Handbook of algorithms for physical design automation part 73.pdf (Sổ tay thuật toán)

Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C035 702 Finals Page 702 24-9-2008 #9 Handbook of Algorithms for Physical Design Automation It can be shown that, when the diffraction pattern of Equation 35.10 is placed at exactly the focal distance in front of the lens, the field at the focal plane (a distance f behind the lens) allows this phase factor to cancel the phase factor in the Fraunhofer diffraction formula, and the field in the image plane becomes E(p, q) ∝ −i ∞ ∞ 1 M(x, y)e−i(2π/λ)(xp+yq)dx dy λf −∞ −∞ (35.12) This form will be recognized as a mathematical representation that corresponds to the 2D Fourier transform [30] of the mask pattern: E(p, q) ∝ FT[M(x, y)] (35.13) To actually form the image of the mask at position (x1 , y1 ), the lens aperture and behavior, represented by a pupil function designated as P(a, b), are multiplied with the diffraction pattern at the focal point. This image in the focal plane is in turn transformed by a second lens at a distance f : E(x1 , y1 ) ∝ FT P(a, b) · E(p · q) = FT P(a, b) · FT M(x, y) (35.14) where P represents the pupil function, encompassing the wavefront transforming behavior of the lens. This is illustrated in Figure 35.5. Pupil functions can be simple mathematical structures, such as P(a, b) = 1, a2 + b2 ≤ r 1, ρ ≤ r = 0, a2 + b2 > r 0, ρ > r (35.15) representing the physical cutoff of the circular lens housing or radius r (shown in both Cartesian (a, b) and polar coordinates (ρ, φ)). However, additional phase behavior of the lens can also be included in the pupil function. Lens aberrations can be represented by an orthonormal set of polynomials called Zernike polynomials, each representing a specific aberration [29]. The Zernike polynomials are generally represented in polar coordinates, following the form Zj (ρ, φ) = amn Rnm (ρ)Yjm (φ) (35.16) Table 35.1 below shows a few of the Zernike polynomials and the corresponding aberration. More detail on these functions can be found in Ref. [29]. System pupil plane Mask First lens Image Second lens FIGURE 35.5 Simplified representation of the optical system of an imaging tool. At the pupil plane, the amplitude of the field represents a two-dimensional Fourier transform of the object, multiplied with the pupil function. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C035 Finals Page 703 24-9-2008 #10 703 Modeling and Computational Lithography TABLE 35.1 First Ten Zernike Polynomials, an Orthogonal Set of Functions That Describe the Lens Aberrations j 1 2 3 4 5 6 7 8 9 10 Note: n 0 1 1 2 2 2 3 3 3 3 m 0 1 1 0 2 2 1 1 3 3 am √n 1 √ 4 √ 4 √ 3 √ 6 √ 6 √ 8 √ 8 √ 8 √ 8 Rmn (ρ) 1 ρ ρ 2ρ 2 − 1 ρ2 ρ2 3ρ 3 − 2ρ 3ρ 3 − 2ρ ρ3 ρ3 Ynm (φ) 1 Cosφ Sinφ 1 Sin2φ Cos(2φ) Sinφ Cosφ Sin3φ Cos3φ Aberration Piston x-Tilt y-Tilt Defocus 45◦ Astigmatism 90◦ Astigmatism (Balanced) y-coma (Balanced) x-coma Shamrock Shamrock More details can be found in Ref. [29]. At this point, the image can be calculated, but the representation is still in terms of the amplitude and phase of the local electric field. Photosensors, whether they be the retinas of the eye, a photoelectric cell, or the molecules of a photoresist, produce a signal in proportion to the amount of energy in the electromagnetic field. The energy is proportional to the image intensity, found by squaring the modulus of the electric field: I(x, y) = E · E ∗ (35.17) where ∗ denotes the complex conjugate operation. 35.2.2.3 Linearity Although actual imaging systems comprise more than two simple phase front transformations, a key theorem on which all lens design is based is that any complex lens can be reduced to a simple Fourier transform, a Pupil function, and an inverse Fourier transform. This is a very powerful result, and is the basis of the entire field of Fourier optics [30]. Regardless of the exact lens structure and configuration, image simulation becomes a simple matter of designating the appropriate coordinate system, computing Fourier transforms and finding the proper representation of the pupil function P. Because Fourier transforms themselves are linear, the optical system is modeled by a linear process. This means that any arbitrary image can be assembled by creating a superposition of images from a suitable set of building blocks, each computed on its own. The linearity of the Fourier transform allows a complex 2D pattern to be decomposed into a Fourier series expansion of different 2D spatial frequencies, each being treated in turn and the final fields summed together. Note also that a nonmonochromatic distribution of wavelengths λ can similarly be computed wavelength by wavelength, and the final results summed as appropriate. This linearity holds as long as the media can be adequately descried by a refractive index, as in Equation 35.8. Note that for some materials, optical properties can change in the presence of strong electric fields, and the refractive index itself becomes an expansion: n = n1 + n2 E 2 + · · · (35.18) Materials in which these effects are significant are called ‘nonlinear optical materials’ [31]. Clearly, these nonlinearities can cause additional complications if they were to be used in imaging Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C035 704 Finals Page 704 24-9-2008 #11 Handbook of Algorithms for Physical Design Automation applications. However, values of n2 are generally very small, even for highly nonlinear materials, and these nonlinear effects are generally only observed using lasers with extremely high power densities. In general, the assumption that a total E field can be represented by a linear superposition of E fields remains valid. 35.2.2.4 Computation by Superposition The mathematics of Equation 35.14 represent that the imaging of any particular mask function is the multiplication of the FT of the mask with the pupil function. Because multiplication in Fourier space corresponds to convolution in position (x, y) space, image simulation reduces to the ability to do the following computational tasks in various combinations: 1. Digitize the mask function into a 2D amplitude and phase pixel array [M(x, y)] 2. Estimate a discrete 2D representation of the pupil function [P(ωx , ωy )] 3. Perform array multiplication (e.g., [P] • [M] in frequency space) 4. Compute discrete Fourier transforms (and inverse transforms as well) 35.2.2.4.1 Pixel Representation of the Mask Creating a pixel representation of the mask is usually fairly straightforward. Mask layouts are generated using polygons, often exclusively with Manhattan geometries. The ability to create an accurate discrete representation of the layout then becomes a question of the resolution desired and the size of array that can be computationally managed. This selection of the address grid can impact the computation and data management properties significantly, so should be done with care. Generally, a grid around 1 nm is selected for contemporary ICs with features as small as 45 nm. 35.2.2.4.2 Pixel Representation of the Pupil Once the pupil function is known, a similar mapping onto a grid is carried out. Here, the resolution of the pupil components need not be nearly as dense as the grid selected for the layout. However, because the transform of the mask and the pupil must be entry-wise multiplied, some care should be taken to ensure that the two grids match well. Although the simplest pupil functions are mathematically easy to represent (e.g., a circular aperture), these functions do not map to a Manhattan grid in the same way most mask functions can. In addition to this, the lens aberrations, also incorporated into the pupil, typically have circular symmetry (Table 35.1). Staircasing of these non-Manhattan functions occurs, and without a very fine grid, the results are less accurate. 35.2.2.4.3 Array Multiplication This is one of the basic computing operations, and is typically straightforward. The matrix multiplication occurs pixel by pixel, and the entries in the corresponding matrices are therefore multiplied entrywise. ⎡ P1,1 M1,1 ⎢ P2,1 M2,1 ⎢ ⎣ P3,1 M3,1 P4,1 M4,1 ⎡ P1,1 ⎢ P2,1 =⎢ ⎣ P3,1 P4,1 P1,2 M1,2 P2,2 M2,2 P3,2 M3,2 P4,2 M4,2 P1,2 P2,2 P3,2 P4,2 P1,3 M1,3 P2,3 M2,3 P3,3 M3,3 P4,3 M4,3 P1,3 P2,3 P3,3 P4,3 P1,4 P2,4 P3,4 P4,4 ⎤ P1,4 M1,4 P2,4 M2,4 ⎥ ⎥ P3,4 M3,4 ⎦ P4,4 M4,4 ⎤ ⎡ M1,1 M1,2 ⎥ ⎢ M2,1 M2,2 ⎥•⎢ ⎦ ⎣ M3,1 M3,2 M4,1 M4,2 M1,3 M2,3 M3,3 M4,3 ⎤ M1,4 M2,4 ⎥ ⎥ M3,4 ⎦ M4,4 (35.19) Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C035 Finals Page 705 24-9-2008 #12 705 Modeling and Computational Lithography where M1,1 represents, for example, a pixel of the Fourier transform of M(x, y), M1,1 = FT M(x, y) Pixel(a 1 ,b1 ) (35.20) P1,1 represents the pupil function at pixel (a1 , b1 ), etc. It is clear from this that the grids of the mask function, the pupil function, and the final image need to be matched to avoid excessive interpolation. 35.2.2.4.4 Fast Fourier Transform The fast Fourier transform (FFT) is one of the best known and widely used computational algorithms [32–34]. Normally, a discrete Fourier transform (DFT) numerically executing the Fourier transform in a brute force manner, would require O(N 2 ) arithmetic operations. However, when the functions to be transformed can be discretized into elements that are a multiple of 2, the DFT can be broken down into a number of smaller DFTs. The final result can be constructed to only have O(N log N) arithmetic operations. In a similar fashion, 2D discrete Fourier transforms can be broken down into a collection of 1D DFTs, each with a similar gain in computational efficiency. Because the mask function M is well behaved (with values of either 0 or 1, depending on the coordinates) and the pupil function P is continuous, both the mask function and pupil function can be digitized into a 2D array of pixels, with the number of pixels on each side being some multiple of 2. The FFT can therefore be used for this computation, and it has become the main engine of image simulation. 35.2.3 RET TOOLS The ability to simulate images quickly with tools such as the FFT and to compose arbitrary images based on the superposition of partial images gives rise to the possibility of EDA tools with dual, complementary capabilities: a database engine, to manage and process layout polygons, and a process simulation engine. The process engine calls on certain layers of data representing portions of the IC layout, transforms them to simulate processing behavior, and returns a representation of the transformed data to the database for further analysis. This is illustrated schematically in Figure 35.6. This combination of data management and simulation is how the entire class of ‘resolution enhancement techniques (RETs) [35] are implemented in an EDA flow. Layout Layer 1: Implant Layer 2: Isolation Layer 3: Gate Layer 4: Contact Layer 5: Metal 1 Layer 6: Via 1 Layer 7: Metal 2 Layer 8: Via 2 … Other simulated layers Single data layer Simulation FIGURE 35.6 Lithographic simulation of a single layer of an IC layout. Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C035 706 Finals Page 706 24-9-2008 #13 Handbook of Algorithms for Physical Design Automation (a) (b) FIGURE 35.7 Iso-dense bias. (a) represents the drawn layout, while (b) illustrates the result on the wafer. For this process, the lines in the dense region are thinner than isolated lines with the same nominal dimension. (Reproduced from Schellenberg, F.M., Zhang, H., and Morrow, J., Optical Microlithography XI, Proceedings of SPIE, 3334, 892, 1998. With permission.) There are three major RETs in use today: Optical and process correction (OPC), phase-shifting masks (PSM), and off-axis illumination (OAI) [35–37]. Each corresponds to control and manipulation of one of the independent variables of the optical wave at the mask: amplitude (OPC), phase (PSM), and direction (OAI). The changes required for OPC and PSM are implemented by changing the layout of the photomask, while OAI is implemented by changing the pattern of light emerging from the illuminator as it falls on the mask. 35.2.3.1 OPC Measured linewidth The acronym ‘OPC’, which is now used as a general term for changing the layout to compensate for process effects (optical and process correction), originally stood for optical proximity correction, and was used to predict and compensate for one-dimensional proximity effects. One example of a 1D effect, ‘iso-dense bias’ [38,39], is illustrated in Figure 35.7 [40]. Here, isolated and dense features of identical dimension on the photomask print at different dimensions on the wafer, depending on the proximity to nearby neighbors. Shown in the ‘pitch curve’ of Figure 35.8 is the characteristic behavior observed for 1D periodic features in a typical optical lithography process [41]. In this case, ‘pitch’ is the 1D sum of line and space dimensions. Some of this can be readily understood as an interaction of the Fourier spectrum of the photomask layout and the low-pass properties of the stepper lens and process: Dense lines have a well-defined Target linewidth Dense Isolated Pitch FIGURE 35.8 Iso-dense pitch curve, quantifying the linewidth changes for nominally identical features (i.e., lines all at a single target dimension) as a function of pitch. (Adapted from Cobb, N.B., Fast optical and process proximity correction algorithms for integrated circuit manufacturing, Ph.D. Dissertation, University of California, Berkeley, California, 1998. With permission.) Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C035 Finals Page 707 24-9-2008 #14 707 Modeling and Computational Lithography (a) (b) (c) (c) FIGURE 35.9 (a) and (b) Line-end pullback and (c) and (d) corner rounding. (Reproduced from Schellenberg, F.M., Zhang, H., and Morris, J., Optical Microlithography XI, Proceedings of SPIE, 3334, 892, 1998. With permission.) pitch and therefore a narrow spectrum, which passes easily through the pupil, while isolated features with sharp edges correspond to a range of spatial frequencies, including many high frequencies that are cut off by the pupil. It is therefore not a surprise that isolated and dense features of the same nominal dimension may have different images on the wafer. Additional effects that can impact the image are line-end pullback and 2D corner rounding, illustrated in Figure 35.9 [40]. These also are interpretable partly through the spectral analysis of the layout. To compensate for the loss in higher spatial frequencies, the positions of the edges in the original layout can be altered and adjusted as appropriate to correct the image in the local environment. [38,42,43] This is illustrated in Figure 35.10. Additional features not present in the original layout, sometimes called ‘scattering bars’ or ‘assist features’ can also be added to the layout [44,45]. These features, with dimensions chosen so that they themselves do not print on the wafer, form a quasi-dense environment around printing features, which would otherwise be isolated. An example is illustrated in Figure 35.11. The overall effect is to make the behavior of the isolated features better match the behavior of dense features on the final wafer. 35.2.3.2 PSM Traditional photomasks are fabricated using a lithography process to etch away portions of a layer of opaque chrome coated on a quartz mask blank [46,47]. The presence or absence of chrome forms the pattern to be reproduced on the wafer. However, the underlying quartz substrate of the mask can be etched as well. Because the refractive index of the quartz and air are different, a relative phase shift between the two neighboring regions can be created. This is illustrated in Figure 35.12. For apertures that are close together, if the light emerging from the apertures has the same phase, the images overlap on the wafer, the fields add, and the spots blur together, as shown in Figure 35.13a. If the phase difference is 180◦ , as shown in Figure 35.13b, however, the wave peaks and troughs sum to zero in the overlap region, and destructive interference occurs. Therefore, for two regions in close Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C035 708 Finals Page 708 24-9-2008 #15 Handbook of Algorithms for Physical Design Automation (a) (c) (b) (d) FIGURE 35.10 (a) Original layout and (b) its simulated wafer result, and (c) layout after modification with OPC and (d) its simulated wafer result. The wafer result for the corrected version is clearly a better match to the original drawn polygon. (Adapted from Maurer, W. and Schellenberg, F.M., Handbook of Photomask Manufacturing Technology, S. Rizvi, Eds., CRC Press, Boca Raton, Florida, 2005. With permission.) FIGURE 35.11 Example of a contemporary layout with printing features and SRAF. (Reproduced, Courtesy Mentor Graphics.) proximity, a dark fringe forms, allowing the images to remain distinct. [48–50] This is often called an ‘alternating’ PSM, because the phase alternates between apertures. Careful assignment of the mask regions to be etched, or phase-shifted, can lead to enhanced resolution for an IC layout. This can lead to problems, however, if polygons that require phase shifting Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C035 Finals Page 709 24-9-2008 #16 709 Modeling and Computational Lithography Quartz substrate Chrome 0⬚ 180⬚ Mask profile E-field Intensity Mask profile Intensity E-field FIGURE 35.12 Cross-section view of an optical wave passing through two apertures of a photomask. Etching the mask substrate for one of the apertures can produce a phase shift of 180◦ . (a) (b) 0⬚ 180⬚ FIGURE 35.13 Amplitude and intensity for (a) a conventional mask, and (b) a mask with a 180◦ phase shift. Contrast for neighboring apertures is clearly enhanced for the phase-0 shifting mask. (Adapted from Maurer, W. and Schellenberg, F.M., Handbook of Photomask Manufacturing Technology, S. Rizvi, Ed., CRC Press, Boca Raton, Florida, 2004. With permission.) in one area of the chip are contiguous with polygons in other regions that require the opposite phase. These topological constraints, illustrated in Figure 35.14, are called ‘phase conflicts’, and can place additional design rule restrictions on layouts [51–54]. Several variations on phase-shifting techniques have been adopted. The most common is a hybrid phase shifter, called an ‘attenuated PSM’ [55]. Here, the opaque chrome material of a conventional photomask is replaced with an attenuating but partially transmitting material (typically a MoSi film with 6 percent transmission [56]), with properties selected such that the light weakly transmitted through the film emerges with a phase shift of 180◦ . This improves contrast between light and dark regions, because the E-field (and therefore intensity) must be zero somewhere near the edge between the clear region and the phase shifted, darker region. However, fabrication techniques are similar to regular chrome mask processing, and no additional quartz etch step is required. There are also several double exposure techniques, in which certain phase-shifted features are created on a first photomask, while a second mask is used to trim or otherwise adapt the exposed Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C035 710 Finals Page 710 24-9-2008 #17 Handbook of Algorithms for Physical Design Automation Desired layout Phase conflict regions 180⬚ phase shift FIGURE 35.14 Examples of layouts that have phase conflicts. region to complete the exposure. [57–60] In this way, some of the unwanted artifacts of the phaseshifting structures can be eliminated in the second exposure. More details on various PSM techniques can be found in the literature. [35,37] 35.2.3.3 OAI For light falling at or near normal incidence to the photomask (on-axis illumination), the diffraction spectrum is straightforward to interpret. For light entering at an angle (i.e., using off-axis illumination [OAI]), the spectrum is shifted [61], as shown in Figure 35.15. Clearly, depending on the layout on the mask and the imaging properties of the lens system, the spectral content of the image can be significantly affected. FIGURE 35.15 Spectrum for an off-axis ray (left) and spectrum for an annular cone of off-axis rays (right). Alpert/Handbook of Algorithms for Physical Design Automation AU7242_C035 Finals Page 711 24-9-2008 #18 711 Modeling and Computational Lithography −1 orders +1 orders 0 orders − + −1 +1 FIGURE 35.16 Spectrum for conventional illumination (left), and for off-axis annular illumination (right), in which the annulus has been chosen to coincide with the diffracted orders of the pattern on the photomask. Typical illuminators shape the light to be uniform and to illuminate the photomask with a fairly narrow range of angles. The spectrum of illumination then corresponds to a circle. By using illumination with a specific angle of incidence, represented, for example, by the annulus in Figure 35.16, certain pitches can be emphasized and their imaging contrast enhanced, but only at the expense of lower contrast for other spatial frequencies [61,62]. For IC layouts with a large proportion of periodic patterns, such as memories, a suitable choice of illuminator pattern that matches the spatial frequencies of the layout can enhance imaging performance significantly [62]. An example of this is shown in Figure 35.17, in which a quadrupole-like illuminator was used in combination with subresolution assist features (SRAFs) to overcome certain “forbidden pitches” of low contrast [63]. More elaborate interactions between the spectrum of source angles and the photomask layout are possible. Shown in Figure 35.18 is an example of an IC cell and a source spectrum created through mask/source co-optimization. There are several methods demonstrated to achieve this goal [64–67]. 35.2.3.4 RET Combinations Although each of these techniques can enhance lithographic performance in and of itself, it is in combinations that dramatic improvements in imaging performance are achieved. For example, phase-mask images may have higher contrast, but still suffer from iso/dense bias, requiring OPC [59]. Likewise, combinations of OAI tuned for photomasks with OPC layouts can be very effective [68–70]. In some cases, all three techniques have been used together to create the best lithographic performance [35,71,72]. Success with developing processes using these combinations, with choices tuned to the unique combinations of skills present in individual companies, is a lively source of competition among IC makers.

Handbook of algorithms for physical design automation part 73

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