Methods for Benchmarking Photolithography Simulators: Part V Trey Graves, Mark D. Smith, and Sanjay Kapasi KLA-Tencor Corp., ABSTRACT As the semiconductor industry moves to double patterning solutions for smaller feature sizes, photolithography simulators will be required to model the effects of non-planar film stacks in the lithography process. This presents new computational challenges for modeling the exposure, post-exposure bake (PEB), and development steps. The algorithms are more complex, sometimes requiring very different formulations than in the all-planar film stack case. It is important that the level of accuracy of the models be assessed. For these reasons, we have extended our previous papers in which we proposed standard benchmark problems for computations such as rigorous EMF mask diffraction, optical imaging, PEB, and development [1-4]. In this paper, we evaluate the accuracy of the new PROLITH wafer topography models. The benchmarks presented here pertain to the models (and their associated outputs) most affected by the switch to non-planar film stacks: imaging at the wafer (image intensity in-media) and PEB (blocked polymer concentration). Closed-form solutions are formulated with the same assumptions used in the model implementation. These solutions can be used as an absolute standard and compared against a simulator. The benchmark can then be used to judge the simulator, in particular as it applies to speed vs. accuracy tradeoffs. Keywords: Lithography simulation, numerical accuracy, image intensity, post-exposure bake, PROLITH
1. INTRODUCTION Single patterning lithography has pushed k1 values near the theoretical limit (0.25). Double patterning will be the method used to print features at the 32nm and 22nm nodes and possibly beyond. Double patterning presents some new challenges for lithography simulation. In the past, it was a good assumption that the wafer stack was made of planar homogeneous films. The first patterning step in a double patterning process is modeled well under this assumption. However, in the second step this is no longer the case. Imaging, PEB, and develop computations must handle the non-planar topography introduced by processes such as etch, spin coating, deposition, etc. The algorithms that do these computations are more complex, sometimes requiring very different formulations than in the all-planar film stack case The accuracy of simulators is important, especially if lithographers are to use these simulators to make quantitative assessments that lead to critical decision making. It is also important to realize that there are many ways to model various steps in the photolithographic process. FDTD, RCWA, and FEM are just a few of the methods that can be used to determine the image-in-media in a non-planar film stack. Regardless of the implementation of the model, the algorithm should have good characteristics in terms of accuracy and convergence. It is also necessary that any numerical implementation of a model in a simulator be reasonably fast, as well as free of bugs or algorithm problems. For these reasons, we have extended our previous papers [1,2,3,4] where we proposed standard benchmark problems for aerial image calculations, image in resist calculations, EMF mask topography effects, PEB, and development. The benchmarks presented here are for image-in-media and post-exposure bake (PEB) under wafer topography conditions. We will use closed-form solutions as an absolute standard to judge the accuracy of a simulator. The benchmark can then be used to judge the simulator as it applies to speed vs. accuracy tradeoffs.
Optical Microlithography XXIII, edited by Mircea V. Dusa, Will Conley, Proc. of SPIE Vol. 7640, 764033 路 漏 2010 SPIE 路 CCC code: 0277-786X/10/$18 路 doi: 10.1117/12.846376
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2. IMAGE-IN-MEDIA BENCHMARK We are interested in determining the light intensity in materials such as resist, hard masks, anti-reflective coatings, etc. under wafer topography conditions. We call this the “image-in-media” to distinguish it from the image-in-resist which is of primary importance in a lithography simulator. This benchmark problem follows a method presented by Botten et al. [5], except that here we have adapted the method so that the incident beam passes through the immersion fluid (water) instead of air. The algebraic details of the solution will not be presented here, but a brief outline of the method is as follows. First, the eigenfunctions for the electric field (S polarization) or for the magnetic field (P polarization) are found inside the grating material. These eigenfunctions can be found analytically but the eigenvalues must be found numerically by solving a transcendental equation. The eigenfunctions represent plane waves propagating in the oxide and resist which are periodic and continuous across the material interfaces. Because the grating is lossless, the eigenfunctions have the properties that they are self-adjoint, and they form a complete, orthogonal basis for continuous functions [5]. This means that the electric field (or magnetic field) inside the grating can be represented by an infinite series of eigenfunctions. The electric (or magnetic) fields in the regions above and below the grating are represented by Rayleigh expansions (sets of plane waves). We find the coefficients in the eigenfunction expansion and in the Rayleigh expansions by using the method of moments to enforce the boundary conditions. This leads to a large set of algebraic equations that can be solved using any standard linear algebra package. Of course, the infinite series that is used to represent the fields in the grating and in the regions above and below the grating must be truncated in order to find a numerical solution to the problem. We chose expansions with 31 eigenfunctions and 83 Raleigh modes for the transmitted and reflected fields. Finally, we used MATLAB to find the eigenvalues and to solve the set of linear algebraic equations. Reference 5 assumes that the imaginary part of the index of refraction is zero. In the following comparison, we use only the real part of the refractive index. The immersion fluid is water (n = 1.44), the grating is made of “resist” (n = 1.7) and “oxide” (n = 1.563). The substrate is “silicon” (n = 0.88). The periodic grating of the wafer pattern uses a 1:1 duty with 150nm of resist and 150 nm of oxide (giving a 300nm pitch). The depth of this grating is 120 nm. The wavelength is 193nm. Normal incidence is used. The setup is shown in Figure 1.
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Substrat Figure 1. Dimensions and geometry of image-in-media benchmark. By using the methods described above, we calculated the intensity of the electric field (electric field squared) in the resist and oxide. We can also calculate the results using the PROLITH Wafer Topography
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simulator. Results are shown in Figures 2 and 3. The results indicate that even at low truncation order, PROLITH has good accuracy. It also converges very well with increasing truncation order.
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Figure 2. The upper plot shows the image intensity in resist and oxide for the closed-form method. The middle picture is the result from PROLITH with a maximum truncation order of 40. In the lower plot the difference is plotted and is seen to be very small. The regions with slightly larger error at 75 nm and 225 nm in x are the interfaces of the resist and oxide.
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RMS Error (Normalized Intensity)
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Figure 3. RMS error of the electric field intensity vs. truncation order.
3. POST-EXPOSURE BAKE BENCHMARK During PEB, the photoacid (H), created during the exposure process, diffuses through the resist and catalytically reacts with blocked polymer sites, M. At the same time, the base quencher, Q, also diffuses through the resist and neutralizes the acid [6]. Mathematically, we model this as:
dM = - ka ⋅ H ⋅ M dt
dH = - k loss ⋅ H - k Q ⋅ H ⋅ Q + DH ∇ 2 H dt dQ = - k Q ⋅ H ⋅ Q + DQ ∇ 2 Q . dt
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Here DH is the diffusivity of the acid, kloss is the acid loss reaction rate constant, kQ is the acid-base quench rate constant, DQ is the diffusivity of the base quencher, and ka is the deblocking reaction rate constant.
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3.1. POST-EXOSURE BAKE SOLUTION The neutralizing of acid and base makes an analytical solution of equations (1)-(3) difficult to impossible. However, a simplified solution is possible with no quencher. Setting Q=0, equations (1)-(3) reduce to two equations:
dM = - ka ⋅ H ⋅ M dt dH = - k loss ⋅ H + D∇ 2 H dt
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D now represents the diffusivity of the acid. Assuming periodic boundary conditions, the following solutions for H and M can be shown to satisfy (4) and (5):
⎛ ⎡ 2πn ⎤ 2 ⎞ ⎛ 2πnx ⎞ ⎟ ˆ ⎟⎟ exp⎜ − ⎢ H = ∑ H n Cos⎜⎜ ⎥ Dt ⎟ ⎜ p p n ⎝ ⎠ ⎦ ⎝ ⎣ ⎠
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⎡ ⎛ ⎡ 2πn ⎤ 2 ⎞ ⎤ ⎢ 1 − exp⎜ − ⎢ Dt ⎟ ⎥ ⎥ ⎟⎥ ⎜ ⎢ ⎣ p ⎦ ⎛ 2πnx ⎞ ⎠ . ⎝ ˆ ⎟⎟ M = exp ⎢− k a ∑ H n Cos⎜⎜ ⎥ 2 n ⎝ p ⎠ ⎡ 2πn ⎤ ⎥ ⎢ ⎥ ⎢ ⎢ p ⎥ D ⎣ ⎦ ⎦⎥ ⎣⎢
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x is the spatial coordinate in this 1D test problem. The pitch is labeled by p. Hˆ n is the Fourier coefficient of the initial acid concentration. In the following section a solution will be derived for Hˆ n for a specific test problem. Another solution to (4) and (5) can be found for an infinite domain instead of periodic boundaries. The solution is obtained with a Green’s function approach. This is the “resist blur” function from the IBM group [7].
3.2. POST-EXPOSURE BAKE SOLUTION VERIFICATION In a previous benchmarking paper, we chose a set of conditions that gave a closed-form solution for Hˆ n [4]. The illuminator and mask settings were set up to provide 2-beam interference at the wafer. This choice allowed Hˆ n to be written in terms of Bessel functions. This can still be done; however, in this study we have determined Hˆ n by Fourier transforming the initial acid concentration. The mask is alternating PSM with 65nm lines and spaces. For the imaging system, we use a 193nm exposure wavelength, NA =1.2, coherent illumination (σ = 0), unpolarized light and a reduction ratio = 4.0. The film stack has all optical properties set to the properties of a non-absorbing “resist” (n = 1.72, k = 0). The exposure dose is 2 mJ/cm2 with the Dill C parameter set to 0.06 cm2/mJ. Post-exposure bake time is 60 seconds. The diffusion coefficient for acid is 4.482 nm2/s. An amplification rate of 0.3678 sec-1 is used.
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The region of interest is T-shaped, as shown in Figure 4. The image-in-media is calculated with the same refractive indeces inside and outside the T. The PEB solution is then computed only in the T-shaped domain. The above solution is valid as a benchmark in this case because the no-flux boundary conditions are implemented in PROLITH at all edges of the domain. As can be seen in Figure 5, the PROLITH model has very good convergence. . Topography
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Figure 4. The left figure shows the topography of the problem. The middle and right figures show the image-in-media used as input for the problem. The figure on the right shows the acid concentration after PEB.
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Figure 5. The L2-norm is shown as a function of grid size. The convergence is very good. The tapering off of the error at small grid size is due to 6 digits of output in PROLITH results. The straight line is shown as a guide to the eye to indicate that the convergence is quadratic.
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4. CONCLUSION We have reviewed closed form solutions for image-in-media and PEB when the wafer film stack is no longer made of all-planar films. These solutions can be used to quantitatively determine the numerical accuracy of a wafer topography simulator, which is currently highly relevant to optical lithography. PROLITH’s image-in-media and PEB models converge rapidly to the exact solutions.
5. REFERENCES 1. M.D. Smith, C.A. Mack, “Methods for Benchmarking Photolithography Simulators”, Proc. SPIE, Vol. 5040 (2003) pp. 57-68. 2. M.D. Smith, J. D. Byers, C.A. Mack, “Methods for Benchmarking Photolithography Simulators: Part II”, Proc. SPIE, Vol. 5377 (2004) pp. 1475-1486. 3. Mark D. Smith, Trey Graves, Jeffrey D. Byers, Chris A. Mack, “Methods for Benchmarking Photolithography Simulators: Part III,” Optical Microlithography XVIII, Proc., SPIE Vol. 5754-99 (2005). 4. Trey Graves, Mark D. Smith, C. A. Mack, “Methods for Benchmarking Photolithography Simulators: Part IV,” Optical Microlithography XIX, Proc., SPIE Vol. 6154 (2006). 5. I.C. Botten, M.S. Craig, R.C. McPhedran, J.L. Adams, J.R. Andrewartha, “The dielectric lamellar diffraction grating”, Optica Acta, Vol. 28 (1981) pp. 413-428. 6. M.D. Smith, J. D. Byers, C.A. Mack, “The lithographic impact of resist model parameters”, Proc. SPIE, Vol. 5376 (2004) pp. 322-332. 7. Hinsberg, et al. “Extendibility of chemically amplified resists: another brick wall?”, Proc. SPIE, Vol. 5039 (2003) pp. 1-14.
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