads, the problem of thread competition is considered, tasks and memory resources are reasonably divided, the condition that a plurality of threads read and write the same memory address is avoided as much as possible, and program interruption or breakdown caused by race conditions is prevented. In one embodiment of the invention, the stability of the algorithm can be improved as much as possible by calling the existing functions in the openMP library on the premise of ensuring that the performance of the algorithm is not lost too much. In one embodiment of the invention, the matrix operation is completed by selecting a proper high-performance computing library, so that the matrix calculation efficiency is effectively improved, and the matrix operation is performed by the mature high-performance computing library such as MKL, openBlas and the like at a higher speed through testing. It should be appreciated that in other embodiments of the invention, other parallel computing libraries may be employed, such as MPI, TBB, pthreads, etc.; and
finally, at step 204, the results of the calculations are combined. And after all the subtasks are completed, collecting calculation results of all the wavelength points to obtain complete reflection and transmission coefficient data for subsequent calculation so as to determine the theoretical spectrum of the sample.
In order to verify the effect of the method as described above, the method as described above is performed using a certain multi-core CPU, where the maximum number of threads of the CPU is 104, the wavelength range to be calculated is 250 to 750nm, the mesh width is 5nm, 101 wavelength points are included in total, the number of sample layers is 10, and the number of terms of fourier series expansion is 20. The reflection and transmission coefficients of the wavelength points are calculated by adopting a single-thread calculation method and the method in the embodiment of the invention. Fig. 3 shows a schematic diagram of a method of acquiring a theoretical spectrum in comparison to the calculation time required by the prior art according to an embodiment of the present invention. It can be seen that the time consumption (5.37356 s) of the method in the embodiment of the present invention is significantly lower than that of the existing single-threaded method (27.06496 s) under the condition that the calculation accuracy is the same (both Chi square are 0.000344). The number of sample layers is continuously increased from 10 to 90, the calculated amount is continuously increased, the actual physical field is more and more complex, and it can be seen that even though the method is a complex physical model, the method in the embodiment of the invention always has better acceleration effect, when the single-thread speed is close to 10 4 Second, parallel computation can still be kept at 10 3 Within seconds.
It should be appreciated that in order to further increase computational power, improve algorithm performance, and shorten computation time, in other embodiments of the present invention, a high-performance computer cluster may also be employed to implement the methods described above.
Furthermore, embodiments may be provided as a computer program product that may include one or more machine-readable media having stored thereon machine-executable instructions that, when executed by one or more machines, such as a computer, computer network, or other electronic device, may result in the one or more machines performing operations in accordance with embodiments of the present invention. The machine-readable medium may include, but is not limited to, floppy diskettes, optical disks, CD-ROMs (compact disk read-only memory), and magneto-optical disks, ROMs (read-only memory), RAMs (random access memory), EPROMs (erasable programmable read-only memory), EEPROMs (electrically erasable programmable read-only memory), magnetic or optical cards, flash memory, or other type of media/machine-readable medium suitable for storing machine-executable instructions.
Furthermore, embodiments may be downloaded as a computer program product, wherein the program may be transferred from a remote computer (e.g., a server) to a requesting computer (e.g., a client) by way of one or more data signals embodied in and/or modulated by a carrier wave or other propagation medium via a communication link (e.g., a modem and/or network connection). Accordingly, a machine-readable medium as used herein may include such a carrier wave, but is not required.
While various embodiments of the present invention have been described above, it should be understood that they have been presented by way of example only, and not limitation. It will be apparent to those skilled in the relevant art that various combinations, modifications, and variations can be made therein without departing from the spirit and scope of the invention. Thus, the breadth and scope of the present invention as disclosed herein should not be limited by any of the above-described exemplary embodiments, but should be defined only in accordance with the following claims and their equivalents.
Claims (10)
1. A method of obtaining a theoretical spectrum comprising the steps of:
performing Fourier series expansion on different optical characteristic parameters of the sample, and substituting the different optical characteristic parameters into a Maxwell equation set;
calculating the reflection and transmission coefficients of each wavelength point through an operation component of a multi-core central processing unit, wherein the operation component comprises a plurality of parallel operation modules, and each operation module calculates the reflection and transmission coefficients of one wavelength point at a time; and
and determining the theoretical spectrum of the sample according to the reflection and transmission coefficients of all the wavelength points.
2. The method of claim 1, wherein the optical characteristic of the sample comprises a geometry, a boundary condition, and a characteristic of incident light.
3. The method of claim 2, wherein the geometric shape, boundary conditions include the size, shape, number of layers, and surface periodicity of the sample; and/or
The characteristic parameters of the incident light include wavelength, angle and polarization state.
4. The method of claim 1, wherein the number of terms of the fourier series expansion is determined based on a required calculation accuracy and/or an expected calculation duration.
5. The method of claim 1, wherein the number of operational modules is greater than or equal to the number of wavelength points, each operational module performing a calculation simultaneously.
6. The method of claim 1, wherein the number of computation modules M is less than the number of wavelength points N, each computation module computes reflection and transmission coefficients for ⌊ N/M ⌋ wavelength points, and one or more computation modules that preferentially do the computation compute reflection and transmission coefficients for the remaining N-M ⌊ N/M ⌋ wavelength points.
7. The method of claim 1, wherein the calculation of the reflection and transmission coefficients for each wavelength point comprises the steps of:
solving maxwell's equation set, calculating the electromagnetic field distribution of reflection and transmission to obtain a scattering matrix; and
and obtaining the reflection coefficient and the transmission coefficient through matrix operation.
8. The method of claim 1, wherein the compute component implements parallel computation based on openMP.
9. An electronic device for acquiring a theoretical spectrum, characterized by comprising a memory and a processor, wherein the memory is configured to store a computer program which, when run by the processor, performs the method of any of claims 1 to 8.
10. A computer-readable storage medium for acquiring theoretical spectra, characterized in that a computer program is stored, which, when run on a processor, performs the method according to any of claims 1 to 8.
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| US20030204326A1 (en) * | 2001-07-16 | 2003-10-30 | Jon Opsal | Real time analysis of periodic structures on semiconductors |
| US9915522B1 (en) * | 2013-06-03 | 2018-03-13 | Kla-Tencor Corporation | Optimized spatial modeling for optical CD metrology |
| CN113343492A (en) * | 2021-06-30 | 2021-09-03 | 上海精测半导体技术有限公司 | Theoretical spectral data optimization method and system and optical measurement method |
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| 位旭龙: "DSP在OCD模拟平台加速中的应用研究", 中国优秀硕士学位论文全文数据库信息科技辑, 15 January 2014 (2014-01-15), pages 135 - 153 * |
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