JP2015162060A - Electromagnetic field analysis system, electromagnetic field analysis method, electromagnetic field analysis program and record medium recording this program - Google Patents

Abstract

PROBLEM TO BE SOLVED: To provide an electromagnetic field analysis system that can efficiently simulate a scattering phenomenon of a wave incident on a target having both a fine defect structure and a large-sized multilayer structure, is applicable even if the defect structure is non-periodic, prevents memory consumption of a computer from increasing even if the defect structure is distributed widely and prevents calculation time and the memory consumption from increasing even if the multilayer structure is large-sized.SOLUTION: An electromagnetic field analysis system comprises: boundary setting means that sets a virtual boundary separating a defect structure and a multilayer structure from each other; boundary condition calculation means that calculates a virtual boundary condition at least based on an incident wave reflectivity on the set virtual boundary when being viewed from the defect structure side; and defect-structure electromagnetic field calculation means that resolves the defect structure into multiple local defects, and then calculates an electromagnetic field of the defect structure by a multistep difference field boundary element method of repeatedly solving an integral equation which is obtained from both boundaries around a local defect obtained by adding at least one of these resolved local defects and the virtual boundary condition.

Description

  The present invention relates to an electromagnetic field analysis system, an electromagnetic field analysis method, an electromagnetic field analysis program, and a recording medium on which the program is recorded, which analyzes a scattering phenomenon of electromagnetic waves (light) incident on an object such as an optical element.

  Many photomasks and optical elements such as optical media, diffraction gratings, holographic elements, or antireflection films have a multilayer structure made of a metal material or a dielectric material, and micro defects or non-periodic uneven patterns on the surface of the multilayer structure. And an irregular defect structure. In an element having such a multilayer structure and a defect structure on the surface thereof, the quality greatly depends on the size and shape of the defect structure on the surface, so that the accuracy is less than the wavelength of light (several nm to several hundred nm). It is required to design and manufacture. For this reason, when designing this type of device, repeat the simulation of how the multilayer structure and the defect structure on its surface reflect and scatter electromagnetic waves (light) so that it approaches the target characteristics. It is necessary to determine the dimensions of each part.

As such a simulation, the following methods have already been developed.
Rigorous coupled wave analysis (RCWA) method: An optimization method that combines optical analysis based on a rigorous coupled wave theory and a simulated annealing method, and is used for designing a diffraction grating (for example, see Non-Patent Document 1). .
Time domain finite difference (FDTD) method: A method that divides the analysis region into small regions and applies Maxwell's equations to the difference in terms of time and space, and is used for light scattering analysis of optical media. (For example, refer nonpatent literature 2).
Finite / Boundary Element Coupling Method: A method that formulates the inside of a multilayer structure by the finite element method and the surface irregularities by the boundary element method, and is used for light scattering analysis of optical media (for example, non-patent literature) 3).
Finite element-boundary element hybrid analysis method: A combined method that applies the finite element method only to the inside of a multilayer structure to obtain a line form with unknowns, and uses the boundary element method to apply the surface integral formula by the Green function to the boundary part. Yes (see Non-Patent Document 4, for example).

Shiozaki et al., IEICE Technical Report, Vol. 103, No. 613, pp. 47-50 (2004) Fukai et al., IEICE Technical Report, Vol. 98, No. 510, pp. 73-78 (1999) Tanaka et al., Proceedings of the IEICE General Conference, p. 45 (1995) B. Alavikia et al. , J .; Opt. Soc. Am. A, Vol. 28, N0.10 (2011)

  However, the rigorous coupled wave analysis (RCWA) method disclosed in Non-Patent Document 1 can be applied to a large-scale multilayer structure, but the defect structure must be periodic (repetitively having the same shape spatially). There is a restriction that it cannot be applied.

  Further, the time domain finite difference (FDTD) method disclosed in Non-Patent Document 2 is highly versatile because it can be used even if the defect structure is not periodic, but the memory consumption of the computer compared to the area of the element Is extremely large and cannot be applied to a large-scale multilayer structure when the defect structure is distributed over a wide range, and is limited to the case where the thickness of the multilayer structure is at most several tens of wavelengths. End up.

  Furthermore, although the finite / boundary element coupling method disclosed in Non-Patent Document 3 can reduce the memory consumption of a computer, the incident wave to the object is not a focused beam (limited) due to the nature of the equation used. The light must be distributed only in a narrow range.

  Furthermore, the finite element-boundary element hybrid analysis method disclosed in Non-Patent Document 4 can be used even if the defect structure is not periodic, and can cope with a large-scale multilayer structure. Since a large amount of computation is required for processing, a very long processing time is required particularly when performing computation while maintaining high accuracy.

  Accordingly, an object of the present invention is to provide an electromagnetic field analysis system, an electromagnetic field analysis method, and an electromagnetic field analysis capable of efficiently simulating the scattering phenomenon of incident waves on an object having both a fine defect structure and a large-scale multilayer structure. A program and a recording medium on which the program is recorded are provided.

  Another object of the present invention is applicable to an electromagnetic field analysis system and an electromagnetic field analysis method that can be applied even if the defect structure is aperiodic and does not increase the memory consumption of the computer even if the defect structure is distributed over a wide range. Another object is to provide an electromagnetic field analysis program and a recording medium on which the program is recorded.

  Still another object of the present invention is to provide an electromagnetic field analysis system, an electromagnetic field analysis method, an electromagnetic field analysis program, and a recording medium on which this program is recorded, which does not increase computation time and memory consumption even when the multilayer structure is large-scale. Is to provide.

  According to the present invention, there is provided an electromagnetic field analysis system for analyzing an electromagnetic field when an incident wave is applied to an object having a multilayer structure and an irregular defect structure located on a surface side of the multilayer structure. Boundary setting means for setting a virtual boundary for dividing the defect structure and the multilayer structure from each other, and a boundary condition calculation for calculating a virtual boundary condition based at least on the reflectance of the incident wave viewed from the defect structure side on the set virtual boundary And solving an integral equation obtained from a boundary and a virtual boundary condition around the local defect obtained by decomposing the defect structure into a plurality of local defects and adding at least one of the decomposed local defects. There is provided an electromagnetic field analysis system including defect structure electromagnetic field calculation means for calculating an electromagnetic field of a defect structure by a multi-stage difference field boundary element method that repeats the above.

  The boundary setting means sets a virtual boundary that divides the defect structure and the multilayer structure at appropriate positions. The appropriate position in this case is a position where most of the region of the object is on the multilayer structure side and the defect structure side is a very narrow region. The boundary condition calculation means obtains a virtual boundary condition based on at least the reflectance of the incident wave when the multilayer structure side is viewed from the defect structure side on the virtual boundary by solving a recurrence formula. The defect structure electromagnetic field calculation means adds one of the local defects obtained by decomposing the defect structure, solves the integral equation obtained from the boundary around the local defect and the virtual boundary condition, and further adds another local defect. The electromagnetic field of this defect structure is calculated by a multi-step differential field boundary element method that repeats solving an integral equation obtained by adding a physical defect. Since the multilayer structure is replaced with a virtual boundary, an object with a large area of “defect structure + multilayer structure” is replaced with an object with a small area of “defect structure + virtual boundary”. Electromagnetic fields in defect structures with boundary conditions include, for example, Sugisaka et al., IEICE General Conference Proceedings, C-1-13 (2012), Sugisaka et al. It can be obtained only by solving a small-scale integral equation by the multi-stage difference field boundary element method known in the Proceedings, 17a-A2-2 (2012). As a result, it is possible to efficiently simulate an incident wave scattering phenomenon on an object having both a fine defect structure and a large-scale multilayer structure. Further, the present invention can be applied even if the defect structure is aperiodic, and the memory consumption of the computer does not increase even if the defect structure is distributed over a wide range. Furthermore, the part of the multilayer structure is incorporated into the integral equation in the form of boundary conditions when calculating the defect structure, and as will be described later, the electromagnetic field in the multilayer structure uses a part of the solution of the integral equation as the initial value. Therefore, even if the multilayer structure is large-scale, the calculation time and memory consumption do not increase.

  The defect structure electromagnetic field calculation means includes a decomposition part that decomposes the defect structure into a plurality of local defects, a defect addition part that adds one of the decomposed local defects to the original position, and a periphery of the added local defect Boundary approximation part that approximates the boundary with a set of boundary elements, the electromagnetic field before adding the local defect, the difference between the electromagnetic field before and after adding the local defect, the approximate boundary element, the calculated virtual boundary condition, and It is preferable to include an solving unit that solves an integral equation including at least a constant term, and an electromagnetic field updating unit that calculates an electromagnetic field after adding a local defect from the solution of the integral equation and updates the electromagnetic field. .

  The defect structure electromagnetic field calculating means obtains the electromagnetic field at the junction between the added local defect and the substrate and the electromagnetic field at the surface of the added local defect before adding the resolved local defect, and a constant term. It is also preferable to further include a constant term updating unit for updating

  Repeated operation in which the defect structure electromagnetic field calculation means repeatedly performs the processing operations of the defect adding unit, boundary approximating unit, solution finding unit, electromagnetic field updating unit, and constant term updating unit until all decomposed local defects are added. It is also preferable to further include a portion.

  It is also preferable to further include a multilayer structure electromagnetic field calculation means for calculating an electromagnetic field in the multilayer structure by a recurrence formula with a part of the solution of the integral equation as an initial value.

  According to the present invention, there is further provided an electromagnetic field analysis method for analyzing an electromagnetic field when an incident wave is applied to an object having a multilayer structure and an irregular defect structure located on a surface side of the multilayer structure. And setting a virtual boundary that divides the defect structure and the multilayer structure from each other, while calculating a virtual boundary condition based at least on the reflectance of the incident wave viewed from the defect structure side on the set virtual boundary, A multi-stage difference that is decomposed into a plurality of local defects and repeatedly solves the integral equation obtained from the boundary and virtual boundary conditions around the local defect obtained by adding at least one of the decomposed local defects An electromagnetic field analysis method for calculating an electromagnetic field of a defect structure by a field boundary element method is provided.

  A virtual boundary that divides the defect structure and the multilayer structure at an appropriate position is set. The appropriate position in this case is a position where most of the region of the object is on the multilayer structure side and the defect structure side is a very narrow region. On the other hand, a virtual boundary condition based on at least the reflectance of the incident wave when the multilayer structure side is viewed from the defect structure side on the virtual boundary is obtained by solving a recurrence formula. It is obtained by adding one of local defects obtained by decomposing the defect structure, solving the integral equation obtained from the boundary and virtual boundary conditions around the local defect, and adding other local defects The electromagnetic field of this defect structure is calculated by a multistage difference field boundary element method that repeats solving the integral equation. Since the multilayer structure is replaced with a virtual boundary, an object with a large area of “defect structure + multilayer structure” is replaced with an object with a small area of “defect structure + virtual boundary”. The electromagnetic field in a defect structure with boundary conditions can be obtained by simply solving a small integral equation by the multi-step differential field boundary element method. As a result, it is possible to efficiently simulate an incident wave scattering phenomenon on an object having both a fine defect structure and a large-scale multilayer structure. In addition, since the multi-stage difference field boundary element method is used, it is applicable even if the defect structure is aperiodic. Since the calculation is performed by adding local defects one by one, the defect structure Even if distributed widely, the memory consumption of the computer does not increase. In addition, the multilayer structure part is incorporated into the integral equation in the form of boundary conditions when calculating the defect structure, and as will be described later, the electromagnetic field in the multilayer structure is part of the solution of the integral equation. Since the initial value can be calculated by a recurrence formula, the calculation time and memory consumption do not increase even if the multi-layer structure is large.

  The calculation of the electromagnetic field of the defect structure decomposes the defect structure into a plurality of local defects, adds one of the decomposed local defects to the original position, and sets the boundary around the added local defect as a set of boundary elements Solve the integral equation including at least the electromagnetic field before adding the local defect, the difference between the electromagnetic field before and after adding the local defect, the approximate boundary element, the calculated virtual boundary condition, and the constant term, It preferably includes calculating an electromagnetic field after adding a local defect from the solution of the integral equation.

  The calculation of the electromagnetic field of the defect structure is a constant obtained by calculating the electromagnetic field at the junction between the added local defect and the substrate and the surface field of the added local defect before adding the resolved local defect. It preferably also includes updating the term.

  Until the calculation of the electromagnetic field of the defect structure adds all the decomposed local defects, the process of adding to the original position of one of the local defects, the approximation process by the set of boundary elements of the boundary around the added local defect It is also preferable that the method further includes repeatedly performing the process of solving the integral equation, the process of updating the electromagnetic field, and the process of updating the constant term.

  It is preferable that the method further includes calculating an electromagnetic field in the multilayer structure by a recurrence formula with a part of the solution of the integral equation as an initial value.

  According to the present invention, furthermore, an electromagnetic field analysis program for analyzing an electromagnetic field when an incident wave is applied to an object having a multilayer structure and an irregular defect structure located on the surface side of the multilayer structure. A boundary setting procedure for setting a virtual boundary for dividing the defect structure and the multilayer structure from each other and a virtual boundary condition based on at least the reflectance of the incident wave viewed from the defect structure side on the set virtual boundary are calculated. Integral equation obtained from boundary condition calculation procedure, boundary around virtual defect obtained by decomposing defect structure into multiple local defects and adding at least one of the decomposed local defects Electromagnetic field analysis program comprising defect structure electromagnetic field calculation procedure for calculating electromagnetic field of defect structure by multi-step difference field boundary element method that repeats solving Computer readable recording medium storing an electromagnetic field analysis program for row is provided.

  According to the present invention, it is possible to efficiently simulate an incident wave scattering phenomenon on an object having both a fine defect structure and a large-scale multilayer structure. Further, the present invention can be applied even if the defect structure is aperiodic, and the memory consumption of the computer does not increase even if the defect structure is distributed over a wide range. Furthermore, even if the multilayer structure is large-scale, the calculation time and memory consumption do not increase.

It is sectional drawing which shows the example of the target object which should be analyzed to which the electromagnetic field analysis system of this invention is applied. It is a figure which shows the principle of the electromagnetic field analysis system of this invention roughly. 1 is a block diagram schematically showing an overall configuration of an electromagnetic field analysis system as one embodiment of the present invention. FIG. It is a flowchart explaining operation | movement of the principal part of the electromagnetic field analysis system in embodiment of FIG. It is a block diagram which shows roughly the structure of the principal part of the electromagnetic field analysis system in embodiment of FIG. It is sectional drawing for demonstrating operation | movement of the electromagnetic field analysis system in embodiment of FIG. It is sectional drawing for demonstrating operation | movement of the electromagnetic field analysis system in embodiment of FIG. It is sectional drawing for demonstrating operation | movement of the electromagnetic field analysis system in embodiment of FIG. It is sectional drawing for demonstrating operation | movement of the electromagnetic field analysis system in embodiment of FIG. It is sectional drawing for demonstrating operation | movement of the electromagnetic field analysis system in embodiment of FIG. It is sectional drawing for demonstrating operation | movement of the electromagnetic field analysis system in embodiment of FIG. It is sectional drawing for demonstrating operation | movement of the electromagnetic field analysis system in embodiment of FIG. It is sectional drawing for demonstrating operation | movement of the electromagnetic field analysis system in embodiment of FIG. It is a figure for demonstrating operation | movement of the electromagnetic field analysis system in embodiment of FIG. It is sectional drawing for demonstrating operation | movement of the electromagnetic field analysis system in embodiment of FIG. It is sectional drawing for demonstrating operation | movement of the electromagnetic field analysis system in embodiment of FIG. It is sectional drawing for demonstrating operation | movement of the electromagnetic field analysis system in embodiment of FIG. It is sectional drawing for demonstrating operation | movement of the electromagnetic field analysis system in embodiment of FIG. FIG. 4 is a diagram for explaining a state in which an electromagnetic field is updated by obtaining a differential electromagnetic field in the embodiment of FIG. 3. In the embodiment of FIG. 3, it is a figure for demonstrating electromagnetic field distribution when the reflection in the back surface of a target object is considered. In the embodiment of FIG. 3, it is a figure for demonstrating the relationship between the number of layers of a multilayer structure, and calculation time. In the embodiment of FIG. 3, it is a figure for demonstrating the relationship between the layer thickness of a multilayered structure, and calculation time.

  FIG. 1 shows an example of an object to be analyzed to which the electromagnetic field analysis system of the present invention is applied, and FIG. 2 schematically shows the principle of the electromagnetic field analysis system of the present invention.

  The electromagnetic field analysis system of the present invention can analyze an electromagnetic field in an object having a surface defect structure and a multilayer structure such as a photomask for EUV (extreme ultraviolet) lithography and a diffraction grating having defects. As shown in FIG. 1A, an EUV mask has a defect structure (uneven structure) formed by a light shielding film 11 formed by EUV lithography on the surface side of a multilayer structure (substrate) made of an optical multilayer film such as a multilayer mirror 10. Exists. Further, as shown in FIG. 1B, the diffraction grating having defects has a large number of regular ridges 14 parallel to each other on the surface side of the multilayer structure composed of the glass substrate 13 having the coating layer 12 laminated on the back surface. There is a defect structure (uneven structure) due to the defects 15 such as dust and scratches. In the present invention, it is possible to analyze an electromagnetic field formed by a scattering phenomenon when electromagnetic waves are incident on an object of this type.

  That is, according to the present invention, as shown in FIG. 2A, when an incident wave is applied to an object having the surface defect structure 20 and the multilayer structure 21 from the surface defect structure 20 side, The virtual boundary 22 which divides | segments these in the appropriate position between the multilayer structures 21 is set. An appropriate position in this case is a position where most of the region of the object is on the multilayer structure 21 side and the defect structure 20 side is a very narrow region. The relationship between the transmitted wave from the upper side (defect structure 21 side) of the virtual boundary 22 to the multilayer structure 20 and the reflected wave from the multilayer structure 20 to the defect structure 21 side is obtained by solving a recurrence formula, and the virtual boundary condition is determined from this relationship. Set. That is, as shown in FIG. 2A, the surface defect structure 20 and the multilayer structure 21 are replaced with a surface defect structure having a virtual boundary with boundary conditions.

  Therefore, according to the present invention, as shown in FIG. 2B, the electromagnetic field around the defect structure can be obtained only by solving a small-scale integral equation relating to the surface defect structure having a virtual boundary with boundary conditions. it can.

  Next, the configuration of the electromagnetic field analysis system will be described.

  FIG. 3 schematically shows the entire configuration of the electromagnetic field analysis system as one embodiment of the present invention, FIG. 4 illustrates the operation of the main part of the electromagnetic field analysis system in the present embodiment, and FIG. 1 schematically shows a configuration of a main part of an electromagnetic field analysis system in the present embodiment.

  As shown in FIG. 3, the electrical configuration of the electromagnetic field analysis system in the present embodiment includes a central processing unit (CPU) 31, a read-only memory (ROM) 32, and a random access connected to each other via a bus 30. The computer includes a memory (RAM) 33, a hard disk drive (HDD) 34, an image processing unit 35, an external memory drive 36, and an input / output interface 37, and a program for operating the computer.

  The image processing unit 35 is connected to a display 38, and the external memory driving device 36 can be loaded with, for example, a Blu-ray disc / digital versatile disc / compact disc (BD / DVD / CD) 39 or a memory card. A keyboard 40 and a mouse 41 are connected to the output interface 37.

  The CPU 31 executes a program stored in the RAM 33 according to a basic program such as an operation system (OS) or a boot program stored in the ROM 32 to perform the processing of this embodiment. Further, the CPU 31 controls operations of the RAM 33, HDD 34, image processing unit 35, external memory driving device 36, and input / output interface 37.

  The RAM 33 is used as a main memory of the electromagnetic field analysis system, and stores programs and data transferred from the HDD 34 and the external memory driving device 36. The RAM 33 is also used as a work area for temporarily storing various data during program execution.

  The HDD 34 stores programs and data in advance.

  The image processing unit 35 performs graphic processing according to an instruction from the CPU 31 to generate image data. The generated image data is output to the display display 38.

  The external memory drive device 36 reads a program and data from the set external memory such as a BD / DVD / CD 39 or a memory card in accordance with an instruction from the CPU 31, and transfers it to the RAM 33. It is also possible to write programs and data to the set external memory.

  The input / output interface 37 controls data exchange between the keyboard 40 and mouse 41 and the CPU 31 or RAM 33.

  In the electromagnetic field analysis system having such a configuration, when operating, the CPU 31 first secures a program storage area, a data storage area, and a work area in the RAM 33, and loads the program and data from the HDD 34 or the outside to store the program. Store in the area and data storage area. Next, the processing shown in FIG. 4 is executed based on the program stored in the program storage area. When the CPU 31 executes the program, an electromagnetic field analysis system 50 as schematically shown in FIG. 5 is constructed.

  In FIG. 5, 51 is a boundary setting means for setting a virtual boundary that divides the defect structure and multilayer structure of the object from each other, 52 is a boundary condition calculation means for calculating the boundary condition of the set virtual boundary, and 53 is the defect structure. The multilayer structure electromagnetic field calculation means 54 for calculating the electromagnetic field in the removed multilayer structure is obtained from the boundary around the local defect obtained by adding one of the local defects, the virtual boundary condition, and the constant term. The defect structure electromagnetic field calculation means for calculating the electromagnetic field of the defect structure by a multi-stage difference field boundary element method that repeatedly solves the integral equation is shown.

  The defect structure electromagnetic field calculation means 54 includes a decomposition unit 54a that decomposes the defect structure into a plurality of local defects, a defect addition unit 54b that adds one of the decomposed local defects to the original position, and an added local A boundary approximating unit 54c that approximates the boundary around the defect with a set of boundary elements, the electromagnetic field before adding the local defect, the difference between the electromagnetic field before and after adding the local defect, the approximate boundary element, the calculated virtual element The solving unit 54d that solves the integral equation including at least the boundary condition and the calculated constant term, and the electromagnetic field updating unit that calculates and updates the electromagnetic field when the local defect at that time is added from the solution of the integral equation 54e, a constant term update unit 54f that calculates a constant term that is an electromagnetic field before the next defect addition and updates the constant term of the integral equation, and a constant term update unit 54f until all the resolved local defects are added. Defect addition unit 54 Includes boundary approximation unit 54c, solution finding unit 54d, and a repeating operation portion 54g which performed by repeating the processing operation of the electromagnetic field update unit 54e.

  Hereinafter, the configuration and operation of the electromagnetic field analysis system 50 will be described in detail using the flowchart shown in FIG.

  The boundary setting means 51 of the electromagnetic field analysis system 50 first sets a virtual boundary S2 (step A1 in FIG. 4).

  The structure of the analysis object is an object having a cross-sectional structure as shown in FIG. 6, and both the structure and the electromagnetic field are constant in the direction perpendicular to the paper surface of FIG. There are a plurality of layered structures (multilayered structures) made of different dielectrics inside the object, and it continues infinitely in the horizontal direction of the drawing. There is no restriction on the thickness of each layer. Irregular local defects on the surface of the object (defects here are dielectric structures that disturb the ideal multilayer structure consisting only of parallel straight boundaries, or that disturb the periodic uneven structure of the surface. ). The local defects are protrusions on the surface of the object or particulate foreign substances embedded in the object, and the number thereof may be any number as long as it is finite. The electromagnetic field analysis system 50 obtains electromagnetic fields inside and around the object when an incident wave propagates from above to such an object.

  Although the incident wave is assumed to be a plane wave, incident waves of other shapes are also represented by a superposition of plane waves, and can be applied to any incident wave. Further, electromagnetic waves have a polarization state, and in the following, cases of p-polarized light and s-polarized light will be described. Other polarized light such as elliptically polarized light is expressed by a linear combination of p-polarized light and s-polarized light, and therefore any polarization state may be used as long as it is completely polarized light.

  The boundary setting means 51 draws a virtual boundary S5 with respect to the cross-sectional structure so as to be parallel to the boundary line (for example, S4) inside the multilayer structure. The upper region S6 of the virtual boundary S5 is a “surface defect structure”, and the lower region S7 of the virtual boundary S5 is a “multilayer structure”. The multilayer structure S7 is prevented from entering the surface defect structure S6, and the virtual boundary S5 and other dielectric boundaries are prevented from intersecting. The virtual boundary S5 is set as close as possible to the surface S8 of the object.

  As shown in FIG. 7, the dielectric constant inside the local defect may be different from the surroundings, and a plurality of types of dielectrics may exist in the surface defect structure S6. However, the virtual boundary S5 is set as close as possible to the surface and parallel to the multilayer structure so as not to cross other dielectric boundaries. Any layer of the multilayer structure S7 may be very thick like an element having a coating layer (dielectric thin film) S9 formed on the back surface. When considering the back surface S9 of the element as in this example, it is assumed that an air layer S10 having an infinite thickness exists at the lower part of the element, and the multilayer structure S7 includes the air layer S10.

  Further, the local defect is not limited to the unevenness of the surface. For example, as shown in FIG. 8, the particulate foreign matter S11 embedded near the surface of the multilayer structure can be regarded as the local defect. In this case, the virtual boundary S5 is set so as to pass between the foreign matter S11 that is a local defect and the multilayer structure therebelow and does not cross all dielectric boundaries including the surface of the foreign matter S11. Since the multilayer structure may be large-scale, the virtual boundary S5 is made as close to the foreign substance S11 as possible.

  Next, the decomposition unit 54a of the defect structure electromagnetic field calculation means 54 decomposes the defect structure on the surface into a plurality of local defects (step A2 in FIG. 4).

  As shown in FIG. 9, the surface defect structure is separated into a plurality of local defects S12 to S14 and a planar structure. The remaining structure after the local defect is separated is only a virtual boundary S5 and a linear dielectric boundary S15 parallel to the virtual boundary S5.

  As shown in FIG. 10, even in the case of a dielectric having different protrusions, only the virtual boundary S5 and the linear dielectric boundary S19 parallel to the virtual boundary S5 remain after the protrusions S16 to S18 are separated. .

  In the case of a buried local defect, the particulate foreign matter S11 (see FIG. 8) is removed. When the concavo-convex structure is included at the same time, after removing the embedded foreign matter, the protrusion is separated as shown in FIG. 9 or FIG.

  Next, the defect adding unit 54b of the defect structure electromagnetic field calculating unit 54 adds a position based on one of the local defects (step A3 in FIG. 4).

  For example, as shown in FIG. 11, it is assumed that the local defect S12 has already been added and the local defects S13 and S14 have not yet been added. One local defect that has not yet been added is selected (any one may be selected, but here, local defect S13 is selected), and this local defect S13 is added at the same position as before separation. For the local defect S12 that has already been added and the local defect S14 that has not yet been added, no operation is performed here.

  Next, the boundary approximating unit 54c of the defect structure electromagnetic field calculating unit 54 approximates the dielectric boundary around the added local defect with a set of boundary elements (step A4 in FIG. 4).

Here, as shown in FIG. 12, there are the following four types of boundaries approximated by boundary elements.
(1) Boundary C0: a dielectric boundary (a boundary between the substrate S20 and the external region S21) before the local defect S13 is added. However, a boundary far from the local defect S13 is not necessary. If there is a local defect that has already been added, an element is set along the surface of the local defect.
(2) Boundary C1: A junction between the added local defect S13 and the substrate S20. When the dielectric constants of the substrate S20 and the local defect S13 are the same, it is not a dielectric boundary, but is always set for convenience of calculation.
(3) Boundary C2: The surface of the added local defect S13.
(4) Boundary C3: A virtual boundary. However, it becomes unnecessary at a distance sufficiently far from the local defect S13. A range (total length) obtained by dividing the element is denoted by Λ. If the boundary C3 is sufficiently close to the local defect S13, Λ may be shortened (may be the same as the boundary C0), but it needs to be longer as the distance from the local defect S13 increases.

  The boundary element is generally a short line segment, but if the defect surface is a curve, an arc may be used instead of the line segment.

  Next, the solving unit 54d of the defect structure electromagnetic field calculation means 54 solves the integral equation and calculates the electromagnetic fields on the boundaries C0 to C3 (step A5 in FIG. 4).

First, as shown in FIG. 13, the electromagnetic fields in the region S1 and the region S2 before the local defect addition are represented as f ₁ (ρ) and f ₂ (ρ). However, Δf (ρ) represents the difference in the electromagnetic field before and after the defect addition. The electromagnetic field f (ρ) changes in the electromagnetic field variable depending on the polarization state of the incident wave, and represents a magnetic field in the z direction when p-polarized light and an electric field in the z direction when s-polarized light. Since f ₁ (ρ) and f ₂ (ρ) are obtained when a previous local defect is added, they are assumed to be known values here. That is, the unknown variables are Δf ₁ (ρ), Δf ₂ (ρ), and f ₃ (ρ).

The electromagnetic field Δf at the boundary C3 shown in FIG. That is, the electromagnetic field on this boundary is represented by a superposition of plane waves in various directions. The absolute value of the angle θ formed between the propagation direction of each plane wave and the x axis in FIG. 13 is designated by p. Depending on the sign of θ, there are two components: a component propagating in the −y direction and a component propagating in the + y direction. With unknown variables xi] _p, represents the upper boundary C3 amplitudes xi] _p of plane wave components propagating in -y direction, the amplitude of the plane wave components propagating in the + y direction as ξ _p α _p. α _p is the amplitude ratio of the plane wave propagating in the −y direction to the plane wave propagating in the + y direction, and is a (Robin type) boundary condition imposed on the boundary C3. This value α _p is separately calculated from the multilayer structure portion (boundary condition calculation means 52, step C1 in FIG. 4).

By solving the integral equation of the following equation (Equation 2), the unknown electromagnetic fields Δf ₁ and f ₃ on the boundaries C0 to C3 shown in FIG. 12 are obtained. Here, ρ is a two-dimensional position vector representing coordinates in the cross-sectional structure. The left side of these equations includes unknown variables, and the right side is a known electromagnetic field or a constant term obtained by integrating it.

In the above equation (Equation 2), the definitions of the integration operators L, L ′ and F are as shown in the following equation (Equation 3). However, η _m represents the dielectric constant (when p-polarized light) or the magnetic permeability (when s-polarized light) of the region S _m (m = 1, 2, 3, see FIG. 12). The direction of integration is performed along the arrow in FIG.

  As shown in FIG. 14, n and l in the above-described integral equation are a local coordinate system with (n, l, z) as an axis, and change depending on the direction of the boundary element. l represents the direction along the boundary, and n represents the normal direction of the boundary.

G in the integration operator is a free space Green's function, and is expressed by the following equation (Equation 4) using a second kind 0th-order Hankel function. Here, _nm is the refractive index of the region _Sm (see FIG. 12), c is the speed of light in vacuum, ω is the angular frequency of the incident wave, and j is an imaginary unit.

ここで、Ｃ_ｑは境界Ｃのｑ番目の境界要素を表し、ｆ_ｑはその境界要素上の電磁界を表す。 In order to solve the integral equation by a computer, the unknown variables Δf ₁ and f ₃ expressed in the form of functions are discretized. A discretization method similar to the general boundary element method can be applied here. As an example, if Δf ₁ and f ₃ are approximated to have a constant value within one boundary element, for example, the integration operator L is expressed by the following equation (Equation 5).

Here, C _q represents the q-th boundary element of the boundary C, and f _q represents the electromagnetic field on the boundary element.

Here, the partial differentiation of f _q does not mean that a differential operation is performed using the obtained f _q , but represents one unknown variable obtained independently of f _q . Note that the boundary C3 is already discretized by the discrete unknown variable ξ, and the above processing is not necessary.

  By the above discretization process, the above-mentioned integral equation (Equation 2) is expressed by a linear combination of unknown variables. A solution is obtained by a computer using these as simultaneous equations. As a solution method, a Gaussian elimination method, a direct method such as LU decomposition, or an iterative method such as GMRES or BiCGSTAB method can be applied.

  Thereafter, the electromagnetic field update unit 54e of the defect structure electromagnetic field calculation means 54 calculates the electromagnetic field after adding local defects at any point of the defect structure from the solution of the integral equation, and updates the electromagnetic field. (Step A6 in FIG. 4).

That is, Δf ₁ , Δf _2, and f ₃ at an arbitrary point in the surface defect structure portion are obtained from the following equation (Equation 6) using Δf ₁ and f ₃ on the boundary, which is the solution of the integral equation. As for Δf ₁ and Δf ₂ , by taking the sum of f ₁ and f ₂ , respectively, an electromagnetic field after adding a local defect at that time can be obtained. f ₃ is itself is the electromagnetic field after local defects added at that time. S1 to S3 in this case correspond to FIG.

  A case of a buried defect will be described. In this case, the boundary and electromagnetic field before and after the defect addition are set as shown in FIG. The line integral of C0 and the virtual boundary C3 is calculated in the right direction. C1 integrates clockwise. When discretizing into boundary elements, C0 and C3 need only be near the defect S3.

The integral equation in this case is as shown in the following equation (Equation 7).

From the solution of the integral equation of embedded defects shown in the above equation (Equation 7), the electromagnetic field of each region is given by the following equation (Equation 8).

  Next, the repetitive operation unit 54g of the defect structure electromagnetic field calculation unit 54 determines whether all local defects have been added (step D1 in FIG. 4). If all local defects have been added (in the case of YES), this process is terminated. If there is a local defect that has not been added (in the case of NO), the process in the next step A7 (FIG. 4). I do. That is, the repetitive operation unit 54g repeats the processing operations of the constant term update unit 54f, the defect addition unit 54b, the boundary approximation unit 54c, the solution finding unit 54d, and the electromagnetic field update unit 54e until all the decomposed local defects are added. To implement.

  The constant term update unit 54f of the defect structure electromagnetic field calculation means 54 calculates a constant term that is an electromagnetic field before the next defect addition and updates the constant term of the integral equation (step A7 in FIG. 4).

In order to solve the integral equation, (A) the electromagnetic field on C1 before addition of local defects, (B) the normal differential value of the electromagnetic field on C1, and (C) the electromagnetic field on C2 as constant terms is necessary. Therefore, the electromagnetic fields at positions (B1 and B2 in FIG. 16) corresponding to C1 and C2 when the next local defect is added are obtained in advance. Since these boundaries are at the current C0 and S2, respectively, the electromagnetic field Δf ₁ on C0 (part of the solution of the number of integral equations of Equation 2) and the electromagnetic field Δf ₂ within S2 (calculated from the equation of Equation 8) ) And save the values of f ₁ + Δf ₁ and f ₂ + Δf ₂ , respectively. However, when B1 is not within the current range of C0, Δf ₁ = 0.

  On the other hand, the multilayer structure electromagnetic field calculation means 53 calculates the electromagnetic field of the structure from which all surface defects have been removed (step B1 in FIG. 4). When all the defects are removed, if the surface has a periodic uneven structure (a multilayer structure may be included), the electromagnetic field is calculated using strict coupling wave analysis (RCWA) or the like. On the other hand, when the structure when all the defects are removed is a multilayer structure consisting of only parallel straight boundaries, the electromagnetic field is calculated by the following procedure.

As shown in FIG. 17, the incident wave is a plane wave that propagates in the y direction with an amplitude a ₁ in the external region s ₁ (considered as an infinite thickness air layer). It is assumed that the reflected wave of s ₁ propagates in the −y direction with an amplitude b ₁ . Similarly, for the multilayer structures s _{2 to} s _N inside the element, components that propagate in the + y direction are a _{2 to} a _N , and components that propagate in the −y direction are b _{2 to} b _N. The upper and lower boundary coordinate layer _{s m y m,} and _{y m + 1.}

In this case, the electromagnetic field inside the layer s _m (m = 1, 2,..., N) is expressed by the following equation (Equation 9). If the amplitudes a _m and b _m are obtained, the electromagnetic field inside s _m is given by this equation. θ is the angle between the traveling direction of the incident wave at s ₁ and the x axis.

Let c _m be the ratio of b _m to a _m . b _N are (no incident light from the back side of the device) because it is 0, a c _N = 0. Using this as the first term, the following recursion formula (Equation 10) is used to obtain c _{1 in} order.

However, when s-polarized light, _{r m} and r _{'m + 1} is given by the following expression (11). d _m is the thickness of the _{s m} layer.

Also, when p-polarized light, _{r m} and r _{'m + 1} is given by the following expression (Expression 12). n _m is the refractive index of _{s m} layer.

If Motomare c _m are all from the value of _{a 1} (known. The amplitude of the incident wave), all of _{a m} is obtained by the following recurrence formula (number 13). Finally, b _m is obtained from _cm and a _m

For s-polarized light, t _m is given by the following equation (Formula 14).

For p-polarized light, t _m is given by the following equation (Equation 15).

  On the other hand, the boundary condition calculation means 52 calculates the boundary condition of the set virtual boundary (step C1 in FIG. 4).

That is, α _p used in the integration calculation F is obtained. As shown in FIG. 18, the alpha _p is a plane wave propagating downward on the virtual boundary B1, a ratio of a plane wave propagating upward, when the virtual boundary B1 is to be in s _2, above This corresponds to c ₂ in the equation (10). This value is obtained sequentially from the recurrence formula (Equation 10) and c _N = 0. Since it is necessary to more calculated for different p, (varies the value of p by _{k m) k xp} into equation of _{k x} number 9 of equation (1) described above, _{c 2} (alpha _p ) Is repeated. During calculation, the upper structure from the virtual boundary B1 is not necessary to consider, consider a layer s ₂ is followed indefinitely. α _p represents the reflectance when the downward direction is viewed from the virtual boundary B1, and the multilayer structure s _{3 to} s _{N in} which complex multiple reflection occurs is replaced with a simple boundary having the reflectance α _p. it can. As a result, the boundary integral of the multilayer structure portion is not necessary in the integral equation (Equation 2), and the amount of calculation can be greatly reduced.

  As described above in detail, according to the present embodiment, for example, a surface defect structure portion that is a concavo-convex structure or a foreign substance is subjected to a multi-step (multistep) differential field boundary element method and an incident wave to a multilayer structure portion. The relationship with the reflected wave is obtained from the transmission matrix. Due to the characteristics of the multi-stage difference field boundary element method, it can be applied even if the defect structure is aperiodic.

  FIG. 19 shows how the electromagnetic field is updated by obtaining the differential electromagnetic field in this embodiment. In this embodiment, as shown in the figure, the electromagnetic field is obtained and updated while adding local defects (protrusions and foreign matter) in the defect structure portion one by one in consideration of the interaction with the surroundings. For this reason, even if defects are distributed over a wide range, the amount of memory consumed once by the computer does not increase.

  Furthermore, in the present embodiment, the multilayer structure portion is incorporated in the integral equation in the form of boundary conditions when calculating the defect structure, and the electromagnetic field in the multilayer structure is a part of the solution of the integral equation. Since the initial value can be calculated by a recurrence formula, the calculation time and memory consumption do not increase even if the multi-layer structure is large.

  FIG. 20 illustrates the electromagnetic field distribution in the present embodiment when reflection on the back surface of the object is taken into consideration. For example, as shown in the figure, a dielectric layer having a convex local defect having a refractive index n = 2.0 on the surface side and having a refractive index n = 1.5 and n = 2.0 below the convex local defect. When an incident wave (600 nm), which is a plane wave of p-polarized light, is incident at an incident angle of 45 ° on a multi-layered structure having a thickness of 2.0 mm (more than 3300 times the wavelength) in which the layers are alternately stacked, calculation by a computer The time was several seconds, and it could be processed sufficiently with one personal computer. When the conventional time domain finite difference (FDTD) method is used, the memory consumption is enormous, and it is essential to use a large computer. Note that the color map in FIG. 20 represents the instantaneous value of the magnetic field in the direction perpendicular to the paper, where red represents a positive magnetic field and blue represents a negative magnetic field.

  FIG. 21 shows the relationship between the number of layers of the multilayer structure and calculation time in this embodiment, and FIG. 22 shows the relationship between the layer thickness of the multilayer structure and calculation time in this embodiment. As shown in FIG. 21, since the number of unknowns in the integral equation is independent of the number of layers, even if the number of layers increases, the amount of increase in calculation time is extremely small. Even when the number of layers was 1000, the increase in calculation time was less than 5%. Further, as shown in FIG. 22, since the number of unknowns of the integral equation does not depend on the layer thickness, the calculation time is constant with respect to the change in the layer thickness. That is, even a device having a multi-layer film exceeding 1000 layers or a large-scale multi-layer structure having a thickness several thousand times the wavelength can be processed in a few seconds using a single personal computer.

  Furthermore, in order to solve the integral equation according to the present embodiment, it is sufficient that the amplitude of the incident wave is known, so there is no restriction on the shape of the incident wave.

  In this embodiment, as in the case of the finite element-boundary element hybrid analysis, boundary integration of the Green function is necessary. However, the Green function used in this embodiment is only the free space Green function, and the multi-layered Green function is It is unnecessary. For this reason, time-consuming processing is not performed with respect to the Green function, and the speed can be increased.

  All the embodiments described above are illustrative of the present invention and are not intended to be limiting, and the present invention can be implemented in other various modifications and changes. Therefore, the scope of the present invention is defined only by the claims and their equivalents.

  The electromagnetic field analysis system and the electromagnetic field analysis method of the present invention are applicable to, for example, photomask design for photolithography and EUV photolithography. In this case, the circuit pattern on the substrate and the surface of the photomask corresponds to the multilayer structure and the defect structure calculated in the present invention, respectively. It can be used to confirm by simulation whether exposure can be performed correctly with the designed mask. It can also be used when estimating in advance by simulation how foreign matter or the like mixed during the manufacture of the mask affects the exposure result. Furthermore, the electromagnetic field analysis system and electromagnetic field analysis method of the present invention can be applied to the design of optical elements, for example. When designing an element with no periodicity in an uneven structure such as an optical medium, a diffractive lens, or a hologram, the optical characteristics are targeted by repeating simulation while considering back reflection of the glass substrate and multiple reflection inside the antireflection film. Can also be used when approaching.

DESCRIPTION OF SYMBOLS 10 Multilayer mirror 11 Light-shielding film 12 Coating layer 13 Glass substrate 14 Projection 15 Defect 20 Defect structure 21 Multilayer structure 22 Virtual boundary with boundary condition 30 Bus 31 CPU
32 ROM
33 RAM
34 HDD
35 Image Processing Unit 36 External Memory Drive Device 37 Input / Output Interface 38 Display Display 39 BD / DVD / CD
DESCRIPTION OF SYMBOLS 40 Keyboard 41 Mouse 50 Electromagnetic field analysis system 51 Boundary setting means 52 Boundary condition calculation means 53 Multilayer structure electromagnetic field calculation means 54 Defect structure electromagnetic field calculation means 54a Decomposition part 54b Defect addition part 54c Boundary approximation part 54d Solving part 54e Electromagnetic field update Part 54f constant term update part 54g repetitive operation part

Claims (12)

An electromagnetic field analysis system for analyzing an electromagnetic field when an incident wave is applied to an object having a multilayer structure and an irregular defect structure located on a surface side of the multilayer structure,
Boundary setting means for setting a virtual boundary for dividing the defect structure and the multilayer structure from each other, and a virtual boundary condition based on at least the reflectance of the incident wave viewed from the defect structure side on the set virtual boundary Obtained from boundary condition calculation means, a boundary around the local defect obtained by decomposing the defect structure into a plurality of local defects, and adding at least one of the decomposed local defects, and the virtual boundary condition An electromagnetic field analysis system comprising: defect structure electromagnetic field calculation means for calculating an electromagnetic field of the defect structure by a multistage difference field boundary element method that repeatedly solves the integral equation.   The defect structure electromagnetic field calculation means includes a decomposition unit that decomposes the defect structure into a plurality of local defects, a defect addition unit that adds one of the decomposed local defects to an original position, and the added local A boundary approximation unit that approximates a boundary around a local defect with a set of boundary elements; an electromagnetic field before adding the local defect; a difference between electromagnetic fields before and after adding the local defect; the approximate boundary element; Solving unit for solving the calculated virtual boundary condition and an integral equation including at least a constant term, and an electromagnetic field update for updating the electromagnetic field by calculating the electromagnetic field after adding the local defect from the solution of the integral equation The electromagnetic field analysis system according to claim 1, further comprising a unit.   The defect structure electromagnetic field calculation means obtains an electromagnetic field at a junction between the local defect to be added and the substrate before adding the decomposed local defect and an electromagnetic field on the surface of the local defect to be added. The electromagnetic field analysis system according to claim 2, further comprising a constant term update unit that updates the constant term.   The processing operations of the defect adding unit, the boundary approximating unit, the solution finding unit, the electromagnetic field updating unit, and the constant term updating unit until the defect structure electromagnetic field calculating unit adds all the decomposed local defects. The electromagnetic field analysis system according to claim 3, further comprising a repetitive operation unit that repeatedly performs the operation.   5. The multilayer structure electromagnetic field calculation means for calculating an electromagnetic field in the multilayer structure by a recurrence formula with a part of the solution of the integral equation as an initial value. The electromagnetic field analysis system according to item 1. An electromagnetic field analysis method for analyzing an electromagnetic field when an incident wave is applied to an object having a multilayer structure and an irregular defect structure located on a surface side of the multilayer structure,
Setting a virtual boundary that divides the defect structure and the multilayer structure from each other, and calculating a virtual boundary condition based on at least a reflectance of an incident wave viewed from the defect structure side on the set virtual boundary; Decomposing the structure into a plurality of local defects, and at least adding one of the decomposed local defects to solve the boundary around the local defect and solving the integral equation obtained from the virtual boundary condition An electromagnetic field analysis method, wherein an electromagnetic field of the defect structure is calculated by a multi-stage difference field boundary element method.   The calculation of the electromagnetic field of the defect structure decomposes the defect structure into a plurality of local defects, adds one of the decomposed local defects to the original position, and defines a boundary around the added local defect. Approximation with a set of boundary elements and electromagnetic field before adding the local defect, difference between electromagnetic fields before and after adding the local defect, the approximate boundary element, the calculated virtual boundary condition, and a constant term The method further comprises: calculating an electromagnetic field after adding the local defect from the solution of the integral equation and updating the electromagnetic field by solving an integral equation including at least Electromagnetic field analysis method.   The calculation of the electromagnetic field of the defect structure determines the electromagnetic field of the junction between the added local defect and the substrate and the electromagnetic field of the surface of the added local defect before adding the decomposed local defect. The electromagnetic field analysis method according to claim 7, further comprising updating the constant term.   Until the calculation of the electromagnetic field of the defect structure adds all the decomposed local defects, the process of adding to one original position of the local defects, the boundary element of the boundary around the added local defects 9. The method according to claim 8, further comprising: repeatedly performing an approximation process by a set, a process of solving the integral equation, a process of updating the electromagnetic field, and a process of updating the constant term. Electromagnetic field analysis method.   10. The method according to claim 6, further comprising calculating an electromagnetic field in the multilayer structure by a recurrence formula using a part of the solution of the integral equation as an initial value. 10. Electromagnetic field analysis method. An electromagnetic field analysis program for analyzing an electromagnetic field when an incident wave is applied to an object having a multilayer structure and an irregular defect structure located on a surface side of the multilayer structure,
A boundary setting procedure for setting a virtual boundary for dividing the defect structure and the multilayer structure from each other, and a virtual boundary condition based on at least the reflectance of the incident wave viewed from the defect structure side on the set virtual boundary are calculated. Obtained from a boundary condition calculation procedure, a boundary around the local defect obtained by decomposing the defect structure into a plurality of local defects, and adding at least one of the decomposed local defects, and the virtual boundary condition An electromagnetic field analysis program comprising: a defect structure electromagnetic field calculation procedure for calculating an electromagnetic field of the defect structure by a multi-stage difference field boundary element method that repeatedly solves the integral equation.   An electromagnetic field analysis program for analyzing an electromagnetic field when an incident wave is applied to an object having a multilayer structure and an irregular defect structure located on a surface side of the multilayer structure, the defect structure and the defect structure A boundary setting procedure for setting a virtual boundary for dividing the multilayer structure from each other; and a boundary condition calculation procedure for calculating a virtual boundary condition based at least on the reflectance of the incident wave viewed from the defect structure side on the set virtual boundary; The defect structure is decomposed into a plurality of local defects, and at least one of the decomposed local defects is added to solve the integral equation obtained from the boundary around the local defect and the virtual boundary condition. A computer having recorded thereon an electromagnetic field analysis program for causing a computer to execute a defect structure electromagnetic field calculation procedure for calculating an electromagnetic field of the defect structure by a multi-step differential field boundary element method Readable recording medium.

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