Lithographic source and mask optimization with narrow-band level-set method

Yijiang Shen

doi:10.1364/OE.26.010065

1. Introduction

The ever growing integration density of semiconductor devices in the sub-22 nm technology node and low k₁ regime is appealing for more aggressive techniques with new physical lithographical tools and computational strategies. Along the way, the continuous technology development of shorter exposure wavelength and hyper numerical apertures (NAs) join hands with resolution enhancement techniques (RETs) [1, 2] to increase the resolution capacity of the lithography systems to achieve small minimum printed feature size. As an integral part of advanced inverse lithography technique (ILT) [3], source and mask optimization (SMO) expands the solution space of the source and mask patterns by the joint optimization of the illumination and mask shapes, whereas mask optimization (MO) procedure has limited degree of freedom with fixed source patterns.

The main goal of the SMO approach is to achieve a pair of optimized source and mask combined to ensure a higher image fidelity. Early SMO methods [4–7] are not pixel-based and computationally expensive. With the continuous shrink of critical dimension (CD), lithography simulators have extended from low-NA and high-NA for dry lithography to hyper-NA for immersion lithography, from scalar optics to vector optics including the polarization effect; complying with such extension, the combined effort of the developments of hyper-NA projection optics [8] and illumination source representations from the arc decomposition [5], the meshpoint representation [9] to pixelated sources by customized diffractive optical element (DOE) [10], have offered significant algorithmic insights into pixel-based method [11], process robustness [12–14], pupil and mask topology compensation [15, 16], source representation with Zernike polynomial functions [17] and optimization approaches including gradient methods [12,18,19], augmented Lagrangian [20] and alike. Meanwhile, level-set approaches widely applied in nonlinear inverse imaging problems [21, 22] and dynamic implicit surfaces [23], have been actively explored in [24, 25] solving inverse lithography problems, and systematic level-set implementations with appropriate finite-difference schemes are presented in scalar [26–28] and vector imaging systems [29, 30]. The above techniques generally update all optimization variables that is every source and mask pixel in every iteration, therefore computationally intensive with slow convergence. In this regard, modified level-set schemes are localized to reduce the computation labor, including fast marching [31,32] which requires a monotonically advancing front therefore not applicable to ILT and narrow-band methods [33, 34]; these methods are plagued with the irregularities of the level-set functions which should be numerically remedied by reinitialization [23, 35] introducing fundamental problems yet to be solved such as when and how to apply the reinitialization [36]. In [37,38], a distance regularized level-set evolution (DRLSE) is established to eliminate the need for reinitialization ensuring accurate computation and level-set evolution. To this end, a narrow-band variational level-set formulation is used in this work to further improve lithographic SMO performances.

Level set as a mathematical technique is pioneered by Osher [39] and Sethian. The advantages of applying the level-set perspective to inverse problems are at least threefold: the zero level surface corresponding to the propagating hypersurface may change topology or form sharp corners; discretization and numerical schemes can be devised to approximate the solution; intrinsic geometrical properties such as the normal vector and the curvature are easily determined from the level set function. This paper focuses on the application of localized narrow-band methods in lithographic SMO which significantly reduces optimization complexity. A complete set of variational level-set formulations of the optimization problem is established incorporating a distance regularization term and an external energy term that drives the zero level set toward desired mask features minimizing the pattern difference between the printed wafer and the desired pattern. The distance regularization term maintains a desired signed-distance profile near its zero level set, not only enabling a stable level-set evolution eliminating the need for reinitialization in a principle way, but also allowing a more simple and efficient narrow-band implementation. Consequently, the optimization work is conducted only in a neighborhood of the zero level set, providing a significant computation cost and memory usage reduction. Moreover, relative large Euler time steps dictated by the Courant-Friedrichs-Lewy (CFL) condition can be used to further reduce the number of iterations and computation time, achieving accelerated convergence.

2. Wafer image formation

An optical lithography imaging system is depicted in Fig. 1. Let E₀ ∈ ℝ^N×N being a matrix of size N × N with real number entry values, be the emitting electrical field components in the spatial coordinate (x, y, z) from a propagating monochromatic wave in the direction k embedding the direct cosine (α, β, γ)^T on the mask side emanated by a point source (α_s, β_s), and M ∈ ℝ^N×N denotes the mask pattern. Thus, the mask near field can be approximated using the constant scattering coefficient assumption (CSCA) [40] as E₀ ⊙ B ⊙ M, in which ⊙ is the entry-by-entry multiplication, and B is the diffraction matrix and the entry in B is defined as $B (m, n) = e^{\frac{j \times 2 π \times β_{s} \times m}{N}} \times e^{\frac{j \times 2 π \times α_{s} \times n}{N}}$ , m, n = 0, 1, · · ·, N − 1, where the discretion of defining matrix B is explained in [30]. Assuming E₀ has identical entries, following fourier optics [41] and Abbe method [42], the aerial image under a partially coherent illumination can be described as [8,19]

I_{a} = \frac{1}{\sum_{α_{s}, β_{s}} J} \sum_{α_{s}} \sum_{β_{s}} J (α_{s}, β_{s}) \sum_{p = x, y, z} {‖ H_{p} (α_{s}, β_{s}) \otimes (B (α_{s}, β_{s}) ⊙ M) ‖}^{2},

where H_p, B are functions of (α_s, β_s) and ⊗ denotes convolution operation, J is a N_s × N_s scalar matrix representing the source pattern distribution and

\sum_{α_{s}, β_{s}}

J is the sum of nonzero source intensities. H_p(α_s, β_s), p = x, y, z are referred to as the equivalent filters of the x, y, z components and are computed as

H_{p} = ℱ^{- 1} {\frac{n^{'}}{R} \sqrt{\frac{n^{'} γ}{γ^{'}}} h (α^{'}, β^{'}) V_{p} (α^{'}, β^{'}, γ^{'})}

, p = x, y, z, in which, ℱ⁻¹ denotes inverse Fourier Transform operation, R denotes the transverse magnification, n′ and (α′, β′, γ′)^T are the refractive index and the light propagation direction cosine in the wafer side, respectively, V characterizes the rotating factor in a hyper-NA system [2] and h denotes a low pass pupil function of the projection lens.

Fig. 1 Projection optics in a vector imaging model.

Download Full Size | PDF

The resist effect can be approximated using a logarithmic sigmoid function $sig (x) = \frac{1}{1 + e^{- a (x - t_{r})}}$ with a being the steepness of the sigmoid function and t_r being the threshold. Putting together, we can write the wafer imaging formation 𝒯{·} as

I = 𝒯 {M} = sig (I_{a}) .

In what follows, we will drop the argument (α_s, β_s) when there is no ambiguity. Due to optical diffraction, I is necessarily a distorted version of a given target pattern, which has to be compensated by computational lithography techniques, such as the SMO with designed mask and source patterns so as to achieve a desired printed pattern.

3. Level-set based optimization framework

Given a target pattern I₀ ∈ ℝ^N×N, the goal of the SMO is to find the optimal source Ĵ ∈ ℝ^N_s×N_s and mask pattern M̃ ∈ ℝ^N×N, which minimizes a designed distance between 𝒯{·} and I₀, namely

(\hat{J}, \hat{M}) = \underset{J \in ℛ^{N_{s} \times N_{s}}}{argmin} \underset{M \in ℛ^{N \times N}}{argmin} d {I_{0}, 𝒯 {J, M}},

with d(·, ·) being the sum of the mismatches between the wafer image and the desired one over all locations.

If level-set descriptions ψ and ϕ are given to J and M respectively by defining

\begin{array}{l} J = {\begin{array}{l} j_{int} & for {r : ψ (r) < 0} \\ j_{ext} & for {r : ψ (r) > 0} \end{array} & , and \end{array} M = {\begin{array}{l} m_{int} & for {r : ϕ (r) < 0} \\ m_{ext} & for {r : ϕ (r) > 0} \end{array},

where r denotes spatial coordinate (x, y), the inverse lithography problem is reformulated to handle the level-set functions ψ and ϕ instead of the pixelated source J and mask M. Constants j_int = 1, j_ext = 0 and m_int = 1, m_ext = 0 when a binary mask and source patterns are considered. The boundaries of the subregions in M and J are zero level sets of ϕ(r) and ψ(r), namely ϕ(r) = 0 and ψ(r) = 0. With a distance function d_M(r) defined as

d_{M} (r) = min (| r - \partial M (r) |),

where ∂M(r) denotes the boundaries of M(r), ϕ(r) is defined as the signed distance function related to d_M(r)

ϕ (r) = {\begin{array}{l} - d_{M} (r) & r \in \partial M_{-} \\ 0 & r \in \partial M (r) \\ d_{M} (r) & r \in \partial M_{+} \end{array},

where ∂M₋ is the region occupied by the mask pattern M and ∂M₊ is the void region. ψ(r) can be similarly defined as a signed distance function. Subsequently, solving Eq. (3) with a least squares leads to minimizing

F (J, M) = \frac{1}{2} {‖ 𝒯 {J, M} - I_{0} ‖}^{2},

where ‖ · ‖ stands for the l₂ norm. The source and mask co-optimization approach updates both J and M in each iteration, which is explicitly revealed by the evolution of level surfaces represented by the time-dependent models with respect to ψ and ϕ. Following the derivations in [26], we arrive at the Hamilton-Jacobi equations

\frac{\partial ψ}{\partial t} = - | \nabla ψ | v_{ψ} (r, t), and \frac{\partial ϕ}{\partial t} = - | \nabla ϕ | v_{ϕ} (r, t),

in which ∇ denotes the gradient, t is the artificial time, v_ψ(r, t) and v_ϕ(r, t) are the velocity functions normal to the surfaces of ψ and ϕ, respectively, and are defined as

\begin{array}{l} v_{ψ} (r, t) & = - 𝒥 {J}^{T} (𝒯 {J} - I_{0}) = \frac{1}{2} \frac{\partial}{\partial J} {(I - I_{0})}^{2} \\ = 2 a \sum_{α_{s}, β_{s}} \frac{\sum_{p = x, y, z} {‖ E_{p} (α_{s}, β_{s}) ‖}^{2} - I_{a}}{\sum_{α_{s}, β_{s}} J} ⊙ (I_{0} - I) ⊙ I ⊙ (1 - I), \end{array}

and

\begin{matrix} v_{ϕ} (r, t) = - 𝒥 {M}^{T} (𝒯 {M} - I_{0}) = \frac{1}{2} \frac{\partial}{\partial M} {(I - I_{0})}^{2} = - \frac{2 a}{\sum_{α_{s}, β_{s}} J} \sum_{α_{s}, β_{s}} \sum_{p = x, y, z} J (α_{s}, β_{s}) \\ Real [{(B)}^{*} ⊙ ({(H_{p})}^{*^{○}} \otimes {E_{p} (α_{s}, β_{s}) ⊙ (I_{0} - I) ⊙ I ⊙ (1 - I)})], \end{matrix}

with * being the conjugate operation, ○ flipping the matrix in the argument in both up-down and right-left directions, 1 ∈ ℝ^N×N being the all-ones matrix and E_p(α_s, β_s) = H_p(α_s, β_s) ⊗ (B(α_s, β_s) ⊙ M). It is observed that there are three convolutions in the computation of E_p, the resist image, and Eq. (8). The computation complexity of convolution is superlinear with respect to N which is very time consuming and applying Fast Fourier Transform (FFT) to remove the convolution operations will effectively reduce the computation intensity. Consequently, E_p is computed as ℱ⁻¹ [ℱ(H_p) ⊙ ℱ(B ⊙ M)] and Eq. (9) is rewritten as

\begin{array}{l} v_{ϕ} & = - \frac{2 a}{\sum_{α_{s}, β_{s}} J} \sum_{α_{s}, β_{s}} \sum_{p = x, y, z} J (α_{s}, β_{s}) \\ Real [{(B)}^{*} ⊙ ℱ^{- 1} {ℱ [({(H_{p})}^{*^{°}}] ⊙ ℱ [E_{p} ⊙ (I_{0} - I) ⊙ I ⊙ (1 - I)])}] . \end{array}

Another way to lighten the computation load is referred to as the electrical-field caching technique (EFCT) [19,30], which means that E_p is computed only once in each iteration and repeatedly used in Eqs. (8) and (9).

It should be noted that although the contour of interest is embedded as the zero level set of an level set function (LSF), it is necessary to maintain the LSF in good condition securing a stable evolution and accurate numerical computation. This requires the LSF to be smooth and not too steep or too flat (at least in a vicinity of the zero level set), which is well satisfied by signed distance properties |∇ϕ| = 1 and |∇ψ| = 1, with respect to ϕ and ψ [38]. Therefore reinitialization has been widely used in level set methods, yet with fundamental problems as when and how with regard to the practice of reinitialization [36]. Besides, current SMO approaches optimize the source pattern over all its degrees of freedom, which includes computing the evolution of all the level sets, not simply the zero level set corresponding to the front itself. These two issues will be addressed in the next session.

4. Narrow-band methods

The standard re-initialization method is to keep the evolving ψ and ϕ as signed distance functions during the evolution, especially in a neighborhood of the zero level set. Consequently, let ς = ψ or ϕ, the distance regularized level set (DRLS) term, $ℛ_{p} (ς) ≜ \sum_{Ω \in ℝ^{N \times N}} p (| \nabla ς |)$ where p(s) : [0, ∞] → ℝ is a potential function [37,38], is inserted into the cost function. The Gâteaux derivative of the functional ℛ_p(ς) is

\frac{\partial ℛ_{p}}{\partial ς} = - \nabla \cdot (d_{p} (| \nabla ς |) \nabla ς),

where

d_{p} (s) ≜ \frac{p^{'} (s)}{s}

. In this work,

p (s) ≜ \frac{1}{2} {(s - 1)}^{2}

will be used to maintain the signed distance property |∇ς| = 1 of the LSF, therefore ℛ_p(ς) is explicitly express as

ℛ_{p} (ς) = \frac{1}{2} \sum_{Ω} {(| \nabla ς | - 1)}^{2},

and

d_{p} (s) = 1 - \frac{1}{s}

. Combining the DLSR terms and the external energy term in (6) with a multiplier μ_ς yields

\frac{\partial ς}{\partial t} = - | \nabla ς | v_{ς} - μ_{ς} | \nabla ς | [Δ ς - \nabla \cdot (\frac{\nabla ς}{| \nabla ς |})] = - | \nabla ς | g_{ς} (r, t),

where Δ; is the Laplacian operator and

g_{ς} (r, t) = v_{ς} + μ_{ς} [Δ ς - \nabla \cdot (\frac{\nabla ς}{| \nabla ς |})]

, with respect to ς = ψ or ϕ. It should be pointed out that in this paper we will restrict our discussion using

p (s) ≜ \frac{1}{2} {(s - 1)}^{2}

and leave the investigation of other available options of p(s) in our future work.

The level-set formulation in Eq. (12) has an intrinsic capability of maintaining regularity of the level set function, ensuring a stable evolution of ς without the necessity of reinitialization, which is a iterative process and could be time consuming. Moreover, the construction of the narrow band is simple and straightforward allowing for updating ς only in the vicinity of the zero level set corresponding to the moving front, while in conventional level-set formulations [26–29], narrow-band implementation would require even more frequent reinitialization [33, 34] or additional velocity extension [43]. If the set of all the zero crossing points of ψ is denoted by Z, the narrow band is constructed as

B_{r} = \underset{(m, n) \in Z}{\cup} 𝒩_{(m, n)}^{r},

where

𝒩_{(m, n)}^{r}

is a square block centered at point (m, n) and r is the preset maximum distance from Z. Hence, instead of performing updates in the full domain, the proposed narrow-band technique operates over a significantly small number of optimization variables, thus efficiently improving the computation speed. Generally, for convergence and stability, Courant-Friedrichs-Lewy (CFL) condition is often applied asserting that the numerical wave speed must be at least as fast as the physical wave [23], giving

δ t_{ς} \cdot max {\frac{| g_{x} |}{δ x} + \frac{| g_{y} |}{δ y}} = ∊,

where δx and δy are the grid size of the discrete Cartesian grid, g_x and g_y are the components of g_ς in the x and y direction, 0 < ∊ < 1 is the CFL number and δtς is the Euler time step. The narrow-band implementation of the level-set formulation in Eq. (12) allows the use of a larger time step in the finite difference scheme to reduce the number of iterations and computation time compared to its full domain version. The main workflow of the proposed approach is described in Algorithm (1).

Algorithm 1. SMO with narrow-band level-set methods

View Table | View all tables in this article

5. Numerical results

The stable time-dependent model in Eq. (12) is a partial differential equation, which can be solved by readily available first-order temporal accurate and second-order accurate spatial finite-difference schemes [26,28]. The Lagrangian multiplier μ_ς is set to be 0.1. The imaging parameters of a 45 technology node system used in the numerical simulations are: CD 40nm, λ = 193nm, NA = 1.35, resolution δx = δy = 4nm/pixel, steepness of the sigmoid function a = 0.85, threshold t_r = 0.3, the original illumination source J₀ is a dipole source, which is given in Fig. 2(a). Figure 2(b) depicts the desired mask pattern I₀, and Figs. 2(c) and 2(d) show the aerial image I_a and the resist image I of I₀ illuminated by J₀, respectively, where severe pattern distortion incurred by the low pass nature of the pupil function h is observed. With pattern error (PE) defined as the square of the L₂ norm of the difference between the target pattern I₀ and the resist image I, Fig. (2)(d) bears a PE of 2742, which has to be compensated by computational lithography techniques.

Fig. 2 (a) The dipole source J₀. (b) The desired mask pattern I₀. (c) The aerial image I_a of I₀ illuminated by J₀. (d) The resist image I of I₀ illuminated by J₀, with PE 2742.

Download Full Size | PDF

From [30], it is noted that the edge placement error (EPE) has a positive relationship with the PE in general, however, explicit incorporation of the EPE into the cost function is often too complicated, therefore the minimization of PE is applied in the simulations. In Fig. 3, synthesized source and mask patterns derived using the steepest-descent (SD) and the conventional level-set method are presented. Aerial and resist images in Figs. 3(c)–3(d) and Figs. 3(g)–3(h) are simulated with the optimized masks in Figs. 3(b) and 3(f) illuminated by the optimized source patterns in Figs. 3(a) and 3(e) using the SD method and the conventional level-set method, respectively, which by simple observation, greatly improve the pattern fidelity from 2742 in Fig. 2(d) to under 600 with the conventional level-set method slightly outperforming the SD method which is consistent with the observations from [26]. One issue to clarify is that simulation using the SD method is performed on free-from source patterns leading to a gray-scale representation instead of a binary one in the level-set method. For fairness of comparison, a parametric transformation

J = \frac{1 + cos θ_{J}}{2} and M = \frac{1 + cos θ_{M}}{2}

where θ_J ∈ ℝ^N_s×N_s and θ_M ∈ ℝ^N×N, is applied to reduce the binary-constrained optimization problem to an unconstrained one in the SD method; the updating time step is also defined by the CFL condition in Eq. (14). Another set of simulations is given in Fig. 4 where the proposed narrow-band level-set SMO is applied, in which from left to right lies the optimized sources Ĵ, the optimized masks M̂, the aerial images I_a and the resist images I and from top to bottom the simulation results with the preset maximum distance r = 1, 2, 3 and 4, respectively. It is concluded that the performance of the proposed SMO method is at least comparable to the SD method and conventional level-set method in terms of pattern fidelity, if not better.

Fig. 3 Simulation of lithographic imaging with source and mask patterns using the steepest descent (SD) method and the level-set based method. (a) and (e), (b) and (f), (c) and (g), (d) with PE 584 and (h) with PE 517 represent synthesized source patterns, synthesized mask patterns, the corresponding aerial images and resist images, using the SD method and the level-set based method, respectively.

Download Full Size | PDF

Fig. 4 Simulation of lithographic imaging using the proposed narrow-band level-set method with different r. Columns Ĵ, M̂, I_a and I represent synthesized source patterns, synthesized mask patterns, the corresponding aerial images and resist images, with r = 1, 2, 3 and 4, respectively. The resist images (d), (h), (l) and (p) with input mask patterns (b), (f), (j) and (n) illuminated by (a), (e), (i) and (m), bear PEs of 643, 506, 487 and 563, respectively.

Download Full Size | PDF

Figure. 5 compares the PE convergence for the simulations in Fig. 3 and Fig. 4 with respect to simulation time. All the simulations stop after 100 iterations. Considerably stable convergence is observed in all the simulations with small oscillations after the local minimum is achieved, which is especially important for level set methods to maintain regularity of the LSFs, particularly the desirable signed distance property in a vicinity of the zero level set, ensuring accurate computation and stable level set evolution. Also, while the conventional level-set method is slightly faster than the SD method by including the gradient magnitude |∇ς| in Eq. (12) [26], the proposed narrow-band level-set SMO greatly accelerates the convergence for r = 1, 2, 3 and 4, which is intuitively observed in Fig. 5. A more quantitative comparison of the runtimes of the simulations in Fig. 3 and Fig. 4 is given in Table 1. All the computations are carried out using MATLAB on an Intel(R) Core(TM) i7-6700 CPU, 3.40 GHz, 8.00 GB of RAM. The SD and the conventional level-set method will take around 280 mins to complete the 100 iterations. On the other hand, the narrow-band level-set method takes only 36.7044, 49.2315, 72.0964 and 92.2435 mins for all 100 iterations, with r = 1, 2, 3 and 4, thus achieving a respective 7.714-, 5.7492-, 3.9259- and 3.0687-fold speedup.

Fig. 5 Pattern error versus simulation time for the desired pattern in Fig. 2(b).

Download Full Size | PDF

Table 1. Runtime (minutes) in Fig. 3 and Fig. 4.

View Table | View all tables in this article

The runtimes of the simulations in Fig. 3 and Fig. 4 with respect to iteration number are illustrated in Fig. 6, suggesting computation linearity for all the simulations and a much faster speed of the proposed narrow-band level-set method than that of the SD method and the conventional level-set method. From Eqs. (8), (9) and (12), it can be seen that the computation load is primarily governed by the number of convolution operations in the iterations. For conventional method which updates all the source points, the coherent aerial image $\sum_{p = x, y, z} {‖ H_{p} (α_{s}, β_{s}) \otimes (B) (α_{s}, β_{s}) ⊙ M ‖}^{2}$ generated by every source point has to be computed summing up to $3 \times N_{s}^{2}$ convolutions in one iteration, and another 3 × N_nz convolutions are incurred by the computation of v_ϕ(r, t) in Eq. (9), where N_nz denotes the number of non-zero source points. The total sampling number on the mask M is usually much larger than that of the source pattern J, which means N_s ≪ N. For the simulations in Fig. 3 and Fig. 4, N_s = 29 and N = 257, therefore, for the conventional level-set method, around 2700 convolutions are calculated in one iteration. The proposed narrow-band method, however, takes the perspective of source and mask optimization as the LSF evolution in the vicinity of the zero level set corresponding to the moving front itself, hence only updating the source points in the narrow-band B_r of ψ will contribute to the computation load amounting to 3 × (N_nz + N_B − N_nz ∪ N_B) convolution operations where N_B denotes the number of source points in B_r. It should be noted that there is overlapping between B_r and the non-zeros source points, meaning N_nz ∪ N_B > 0. Number of convolution operations and number of updated source points with respect to iterations are depicted in Figs. 7(a) and 7(b), respectively. Figure. 7(b) shows that since N_B is generally much smaller than $N_{s}^{2}$ , there are much less source points in need of updating, which greatly reduce the optimization complexity, leading to significantly smaller number of convolution operations in Fig. 7(a). Table 2 quantitatively demonstrates in the second row, that average convolution operation numbers in one iteration drop dramatically from 2670 for the conventional level-set method to 318, 432, 647 and 835 when the proposed narrow-band technique is applied with r = 1, 2, 3 and 4 and in the third row, average updated source points in one iteration drop from 841 to 35.6, 98.9, 158.5 and 220.6 as accordingly.

Fig. 6 Runtime for the simulations in Fig. 3 and Fig. 4.

Download Full Size | PDF

Fig. 7 (a) Computation load in terms of number of convolutions in every iteration. (b) Optimization complexity in terms of number of source points updated in every iteration.

Download Full Size | PDF

Table 2. Average number of convolution operations and average number of updated source points.

View Table | View all tables in this article

The CFL condition of the finite difference schemes states that the Euler time step is confined by the boundedness of |g_ψ| in the x and y direction in Eq. (14). For conventional level-set method, with δx and δy fixed, the Euler time step of updating the source pattern δt_ψ is dictated by max {|g_x| + |g_y|} defined on all the source points, while for the proposed narrow-band method, only the source points in the narrow-band B_r are accounted for, therefore a relatively larger δt_ψ is allowed to speed up the curve evolution. Euler time step δt_ψ for updating Ĵ is presented in Fig. 8, with a respective average Euler time step of 0.0070, 0.0096, 0.0078 and 0.0052 for the proposed narrow-band SMO approach with r = 1, 2, 3 and 4, which is larger than that of 0.0050 for the conventional level-set method.

Fig. 8 Euler time step δt_ψ for updating Ĵ in every iteration.

Download Full Size | PDF

Another advantage of the proposed SMO approach is the lower memory requirement of the electric-field caching technique (EFCT) which, when calculating the aerial image I_a, specifically keeps the caching of the coherent aerial images $\sum_{p = x, y, z} {‖ E_{p} (α_{s}, β_{s}) ‖}^{2}$ generated by the source points in the narrow band B_r and the electric-field components E_p(α_s, β_s) = H_p(α_s, β_s)⊗(B(α_s, β_s)⊙M), p = x, y, z generated by non-zero source points for later computation of v_ψ(r, t) and v_ϕ(r, t). Therefore, the number of $\sum_{p = x, y, z} {‖ E_{p} (α_{s}, β_{s}) ‖}^{2}$ and E_p(α_s, β_s) in need of caching, both of which are matrices of size N × N, amounts to N_B + 3 × N_nz, which is a considerable reduction comparing to that of the full domain counterpart which is $N_{s}^{2} + 3 \times N_{n z}$ , because in general, $N_{B} ≪ N_{s}^{2}$ justified by Fig. 7(b) and Table 2.

Simulations are also performed on the desired mask pattern I₀ in Fig. 9(b) (the same I₀ in Fig. 2(b)) illuminated by an original annular source J₁ with σ_in = 0.6 and σ_out = 0.9 in Fig. 9(a). It is observed that the annular source J₁ outperforms the dipole source J₀ with resist image PE 2361 in Fig. 9(d) compared to PE 2742 in Fig. 2(d) when no OPC is involved. Figures. 9(e) and 9(f) present the synthesized source and mask patterns using the proposed narrow-band level-set based SMO method with r = 3 to improve the pattern fidelity with resist image PE 486 in Fig. 9(h). Despite the different synthesized source and mask patterns in Fig. 4 and Fig. 9, the proposed method converges to very similar local minima with different original illumination sources J₀ and J₁. It should be noted that the inverse lithography problem is non-convex with multiple local minima. With the time-dependent model in Eq. (12) solved with finite-difference schemes, there is no guarantee of reaching global minimum. However, ILT is an ill-posed problem and a global minimum is often not required. Any good local minimum (where goodness is defined by data fidelity and user-defined properties) will suffice as an acceptable solution.

Fig. 9 (a) The annular source J₁. (b) The desired mask pattern I₀. (c) The aerial image I_a of I₀ illuminated by J₁. (d) The resist image I of I₀ illuminated by J₁, with PE 2361. (e) and (f) The synthesized source and mask patterns using the proposed narrow-band level-set method with r = 3. (g) and (h) The aerial image and the resist image with PE 486 with (f) as input illuminated by (e), respectively.

Download Full Size | PDF

The proposed SMO approach is further applied to another desired mask pattern I₁ in Fig. 10(b) illuminated by the annular source J₁ in Fig. 10(a). The pattern fidelity is notably improved from PE 3091 in the resist image (Fig. 10(d)) without OPC to PE 346 in the resist image (Fig. 10(h)) illuminated by the synthesized source pattern in Fig. 10(e) with the synthesized mask pattern in Fig. 10(f) as input. Improvements in simulation runtimes, computational load, Euler time step and caching memory requirement are similarly observed for simulations in Fig. 9 and Fig. 10.

Fig. 10 (a) The annular source J₁. (b) The desired mask pattern I₁. (c) The aerial image I_a of I₁ illuminated by J₁. (d) The resist image I of I₁ illuminated by J₁, with PE 3091. (e) and (f) The synthesized source and mask patterns using the proposed narrow-band level-set method with r = 3. (g) and (h) The aerial image and the resist image with PE 346 with (f) as input illuminated by (e), respectively.

Download Full Size | PDF

6. Conclusion

In this paper, we investigate the narrow-band level-set formulations for source and mask synthesis in optical lithography. The inverse problem is reformulated to address the source and mask optimization problem by tracking the evolution of LSFs embedding the level-set representation of source and mask patterns. Also, a distance regularized level set (DRLS) term is incorporated into the level-set formulation which not only enables a stable evolution of the LSFs and accurate computation by maintaining a designed signed-distance property, but also a simpler and more efficient numerical narrow-band implementation than the conventional full domain level-set methods. The proposed narrow-band method performs evolution in the vicinity of the zero level-set (the narrow band) instead of all the level sets, effectively reducing optimization complexity, resulting in significant improvement in pattern fidelity convergence in terms of runtime, computation load, Euler time step and caching memory requirement, which are merited by the numerical simulations. The theoretic and numerical analysis of the proposed narrow-band level-set SMO approach enhance the algorithmic understanding of the applicability of fast level-set techniques to source and mask optimization problems in next-generation immersion lithography.

Funding

Natural Science Foundation of Guangdong Province, China (2016A030313709, 2015A030310290); Guangzhou Science and Technology Project, China (201607010180).

References and links

1. A. K.-K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE Press, 2001). [CrossRef]

2. A. K.-K. Wong, Optical Imaging in Projection Lithography (SPIE Press, 2005). [CrossRef]

3. L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): a natural solution for model-based SRAF at 45nm and 32nm,” Proc. SPIE , 6607, 660739 (2007). [CrossRef]

4. T. S. Gau, R. G. Liu, C. M. Lai, and F. J. Liang, “The customized illumination aperture filter for low k1 photolithography process,” Proc. SPIE , 4000, 271–282 (2000). [CrossRef]

5. A. E. Rosenbluth, S. J. Bukofsky, C. A. Fonseca, M. S. Hibbs, K. Lai, R. N. Singh, and A. K. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlithogr. Microfabr. Microsyst. 1(1), 13–30 (2002).

6. C. Progler, W. Conley, B. Socha, and Y. Ham, “Layout and source dependent transmission tuning,” Proc. SPIE , 5454, 315–326 (2004).

7. R. Socha, X. Shi, and D. Lehoty, “Simultaneous source mask optimization (SMO),” Proc. SPIE , 5853, pp. 180–193 (2005). [CrossRef]

8. D. Peng, P. Hu, V. Tolani, T. Dam, J. Tyminski, and S. Slonaker, “Toward a consistent and accurate approach to modeling projection optics,” Proc. SPIE , 7640, 76402Y (2010). [CrossRef]

9. T. Fühner, A. Erdmann, and Sebastian Seifert, “Direct optimization approach for lithographic process conditions,” J. Microlithogr. Microfabr. Microsyst. 6(3), 031006 (2007).

10. K. Lai, A. E. Rosenbluth, S. Bagheri, J. Hoffnagle, K. Tian, D. Melville, J. Tirapu-Azpiroz, M. Fakhry, Y. Kim, and S. Halle, “Experimental result and simulation analysis for the use of pixelated illumination from source mask optimization for 22nm logic lithography process,” Proc. SPIE , 7274, 72740A (2009). [CrossRef]

11. X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Opt. Express 17(7), 5783 (2009). [CrossRef] [PubMed]

12. J. C. Yu and P. Yu, “Gradient-based fast source mask optimization (SMO),” Proc. SPIE , 7973, 23067–23077 (2011).

13. N. Jia and E. Y. Lam, “Pixelated source mask optimization for process robustness in optical lithography,” Opt. Express 19(20), 19384–19398 (2011). [CrossRef] [PubMed]

14. J. Li, Y. Shen, and E. Y. Lam, “Hotspot-aware fast source and mask optimization,” Opt. Express 20(19), 21792–21804 (2012). [CrossRef] [PubMed]

15. J. Li and E. Y. Lam, “Joint optimization of source, mask, and pupil in optical lithography,” Proc. SPIE , 9052, 90520S (2014). [CrossRef]

16. J. Li and E. Y. Lam, “Robust source and mask optimization compensating for mask topography effects in computational lithography,” Opt. Express 22(8), 9471–9485 (2014). [CrossRef] [PubMed]

17. X. Wu, S. Liu, J. Li, and E. Y. Lam, “Efficient source mask optimization with Zernike polynomial functions for source representation,” Opt. Express 22(4), 3924–3937 (2014). [CrossRef] [PubMed]

18. Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-Based Source and Mask Optimization in Optical Lithography,” IEEE. T. Image. Process 20(10), 2856–2864 (2011). [CrossRef]

19. X. Ma, C. Han, Y. Li, L. Dong, and G. R. Arce, “Pixelated source and mask optimization for immersion lithography,” J. Opt. Soc. Am. A 30(1), 112–123 (2013). [CrossRef]

20. J. Li, S. Liu, and E. Y. Lam, “Efficient source and mask optimization with augmented Lagrangian methods in optical lithography,” Opt. Express 21(7), 8076–8090 (2013). [CrossRef] [PubMed]

21. L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60(1–4), 259–268 (1992). [CrossRef]

22. S. Osher and N. Paragios, Geometric Level Set Methods in Imaging, Vision, and Graphics (Springer, 2003).

23. S. Osher and R. Fedkiw, Level Set Methods and Dynamic Implicit Surfaces (Springer, 2003). [CrossRef]

24. L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, and K. H. Baik, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” Proc. SPIE , 7520, 75200X (2009). [CrossRef]

25. V. Tolani, P. Hu, D. Peng, T. Cecil, R. Sinn, L. Pang, and B. Gleason, “Source-mask co-optimization (SMO) using level set methods,” Proc. SPIE , 7488, 74880Y (2009). [CrossRef]

26. Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express 17(26), 23690–23701 (2009). [CrossRef]

27. Y. Shen, N. Wong, and E. Y. Lam, “Aberration-aware robust mask design with level-set-based inverse lithography,” Proc. SPIE , 7748, 1–8 (2010).

28. Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust level-set-based inverse lithography,” Opt. Express 19(6), 5511–5521 (2011). [CrossRef] [PubMed]

29. Y. Shen, “Level-set based ILT with a vector imaging model,” in Proceedings of IEEE Conference on Semiconductor Technology International (IEEE2017), pp. 1–3.

30. Y. Shen, “Level-set based mask synthesis with a vector imaging model,” Opt. Express 25(18), 21775 (2017). [CrossRef] [PubMed]

31. J. A. Sethian, “A fast marching level set method for monotonically advancing fronts,” Proc. Natl. Acad. Sci. U. S. A. 93(4), pp. 1591–1595 (1996). [CrossRef] [PubMed]

32. J. A. Sethian, Level Set Methods and Fast Marching Methods (Cambridge University, 1999).

33. D. Adalsteinsson and J. A. Sethian, “A Fast Level Set Method for Propagating Interfaces,” J. Comput. Phys. 118(2), 269–277 (1995). [CrossRef]

34. D. Peng, B. Merriman, S. Osher, H. Zhao, and M. Kang, “A PDE-Based Fast Local Level Set Method,” J. Comput. Phys. 155(2), 410–438 (1999). [CrossRef]

35. M. Sussman and E. Fatemi, “An Efficient Interface-Preserving Level Set Redistancing Algorithm and Its Application to Interfacial Incompressible Fluid Flow,” SIAM J. Sci. Comput. 20(4), 1165–1191 (1999). [CrossRef]

36. J. Gomes and O. Faugeras, “Reconciling Distance Functions and Level Sets,” J. Vis. Commun. Image Represent. 11(2), 209–223 (2000). [CrossRef]

37. C. Li, C. Xu, C. Gui, and M. D. Fox, “Level set evolution without re-initialization: A new variational formulation,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE2005), pp. 430–436.

38. C. Li, C. Xu, C. Gui, and M. D. Fox, “Distance Regularized Level Set Evolution and Its Application to Image Segmentation,” IEEE. T. Image. Process 19(12), 3243(2010). [CrossRef]

39. S. Osher and R. P. Fedkiw, “Level set methods: an overview and some recent results,” J. Comput. Phys. 169(2), 463–502 (2001). [CrossRef]

40. T. V. Pistor, A. R. Neureuther, and R. J. Socha, “Modeling oblique incidence effects in photomasks,” Proc. SPIE , 4000, 228–237 (2000). [CrossRef]

41. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill Science, 1996).

42. M. Born and E. Wolf, Principle of Optics (Cambridge University, 1999). [CrossRef]

43. D. Adalsteinsson and J. A. Sethian, “The fast construction of extension velocities in level set methods,” J. Comput. Phys. 108(1), 2–22 (1999). [CrossRef]

Lithographic source and mask optimization with narrow-band level-set method

Abstract

1. Introduction

2. Wafer image formation

3. Level-set based optimization framework

4. Narrow-band methods

5. Numerical results

6. Conclusion

Funding

References and links

Cited By

Figures (10)

Tables (3)

Equations (17)

Optics Express

	SD	Level-set	r = 1	r = 2	r = 3	r = 4
Total time	283.0414	282.3885	36.7044	49.2315	72.0964	92.2435
Speedup ratio	1	1.0023	7.714	5.7492	3.9259	3.0687

	Level-set	r = 1	r = 2	r = 3	r = 4
Average convolution number	2670	318	432	647	835
Average updated source points	841	35.6	98.9	158.5	220.6