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Abstract
We present an optical proximity correction (OPC) method based on a genetic algorithm for reducing the optical proximity effect-induced pattern distortion in digital micromirror device (DMD) maskless lithography. Via this algorithm-assisted grayscale modulation of the initial mask at the pixel level, the exposure pattern can be enhanced significantly. Actual exposure experiments revealed that the rate of matching between the final exposure pattern and the mask pattern can be increased by up to 20%. This method's applicability to complex masks further demonstrates its universality for mask pattern optimization. We believe that our algorithm-assisted OPC could be highly helpful for high-fidelity and efficient DMD maskless lithography for microfabrication.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Owing to its versatile photomask realization in a digital way, the optical maskless lithography has attracted great interest [1]. Particularly, by virtue of high refresh rate and high contrast, the digital micromirrors device (DMD) based maskless lithography [2–8] has been widely used in micro-nano fabrication for a variety of applications, such as printed circuit boards (PCBs) [9,10], micro-electromechanical systems (MEMS) [11,12], micro-nano optical elements [13–16], and 3D lithography [17–20]. As a projection lithography technique, the exposure pattern distortion caused by the optical proximity effect (OPE) is unavoidable, which results in a mismatch between the fabricated microstructure and the initial design. In conventional projection lithography, the optical field distribution on the photoresist is corrected by predistortion and transmittance adjustment of the physical mask to reduce the OPE, which is also called optical proximity correction (OPC). Based on the pixel-level grayscale modulation of the DMD, OPC has also been applied to the DMD maskless lithography [21,22]. However, the reported methods are essentially rule-based OPC methods, which require time-consuming and complex iteration exposure experiments to establish a mask correction library, leading to an inefficient mask optimization process.
In this work, a genetic algorithm-based OPC method for DMD maskless lithography was proposed. This method optimizes the pixel grayscale of the initial binary mask using the algorithm. In section 2, we describe the DMD maskless lithography system and process flow utilized in this study. Next, in section 3, we present a model for the pixel-combination lithography process and demonstrate its accuracy through exposure experiments. In section 4, we discuss the fundamental principle of the genetic algorithm and its implementation for mask optimization. Then, using an “L” mask as an illustration, we demonstrate that the simulation results revealed a significant pattern error reduction from 1942 to 222 after optimization, and that the algorithm demonstrated robust convergence. In section 5, the results of the exposure experiment further showed that the exposure pattern of the optimized “L” mask is more accurate. After optimization, the matching rate between the final exposure pattern and the mask pattern could be increased from 48% to 71%, according to the quantization results. The universality of the method for mask optimization was also demonstrated by the increased fidelity of the exposure pattern based on genetic algorithm-assisted lithography compared to conventional lithography when using a more complex mask pattern.
2. Experimental setup and methods
A schematic of the DMD maskless lithography system is shown in Fig. 1. The system includes a UV-light source (center wavelength λ = 405 nm, LED) for exposure, a red-light source for alignment (center wavelength λ = 625 nm, LED), a yellow-light source (center wavelength λ = 595 nm, LED) for sample observation, DMD chip (DLP7000, Texas Instruments, USA), a projection system, and a X-Y-Z sample stage. The DMD chip, a 1024 × 768 array of square micromirrors with the micromirror pitch of 13.68 µm, is used to generate a digital mask and reflect the UV light carrying mask information into the projection system. We use an infinity-corrected optical system for light projection, which includes a tube lens (TTL200, Thorlabs, USA) with focal length of 200 mm and an objective lens (UPLFLN 10×, Olympus, Japan, NA=0.3) with focal length of 18 mm. After reduction by the projection system, the optimal dimension of a single micromirror pattern on photoresist is approximately 1.2 µm.
We employ a spin-coated positive resist of AR-P-5350 (Allresisit, Germany) with thickness of ∼1.5 µm on the Si/SiO2 substrate for exposure verification. In the exposure process, we find that the exposure time corresponding to the optimized grayscale mask is 2.3 s with a UV intensity of 81 mW/cm2, while the binary mask without optimization is 1.4 s. This may be related to (1) the loss of UV energy in the process of grayscale modulation of DMD; and (2) the selection error of photoresist threshold during simulation. Therefore, we use an intensity meter (PM100D with S130VC power sensor, Thorlabs, USA) to measure the UV intensity of digital mask, all DMD micromirrors work at “on” state, filled by different grayscale. The experimental results are shown in Fig. 2. Clearly, the relationship between the average UV intensity and the grayscale of pixels on DMD is not ideal linear. Especially, when the grayscale is greater than 127, there will be a loss of exposure dose, which verifies our speculation (1). After the exposure process, the sample was immersed in the AR300-26 developer for about 45 s for the pattern formation. Finally, the substrate was rinsed with deionized water and dried with nitrogen gas.
3. Lithography simulation
To perform a simulation-based analysis of the lithographic procedure, it is essential to model the exposure process and the photoresist. To simplify the modeling procedure, we assume that the UV light intensity passing through the photoresist remains approximately constant. The model is detailed described below.
3.1. Exposure modeling
The imaging model based on Fraunhofer diffraction theory can effectively describe the diffraction characteristics in DMD projection system [23–25]. Herein, the light intensity distribution ${I_{(m,n)}}(x,y)$ of a single micromirror on the photoresist can be regarded as Fraunhofer rectangular aperture diffraction:
in whichSince the UV light source used in this article is an incoherent light source, when multiple DMD micromirrors operate simultaneously, the light intensity distribution $I(x,y)$ on the photoresist can be viewed as a simple superposition of the light intensity distribution of each micromirror operating alone, rather than taking the coherence factor into account. Therefore, $I(x,y)$ can be written as:
3.2 Photoresist exposing simulation
In order to reduce the computational complexity of the optimization process, we use a simplified photoresist model [26]. The functional relationship between the etching rate $Z(x,y)$ at any position on the photoresist after development process and the exposure dose $D(x,y)$ can be described as:
Fig. 3. Exposure and simulation experiments results of the “L” mask without optimization. (a) The exposure pattern of the “L” mask, the enlarged image in the upper right corner shows the binary image of the exposure pattern, and (b) The binary simulated pattern of the “L” mask.
To verify the correctness of the above-described model, an “L” mask was used for exposure and simulation experiments. As shown in the upper left corner of Fig. 3(a), its minimum line-width is corresponding to 2 pixels. Figure 3(a) shows the exposure pattern of the “L” mask. It can be seen that due to the existence of OPE, line-width variation and corner-rounding phenomenon appear in exposure pattern. In addition, the asymmetry in the horizontal and vertical directions is shown in the exposure pattern due to the aberration of the optical elements [22]. The binary simulated pattern of the “L” mask according to Eqs. (4) and (5) is depicted in Fig. 3(b), where it can be seen that the simulation results reproduce the distortion characteristics of the pattern very well. Obviously, the established model can well simulate the lithography process.
4. Mask optimization based on genetic algorithm
We choose the genetic algorithm for mask optimization in DMD maskless lithography, which is also used for the hard mask optimization [27,28]. The optimization methods proposed in [27] and [28] improve the fidelity of exposure patterns by modifying the geometry of the initial mask, where genes are used to encode the movement of segments along the mask boundary. However, this method is not suitable for DMD maskless lithography (see [22]), for its digital mask consists of micromirror pixels, each micromirror size is generally around 10 µm. Fortunately, we can modulate the grayscale of pixels to optimize the mask. In our work, genes are used to encode the change of grayscale of the corresponding pixels.
The workflow of the mask optimization algorithm is shown in Fig. 4. Genetic algorithm is a computational model that simulates natural selection and genetic mechanisms in biological evolution. In this algorithm, an chromosome represents a possible solution to the given problem, and a group of chromosomes forms a population. After completing the initialization of the population, the algorithm enters the cycle of “evaluation-sorting and selection-crossover-mutation” until the stop criteria (the maximum number of generations or a target fitness value) is satisfied, where the fitness function returns the fitness value (the basis for sorting and selection) of the chromosome.
4.1 Encoding method
In this article, firstly, the chromosomes in the algorithm are defined as follows: the binary matrix of the initial mask is column-vectorized to obtain the corresponding chromosome, and the genes in the chromosome correspond to the binary information of the initial mask one by one. Taking the initial mask ${M_0}$ as an example, the column-vectorization process can be expressed as:
4.2 Fitness function
The return value ${f_{fitness}}$ of the fitness function is set as the pattern error between the binary image P of the simulated pattern (obtained by the model described in Section 3) corresponding to the intermediate mask during optimization and the binary image P0 of the mask pattern corresponding to the initial mask:
where (i, j) is the coordinate of the pattern’s pixels, the pattern sampling interval is 0.1 µm, and the simulation process is completed on MATLAB. Clearly, the lesser the pattern error, the greater the chromosome's fitness.4.3 Selection method
After the fitness value of each chromosome is obtained, a selection method will select the appropriate chromosomes as the next generation population to breed offspring. In our algorithm, to retain high-quality chromosomes as much as possible, the selection method is set as follows: (1) N parent chromosomes and N offspring chromosomes are sorted in ascending order according to their fitness values, where N is the population size; then (2) Select the first N chromosomes to form the next generation population.
4.4 Crossover scheme and mutation scheme
In this work, we choose the uniform crossover as our crossover scheme. Firstly, crossover can only occur between parent chromosomes with adjacent numbers after sorting (the crossover probability of two parent chromosomes is 0.9), such as No. 1 and No. 2, No. 3 and No. 4, and so on. Then, the same indexed genes of two parent chromosomes are crossover with a probability of 0.5 (the default value of the uniform crossover [29,30]). After crossover operation is completed, two new offspring chromosomes are obtained.
Uniform mutation was chosen as our variation scheme for its simplicity. To accelerate the emergence of the optimal chromosome, we set the initial mutation probability as 0.1, and then it will linearly decrease to 0 with the increase of the number of generations. Moreover, our genetic algorithm-based OPC method only modulated effective pixels of the initial binary mask (that is, only “1” pixels participated in mutation operation, while “0” pixels did not). Therefore, when an indexed gene mutates to a value of 0, it will keep the original value.
4.5 Simulation settings and results
We use “L” mask (see Fig. 3(a)) as an example to show the optimization effect of the genetic algorithm. Table 1 lists the iteration parameters of the genetic algorithm. The size of the initial mask is 20 × 20 pixels (that is, the number of genes in each chromosome is 400). The size of the population is 50. The optimization process would be terminated when one of these following criterions is satisfied: (1) the loop count reaches the maximum (1000) of generations; and (2) The optimal fitness value of the population decreases by less than 10 within 100 generations.
Table 1. Specifications for iteration parameters of the genetic algorithm
Figure 5(a) depicts a convergence curve that demonstrates the algorithm's robust convergence. It reveals that the pattern error of the “L” mask decreases from 1942 to 222 after optimization. This iteration process took 0.7 hour using an Intel i7-11700F CPU (2.5 GHZ) and a 32GB memory. As shown in Fig. 5(d), the line-width of the simulated pattern of the optimized “L” mask is closer to the theoretical value and the corner-rounding phenomenon is improved. Figures 5(b) and 5(c) depict, respectively, the grayscale image and the grayscale distribution of the optimized “L” mask. Consistent with the expectation of mask optimization, the higher grayscale in the mask is primarily distributed at the convex corner while the corresponding grayscale at the concave corner is smaller, as shown by the optimization results.
Fig. 5. Simulation optimization results of the “L” mask. (a) The convergence curve of the iteration process, (b) 8-bit grayscale mask of “L” pattern with optimization, (c) Grayscale distribution of the optimized “L” mask, (d) Simulated pattern of the optimized “L” mask.
5. Results and discussions
Figure 6 shows the patterning results of using the optimized “L” grayscale mask. Qualitatively, compared with the exposure pattern of the initial “L” binary mask in Fig. 3(a), the optimized exposure pattern has sharper pattern features. The line-width is closer to the optimal value, and the phenomenon of corner-rounding is also reduced. This demonstrates that the proposed OPC method based on genetic algorithms can effectively correct pattern distortion induced by OPE in DMD maskless lithography.
Fig. 6. Exposure pattern of the optimized “L” mask. The enlarged image shows the binary image of the exposure pattern.
To quantitatively analyze the optimization effect of the exposure pattern, the image subtraction technique [22] was applied to calculate the matching rate between the exposure pattern and the mask pattern, the matching rate MR can be defined as:
The calculation results of the matching rate are displayed in Fig. 7. Figure 7(a) shows the binary image of the mask pattern. Figures 7(b) and 7(c) show the calculation results without and with optimization according to Eq. (8), respectively. It shows that the matching rates of the exposure patterns without and with optimization are 48% and 71%, respectively.
Fig. 7. Calculation of matching rate based on image subtraction. (a) The binary image of the “L” mask pattern, (b) and (c) The calculation results of the exposure patterns without and with optimization, respectively. The image sampling interval is 0.1 µm.
Finally, to demonstrate the universality of the method for mask optimization, we conducted simulation and exposure experiments for the “UESTC” mask. The results are shown in Figs. 8(a) and 8(b). It demonstrates that the optimization effect of the “UESTC” mask is consistent with the prior one, with an improvement of exact matching rate from 52.8% to 72.8%. In addition, concave corners of the “E” and “S” patterns exhibit a better correction effect. We note that due to the pattern in these areas is denser, more diffraction light was superimposed, resulting in a more pronounced corner-rounding phenomenon than other regions. However, the correction effect of the method is more evident for the concave corner regions. This result further confirms the effectiveness of the genetic algorithm-assisted OPC for the lithography.
Fig. 8. Comparison of the simulation and exposure experiments results of the “UESTC” mask without and with optimization. (a) The results without optimization: the initial mask of the “UESTC” mask (upper left), the simulated pattern using the initial mask (upper right), and the exposure pattern of the initial mask (lower). (b) The results with optimization: the grayscale mask of the “UESTC” mask with optimization (upper left), the simulated pattern using the grayscale mask (upper right), and the exposure pattern of the grayscale mask (lower).
6. Conclusion
We proposed an OPC method based on genetic algorithm for DMD maskless lithography. This method employs a computer to perform grayscale modulation at the pixel level on the initial binary mask to reduce pattern distortion caused by OPE. Using an “L” mask as an example, the simulation results demonstrated the algorithm's robust convergence, and optimization reduced the pattern error of the simulated pattern from 1942 to 222. The optimized “L” grayscale mask was then utilized for the exposure experiment. The results demonstrated that the optimized exposure pattern has sharper pattern characteristics, such as line-width that is closer to the optimal value and the suppression of the phenomenon of corner-rounding. In addition, the quantization results showed that the exposure pattern matching rates are 48% without optimization and 71% with optimization. Results from the exposure experiment of optimization for more complex mask suggest that this is a universal method for OPC in DMD lithography. We believe that pattern distortion correction could be more effective and efficient if the model and algorithm were optimized further.
Funding
National Key Research and Development Program of China (2019YFB2203504); National Natural Science Foundation of China (61975024, 62074024); Sichuan Province Science and Technology Support Program (2023YFH0090, 2023NSFSC0365).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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