WO2022126692A1 - Su-8胶紫外光背面光刻工艺的二维光强分布模拟方法 - Google Patents
Su-8胶紫外光背面光刻工艺的二维光强分布模拟方法 Download PDFInfo
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- 238000000034 method Methods 0.000 title claims abstract description 44
- 229920001486 SU-8 photoresist Polymers 0.000 title claims abstract description 42
- 238000004088 simulation Methods 0.000 title claims abstract description 36
- 238000001459 lithography Methods 0.000 title claims abstract description 30
- 230000005672 electromagnetic field Effects 0.000 claims abstract description 37
- 239000000463 material Substances 0.000 claims abstract description 16
- 239000003292 glue Substances 0.000 claims description 31
- 239000011159 matrix material Substances 0.000 claims description 27
- 229920002120 photoresistant polymer Polymers 0.000 claims description 19
- 230000005684 electric field Effects 0.000 claims description 13
- 238000012876 topography Methods 0.000 claims description 12
- 239000000853 adhesive Substances 0.000 claims description 4
- 230000001070 adhesive effect Effects 0.000 claims description 4
- 238000004364 calculation method Methods 0.000 claims description 4
- 230000003287 optical effect Effects 0.000 claims description 4
- 238000005286 illumination Methods 0.000 claims description 3
- 238000013178 mathematical model Methods 0.000 claims description 3
- 238000000926 separation method Methods 0.000 abstract description 2
- 239000006185 dispersion Substances 0.000 abstract 1
- 238000000206 photolithography Methods 0.000 description 8
- 239000000758 substrate Substances 0.000 description 7
- 238000010586 diagram Methods 0.000 description 6
- 238000011161 development Methods 0.000 description 5
- 230000000694 effects Effects 0.000 description 5
- 238000005516 engineering process Methods 0.000 description 4
- 238000004519 manufacturing process Methods 0.000 description 3
- 238000012545 processing Methods 0.000 description 3
- 238000005094 computer simulation Methods 0.000 description 2
- 239000011324 bead Substances 0.000 description 1
- 230000009286 beneficial effect Effects 0.000 description 1
- 238000000576 coating method Methods 0.000 description 1
- 238000013461 design Methods 0.000 description 1
- 238000002474 experimental method Methods 0.000 description 1
- 238000005290 field theory Methods 0.000 description 1
- 239000011521 glass Substances 0.000 description 1
- 238000012986 modification Methods 0.000 description 1
- 230000004048 modification Effects 0.000 description 1
- 238000013386 optimize process Methods 0.000 description 1
- 238000011160 research Methods 0.000 description 1
- 238000012795 verification Methods 0.000 description 1
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- G—PHYSICS
- G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
- G03F—PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR
- G03F7/00—Photomechanical, e.g. photolithographic, production of textured or patterned surfaces, e.g. printing surfaces; Materials therefor, e.g. comprising photoresists; Apparatus specially adapted therefor
- G03F7/70—Microphotolithographic exposure; Apparatus therefor
- G03F7/70483—Information management; Active and passive control; Testing; Wafer monitoring, e.g. pattern monitoring
- G03F7/70491—Information management, e.g. software; Active and passive control, e.g. details of controlling exposure processes or exposure tool monitoring processes
- G03F7/705—Modelling or simulating from physical phenomena up to complete wafer processes or whole workflow in wafer productions
-
- G—PHYSICS
- G06—COMPUTING; CALCULATING OR COUNTING
- G06F—ELECTRIC DIGITAL DATA PROCESSING
- G06F30/00—Computer-aided design [CAD]
- G06F30/20—Design optimisation, verification or simulation
Definitions
- the invention relates to a two-dimensional light intensity distribution simulation method of SU-8 thick photoresist ultraviolet light incident photolithography process on the back surface, and belongs to the field of computer simulation of micro-electromechanical system (MEMS) processing technology.
- MEMS micro-electromechanical system
- UV lithography technology using SU-8 thick glue is an important microfabrication technology in the field of MEMS. It overcomes the problem of insufficient aspect ratio of ordinary photoresist lithography, and is very suitable for the manufacture of ultra-thick and high aspect ratio. than the MEMS microstructure.
- the traditional SU-8 glue UV vertical incident photolithography process can only produce vertical SU-8 glue microstructures. With the increasing abundance of MEMS devices, inclined structures such as embedded channels, V-shaped grooves and inclined cylinders have appeared. The oblique incidence lithography process can get rid of the limitation of vertical incidence and manufacture various complex SU-8 glue microstructure.
- the SU-8 glue at the edge of the substrate will be thicker than the SU-8 glue in the center of the substrate.
- the phenomenon of over-exposure at the top of the photoresist and under-exposure at the bottom of the photoresist will occur, which affects the size of the final development.
- an air gap will inevitably appear between the reticle and the SU-8 glue.
- SU-8 glue UV backside lithography process to fabricate high aspect ratio SU-8 glue microstructures.
- the reticle is directly used as the substrate, and the SU-8 glue is directly coated on the reticle.
- the incident ultraviolet light directly exposes the SU-8 glue through the back of the reticle.
- the process equipment used in the method is simple, the processing cost is low, the influence of the diffraction effect caused by the uneven surface of the SU-8 glue is avoided, and the influence of the substrate reflection on the microstructure of the SU-8 glue is also avoided. Processing high aspect ratio SU-8 glue microstructures.
- Step 1 spatially discretize the lithography simulation area that needs to perform light intensity distribution simulation, subdivide it into a two-dimensional array composed of grids, and use a two-dimensional matrix to represent the two-dimensional array;
- Step 2 Divide the entire lithography simulation area into several rectangular thin layers along the horizontal direction, and the rectangular thin layers have continuous optical properties in the Z-axis direction;
- Step 3 Establish Maxwell's equations for each of the rectangular thin layers, and separate variables for the Maxwell's equations to obtain an eigenvalue problem, and then use the Fourier transform for the material parameters and electromagnetic fields in each of the rectangular thin layers. Series expansion, the electromagnetic field distribution in each of the rectangular thin layers is obtained by numerically solving the eigenvalue problem;
- Step 4 According to the continuity condition, the electromagnetic field boundary condition is applied to couple each of the rectangular thin layers, and the electromagnetic field distribution of the light after passing through the reticle is obtained;
- Step 5 Calculate the simulation result of the two-dimensional light intensity distribution of the incident ultraviolet light on the back of the SU-8 adhesive according to the electromagnetic field distribution.
- step 3 includes the following steps:
- Step 3-1 Establish Maxwell's equations for each of the rectangular thin layers, separate the electromagnetic field E into two variables, X(x) and Z(z), and substitute them into Maxwell's equations, so as to decompose Maxwell's equations into the following formula ( 1)
- the two differential equations shown, the differential equations contain complex potential k 2 ⁇ and eigenvalue ⁇ 2 ;
- ⁇ is the material permittivity of each of the rectangular thin layers, and k is the wave number;
- Step 3-2 The mask structure is repeatedly arranged in the x direction with a length d as a period, and the dielectric constant ⁇ is expanded by a Fourier series, as shown in formula (2);
- ⁇ j (x) represents the dielectric constant of the jth thin rectangular layer, is the coefficient of the qth term after the Fourier series expansion, L is the Fourier expansion series, i is a complex number, and b is the reciprocal of d;
- Step 3-3 Calculate the coefficient of each term of the Fourier series expansion through the inverse Fourier transform as shown in equation (3):
- Step 3-4 Perform Fourier transform on the variable X(x) as shown in formula (4);
- B l is the lth coefficient after Fourier expansion
- Step 3-5 Substitute the Fourier expansion series into the first differential equation in equation (1) to obtain the eigenvalue matrix equation, as shown in equation (5);
- B is the eigenvector of matrix D, D l, m is the element of the l-th row and m-th column of matrix D, ⁇ lm is the (lm)-th dielectric constant value after Fourier expansion;
- Step 3-6 establish a mathematical model of the electromagnetic field of the j-th rectangular thin layer as shown in formula (6);
- zj represents the coordinates of the jth layer of rectangular thin layer; is the x-direction component of the magnetic field of the jth rectangular thin layer, is the z-direction component of the magnetic field of the jth rectangular thin layer.
- the interface between the air and the first layer is expressed in matrix form according to the continuous boundary condition of the electromagnetic field, and the boundary condition equation shown in formula (7) is obtained;
- a 1 and A' 1 represent the diffraction result information of the first layer
- the matrix R represents the illumination information
- R l represents the element of the lth column of the matrix R
- l 0 is the incident wave order at oblique incidence
- ⁇ 0 is the wavelength of incident light
- ⁇ represents the incident angle
- a j and A' j are the diffraction result information of the j-th layer
- a j+1 and A' j+1 are the diffraction result information of the (j+1)-th layer, is the electric field value of the (j+1)th layer
- the A j and A' j matrices are obtained, and they are substituted into formula (6) for integral calculation to obtain the electromagnetic field value of the jth thin rectangular layer.
- step 5 the simulation result of the two-dimensional light intensity distribution of the incident ultraviolet light on the back of the SU-8 glue is shown in formula (11);
- I l, m is the light intensity value at the coordinates (l, m) in the two-dimensional array
- n r is the real part of the photoresist refractive index
- the present invention adopts the waveguide method based on strict electromagnetic field theory to calculate the light intensity distribution in the photoresist, because the backside incident directly uses the reticle as the substrate, and the SU-8 glue is directly coated on the reticle, without the need for The effects of air gaps are considered, while the effects of substrate reflections are also avoided.
- the influence of different parameters on the light intensity distribution is also comprehensively considered, such as the depth of the photoresist and the incident angle at oblique incidence. Comparing the simulation results with the actual experimental results to verify the accuracy of the model, the method of the present invention can accurately simulate the light intensity distribution inside the SU-8 glue during the photolithography process of incident ultraviolet light on the backside.
- FIG. 1 is a schematic diagram of a lithography simulation model based on a two-dimensional waveguide method
- FIG. 2 is a light intensity distribution curve diagram and a corresponding light intensity contour diagram at different photoresist depths under vertical incidence.
- Figure 1 is a schematic diagram of the lithography simulation model based on the two-dimensional waveguide method.
- the incident light has two dimensions of information, the incident light intensity I 0 and the incident angle ⁇ .
- the establishment of the coordinate system is shown in the figure, where along the mask
- the x-axis is established horizontally on the reticle and the z-axis is established along the direction perpendicular to the reticle, and each layer has continuous optical properties in the z-axis direction.
- the coordinate definitions of the opaque area and the transparent area of the mask can be clearly seen from the figure.
- the reticle is used as the substrate in the photolithography process, the SU-8 glue is directly coated on the reticle, and the incident ultraviolet light directly passes through the backside of the reticle to expose the SU-8 glue. Therefore, in the process of dividing the rectangular thin layer, the glass is defined as the first layer material, which is the material c in the figure, the mask is used as the second layer material, which is the material b in the figure, and the SU-8 photoresist is the first layer material.
- the two-dimensional light intensity distribution simulation method of SU-8 glue ultraviolet light backside lithography process includes the following steps:
- Step 1 The lithography simulation area that needs to perform light intensity distribution simulation is spatially discretized, subdivided into a two-dimensional array composed of grids, and a two-dimensional matrix is used to represent the two-dimensional array.
- Step 2 Divide the entire lithography simulation area into several rectangular thin layers along the horizontal direction, and each rectangular thin layer has continuous optical properties in the Z-axis direction.
- Step 3 Establish Maxwell's equations for each rectangular thin layer separately, and separate the variables of Maxwell's equation to obtain an eigenvalue problem, and then expand the material parameters and electromagnetic fields in each rectangular thin layer with Fourier series, through the numerical value Solve the eigenvalue problem to get the electromagnetic field distribution in each rectangular thin layer.
- Step 4 According to the continuity condition, the electromagnetic field boundary condition is applied to couple each rectangular thin layer, and the electromagnetic field distribution of the light after passing through the reticle is obtained.
- Step 5 Calculate the simulation result of the two-dimensional light intensity distribution of the incident ultraviolet light on the back of the SU-8 adhesive according to the electromagnetic field distribution.
- step 3 includes the following steps:
- Step 3-1 Establish Maxwell's equations for each rectangular thin layer, and use the idea of the separation of variables method to separate the electromagnetic field E into two variables, X(x) and Z(z), and substitute them into Maxwell's equations, thereby converting Maxwell's equations is decomposed into two differential equations as shown in equation (1), the differential equations contain complex potential k 2 ⁇ and eigenvalue ⁇ 2 ;
- ⁇ is the material permittivity of each of the rectangular thin layers
- X(x) and Z(z) are separated variables
- k is the wave number
- Step 3-2 The mask structure is repeatedly arranged in the x direction with a length d as a period, and the dielectric constant ⁇ is expanded by a Fourier series, as shown in formula (2);
- ⁇ j (x) represents the dielectric constant of the jth thin rectangular layer
- L is the Fourier expansion series
- i is a complex number
- b is the reciprocal of d.
- Step 3-3 Calculate the coefficient of each term after Fourier series expansion by inverse Fourier transform as shown in equation (3).
- Step 3-4 Perform Fourier transform on the variable X(x) as shown in formula (4);
- B l is the lth coefficient after Fourier expansion.
- Step 3-5 Substitute the Fourier expansion series into the first differential equation in equation (1) to obtain the eigenvalue matrix equation, as shown in equation (5);
- B is the eigenvector of matrix D
- D l,m is the element in the lth row and mth column of matrix D
- ⁇ lm is the ( lm )th dielectric constant value after Fourier expansion.
- Step 3-6 Calculate the value of the electromagnetic field expression according to the X(x) expression in Equation (4) and the Z(z) expression obtained by solving the second differential equation of Equation (1), thereby establishing the formula (6) Mathematical model of the electromagnetic field of the j-th rectangular thin layer shown;
- step 4 the interface between the air and the first layer is expressed in matrix form according to the continuous boundary condition of the electromagnetic field, and the lower boundary condition equation shown in formula (7) is obtained;
- a 1 and A' 1 represent the diffraction result information of the first layer
- the matrix R represents the illumination information
- R l represents the element of the lth column of the matrix R
- l 0 is the incident wave order at oblique incidence
- ⁇ 0 is the wavelength of incident light
- ⁇ represents the initial light intensity
- ⁇ represents the incident angle
- a j and A' j are the diffraction result information of the j-th layer
- a j+1 and A' j+1 are the diffraction result information of the (j+1)-th layer, is the electric field value of the (j+1)th layer, represents the element of the lth row and the mth column of the topography information of the jth layer, The element in the lth row and the mth column representing the electric field value of the (j+1)th layer.
- the A j and A' j matrices are obtained, and they are substituted into the formula (6) for integral calculation to obtain the electromagnetic field value of the jth thin rectangular layer.
- step 5 the simulation result of the two-dimensional light intensity distribution of the incident ultraviolet light on the back of the SU-8 adhesive is shown in formula (23);
- I l, m is the light intensity value at the coordinates (l, m) in the two-dimensional array
- n r is the real part of the photoresist refractive index.
- FIG. 2 is a light intensity distribution curve diagram and a corresponding light intensity contour diagram at different photoresist depths under vertical incidence.
- the initial incident light intensity was 2.6 mW/cm2
- the incident light wavelength was 365 nm
- the photoresist thickness was 300 ⁇ m
- the mask length was 200 ⁇ m
- the mask hole size was 100 ⁇ m.
- Figure 2(a) is a graph of the light intensity distribution at different photoresist depths at vertical incidence. The photoresist depths of the curve are 5 ⁇ m, 100 ⁇ m, 200 ⁇ m and 300 ⁇ m in sequence from top to bottom.
- Figure 2(b) is the corresponding light intensity contour map.
- the present invention compares the simulation results with the actual experimental results to verify the accuracy of the model. After verification, it is found that the simulation results are consistent with the experimental results, and can be used for the two-dimensional simulation of the SU-8 glue UV back incident lithography process.
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Abstract
本发明公开了一种SU-8胶紫外光背面光刻工艺的二维光强分布模拟方法，步骤1：将需要进行光强分布模拟的光刻仿真区域进行空间离散，细分成网格组成的二维阵列；步骤2：将整个光刻仿真区域沿水平方向划分成若干矩形薄层；步骤3：对每一矩形薄层分别建立麦克斯韦方程，并对麦克斯韦方程进行分离变量得到一个特征值问题，然后将每一矩形薄层中的材料参数和电磁场用傅里叶级数展开，通过数值求解特征值问题得到每一矩形薄层中的电磁场分布情况；步骤4：根据连续性条件施加电磁场边界条件将每一矩形薄层耦合起来，求得光在透过掩模版后的电磁场分布情况；步骤5：根据电磁场分布情况计算得到SU-8胶背面入射紫外光的二维光强分布的模拟结果。
Description
本发明涉及一种SU-8厚光刻胶紫外光背面入射光刻工艺的二维光强分布模拟方法，属于微电子机械系统(MEMS)加工工艺过程计算机模拟领域。
目前采用SU-8厚胶的紫外光(UV)光刻技术是MEMS领域的重要微细加工技术，它克服了普通光刻胶光刻深宽比不足的问题，十分适合于制造超厚、高深宽比的MEMS微结构。传统的SU-8胶紫外光垂直入射光刻工艺只能制造垂直的SU-8胶微结构。随着MEMS器件的日益丰富，出现了如嵌入式沟道、V形槽和倾斜圆柱等这类的倾斜结构，采用斜入射光刻工艺可以摆脱垂直入射的限制，制造各种复杂的SU-8胶微结构。然而，由于SU-8胶在涂覆过程中厚度不均匀以及边珠效应等原因，衬底边缘的SU-8胶会比衬底中央的SU-8胶厚。而且，由于光刻胶顶部和底部曝光剂量的不均匀，会出现光刻胶顶部过度曝光、底部曝光不足的现象，影响最终显影的尺寸。同时，在SU-8胶紫外光斜入射光刻过程中，掩模版与SU-8胶之间不可避免地会出现空气间隙。由于空气间隙产生的衍射效应对光刻精度影响很大，导致难以加工高深宽比的SU-8胶微结构。为了解决这个问题，有学者提出了采用SU-8胶紫外光背面光刻工艺来制造高深宽比的SU-8胶微结构。这种方法直接将掩模版作为衬底，将SU-8胶直接涂覆在掩模版上，光刻过程中，入射紫外光直接透过掩模版背面曝光SU-8胶。该方法使用的工艺设备简单，加工成本低，避免了由于SU-8胶表面不平整产生的衍射效应的影响，同时也避免了衬底反射对SU-8胶微结构的影响，可以更有效地加工高深宽比的SU-8胶微结构。
采用仿真工具来优化工艺性能，既可以避免反复制版、流片、实验所带来的高成本、耗时长问题，也可以利用计算仿真技术寻求最佳工艺条件，大幅提升制造性能，缩短相关MEMS产品的设计周期，降低其开发成本，并且更加深入地了解光刻技术的内在原理。SU-8胶光刻过程中，由于光刻胶显影后的最终形貌很大程度上取决于曝光工艺，它是整个光刻过程中最重要的一步工艺。通过模拟光刻后光刻胶内的光强分布情况，可以对显影后的光刻胶形貌进行预测。因此，进行SU-8厚胶背面入射光刻工艺的光强分布仿真是一项极具发展潜力的研究。
发明内容
发明目的：针对上述现有技术，提出一种SU-8胶紫外光背面光刻工艺的二维光强分布模拟方法，用于模拟光刻后光刻胶内的光强分布情况。
技术方案：SU-8胶紫外光背面光刻工艺的二维光强分布模拟方法，包括如下步骤：
步骤1：将需要进行光强分布模拟的光刻仿真区域进行空间离散，细分成网格组成的二维阵列，并采用二维矩阵代表所述二维阵列；
步骤2：将整个光刻仿真区域沿水平方向划分成若干矩形薄层，所述矩形薄层在Z轴方向上光学性质连续；
步骤3：对每一所述矩形薄层分别建立麦克斯韦方程，并对所述麦克斯韦方程进行分离变量得到一个特征值问题，然后将每一所述矩形薄层中的材料参数和电磁场用傅里叶级数展开，通过数值求解所述特征值问题得到每一所述矩形薄层中的电磁场分布情况；
步骤4：根据连续性条件施加电磁场边界条件将每一所述矩形薄层耦合起来，求得光在透过掩模版后的电磁场分布情况；
步骤5：根据电磁场分布情况计算得到SU-8胶背面入射紫外光的二维光强分布的模拟结果。
进一步的，所述步骤3包括如下步骤：
步骤3-1：对每一所述矩形薄层分别建立麦克斯韦方程，将电磁场E分离成X(x)和Z(z)两个变量，并代入麦克斯韦方程，从而将麦克斯韦方程分解为如式(1)所示的两个微分方程，所述微分方程含有复数势k
2ε和特征值α
2；
其中，ε为每一所述矩形薄层的材料介电常数，k为波数；
步骤3-2：掩模结构在x方向上以长度d为周期重复地排列，将介电常数ε进行傅里叶级数展开，如式(2)所示；
其中，ε
j(x)表示第j层矩形薄层的介电常数，
为傅里叶级数展开后的第q项的系数，L为傅里叶展开级数，i表示复数，b是d的倒数；
步骤3-3：通过如式(3)所示逆傅里叶变换求出傅里叶级数展开后的每一项的系数：
步骤3-4：对变量X(x)进行如式(4)所示傅里叶变换；
其中，B
l为傅里叶展开后第l项系数；
步骤3-5：将傅里叶展开级数代入式(1)中第一个微分方程，得到特征值矩阵方程，如式(5)所示；
其中，B是矩阵D的特征向量，D
l,m为矩阵D中第l行第m列的元素，ε
l-m为经傅里叶展开后的第(l-m)项介电常数值；
步骤3-6：建立如式(6)所示的第j层矩形薄层的电磁场数学模型；
其中，
为第j层矩形薄层的电场y方向分量，
及
表示第j层矩形薄层的第m阶本征模振幅，即矩阵A
j及A'
j的第m列元素；
表示第j层矩形薄层的特征值矩阵的第m列元素，
表示第j层矩形薄层的特征向量矩阵的第l行第m列元素，并通过式(5)求解得到；z
j表示第j层矩形薄层的坐标；
为第j层矩形薄层的磁场x方向分量，
为第j层矩形薄层的磁场z方向分量。
进一步的，所述步骤4中，空气与第一层界面处根据电磁场连续边界条件，用矩阵形式表示，得到如式(7)所示的边界条件方程；
其中，
表示第1层形貌信息，A
1、A'
1表示第1层衍射结果信息，矩阵R表示光照信息，R
l表示矩阵R的第l列元素，l
0为斜入射时入射波阶次，λ
0为入射光波长，
表示初始光照强度，θ表示入射角度；
最后一层界面处根据边界条件得如式(9)所示的边界条件方程：
其中，
为第n层形貌信息，A
n、A'
n为第n层衍射结果信息；
将第j层和第(j+1)层矩形薄层的电场、磁场进行连续性匹配得到：
其中，
为第j层形貌信息，A
j、A'
j为第j层衍射结果信息；A
j+1、A'
j+1为第(j+1)层衍射结果信息，
为第(j+1)层电场值；
根据电磁场边界条件求得A
j及A'
j矩阵，代入式(6)进行积分计算进而得到第j层矩形薄层的电磁场值
进一步的，所述步骤5中，SU-8胶背面入射紫外光的二维光强分布的模拟结果如式(11)所示；
其中，I
l,m为所述二维阵列中坐标(l,m)处光照强度值，
为所述二维阵列中坐标(l,m)处电场
值，n
r为光刻胶折射率实部。
有益效果：本发明采用基于严格电磁场理论的波导法计算光刻胶内的光强分布情 况，由于背面入射直接将掩模版作为衬底，将SU-8胶直接涂覆在掩模版上，不需考虑空气间隙的影响，同时也避免了衬底反射的影响。同时，在背面入射紫外光的二维光强计算模型中，也综合考虑了不同参数对光强分布的影响，如光刻胶的深度、斜入射时的入射角。将模拟结果与实际的实验结果进行对比以验证模型准确性得出，本发明方法可以精确模拟紫外光背面入射这一光刻工艺过程中SU-8胶内部的光强分布情况。
图1是基于二维波导法的光刻仿真模型示意图；
图2是垂直入射时不同光刻胶深度的光强分布曲线图与对应的光强等高线图。
下面结合附图对本发明做更进一步的解释。
如图1所示为基于二维波导法的光刻仿真模型示意图，入射光具有入射光强I
0以及入射角θ这二个维度的信息，坐标系的建立如图所示，其中沿着掩模版水平方向建立x轴，沿着垂直于掩模版方向建立z轴，每一层在z轴方向上都有连续的光学性质。掩模不透光区域和透光区域的坐标定义可从图中明显看出。根据背面入射的特点，光刻过程中将掩模版作为衬底，将SU-8胶直接涂覆在掩模版上，入射紫外光直接透过掩模版背面曝光SU-8胶。因此，在矩形薄层的划分过程中，将玻璃定义为第一层材料，即为图中材料c，掩模版作为第二层材料，即为图中材料b，SU-8光刻胶作为第三层材料，即为图中材料a，每一层在z方向上的坐标值分别是0、z1、z2、z3。这样一来，每一层材料在z方向上的介电常数均满足一致性的原则。
SU-8胶紫外光背面光刻工艺的二维光强分布模拟方法，包括如下步骤：
步骤1：将需要进行光强分布模拟的光刻仿真区域进行空间离散，细分成网格组成的二维阵列，并采用二维矩阵代表该二维阵列。
步骤2：将整个光刻仿真区域沿水平方向划分成若干矩形薄层，每一矩形薄层在Z轴方向上光学性质连续。
步骤3：对每一矩形薄层分别建立麦克斯韦方程，并对麦克斯韦方程进行分离变量得到一个特征值问题，然后将每一矩形薄层中的材料参数和电磁场用傅里叶级数展开，通过数值求解特征值问题得到每一矩形薄层中的电磁场分布情况。
步骤4：根据连续性条件施加电磁场边界条件将每一矩形薄层耦合起来，求得光在透过掩模版后的电磁场分布情况。
步骤5：根据电磁场分布情况计算得到SU-8胶背面入射紫外光的二维光强分布的模拟结果。
具体的，步骤3包括如下步骤：
步骤3-1：对每一矩形薄层分别建立麦克斯韦方程，运用分离变量法的思想，将电磁场E分离成X(x)和Z(z)两个变量，并代入麦克斯韦方程，从而将麦克斯韦方程分解为如式(1)所示的两个微分方程，所述微分方程含有复数势k
2ε和特征值α
2；
其中，ε为每一所述矩形薄层的材料介电常数，X(x)和Z(z)为分离出的变量,k为波数。
步骤3-2：掩模结构在x方向上以长度d为周期重复地排列，将介电常数ε进行傅里叶级数展开，如式(2)所示；
其中，ε
j(x)表示第j层矩形薄层的介电常数，
为傅里叶级数展开后的第q项的系数，L为傅里叶展开级数，i表示复数，b是d的倒数。
步骤3-3：通过如式(3)所示逆傅里叶变换求出傅里叶级数展开后的每一项的系数。
步骤3-4：对变量X(x)进行如式(4)所示傅里叶变换；
其中，B
l为傅里叶展开后第l项系数。
步骤3-5：将傅里叶展开级数代入式(1)中第一个微分方程，得到特征值矩阵方程，如式(5)所示；
其中，B是矩阵D的特征向量，D
l,m为矩阵D中第l行第m列的元素，ε
l-m为经傅里叶展开后的第(l-m)项介电常数值。
步骤3-6：根据式(4)中X(x)表达式和对式(1)第二个微分方程求解得到的Z(z)表达式，计算出电磁场表达式的值，从而建立如式(6)所示的第j层矩形薄层的电磁场数学模型；
其中，
为第j层矩形薄层的电场y方向分量，
及
表示第j层矩形薄层的第m阶本征模振幅，即矩阵A
j及A'
j的第m列元素；
表示第j层矩形薄层的特征值矩阵的第列元素，
表示第j层矩形薄层的特征向量矩阵的第l行第m列元素，并通过式(5)求解得到；z表示z轴坐标，z
j表示第j层矩形薄层的坐标；
为第j层矩形薄层的磁场x方向分量，
为第j层矩形薄层的磁场z方向分量。
步骤4中，空气与第一层界面处根据电磁场连续边界条件，用矩阵形式表示，得到如式(7)所示的下边界条件方程；
其中，
上式中，
表示第1层形貌信息，A
1、A'
1表示第1层衍射结果信息，矩阵R表示光照信息，R
l表示矩阵R的第l列元素，l
0为斜入射时入射波阶次，λ
0为入射光波长，
表示初始光照强度，θ表示入射角度，
表示第1层形貌信息的第l行第m列元素。
最后一层界面处根据边界条件得如式(11)所示的边界条件方程：
其中：
上式中，
为第n层形貌信息，A
n、A'
n为第n层衍射结果信息；
表示第n层形貌信息的第l行第m列元素，ε
s为最后一矩形薄层材料的介电常数，T为最后一矩形薄层的坐标。
将第j层和第(j+1)层矩形薄层的电场、磁场进行连续性匹配得到：
上式中各个元素的表达式如下：
其中，
为第j层形貌信息，A
j、A'
j为第j层衍射结果信息；A
j+1、A'
j+1为第(j+1)层衍射结果信息，
为第(j+1)层电场值，
表示第j层形貌信息的第l行第m列元素，
表示第(j+1)层电场值的第l行第m列元素。
最后，根据电磁场边界条件求得A
j及A'
j矩阵，代入式(6)进行积分计算进而得到第j层矩形薄层的电磁场值
步骤5中，SU-8胶背面入射紫外光的二维光强分布的模拟结果如式(23)所示；
其中，I
l,m为二维阵列中坐标(l,m)处光照强度值，
为二维阵列中坐标(l,m)处电场
值，n
r为光刻胶折射率实部。
图2是垂直入射时不同光刻胶深度的光强分布曲线图与对应的光强等高线图。光刻仿真时，初始入射光强为2.6mW/cm2，入射光波长为365nm，光刻胶厚度为300μm，掩模版长度为200μm，掩模孔大小为100μm。图2(a)为垂直入射时不同光刻胶深度的光强分布曲线图，曲线从上至下光刻胶深度依次为5μm、100μm、200μm和300μm。图2(b)为对应的光强等高线图。本发明将模拟结果与实际的实验结果进行对比，以验证模型准确性。经验证发现模拟结果与实验结果比较一致，可以用于SU-8胶紫外光背面入射光刻过程的二维模拟。
以上所述仅是本发明的优选实施方式，应当指出，对于本技术领域的普通技术人员来说，在不脱离本发明原理的前提下，还可以做出若干改进和润饰，这些改进和润饰也应视为本发明的保护范围。
Claims (4)
- SU-8胶紫外光背面光刻工艺的二维光强分布模拟方法，其特征在于，包括如下步骤：步骤1：将需要进行光强分布模拟的光刻仿真区域进行空间离散，细分成网格组成的二维阵列，并采用二维矩阵代表所述二维阵列；步骤2：将整个光刻仿真区域沿水平方向划分成若干矩形薄层，所述矩形薄层在Z轴方向上光学性质连续；步骤3：对每一所述矩形薄层分别建立麦克斯韦方程，并对所述麦克斯韦方程进行分离变量得到一个特征值问题，然后将每一所述矩形薄层中的材料参数和电磁场用傅里叶级数展开，通过数值求解所述特征值问题得到每一所述矩形薄层中的电磁场分布情况；步骤4：根据连续性条件施加电磁场边界条件将每一所述矩形薄层耦合起来，求得光在透过掩模版后的电磁场分布情况；步骤5：根据电磁场分布情况计算得到SU-8胶背面入射紫外光的二维光强分布的模拟结果。
- 根据权利要求1所述SU-8胶紫外光背面光刻工艺的二维光强分布模拟方法，其特征在于，所述步骤3包括如下步骤：步骤3-1：对每一所述矩形薄层分别建立麦克斯韦方程，将电磁场E分离成X(x)和Z(z)两个变量，并代入麦克斯韦方程，从而将麦克斯韦方程分解为如式(1)所示的两个微分方程，所述微分方程含有复数势k 2ε和特征值α 2；
其中，ε为每一所述矩形薄层的材料介电常数，k为波数；步骤3-2：掩模结构在x方向上以长度d为周期重复地排列，将介电常数ε进行傅里叶级数展开，如式(2)所示；
其中，ε j(x)表示第j层矩形薄层的介电常数，为傅里叶级数展开后的第q项的系 数，L为傅里叶展开级数，i表示复数，b是d的倒数；
步骤3-3：通过如式(3)所示逆傅里叶变换求出傅里叶级数展开后的每一项的系数：
步骤3-4：对变量X(x)进行如式(4)所示傅里叶变换；
其中，B l为傅里叶展开后第l项系数；步骤3-5：将傅里叶展开级数代入式(1)中第一个微分方程，得到特征值矩阵方程，如式(5)所示；DB＝α 2B
其中，B是矩阵D的特征向量，D l,m为矩阵D中第l行第m列的元素，ε l-m为经傅里叶展开后的第(l-m)项介电常数值；步骤3-6：建立如式(6)所示的第j层矩形薄层的电磁场数学模型；
其中，为第j层矩形薄层的电场y方向分量，
及
表示第j层矩形薄层的第m阶本征模振幅，即矩阵A j及A' j的第m列元素；
表示第j层矩形薄层的特征值矩阵的第m列元素，
表示第j层矩形薄层的特征向量矩阵的第l行第m列元素，并通过式(5)求解得到；z j表示第j层矩形薄层的坐标；
为第j层矩形薄层的磁场x方向分量，
为第j层矩形薄层的磁场z方向分量。
- 根据权利要求2所述SU-8胶紫外光背面光刻工艺的二维光强分布模拟方法，其特征在于，所述步骤4中，空气与第一层界面处根据电磁场连续边界条件，用矩阵形式表示，得到如式(7)所示的边界条件方程；
其中，表示第1层形貌信息，A 1、A' 1表示第1层衍射结果信息，矩阵R表示光照信息，R l表示矩阵R的第l列元素，l 0为斜入射时入射波阶次，λ 0为入射光波长，
表示初始光照强度，θ表示入射角度；
最后一层界面处根据边界条件得如式(9)所示的边界条件方程：
其中，为第n层形貌信息，A n、A' n为第n层衍射结果信息；
将第j层和第(j+1)层矩形薄层的电场、磁场进行连续性匹配得到：
其中，为第j层形貌信息，A j、A' j为第j层衍射结果信息；A j+1、A' j+1为第(j+1)层衍射结果信息，
为第(j+1)层电场值；
根据电磁场边界条件求得A j及A' j矩阵，代入式(6)进行积分计算进而得到第j层矩形薄层的电磁场值
- 根据权利要求2所述SU-8胶紫外光背面光刻工艺的二维光强分布模拟方法，其特征在于，所述步骤5中，SU-8胶背面入射紫外光的二维光强分布的模拟结果如式(11)所示；
其中，I l,m为所述二维阵列中坐标(l,m)处光照强度值，为所述二维阵列中坐标(l,m)处电场
值，n r为光刻胶折射率实部。
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