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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 19, Iss. 20 — Sep. 26, 2011
  • pp: 19384–19398
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Pixelated source mask optimization for process robustness in optical lithography

Ningning Jia and Edmund Y. Lam  »View Author Affiliations


Optics Express, Vol. 19, Issue 20, pp. 19384-19398 (2011)
http://dx.doi.org/10.1364/OE.19.019384


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Abstract

Optical lithography has enabled the printing of progressively smaller circuit patterns over the years. However, as the feature size shrinks, the lithographic process variation becomes more pronounced. Source-mask optimization (SMO) is a current technology allowing a co-design of the source and the mask for higher resolution imaging. In this paper, we develop a pixelated SMO using inverse imaging, and incorporate the statistical variations explicitly in an optimization framework. Simulation results demonstrate its efficacy in process robustness enhancement.

© 2011 OSA

1. Introduction

Optical lithography has served the semiconductor industry for decades as the predominant microlithography technology. This is attributed to the continuous technology development for shorter exposure wavelength and larger numerical aperture (NA) to achieve smaller minimum printed feature size [1

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

]. In addition, resolution enhancement techniques (RETs) are developed and become essential for maintaining good printed image quality. Nowadays, as the optical lithography has entered the low-k 1 regime [2

2. M. Rothschild, “A roadmap for optical lithography,” Opt. Photon. News 21(6), 26–31 (2010). [CrossRef]

], printed feature dimensions are highly sensitive to process variations. Traditional RETs are inadequate for the dual task of printing small features and providing enough process margins, which triggers the emergence of more aggressive techniques with new computational strategies. In particular, optimization and image processing techniques participate in more advanced optical proximity correction (OPC) and illumination modification approaches to enrich the lithographers’ arsenal [3

3. E. Y. Lam and A. K. Wong, “Computation lithography: virtual reality and virtual virtuality,” Opt. Express 17(15), 12259–12268 (2009). [CrossRef] [PubMed]

].

Recently, source design and the reticle pattern optimization have been integrated and optimized together. Rosenbluth et al. [13

13. A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsys. 1(1), 13–30 (2002). [CrossRef]

] decomposed the source by arcs, and developed a set of constraints to compute the optimum source and mask with the maximum exposure latitude. Fühner et al. [14

14. T. Fühner, A. Erdmann, and S. Seifert, “Direct optimization approach for lithographic process conditions,” J. Microlith. Microfab. Microsys. 6(3), 031006 (2007).

] adopted a more flexible meshpoint illumination representation defined by track/sector in their genetic optimization framework with the consideration of different process conditions. Currently, customized diffractive optical element (DOE) realizes a pixelated source, where the intensity and shape can be freely adjusted, therefore providing more degrees of freedom for optimization [15

15. K. Lai, S. Bagheri, K. Tian, J. Tirapu-Azpiroz, S. Halle, G. McIntyre, D. Corliss, A. E. Rosenbluth, D. Melville, A. Wagner, M. Burkhardt, J. Hoffnagle, Y. Kim, G. Burr, M. Fakhry, E. Gallagher, T. Faure, M. Hibbs, D. Flagello, J. Zimmermann, B. Kneer, F. Rohmund, F. Hartung, C. Hennerkes, M. Maul, R. Kazinczi, A. Engelen, R. Carpaij, R. Groenendijk, J. Hageman, and C. Russ, “Experimental result and simulation analysis for the use of pixelated illumination from source mask optimization for 22nm logic lithography process,” in Optical Microlithography XXII, vol. 7274 of Proc. SPIE, p. 72740A (2009).

]. Together with the pixelated mask, the so-called free-form source-mask optimization (SMO) fits well into the inverse lithography framework [16

16. L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, K.-H. Baik, and B. Gleason, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” in Lithography Asia 2009, vol. 7520 of Proc. SPIE, p. 75200X (2009).

, 17

17. T. Mülders, V. Domnenko, B. Küchler, T. Klimpel, H.-J. Stock, A. Poonawala, K. N. Taravade, and W. A. Stanton, “Simultaneous source-mask optimization: a numerical combining method,” in Photomask Technology 2010, vol. 7823 of Proc. SPIE, p. 78233X (2010).

]. A gradient-based SMO algorithm was shown to improve the pattern fidelity at the specified imaging conditions [18

18. X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Opt. Express 17(7), 5783–5793 (2009). [CrossRef] [PubMed]

], while another scheme implicitly considered dose sensitivity [19

19. J.-C. Yu and P. Yu, “Gradient-based fast source mask optimization (SMO),” in Optical Microlithography XXIV, vol. 7973 of Proc. SPIE, p. 797320 (2011).

], yet the main focus is still the pattern fidelity at the best process conditions, and their results are obtained from two separate optimization steps: source optimization and the successive mask optimization. In the SMO framework proposed by Peng et al. [20

20. Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-based source and mask optimization in optical lithography,” IEEE Trans. Image Process. 99, 1–10 (2011).

], process robustness is considered by incorporating only one defocus condition rather than the dose-focus matrix.

In this paper, we design robust free-form source and mask patterns with respect to process variations using inverse imaging. To achieve this, we develop a cost function that incorporates not only the pattern fidelity but also the aerial image intensity distribution. We then build a statistical SMO framework and solve it by alternating optimizations of the source and the mask.

2. Lithography imaging model

An optical lithography imaging system is depicted in Fig. 1. The reticle, or the photomask, is illuminated by a light source through a condenser lens L 1. The projection optics then forms an image of the photomask onto the wafer. Due to diffraction and different optical aberrations, however, this is necessarily a distorted image; the goal of inverse lithography is to design, through mathematical modeling and computations, the mask pattern — and sometimes the source as well — so as to achieve a desired printed image.

Fig. 1 Illustration of an optical projection lithography system [1, 3].

Detailed analysis of the lithography imaging system model has been developed over the years. Here, our focus is to introduce to the readers how the image on the wafer is distorted from the mask pattern. Let the former be I(x,y) and the latter be M(x,y). It is also useful to define the frequency domain representation of the mask; hence, we denote the mask pattern spectrum by (f,g), where f and g are normalized frequency variables [21

21. A. K. Wong, Optical Imaging in Projection Microlithography (SPIE, 2005). [CrossRef]

, p. 67]. Two other quantities are also important to describe the lithography system. The first is the optical transfer function denoted by Ĥ(f,g), where its inverse Fourier transform, called the point spread function, is H(x,y). The second is the effective light source Ĵ(f,g) (the Fourier transform of the mutual intensity), which arises because lithography systems involve partially coherent imaging. With these quantities, the intensity distribution at the wafer can be described by [21

21. A. K. Wong, Optical Imaging in Projection Microlithography (SPIE, 2005). [CrossRef]

, Eq. 4.35]
Ia(x,y)=J^(f,g)H^(f+f,g+g)H^(f+f,g+g)M^(f,g)M^(f,g)×ei2π[(ff)x+(gg)y]dfdgdfdgdfdg,
(1)
where † denotes complex conjugate. The six-fold integration can be simplified to
Ia(x,y)=J^(f,g)|H^(f+f,g+g)M^(f,g)ei2π(fx+gy)dfdg|2dfdgf,gJ^(f,g)|M(x,y)*H(x,y;f,g)|2,
(2)
where the approximation is needed for computation in the discrete domain.

The above accounts for the light intensity arriving at the image plane, also known as the aerial image, but this is not what is printed on the wafer. Light reacts with the photoresist, which either increases its development rate with exposure (positive resists) or decreases its rate (negative resists). An image is formed at a particular location when the development is beyond a certain threshold. Thus, I(x,y) is the binarized version of Ia(x, y). For numerical considerations, however, we often avoid using a hard threshold in computing I(x,y). Instead, a smooth transition is preferred. A frequently used model is with the sigmoid function, given by
I(x,y)=sig{Ia(x,y)}=11+eα(Ia(x,y)t),
(3)
where t is the threshold and α controls the steepness of the transition. Combining Eq. (2) and (3), the lithography imaging model is therefore
I(x,y)=sig{f,gJ^(f,g)|M(x,y)*H(x,y;f,g)|2}.
(4)

3. Source mask optimization framework

With a fixed optical setup, we can observe from Eq. (4) that the resulting image on the wafer is controlled by two variables: the mask pattern M(x,y), and the source, which governs the mutual intensity function Ĵ(f,g). In principle, to determine if a pattern can be printed at all, and if so, what the proper light source and mask pattern should be, we should investigate all combinations to see if we can arrive at the desired Ī(x,y). This is the rationale behind SMO.

Unfortunately, we also observe from Eq. (4) that the image is nonlinear in M(x,y) and Ĵ(f,g). Practically, we allow errors in the resulting printed image from our desired pattern; we are content if they do not cause intolerable changes in the resulting circuit’s behavior. Let us denote the desired pattern with Ī(x,y). Furthermore, assume that we are designing a binary mask, so M(x,y) can either be zero or one at any location. We can let it take on other values if we are interested in designing PSMs. Thus, we solve the following optimization problem
minimize𝒟{I(x,y),I¯(x,y)}subjecttoM(x,y){0,1}J^(f,g)0.
(5)
The operator 𝒟{a,b} measures dissimilarity between a and b. Various formulas have been proposed for different specifications [22

22. Y. Granik, “Source optimization for image fidelity and throughput,” J. Microlith. Microfab. Microsys. 3(4), 509–522 (2004). [CrossRef]

, 23

23. J.-C. Yu and P. Yu, “Impacts of cost functions on inverse lithography patterning,” Opt. Express 18(22), 23331–23342 (2010). [CrossRef] [PubMed]

]. Here, we define it to consist of four terms, i.e.,
𝒟{I(x,y),I¯(x,y)}=𝒯{I(x,y),I¯(x,y)}+γ1TV{Ia(x,y)}+γ2aerial{Ia(x,y),I¯(x,y)}+γ3contrast{Ia(x,y),I¯(x,y)}.
(6)
The first term, 𝒯{a,b}, ensures pattern fidelity, while the rest are regularization terms controlled by γ 1,γ 2 and γ 3 respectively. We explain each of them below.

3.1. Pattern fidelity

The pattern fidelity term 𝒯{a,b} is used to count the errors, or mismatches, between a and b, summing over all locations. By putting this as a penalty in the optimization, the printed contour deviation from the desired one is minimized to achieve the smallest accumulated edge placement error (EPE) over the image.

For mathematical convenience, the 1 and the square of the 2 norms are frequently used, i.e.,
(1norm)𝒯{I(x,y),I¯(x,y)}=x,y|I(x,y)I¯(x,y)|
(7)
(2norm)𝒯{I(x,y),I¯(x,y)}=x,y|I(x,y)I¯(x,y)|2.
(8)
In principle, we can use a weighted norm, where the weight is proportional to the extent that an error is allowed at the location. For example, we want to severely penalize any location that may result in bridging two disjoint areas, but an error in an isolated region may often be acceptable. However, this requires further image understanding and analysis and possibly some understanding of the underlying circuit design, and is generally difficult to accomplish. For the experiments described in this manuscript, we use the 2 norm.

3.2. Smoothing

While the pattern fidelity criterion above is concerned with the binarized image printed on the wafer only, we need to take into account the aerial image, Ia(x, y), in the optimization process as well. One important reason is due to process variations. Consider Fig. 2, which plots two possible intensity distributions as a function of the spatial locations. With the same threshold (t 1), the resulting binarized images are identical. However, suppose there exists variations in the exposure, and consequently the threshold is now at t 2. This causes little change in the printed image for (a), where the transition is sharp, but a significant deviation for (b), which the transition is gradual. Thus, it is desirable to have sharp transitions to make the design robust.

Fig. 2 Relationship among intensity distributions, threshold, and the binarized image.

To quantify this, we argue that Ia(x, y) should be piecewise smooth with sharp edges at the transition regions, and therefore the mathematical operation called total variation (TV) is the appropriate metric to use. It is now a common tool in image reconstruction and restoration, and is known to suppress the small-scale noise while preserving the large-scale features [24

24. D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19(6), 165–187 (2003). [CrossRef]

, 25

25. M. K. Ng, H. Shen, E. Y. Lam, and L. Zhang, “A total variation regularization based super-resolution reconstruction algorithm for digital video,” EURASIP Journal on Advances in Signal Processing 2007, Article ID 74,585 (2007). [CrossRef]

]. The TV norm is given by
||Ia(x,y)||TV=|Ia(x,y)|,
(9)
where ∇Ia(x, y) denotes the gradient of Ia(x, y). In numerical implementations, this gradient is approximated by finite difference. Let
xIa(x,y)=Ia(x+1,y)Ia(x1,y)2andyIa(x,y)=Ia(x,y+1)Ia(x,y1)2,
(10)
the gradient ∇Ia(x, y) is then given by
Ia(x,y)[xIa(x,y)]2+[yIa(x,y)]2.
(11)

Fig. 3 shows a one-dimensional example, with real data, to illustrate the effect of TV regularization. This pattern (in green) includes three tightly-packed areas and a relatively isolated one. Without TV regularization, the aerial image intensity (in red) plotted in (a) has many locations with substantial signals where there should be no pattern; consequently, the error margin with the threshold is small. If the threshold reduces somewhat, we would observe spurious areas in the resulting binarized image. We can compare this with (b), where, with TV regularization, such “noise” is significantly reduced.

Fig. 3 Demonstration of TV regularization. The red and green curves represent aerial image intensity and target design, respectively. The dotted line marks the threshold t.

Nevertheless, such TV regularization also comes with some drawbacks. In this example, the signal content of the four features is also reduced, and thus also compromising the robustness of the resulting design, because if the threshold is increased, some features may be lost. In other words, the contrast of the aerial image is reduced. To ameliorate this, we design a weight matrix W(x,y) that mediates the TV function. It takes small values around the transition areas, and large values elsewhere. Mathematically, the transition areas are given by our design pattern, Ī(x,y). We extract its edge by morphology, where the result, E(x,y), is given by
E(x,y)=[I¯(x,y)S(x,y)][I¯(x,y)S(x,y)].
(12)
Here, ⊕ and ⊖ denote dilation and erosion, respectively, and S(x,y) is a 3×3 structure element with all one’s. Morphological dilation expands the shape of the input binary image (Ī(x,y) in this case), while erosion functions in the opposite way. These two operations are common approaches for binary image boundary extraction. Detailed mathematical description of them can be found in [26

26. R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. (Prentice Hall, 2002).

]. The weight matrix W(x,y) is then given by
W(x,y)=[1E(x,y)]*G(x,y),
(13)
where G(x,y) is a blurring function. Consequently, the first regularization term in Eq. (6) is then given by
TV{Ia(x,y)}=x,yW(x,y)|Ia(x,y)|.
(14)

In our experiments we let G(x,y) to be a 5 × 5 Gaussian kernel. This allows a smooth transition from penalty to no penalty, and vice versa. Referring to the earlier example, the weight function is given in Fig. 4(a). In (b) the resulting aerial image intensity curve by applying this weighted TV regularization is shown. The background intensity is smoothed similarly as in Fig. 3(b), while the aerial image contrast at transitions is better preserved.

Fig. 4 The weight function W(x,y) for TV regularization and its effect. The blue curve in (a) plots the weight function, and the red curve in (b) plots the aerial image intensity by using this weighted TV regularization. The green curve denotes the target design in both figures.

3.3. Aerial image and contrast

With the above regularization, how are we going to control the area near the transition areas? In addition, what should be the control? The answer to the first question is straightforward, because the opposite of the above weight function, i.e., 1 – W(x,y), allows us to put the emphasis on the transition areas. The answer to the second question comes in two expressions.

First, our goal is to push intensity values away from the threshold t as far as possible. At places where the design pattern Ī(x,y) = 0, we would like Ia(x, y) ≈ 0; at places where Ī(x,y) = 1, we would like Ia(x, y) ≈ 2t, so the threshold would be mid-way. This is depicted in Fig. 5. We set the threshold mid-way such that the intensity on each side of the nominal threshold could be equally regularized to reduce the contour sensitivity to both higher and lower dose changes.

Fig. 5 Objective of aerial image intensity regularization. The shaded area denotes the vicinity of the target edge.

We can consolidate the two requirements in enforcing Ia(x, y) ≈ 2(x,y). Thus, the second regularization term in Eq. (6), which can also be viewed as a penalty term, is given by
aerial{Ia(x,y),I¯(x,y)}=x,y[1W(x,y)][Ia(x,y)2tI¯(x,y)]2.
(15)

4. Statistical model for process robustness

Our discussion thus far is based on the assumption of an ideal imaging system without any process error [27

27. C. Mack, Fundamental Principles of Optical Lithography: The Science of Microfabrication (Wiley, 2007). [CrossRef]

]. We extend this optimization framework to a robust model by explicitly incorporating process variations, namely, dose variation and focus variation. As a reasonable assumption, we consider the process variations as independent, normally distributed random variables [28

28. N. Jia, A. K. Wong, and E. Y. Lam, “Robust mask design with defocus variation using inverse synthesis,” in Lithography Asia, vol. 7140 of Proc. SPIE, p. 71401W (2008).

]. Specifically, dose variation can be accounted for by varying the threshold t; focus variation, parameterized by β, is modeled by adding a phase term to the optical transfer function as
H^˜(f,g;β)=H^(f,g)eiπβNA2λ(f2+g2),
(17)
where NA is the numerical aperture, λ is the incident light wavelength, and βNA2/λ gives a normalized defocus quantity. A detailed mathematical description of the defocus model can be found in Ref. [10

10. 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]

].

To compute solutions that are robust to process variations, the average wafer performance is optimized by minimizing the expectation of 𝒟 with respect to dose and focus fluctuations. The problem to be solved is thus described by a statistical model as
minimizeE{𝒟{I(x,y),I¯(x,y)}}subjecttoM(x,y){0,1}J^(f,g)0,
(18)
where E{·} takes the expectation operation over t and β. However the expectation integral is difficult to compute due to the nonlinearity of 𝒟. To tackle this problem, we discretize t to take on a set of values tm with probability p(tm), and discretize β to take on a set of values βn with probabilities p(βn). The function E{𝒟{I(x,y),Ī(x,y)}} is computed as
E{𝒟{I(x,y),I¯(x,y)}}m,np(tm)p(βn){𝒟{I(x,y;tm,βn),I¯(x,y)}}.
(19)

5. Optimization procedure

Given the cost function in Eq. (18), we minimize it by iteratively updating the source function and the mask pattern. The optimization procedure consists of multiple functional blocks as shown in the flow diagram in Fig. 6 with labels A to E.

Fig. 6 Optimization procedure of SMO.

Below we explain in details how the mask and source updates are performed. We compute the updates using the nonlinear Hestenes-Stiefel conjugate gradient method [29

29. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

, p. 123]; as such, each update consists of n iterative steps. We use a superscript with brackets to denote the current step. Also, we omit the designation (x,y) for brevity when no confusion arises.

5.1. Mask update

With the approximate objective function in Eq. (19), we first compute the gradient of 𝒟{I(x,y;tm,βn), Ī(x,y)}, denoted as M𝒟m,n(0). The derivation is found in the Appendix. We then sum it for all values of m and n to obtain the gradient of E{𝒟{I(x,y)(x,y)}}, denoted by ∇ME(0), as
ME(0)=m,np(tm)p(βn)M𝒟m,n(0).
(20)
We call this the initial mask update, and assign qM(0)=ME(0).

5.2. Source update

The source can be updated with a similar procedure. Note that the effective source Ĵ(f,g) is a normalized quantity by its total energy [1

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

, p. 45], i.e.,
J^(f,g)=J^(f,g)f,gJ^(f,g).
(25)
In the following, we use Ĵ′(f,g) in the updates.

We first compute the gradient of 𝒟{I(x,y;tm,βn)Ī(x,y)} with respect to the source, denoted as J^𝒟m,n(0). The derivation is again detailed in the Appendix. The initial source update is then
qJ^(0)=J^E(0)=m,np(tm)p(βn)J^𝒟m,n(0).
(26)
The subsequent iterations take the following steps:
  1. Compute the current source using the last update qJ^(k) by
    J^(k+1)=J^(k)+φqJ^(k),
    (27)
    where φ is again a small constant representing the step size.
  2. Calculate the gradient of 𝒟{I(x,y;tm,βn),Ī(x,y)} with respect to Ĵ( k +1), i.e., J^𝒟m,n(k+1), and
    J^E(k+1)=m,np(tm)p(βn)J^𝒟m,n(k+1).
    (28)
  3. Compute an update parameter θJ^(k+1), given by
    θJ^(k+1)=f,g[J^E(k+1)(J^E(k+1)J^E(k))]f,g[qJ^(k)(J^E(k+1)J^E(k))].
    (29)
  4. Compute qJ^(k+1) by
    qJ^(k+1)=J^E(k+1)+θJ^(k+1)qJ^(k).
    (30)

During the source update, symmetry is important to avoid pattern placement error [15

15. K. Lai, S. Bagheri, K. Tian, J. Tirapu-Azpiroz, S. Halle, G. McIntyre, D. Corliss, A. E. Rosenbluth, D. Melville, A. Wagner, M. Burkhardt, J. Hoffnagle, Y. Kim, G. Burr, M. Fakhry, E. Gallagher, T. Faure, M. Hibbs, D. Flagello, J. Zimmermann, B. Kneer, F. Rohmund, F. Hartung, C. Hennerkes, M. Maul, R. Kazinczi, A. Engelen, R. Carpaij, R. Groenendijk, J. Hageman, and C. Russ, “Experimental result and simulation analysis for the use of pixelated illumination from source mask optimization for 22nm logic lithography process,” in Optical Microlithography XXII, vol. 7274 of Proc. SPIE, p. 72740A (2009).

]. Usually a four-fold symmetry is imposed. To meet this specification, we force the gradient with respect to the source to be four-fold symmetric by averaging its four quarters’ components [30

30. N. Jia and E. Y. Lam, “Performance analysis of pixelated source-mask optimization for optical microlithography,” in IEEE International Conference on Electron Devices and Solid-State Circuits (2010).

].

6. Results

Here, we demonstrate the robust SMO algorithm in two distinct test patterns. First is a sparse pattern consisting of two rectangle shapes, as shown in Fig. 7(a). It is represented by a 151 × 151 matrix with a resolution of 10nm × 10nm per pixel. Second is a dense poly pattern shown in Fig. 7(c), represented by a 473 × 473 matrix with a finer grid of 4nm × 4nm per pixel. The imaging system parameters are set to be λ = 193nm and NA = 1.35.

Fig. 7 Test patterns and critical locations for process window calculation. Color lines mark critical locations of the two test patterns.

We compare the performance of SMO with that of mask optimization under a reference annular source. Certainly, with greater flexibility, the former should deliver better results than the latter; our objective here is to quantify how it is better, particular when it pertains to robustness. To do so, for each pattern we measure the feature size at a few critical locations, and then compute the process window according to the measured data. These locations include the main properties (width and length) of a feature, line-ends that are difficult to print, and the minimum feature size such as the space between two rectangles. They are marked by color lines (green if it is inside a feature, pink if outside) in Fig. 7(b) and 7(d) for patterns #1 and #2, respectively.

The size of the process window can be quantitatively measured by two parameters: exposure latitude (EL) and depth of focus (DOF). The former is the range of dose variation (% with respect to the nominal dose) where the feature size is within its tolerance, typically ±10% of its nominal size, at a certain defocus. The latter measures the largest acceptable defocus range (nm) under a fixed dose condition. Detailed descriptions of these quantities can be found in [1

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

, p. 61–69]. The common method of measuring the process window is to examine how large EL or DOF can be when the other quantity is fixed. In the following we compare the DOFs by fixing the EL [31

31. L. Pang, G. Xiao, V. Tolani, P. Hu, T. Cecil, T. Dam, K.-H. Baik, and B. Gleason, “Considering MEEF in inverse lithography technology (ILT) and source mask optimization (SMO),” in Photomask Technology, H. Kawahira and L. S. Zurbrick, eds., vol. 7122 of Proc. SPIE, p. 71221W (2008).

, 32

32. R. J. Socha, D. J. Van Den Broeke, S. D. Hsu, J. F. Chen, T. L. Laidig, N. P. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. E. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML),” in Photomask and Next–Generation Lithography Mask Technology XI, H. Tanabe, ed., vol. 5446 of Proc. SPIE, pp. 516–534 (2004). [PubMed]

]. Note that a larger DOF indicates a more robust performance.

6.1. A sparse pattern

In pattern #1, the two features are identical rectangles with height 110nm and width 60nm, separated by a 50nm space, which is the critical feature size for this pattern. We first assume that we have an annular source with its inner annulus σ inner = 0.7 and outer annulus σ outer = 0.9, as shown in Fig. 8(a). Note that we have applied a Gaussian blur on the annulus to mimic the reality, as a result of which the source does not take the same intensity inside the annulus.

Fig. 8 Simulation results of test pattern #1.

We compute the corresponding optimized mask, together with its simulated output at best focus and at a defocus of 85nm, given in (b) to (d). In the second row, using the robust SMO algorithm presented in this paper, we show the resulting source and mask patterns in (e) and (f). The outputs at best focus and defocus are given in (g) and (h). In terms of pattern error, if we compare the results of mask optimization versus SMO, they give effectively identical output at best focus, but the latter delivers a pattern closer to the design that the former when there is defocus. As for the critical dimension (the 50nm space in between), the former results in a 20nm error at the center, while the latter keeps the nominal size. In other words, SMO gives a more robust design.

The optimized source has a strong component at the horizontal dipole location and four weak poles in the vertical direction. Since the small features in the target design are mainly along the horizontal direction, such a source configuration is more suitable than the circular reference source. We also calculate some numerical results. With a 10% EL, mask optimization with the reference source gains 150nm DOF, while SMO enlarges this number to 170nm for this pattern.

6.2. A dense pattern

Fig. 9 Simulation results of test pattern #2.

7. Conclusion

In this paper, we propose a source-mask optimization method for process robustness enhancement. For this purpose, we introduce a cost function including not only the pattern fidelity term but also regularization terms to adjust the aerial image intensity distribution and its contrast. A statistical model is built by incorporating process variations explicitly into the optimization framework as random variables. Simulation results of sparse and dense patterns show conspicuous process window enlargement.

A. Appendix: Computing the gradients

Given the cost function 𝒟{I(x,y;tm,βn),Ī(x,y)}, we show here how to calculate its gradients with respect to any given mask pattern M and illumination source Ĵ′. With a fixed tm and βn, we denote them as ∇M𝒟 and ∇Ĵ 𝒟 respectively. As with Section 5, we omit the designation (x,y) for brevity.

From Eq. (6), we have
M𝒟=𝒯M+γ1TVM+γ2aerialM+γ3contrastM
(31)
J^𝒟=𝒯J^+γ1TVJ^+γ2aerialJ^+γ3contrastJ^.
(32)
In the following we will present the analytical form of each term in Eq. (31) and (32). The derivation will be omitted when it is straightforward.

A.1. Computing ∇M𝒟

First we calculate the gradient of 𝒯 with respect to the mask M. In Eq. (2), H(x,y; f,g) is replaced by the inverse Fourier transform of Eq. (17). For brevity we denote it as (f,g), and use ′(f,g) to represent (−x,y; f,g,β). Similar to calculating the gradient under coherent imaging system in [9

9. N. Jia and E. Y. Lam, “Machine learning for inverse lithography: using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12(4), 045601 (2010). [CrossRef]

], 𝒯/∂M is computed by
𝒯M=x,y(II¯)2M=f,gJ^(f,g)Re{[2α(II¯)I(1I)(M*H˜(f,g))]*H˜(f,g)}.
(33)
Next we compute the gradient of ℛTV with respect to M. As in Eq. (10), the first derivatives are approximated by finite differences, and performed by matrix multiplication. Given an image U and a shifting matrix D, the first derivative ∇xU is calculated by column operation UD, and ∇yU by row operation DU. So the second term TV/∂M is given by
TVM=x,yW(DIa)2+(IaD)2M=f,gJ^(f,g)Re{[(DT(W[(DIa)2+(IaD)2]12(DIa))+(W[(DIa)2+(IaD)2]12(IaD))DT)(M*H˜(f,g))]*H˜(f,g)}.
(34)
We compute the third term aerial/∂M as
aerialM=x,y(1W)(Ia2tI¯)2M=f,gJ^(f,g)Re{[2(1W)(Ia2tI¯)(M*H˜(f,g))]*H˜(f,g)}.
(35)
The fourth term contrast/∂M is calculated by
contrastM=x,y(1W)[|D(IaI¯)|+|(IaI¯)D|]M=f,gJ^(f,g)Re{[(DT((1W)D(IaI¯)[D(IaI¯)]2)+((1W)(IaI¯)D[(IaI¯)D]2)DT)(M*H˜(f,g))]*H˜(f,g)}.
(36)

A.2. Computing ∇Ĵ′𝒟

Given Eq. (32), we first compute the gradient of 𝒯 with respect to an arbitrary source point Ĵ′(f′,g′) as
𝒯J^(f,g)=x,y[I(x,y)=I¯(x,y)]2J^(f,g)=x,y2α(II¯)I(1I)|M*H˜(f,g)|2Iaf,gJ^(f,g).
(37)
The second term TV/∂Ĵ′(f,g′) in Eq. (32) is given by
TVJ^(f,g)=x,yW(DIa)2+(IaD)2J^(f,g)=x,yW.[(DIa)2+(IaD)2]12[(DIa)D[|M*H˜(f,g)|2Ia]f,gJ^(f,g)+(IaD)[|M*H˜(f.g)|2Ia]Df,gJ^(f,g)].
(38)
We compute the third term aerial/∂Ĵ′(f′,g′) as
aerialJ^(f,g)=x,y(1W)(Ia2tI¯)2J^(f,g)=x,y2(1W)(Ia2tI¯)|M*H˜(f,g)|2Iaf,gJ^(f,g).
(39)
The fourth term contrast/∂Ĵ′(f,g′) is calculated by
contrastJ^(f,g)=x,y(1W)[|D(IaI¯)|+|(IaI¯)D|]J^(f,g)=x,y(1W)[D(IaI¯)[D(IaI¯)]2D[|M*H˜(f,g)|2Ia]f,gJ^(f,g)+(IaI¯)D[(IaI¯)D]2[|M*H˜(f,g)|2Ia]Df,gJ^(f,g)].
(40)

Acknowledgments

This work was supported in part by the University Research Committee of the University of Hong Kong under Project 10400898, by the Research Grants Council of the Hong Kong Special Administrative Region, China, under Projects HKU 7134/08E, and by the UGC Areas of Excellence project Theory, Modeling, and Simulation of Emerging Electronics.

References and links

1.

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

2.

M. Rothschild, “A roadmap for optical lithography,” Opt. Photon. News 21(6), 26–31 (2010). [CrossRef]

3.

E. Y. Lam and A. K. Wong, “Computation lithography: virtual reality and virtual virtuality,” Opt. Express 17(15), 12259–12268 (2009). [CrossRef] [PubMed]

4.

A. Poonawala and P. Milanfar, “Mask design for optical microlithography — an inverse imaging problem,” IEEE Trans. Image Process. 16(3), 774–788 (2007). [CrossRef] [PubMed]

5.

L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): a natural solution for model-based SRAF at 45nm and 32nm,” in Photomask and Next-Generation Lithography Mask Technology XIV, vol. 6607 of Proc. SPIE, p. 660739 (2007).

6.

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

7.

X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express 15(23), 15066–15079 (2007). [CrossRef] [PubMed]

8.

S. H. Chan, A. K. Wong, and E. Y. Lam, “Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography,” Opt. Express 16(19), 14,46–14760 (2008). [CrossRef]

9.

N. Jia and E. Y. Lam, “Machine learning for inverse lithography: using stochastic gradient descent for robust photomask synthesis,” J. Opt. 12(4), 045601 (2010). [CrossRef]

10.

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]

11.

M. Burkhardt, A. Yen, C. Progler, and G. Wells, “Illuminator design for the printing of regular contact patterns,” Microelectron. Eng. 41–42, 91–96 (1998). [CrossRef]

12.

R. Socha, M. Eurlings, F. Nowak, and J. Finders, “Illumination optimization of periodic patterns for maximum process window,” Microelectron. Eng. 61–62, 57–64 (2002). [CrossRef]

13.

A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsys. 1(1), 13–30 (2002). [CrossRef]

14.

T. Fühner, A. Erdmann, and S. Seifert, “Direct optimization approach for lithographic process conditions,” J. Microlith. Microfab. Microsys. 6(3), 031006 (2007).

15.

K. Lai, S. Bagheri, K. Tian, J. Tirapu-Azpiroz, S. Halle, G. McIntyre, D. Corliss, A. E. Rosenbluth, D. Melville, A. Wagner, M. Burkhardt, J. Hoffnagle, Y. Kim, G. Burr, M. Fakhry, E. Gallagher, T. Faure, M. Hibbs, D. Flagello, J. Zimmermann, B. Kneer, F. Rohmund, F. Hartung, C. Hennerkes, M. Maul, R. Kazinczi, A. Engelen, R. Carpaij, R. Groenendijk, J. Hageman, and C. Russ, “Experimental result and simulation analysis for the use of pixelated illumination from source mask optimization for 22nm logic lithography process,” in Optical Microlithography XXII, vol. 7274 of Proc. SPIE, p. 72740A (2009).

16.

L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, K.-H. Baik, and B. Gleason, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” in Lithography Asia 2009, vol. 7520 of Proc. SPIE, p. 75200X (2009).

17.

T. Mülders, V. Domnenko, B. Küchler, T. Klimpel, H.-J. Stock, A. Poonawala, K. N. Taravade, and W. A. Stanton, “Simultaneous source-mask optimization: a numerical combining method,” in Photomask Technology 2010, vol. 7823 of Proc. SPIE, p. 78233X (2010).

18.

X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Opt. Express 17(7), 5783–5793 (2009). [CrossRef] [PubMed]

19.

J.-C. Yu and P. Yu, “Gradient-based fast source mask optimization (SMO),” in Optical Microlithography XXIV, vol. 7973 of Proc. SPIE, p. 797320 (2011).

20.

Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-based source and mask optimization in optical lithography,” IEEE Trans. Image Process. 99, 1–10 (2011).

21.

A. K. Wong, Optical Imaging in Projection Microlithography (SPIE, 2005). [CrossRef]

22.

Y. Granik, “Source optimization for image fidelity and throughput,” J. Microlith. Microfab. Microsys. 3(4), 509–522 (2004). [CrossRef]

23.

J.-C. Yu and P. Yu, “Impacts of cost functions on inverse lithography patterning,” Opt. Express 18(22), 23331–23342 (2010). [CrossRef] [PubMed]

24.

D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl. 19(6), 165–187 (2003). [CrossRef]

25.

M. K. Ng, H. Shen, E. Y. Lam, and L. Zhang, “A total variation regularization based super-resolution reconstruction algorithm for digital video,” EURASIP Journal on Advances in Signal Processing 2007, Article ID 74,585 (2007). [CrossRef]

26.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. (Prentice Hall, 2002).

27.

C. Mack, Fundamental Principles of Optical Lithography: The Science of Microfabrication (Wiley, 2007). [CrossRef]

28.

N. Jia, A. K. Wong, and E. Y. Lam, “Robust mask design with defocus variation using inverse synthesis,” in Lithography Asia, vol. 7140 of Proc. SPIE, p. 71401W (2008).

29.

J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).

30.

N. Jia and E. Y. Lam, “Performance analysis of pixelated source-mask optimization for optical microlithography,” in IEEE International Conference on Electron Devices and Solid-State Circuits (2010).

31.

L. Pang, G. Xiao, V. Tolani, P. Hu, T. Cecil, T. Dam, K.-H. Baik, and B. Gleason, “Considering MEEF in inverse lithography technology (ILT) and source mask optimization (SMO),” in Photomask Technology, H. Kawahira and L. S. Zurbrick, eds., vol. 7122 of Proc. SPIE, p. 71221W (2008).

32.

R. J. Socha, D. J. Van Den Broeke, S. D. Hsu, J. F. Chen, T. L. Laidig, N. P. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. E. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML),” in Photomask and Next–Generation Lithography Mask Technology XI, H. Tanabe, ed., vol. 5446 of Proc. SPIE, pp. 516–534 (2004). [PubMed]

OCIS Codes
(110.3960) Imaging systems : Microlithography
(110.5220) Imaging systems : Photolithography
(110.1758) Imaging systems : Computational imaging

ToC Category:
Imaging Systems

History
Original Manuscript: June 8, 2011
Revised Manuscript: August 26, 2011
Manuscript Accepted: August 30, 2011
Published: September 22, 2011

Citation
Ningning Jia and Edmund Y. Lam, "Pixelated source mask optimization for process robustness in optical lithography," Opt. Express 19, 19384-19398 (2011)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-19-20-19384


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References

  1. A. K. Wong, Resolution Enhancement Techniques in Optical Lithography (SPIE, 2001). [CrossRef]
  2. M. Rothschild, “A roadmap for optical lithography,” Opt. Photon. News21(6), 26–31 (2010). [CrossRef]
  3. E. Y. Lam and A. K. Wong, “Computation lithography: virtual reality and virtual virtuality,” Opt. Express17(15), 12259–12268 (2009). [CrossRef] [PubMed]
  4. A. Poonawala and P. Milanfar, “Mask design for optical microlithography — an inverse imaging problem,” IEEE Trans. Image Process.16(3), 774–788 (2007). [CrossRef] [PubMed]
  5. L. Pang, Y. Liu, and D. Abrams, “Inverse lithography technology (ILT): a natural solution for model-based SRAF at 45nm and 32nm,” in Photomask and Next-Generation Lithography Mask Technology XIV, vol. 6607 of Proc. SPIE, p. 660739 (2007).
  6. Y. Shen, N. Wong, and E. Y. Lam, “Level-set-based inverse lithography for photomask synthesis,” Opt. Express17(26), 23690–23701 (2009). [CrossRef]
  7. X. Ma and G. R. Arce, “Generalized inverse lithography methods for phase-shifting mask design,” Opt. Express15(23), 15066–15079 (2007). [CrossRef] [PubMed]
  8. S. H. Chan, A. K. Wong, and E. Y. Lam, “Initialization for robust inverse synthesis of phase-shifting masks in optical projection lithography,” Opt. Express16(19), 14,46–14760 (2008). [CrossRef]
  9. N. Jia and E. Y. Lam, “Machine learning for inverse lithography: using stochastic gradient descent for robust photomask synthesis,” J. Opt.12(4), 045601 (2010). [CrossRef]
  10. Y. Shen, N. Jia, N. Wong, and E. Y. Lam, “Robust level-set-based inverse lithography,” Opt. Express19(6), 5511–5521 (2011). [CrossRef] [PubMed]
  11. M. Burkhardt, A. Yen, C. Progler, and G. Wells, “Illuminator design for the printing of regular contact patterns,” Microelectron. Eng.41–42, 91–96 (1998). [CrossRef]
  12. R. Socha, M. Eurlings, F. Nowak, and J. Finders, “Illumination optimization of periodic patterns for maximum process window,” Microelectron. Eng.61–62, 57–64 (2002). [CrossRef]
  13. A. E. Rosenbluth, S. Bukofsky, C. Fonseca, M. Hibbs, K. Lai, R. N. Singh, and A. K. Wong, “Optimum mask and source patterns to print a given shape,” J. Microlith. Microfab. Microsys.1(1), 13–30 (2002). [CrossRef]
  14. T. Fühner, A. Erdmann, and S. Seifert, “Direct optimization approach for lithographic process conditions,” J. Microlith. Microfab. Microsys.6(3), 031006 (2007).
  15. K. Lai, S. Bagheri, K. Tian, J. Tirapu-Azpiroz, S. Halle, G. McIntyre, D. Corliss, A. E. Rosenbluth, D. Melville, A. Wagner, M. Burkhardt, J. Hoffnagle, Y. Kim, G. Burr, M. Fakhry, E. Gallagher, T. Faure, M. Hibbs, D. Flagello, J. Zimmermann, B. Kneer, F. Rohmund, F. Hartung, C. Hennerkes, M. Maul, R. Kazinczi, A. Engelen, R. Carpaij, R. Groenendijk, J. Hageman, and C. Russ, “Experimental result and simulation analysis for the use of pixelated illumination from source mask optimization for 22nm logic lithography process,” in Optical Microlithography XXII, vol. 7274 of Proc. SPIE, p. 72740A (2009).
  16. L. Pang, P. Hu, D. Peng, D. Chen, T. Cecil, L. He, G. Xiao, V. Tolani, T. Dam, K.-H. Baik, and B. Gleason, “Source mask optimization (SMO) at full chip scale using inverse lithography technology (ILT) based on level set methods,” in Lithography Asia 2009, vol. 7520 of Proc. SPIE, p. 75200X (2009).
  17. T. Mülders, V. Domnenko, B. Küchler, T. Klimpel, H.-J. Stock, A. Poonawala, K. N. Taravade, and W. A. Stanton, “Simultaneous source-mask optimization: a numerical combining method,” in Photomask Technology 2010, vol. 7823 of Proc. SPIE, p. 78233X (2010).
  18. X. Ma and G. R. Arce, “Pixel-based simultaneous source and mask optimization for resolution enhancement in optical lithography,” Opt. Express17(7), 5783–5793 (2009). [CrossRef] [PubMed]
  19. J.-C. Yu and P. Yu, “Gradient-based fast source mask optimization (SMO),” in Optical Microlithography XXIV, vol. 7973 of Proc. SPIE, p. 797320 (2011).
  20. Y. Peng, J. Zhang, Y. Wang, and Z. Yu, “Gradient-based source and mask optimization in optical lithography,” IEEE Trans. Image Process.99, 1–10 (2011).
  21. A. K. Wong, Optical Imaging in Projection Microlithography (SPIE, 2005). [CrossRef]
  22. Y. Granik, “Source optimization for image fidelity and throughput,” J. Microlith. Microfab. Microsys.3(4), 509–522 (2004). [CrossRef]
  23. J.-C. Yu and P. Yu, “Impacts of cost functions on inverse lithography patterning,” Opt. Express18(22), 23331–23342 (2010). [CrossRef] [PubMed]
  24. D. Strong and T. Chan, “Edge-preserving and scale-dependent properties of total variation regularization,” Inverse Probl.19(6), 165–187 (2003). [CrossRef]
  25. M. K. Ng, H. Shen, E. Y. Lam, and L. Zhang, “A total variation regularization based super-resolution reconstruction algorithm for digital video,” EURASIP Journal on Advances in Signal Processing2007, Article ID 74,585 (2007). [CrossRef]
  26. R. C. Gonzalez and R. E. Woods, Digital Image Processing, 2nd ed. (Prentice Hall, 2002).
  27. C. Mack, Fundamental Principles of Optical Lithography: The Science of Microfabrication (Wiley, 2007). [CrossRef]
  28. N. Jia, A. K. Wong, and E. Y. Lam, “Robust mask design with defocus variation using inverse synthesis,” in Lithography Asia, vol. 7140 of Proc. SPIE, p. 71401W (2008).
  29. J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed. (Springer, 2006).
  30. N. Jia and E. Y. Lam, “Performance analysis of pixelated source-mask optimization for optical microlithography,” in IEEE International Conference on Electron Devices and Solid-State Circuits (2010).
  31. L. Pang, G. Xiao, V. Tolani, P. Hu, T. Cecil, T. Dam, K.-H. Baik, and B. Gleason, “Considering MEEF in inverse lithography technology (ILT) and source mask optimization (SMO),” in Photomask Technology, H. Kawahira and L. S. Zurbrick, eds., vol. 7122 of Proc. SPIE, p. 71221W (2008).
  32. R. J. Socha, D. J. Van Den Broeke, S. D. Hsu, J. F. Chen, T. L. Laidig, N. P. Corcoran, U. Hollerbach, K. E. Wampler, X. Shi, and W. E. Conley, “Contact hole reticle optimization by using interference mapping lithography (IML),” in Photomask and Next–Generation Lithography Mask Technology XI, H. Tanabe, ed., vol. 5446 of Proc. SPIE, pp. 516–534 (2004). [PubMed]

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