Abstract
Plasmonic lithography, which utilizes evanescent near-field imaging that contains high-frequency information, can overcome the diffraction limit of conventional lithography and has emerged as a strong candidate for next-generation integrated circuit (IC) manufacturing. Previous research has primarily focused on improving the resolution limits of plasmonic lithography, while compatibility studies have been largely overlooked. Imaging structure compatibility with mask patterns is an important consideration in lithographic processes for IC fabrication, and plasmonic lithography is particularly sensitive to the pitch of mask patterns—since the pitch determines whether the spatial frequency of the mask's diffraction orders matches the wavevector of surface plasmon polaritons (SPPs), enabling beyond-diffraction imaging. This study focuses on plasmonic imaging lithography structures, establishing a method and workflow for optimizing imaging structure compatibility using a plasmonic lithography optical imaging model based on the rigorous coupled-wave analysis (RCWA) algorithm and particle swarm optimization (PSO). By optimizing imaging structure compatibility, this approach achieves compatibility with various line/space mask patterns from the 7 nm to 28 nm technology nodes. The results are further validated by COMSOL models based on the finite element method (FEM) and coherent transfer function (CTF) analysis. The proposed optimization methodology and workflow establish a solid foundation for enhancing plasmonic lithography compatibility, advancing its path toward industrial application.
In this work, a universal optimization framework combining RCWA and a mixed-variable PSO algorithm is proposed to address the pitch sensitivity and poor compatibility issues inherent in surface plasmonic lithography.
1. Introduction
As a candidate for next-generation lithography technology, plasmonic lithography can break through the diffraction limit in conventional lithography by utilizing evanescent near-field imaging that contains high-frequency information.1 Unlike conventional projection lithography, plasmonic lithography has a localized imaging range in the near field, and its imaging lens is not an ordinary optical lens but a single-layer metal film or a multilayer film with alternating metal and dielectric layers at the nanometer thickness scale.2 Experiments have proved that even with a 365 nm wavelength light source, under a single set of exposure conditions, the resolution can reach approximately 22 nm (∼1/17 of the wavelength of light), and it is possible to improve it further.3,4 This method can achieve large-area, low-cost and high-resolution nanofabrication, and thus has received widespread attention.5 In terms of principle, common structure and effect, plasmonic lithography can be classified into interference lithography, imaging lithography and direct writing lithography.1,2,4–7 Compared with interference lithography which can only produce simple periodic patterns and direct writing lithography which is limited by point-by-point scanning, imaging lithography based on a metal superlens can efficiently achieve optical imaging of complex mask patterns and is one of the alternative solutions for high-resolution lithography imaging at advanced technology nodes.
As the technology nodes in integrated circuit (IC) manufacturing continue to shrink, the pursuit of improving the resolution of lithography has always been a research hotspot, and plasmonic lithography technology is no exception. A large number of studies have focused on the research on improving the resolution of plasmonic lithography,3,8–11 but the research on imaging compatibility has been neglected. However, in IC manufacturing, the compatibility of mask patterns is also an important indicator that should be considered in the lithography process, and plasmonic lithography is more sensitive to the pitch of the mask pattern because the pitch of the mask pattern determines whether the spatial frequency of the mask diffraction order can match the spatial wavevector of SPPs in resonance.12–14 It should be noted that compatibility here refers to the evaluation standard based on light intensity contrast, and the imaging structure used enables the contrast of as many mask patterns as possible to satisfy the requirements. In other words, the more mask patterns whose contrast satisfies the requirements, the better the compatibility of the imaging structure. Taking one-dimensional periodic line/space mask patterns as an example, different pitches or different critical dimensions (CDs) represent different mask patterns. In contrast to the conventional optimization target of resolution improvement, the primary objective of our work is to shift the optimization target to mask pattern compatibility, which, to the best of our knowledge, has not been systematically investigated in the field of plasmonic lithography.
It is obvious that exhaustive scanning of imaging structure parameters (material parameters and thickness of the film layer) can find imaging structures that satisfy the compatibility requirements, but it is time-consuming and has poor portability. This is because the feasible solution space is large but there is no guidance from an optimization algorithm, so the search efficiency is low. Therefore, it is necessary to combine the characteristics of plasmonic lithography imaging structures and, based on the existing plasmonic lithography optical imaging model, develop or improve the existing optimization algorithms to achieve the optimization of imaging structure compatibility and find imaging structures that can be compatible with more mask patterns.
Currently, the common optimization algorithms can be roughly divided into two categories: gradient-based algorithms and heuristic algorithms. Gradient-based algorithms, such as gradient descent (GD), steepest descent (SD), conjugate gradient (CG), and deep learning (DL), have fast convergence and high local optimization accuracy, but they largely rely on gradient information.15 However, there is currently no accurate and analytical plasmonic lithography optical imaging model, so it is impossible to obtain gradient information on evaluation indicators (such as light intensity contrast) for imaging structure parameters, thus making it impossible to use gradient-based optimization algorithms. On the other hand, heuristic algorithms include simulated annealing (SA), ant colony optimization (ACO), genetic algorithm (GA), particle swarm optimization (PSO), and tabu search (TS), which are derived based on the observation of group behavior in ecosystems and have excellent global optimization capabilities and the advantage of not being affected by gradient information.16 The PSO algorithm proposed by Kennedy in 1995 is superior to other heuristic algorithms due to its fewer parameters, simpler principle, and higher convergence efficiency,17 and it has been widely applied in multi-objective control, neural network search, and complex multimodal target optimization fields.18–20 However, the standard PSO algorithm is designed for continuous variables and cannot directly handle the mixed-variable optimization problem encountered here, where film thicknesses are continuous and material permittivities are discrete. Therefore, a secondary objective of our work is to design a hybrid PSO strategy that handles continuous and discrete variables separately, thereby enabling efficient optimization of imaging structure compatibility.
In this paper, taking plasmonic imaging lithography structures as the research object, we employed the plasmonic lithography optical imaging model based on the RCWA algorithm and the PSO algorithm to establish a method and workflow for optimizing the compatibility of plasmonic lithography imaging structures. Through optimization of the thickness and materials of the imaging structure film layers, it is possible to achieve compatibility for different line/space patterns at the specified technology node and larger technology nodes. CTF analysis also confirmed the correctness of the results. The optimization workflow established in this paper is universal and can be extended to other super-resolution imaging designs or imaging performance optimization tasks.
2. Methods
Fig. 1 shows the imaging structure compatibility optimization flow based on the PSO algorithm, which consists of five stages. In the initialization stage, after selecting the light source and the mask set composed of masks with which compatibility is required, N imaging structures composed of film layers with different materials and thicknesses are randomly initialized to generate the particle swarm. In the simulation and evaluation stage, the light intensity contrast at the mid-plane of the photoresist is calculated and used as the threshold metric to count the number of compatible masks for each imaging structure, and the imaging structure with the maximum number of compatible masks is then selected. In the convergence check stage, two sets of termination conditions are used to determine whether the optimization is complete. If neither set of conditions is satisfied, updated imaging structures are generated using the PSO algorithm based on existing structure information and fed back to the simulation and evaluation stage to start the next iteration. When termination conditions are satisfied, the flow outputs the optimized imaging structure, including the optimized thickness and materials of the film layers.
Fig. 1. The imaging structure compatibility optimization flow based on the PSO algorithm.
2.1. Problem formulation
In practical IC manufacturing, there will be diverse layouts with varied mask patterns. Taking the one-dimensional periodic line/space mask pattern as an example, a typical mask is a 1 : 1 periodic pattern with different CDs, as shown in Fig. 2(a). Other masks such as those having 1 : 2, 1 : 3, etc. patterns with the same CD but different pitches are also common, as shown in Fig. 2(b).
Fig. 2. Examples of different mask patterns: (a) 1 : 1 periodic patterns with different CDs, and (b) patterns with the same CD (white) but different pitches.
Fig. 3 shows the 2D and 3D schematic diagrams of the imaging model based on the superlens structure employed in this paper. In the superlens structure, a glass layer is used as a substrate for the mask due to its high transmittance and low thermal expansion coefficient. The mask layer includes the opaque material and the transparent material. The chrome (Cr) objects are patterned on glass as the opaque areas of the mask patterns, and the transparent material, typically air or PMMA, is used to form the transparent areas. Silver layers are chosen due to their negative and wavelength-dependent permittivity, which satisfies the conditions of SPP excitation. The photoresist is coated to record the near-field image for subsequent manufacturing processes such as etching. Notably, the PMMA and Pr layer materials are employed with the same consideration of SPP excitation conditions, and the bottom Ag layer acts as a plasmonic reflector for resonance amplification. The mask layer is fixed as one-dimensional periodic line/space patterns, as this paper focuses on the compatibility of imaging structures with these patterns. A monochromatic transverse magnetic (TM) wave with a wavelength of 365 nm is used as the light source, as TM polarization is more conducive to imaging than transverse electric (TE) polarization.12 The incident light propagates along the z-axis.
Fig. 3. Schematic diagrams of the imaging model: (a) 2D imaging model and (b) 3D imaging model.
In the lithography process, the quality of the final image is contingent on various factors, including line-width roughness (LWR), edge placement error (EPE), and the nonlinearity of the photoresist response. Based on two assumptions (see the SI), this work adopts light intensity contrast as the metric for imaging quality and compatibility. For one-dimensional periodic line/space pattern masks, the light intensity contrast 
significantly influences imaging quality and is directly related to the process window.21 Additionally, since the mid-plane of the photoresist (dashed line in Fig. 3) is usually taken as the reference position for evaluating the imaging quality, the light intensity contrast on this plane is calculated as the metric.
In the imaging structure optimization flow proposed in this work, the glass layer is fixed, since it is the common substrate for masks. As the pattern transfer layer in lithography, the material of layer 3 is fixed as a photoresist to satisfy the exposure requirements. The thickness of layer 4 is set to semi-infinity, since it is the exit region in the rigorous coupled wave analysis (RCWA) algorithm.22,23 Therefore, our optimization flow includes the following optimization variables: permittivity and thickness of the mask layer, layer 1 and layer 2, thickness of layer 3, and permittivity of layer 4.
This study utilizes a light source with a wavelength of 365 nm and a mask set comprising 29 mask patterns with line/space ranging from 1 : 1 to 1 : 10, which include dense, semi-isolated and isolated patterns. Given the light source and mask patterns, the flow generates optimized imaging structures with the maximum number of compatible masks. The optimization problem formulation can be expressed as:
 |
1 |
where ε denotes the permittivity of the mask layer, layer 1, layer 2, and layer 4, and h denotes the thickness of the mask layer, layer 1, layer 2, and layer 3. The objective function Nmask, compatible denotes the number of compatible masks.
2.2. Imaging structure compatibility optimization
Solutions in PSO are represented as particles; these particles hold two vectors, i.e., position vector and velocity vector, representing the evolutionary state of particles in the search space. For a problem with D-dimensional space, particle i holds the position vector Xi = (xi1, xi2, …, xiD) and velocity vector Vi = (vi1, vi2, …, viD). In this paper, as mentioned in the previous section, optimization variables include the thickness and permittivity of film layers. Therefore, different particles represent different imaging structures, and the position vector characterizes the above optimization variables. During each optimization iteration, the film thickness and film permittivity of each imaging structure are updated in accordance with algorithm 1 and algorithm 2 below, respectively, thereby generating the updated imaging structure.
2.2.1. PSO for film thickness
Based on the concepts of PSO, each imaging structure only learns from the personal best (pbest) whose Nmask, compatible is superior to that of its own to balance the diversity and convergence of the imaging structure swarm. Therefore, before each iteration, imaging structures are firstly sorted from the worst to the best in terms of their pbest's Nmask, compatible. The velocity updating rule is shown below.
| Vi(t + 1) = w·Vi(t) + c·rand(pbestr(t) − Xi(t)) | 2 |
where c is set to 2.0 and rand is a random number distributed in [0, 1]. t denotes the number of iterations, and w denotes the inertia weight, whose value follows a Gaussian or Cauchy distribution probabilistically. pbestr(t) is a randomly selected personal best imaging structure thickness vector in the t-th generation, i ≤ r ≤ N, and N is the number of imaging structures in the swarm. Xi and Vi are the thickness vector and the velocity vector of the i-th imaging structure, respectively. Xi is updated as follows, and the overall process of PSO for film thickness is shown in algorithm 1.
| Xi(t + 1) = Xi(t) + Vi(t + 1) | 3 |
2.2.2. PSO for film permittivity
Unlike the continuous variable of thickness, permittivity is a discrete variable. We restrict the permittivity to a pre-defined material database to ensure that materials adopted in the optimized imaging structure are all mature, real and commonly used in semiconductor processes, possessing excellent process compatibility and feasibility. The corresponding algorithm is shown below.24
As shown in algorithm 2, for the j-th permittivity variable, the value of the variable is assigned according to the probability rather than the velocity vector. The available permittivity values with higher probabilities are more likely to be assigned to this variable. The probabilities for available values to be assigned to the j-th variable are initialized at the beginning and then updated by collecting the search information obtained by the whole imaging structure swarm. The probability for an available permittivity value is initialized as follows.
 |
4 |
where nj is the number of the available permittivity values in the database for the j-th variable, Probj,n(0) denotes the probability of the n-th available permittivity value to be assigned to the j-th variable. From line 3 to line 13, the probabilities for available permittivity values are updated by considering the historical search information and the current search information. The equation in line 11 illustrates the updating rule, where the first term denotes the historical search information and the second term denotes the current search information. The tuning parameter α has a function and distribution similar to w. For the n-th available permittivity value of the j-th variable in the t-th iteration, the probability of this value is represented by Probj,n(t), and Countj,n denotes the number of imaging structures that are assigned with the n-th value in the j-th variable. As shown in lines 14 to 16, after the probabilities are updated, each permittivity variable is assigned with a value according to these probabilities.
3. Results and discussion
3.1. Unoptimized imaging structure
Similar to Fig. 3, the details of the unoptimized imaging structure at the wavelength of 365 nm are shown in Table 1.6 The permittivity of the glass layer is set to 2.13. The conductivity of all materials in the imaging structure is set to 0 and the relative permeability is set to 1.
Table 1. The details of the unoptimized imaging structure.
| Film layer | Material | Permittivity | Thickness (nm) |
|---|---|---|---|
| Mask layer | Cr (opaque) | −8.55 + 8.96i | 50 |
| PMMA (transparent) | 2.25 | ||
| Layer 1 | PMMA | 2.25 | 50 |
| Layer 2 | Ag | −2.4 + 0.25i | 20 |
| Layer 3 | Pr | 2.56 | 40 |
| Layer 4 | Ag | −2.4 + 0.25i | — |
In the mask layer, 5 critical dimensions across 4 technology nodes21,25 are considered, and 29 mask patterns at each CD with line/space ranging from 1 : 1 to 1 : 10 and integer pitches are employed to characterize the imaging structure compatibility. Taking the N7 technology node with an 18 nm CD as an example, the details of 29 mask patterns are shown in Table 2. The widths of transparent areas and opaque areas are set to corresponding values to form these masks. RCWA algorithm is used to calculate the light intensity contrast under the above different mask patterns, as shown in Fig. 4.
Table 2. The details of 29 mask patterns at the N7 technology node with an 18 nm CD.
| Line (nm) | Space (nm) | Line/space ratio | Pitch (nm) |
|---|---|---|---|
| 18 | 18 | 1 : 1.0000 | 36 |
| 18 | 22 | 1 : 1.2222 | 40 |
| 18 | 26 | 1 : 1.4444 | 44 |
| 18 | 30 | 1 : 1.6667 | 48 |
| 18 | 34 | 1 : 1.8889 | 52 |
| 18 | 38 | 1 : 2.1111 | 56 |
| 18 | 42 | 1 : 2.3333 | 60 |
| 18 | 46 | 1 : 2.5556 | 64 |
| 18 | 50 | 1 : 2.7778 | 68 |
| 18 | 54 | 1 : 3.0000 | 72 |
| 18 | 61 | 1 : 3.3889 | 79 |
| 18 | 67 | 1 : 3.7222 | 85 |
| 18 | 74 | 1 : 4.1111 | 92 |
| 18 | 81 | 1 : 4.5000 | 99 |
| 18 | 87 | 1 : 4.8333 | 105 |
| 18 | 94 | 1 : 5.2222 | 112 |
| 18 | 100 | 1 : 5.5556 | 118 |
| 18 | 107 | 1 : 5.9444 | 125 |
| 18 | 114 | 1 : 6.3333 | 132 |
| 18 | 120 | 1 : 6.6667 | 138 |
| 18 | 127 | 1 : 7.0556 | 145 |
| 18 | 134 | 1 : 7.4444 | 152 |
| 18 | 140 | 1 : 7.7778 | 158 |
| 18 | 147 | 1 : 8.1667 | 165 |
| 18 | 153 | 1 : 8.5000 | 171 |
| 18 | 160 | 1 : 8.8889 | 178 |
| 18 | 167 | 1 : 9.2778 | 185 |
| 18 | 173 | 1 : 9.6111 | 191 |
| 18 | 180 | 1 : 10.0000 | 198 |
Fig. 4. Contrast curves of the unoptimized imaging structure under different mask patterns at different technology nodes and CDs.
For submicron patterns, positive photoresists typically require a light intensity contrast greater than 0.4 for exposures.3,13,26Fig. 4 clearly demonstrates that isolated patterns with larger pitches have relatively larger light intensity contrast, which means that it has more tolerance for the design of imaging structures for these mask patterns. Thus, it is difficult to design the imaging structures for dense or some semi-isolated patterns with smaller pitches, which is one of the problems we try to solve in this paper.
Taking 0.4 as the threshold metric, we calculate the number of data points above the threshold, which is also the number of compatible masks for the unoptimized imaging structure, as shown in Table 3. Nmask, total is the number of masks in a mask set at each technology node and CD, which is set to 29 in this work. The simulation software COMSOL Multiphysics 6.1 is also used for cross-validation. The results derived from the RCWA algorithm are in excellent agreement with those obtained from COMSOL, which demonstrates that our RCWA algorithm can be employed with the PSO algorithm reliably. Furthermore, as shown in the table, the number of compatible masks for the imaging structure decreases with smaller technology nodes and CDs.
Table 3. The number of compatible masks for the unoptimized imaging structure at different technology nodes and CDs.
| Technology node | CD (nm) | N mask, compatible (RCWA) | N mask, compatible (COMSOL) | N mask, total |
|---|---|---|---|---|
| N28 | 45 | 23 | 23 | 29 |
| N16 | 32 | 19 | 19 | 29 |
| N14 | 24 | 15 | 15 | 29 |
| N7 | 20 | 13 | 13 | 29 |
| N7 | 18 | 11 | 11 | 29 |
3.2. Optimized imaging structure
The imaging structure compatibility optimization flow based on the PSO algorithm is implemented on a Windows server with a 64-core 2.1 GHz Intel Xeon Gold 6430 CPU, NVIDIA GeForce RTX4090 GPU, and 1 TB memory. Algorithms 1 and 2 are used to optimize the film thickness and permittivity of the imaging structure, respectively. The thickness range is shown in Table 4, and the pre-defined material database can be seen in the SI. The maximum number of iterations (iteration threshold) and the number of imaging structures in the swarm are both set to 500 to balance the diversity and convergence of the imaging structure swarm.
Table 4. Thickness range for film layers of the imaging structure.
| Film layer | Thickness range (nm) |
|---|---|
| Mask layer | [20, 50] |
| Layer 1 | [20, 55] |
| Layer 2 | [10, 50] |
| Layer 3 | [20, 45] |
The mask layer is designed as 29 mask patterns with line/space ranging from 1 : 1 to 1 : 10 at N7 with an 18 nm CD. After 38 iterations with a time duration of approximately 40 hours, the details of the optimized imaging structure are shown in Table 5. Additionally, exhaustive scanning is also used to demonstrate the efficiency advantages of our PSO approach, which can be seen in the SI. Furthermore, we also calculate the light intensity contrast of the optimized imaging structure under different mask patterns at different technology nodes and CDs, as shown in Fig. 5.
Table 5. The details of the optimized imaging structure.
| Film layer | Material | Permittivity | Thickness (nm) |
|---|---|---|---|
| Mask layer | Al (opaque) | −14.586 + 2.3647i | 43 |
| Ta2O5 (transparent) | 5.2279 + 0.0009i | ||
| Layer 1 | PMMA | 2.25 | 20 |
| Layer 2 | Ag | −2.4 + 0.25i | 20 |
| Layer 3 | Pr | 2.56 | 20 |
| Layer 4 | Ag | −2.4 + 0.25i | — |
Fig. 5. Contrast curves of the optimized imaging structure under different mask patterns at different technology nodes and CDs.
Fig. 5 shows that the contrast of the optimized imaging structure for all 29 mask patterns at N7 with an 18 nm CD exceeds the threshold metric, indicating excellent compatibility of the imaging structure with these masks. The number of compatible masks for the optimized imaging structure is summarized in Table 6. Interestingly, although the optimized imaging structure is derived from the optimization flow where the mask layer is designed as 29 mask patterns with line/space ranging from 1 : 1 to 1 : 10 at N7 with an 18 nm CD, it is also compatible with all 29 mask patterns at larger technology nodes and CDs. The compatibility comparison is illustrated in Fig. 6. Compared to the unoptimized imaging structure, the optimized imaging structure is compatible with all mask patterns at all technology nodes and CDs, and exhibits compatibility improvements of 26.09%, 52.63%, 93.33%, 123.08% and 163.64%, respectively. Additionally, as a critical parameter for practical engineering applications, the maximum working depths of the unoptimized and optimized imaging structures are also calculated and compared, which can be seen in the SI. It should be noted that throughout the compatibility optimization of the imaging structure, we have consistently maintained the characteristics of imaging lithography, that is, the mask pattern and the imaging pattern satisfy a 1 : 1 geometric mapping relationship (unity-magnification imaging).
Table 6. The number of compatible masks for the optimized imaging structure at different technology nodes and CDs.
| Technology node | CD (nm) | N mask, compatible (RCWA) | N mask, compatible (COMSOL) | N mask, total |
|---|---|---|---|---|
| N28 | 45 | 29 | 29 | 29 |
| N16 | 32 | 29 | 29 | 29 |
| N14 | 24 | 29 | 29 | 29 |
| N7 | 20 | 29 | 29 | 29 |
| N7 | 18 | 29 | 29 | 29 |
Fig. 6. The number of compatible masks and the compatibility improvement ratio for the unoptimized and optimized imaging structures at different technology nodes and CDs.
3.3. Analysis
In this paper, the coherent transfer function is used to analyze imaging structures. CTF is the Fourier transform of the impulse response (point spread function) of the optical system,27 which represents the transfer efficiency of the optical system for light waves at different spatial frequencies, and can be obtained by the transfer matrix method (TMM).28 For the imaging model in Fig. 3, according to the mathematical model of conventional imaging lithography, the electric field at the mid-plane of the photoresist z = zm is approximately expressed as:29
 |
5 |
where xin denotes the x-coordinate in the object space (at the mask), G(kx) is the illumination function of the light source, and CTF(kx) is the CTF of the imaging structure. 
denotes the electric field distribution at the exit surface of the mask, which is rigorously solved by Maxwell's equations, and it is approximated that the mask does not interact with the subsequent imaging model.30 Since the normal incidence with TM polarization is used, G(kx) is 1 and Ey,n is 0.
The light intensity distribution and CTFE are calculated from the electric field:
| I(x, zm) = |E(x, zm)|2 = |FFT−1{FFT[En(x)]CTFE(kx)}|2 | 6 |
 |
7 |
where I(x, zm) denotes the light intensity distribution on the mid-plane of the photoresist, FFT and FFT−1 denote the Fourier transform and the inverse Fourier transform, CTFE is the CTF of the electric field. Notably, CTFE offers intuitive analysis of the transfer efficiency at different spatial frequencies, and the electric field distribution at the imaging plane in k-space can be obtained.
Eqn (6) shows that the light intensity distribution of the imaging plane depends on both the mask pattern and the CTF of the imaging structure. From the perspective of the spatial frequency, the mask pattern determines the range of the spatial frequency, while the CTF not only determines which spatial frequencies can be transferred to the imaging plane, but also regulates the transfer efficiency for specific spatial frequencies. The pupil function (PF) in conventional projection lithography acts similarly to the CTF, and it has a value of 1 inside the cutoff frequency and 0 outside. The evanescent waves carrying the detailed mask pattern information are outside the cutoff frequency, so the relative information cannot be transferred to the imaging plane, leading to the diffraction limit of conventional projection lithography.31
The CTF curves of the unoptimized imaging structure and the optimized imaging structure are shown in Fig. 7. Different kx values in the CTF represent different spatial frequencies of the light wave. When |kx| < nk0, the corresponding wave is a propagating wave; when |kx| = nk0, the corresponding wave is in complete grazing incidence; when |kx| > nk0, the corresponding wave is an evanescent wave. n is the refractive index of the mask incident medium and 
is the wave vector in a vacuum. As shown in Fig. 7, in plasmonic lithography, imaging structures also have certain transfer efficiency in spatial frequencies of evanescent waves, thus overcoming the diffraction limit.
Fig. 7. The CTF curves of the unoptimized and optimized imaging structures.
Moreover, compared to the unoptimized imaging structure, the CTF curve of the optimized imaging structure has two properties: first, the amplitude decreases in spatial frequencies of propagating waves and increases in spatial frequencies of evanescent waves. Second, the optimized imaging structure has a broader high-amplitude bandwidth in evanescent waves. Propagating waves carry the basic mask pattern information, while evanescent waves carry subtle details. The high amplitude of the optimized imaging structure indicates that detailed mask pattern information can be transferred to the photoresist with high efficiency, and the broader high-amplitude bandwidth indicates that more detailed information can be transferred.
When the light source is incident with a TM wave and the pitch of the mask pattern changes, positions of diffraction orders in k-space change accordingly. The spatial frequency of the diffraction order can be expressed as:
 |
8 |
where n is the refractive index of the incident medium, θ is the angle of incidence, m is the diffraction order, and p is the pitch of the mask pattern. Given that the normal incidence (θ = 0) is used in our work, eqn (8) simplifies to:
 |
9 |
To analyze the first-order diffraction (FOD), m is set to ±1, and kx/k0 is expressed as:
 |
10 |
As mentioned in Section 3.1, mask patterns with smaller pitches have lower contrast, so it is difficult to design imaging structures to be compatible with these masks. In this paper, the minimum pitch of the mask pattern is 36 nm, with an 18 nm CD and a 1 : 1 line/space ratio, and its kx/k0 at the FOD is approximately ±10.14. For the unoptimized imaging structure, the CTF amplitude near the FOD is nearly zero, resulting in poor compatibility with the corresponding mask patterns with small pitches. In contrast, with a certain CTF amplitude near the FOD, the optimized imaging structure achieves compatibility with these mask patterns. Additionally, the optimized imaging structure is also compatible with mask patterns at larger technology nodes and CDs which have large pitches, since their diffraction orders fall into the high-amplitude evanescent wave region.
4. Conclusion
This work focuses on plasmonic imaging lithography structures. By using the plasmonic lithography optical imaging model based on the RCWA algorithm and the PSO algorithm, a method and workflow for optimizing the compatibility of plasmonic lithography imaging structures have been established. The innovations of our work are twofold. First, the optimization target is shifted from the conventional resolution improvement to mask pattern compatibility, addressing a previously neglected but practically critical aspect of plasmonic lithography. Second, because film thickness is a continuous variable while material permittivity is a discrete variable, a hybrid PSO strategy is designed to handle these two types of variables separately, thereby enabling efficient mixed-variable optimization of the imaging structure.
Results show that the unoptimized imaging structure cannot fully be compatible with mask patterns in the mask set from the N7 node to the N28 node. However, by optimizing the compatibility of the imaging structure, it is possible to achieve compatibility with different line/space mask patterns from the N7 node to the N28 node. Moreover, the COMSOL model established based on the finite element method and the CTF analysis also confirm the correctness of the results.
It should be noted that although this paper only establishes a compatibility optimization workflow for one-dimensional periodic line/space mask patterns, it is general and can be transferred to other super-resolution imaging designs or the optimization of imaging performance. Additionally, this paper optimizes the plasmonic imaging lithography structures, and the imaging structures before and after optimization both possess the characteristics of plasmonic imaging lithography, tthat is, they maintain a 1 : 1 geometric mapping relationship (unity-magnification imaging) between the mask pattern and the imaging pattern. Moreover, given the same line width, if a one-dimensional line pattern satisfies the imaging requirements, it suggests that a two-dimensional pattern may also be achievable through appropriate resolution enhancement techniques, such as source optimization (SO), optical proximity correction (OPC), and source-mask optimization (SMO). Therefore, the optimization workflow established based on one-dimensional periodic line/space mask patterns provides a solid foundation, and its extension to two-dimensional patterns represents a promising direction for future work. It should also be noted that in the ultrathin-film regime, the thickness and permittivity are physically highly coupled. Integrating the thickness–permittivity coupling into our optimization flow in future work will further refine the optimization results. The optimization workflow proposed in this paper establishes a methodological foundation for improving the compatibility of plasmonic lithography and paves the way for industrial applications of plasmonic lithography.
Conflicts of interest
The authors declare no conflicts of interest.
Supplementary Material
Acknowledgments
The authors would like to acknowledge QM Project B – Research on Resolution Enhancement Technology for Deep Ultraviolet Band Plasmonic Imaging Lithography (E5QM017), the National Natural Science Foundation of China (Grant No. 62574219), and Beijing Nova Program (Grant No. 20250484884). We acknowledge the support from the Joint Laboratory of Microelectronics (JLFS/E-601/24) as well.
Data availability
The 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.
Supplementary information (SI): includes four sections. Section 1 presents the pre-defined material database, detailing the candidate materials and their permittivities at a 365 nm wavelength (Table S1). Section 2 provides a comparison of computational time and performance between the PSO approach and exhaustive scanning (Table S2). Section 3 outlines the two fundamental assumptions regarding the use of contrast as a metric for imaging quality and compatibility. Section 4 presents the maximum working depths of the unoptimized and optimized imaging structures (Fig. S1). See DOI: https://doi.org/10.1039/d6na00582a.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Supplementary Materials
Data Availability Statement
The 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.
Supplementary information (SI): includes four sections. Section 1 presents the pre-defined material database, detailing the candidate materials and their permittivities at a 365 nm wavelength (Table S1). Section 2 provides a comparison of computational time and performance between the PSO approach and exhaustive scanning (Table S2). Section 3 outlines the two fundamental assumptions regarding the use of contrast as a metric for imaging quality and compatibility. Section 4 presents the maximum working depths of the unoptimized and optimized imaging structures (Fig. S1). See DOI: https://doi.org/10.1039/d6na00582a.