Effective simulation for robust inverse lithography using convolutionvariation

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1 Effective simulatio for robust iverse lithography usig covolutiovariatio separatio method We Lv a, Shiyua Liu a, b, *, Xijiag Zhou a, ad Haiqig Wei a a State Key Laboratory of Digital Maufacturig Equipmet ad Techology, Huazhog Uiversity of Sciece ad Techology, Wuha 4374, Chia b Wuha atioal Laboratory for Optoelectroics, Huazhog Uiversity of Sciece ad Techology, Wuha 4374, Chia ABSTRACT As critical dimesio shriks, patter desity of itegrated circuits gets much deser ad lithographic process variatios become more proouced. I order to sythesize masks that are robust to process variatios, the average wafer performace with respect to process fluctuatios is optimized. This approach takes ito accout process variatios explicitly. However, it eeds to calculate a large umber of optical images uder differet process variatios durig its optimizig process ad thus sigificatly icreases the computatioal burde. Most recetly, we proposed a covolutiovariatio separatio (CVS) method for modelig of optical lithography, which separates process variables from the coordiate system ad hece eables fast computatio of optical images through a wide rage of process variatios. I this work, we detail the formulatio of robust iverse lithography makig use of the CVS method, ad further ivestigate the impacts of arbitrary statistical distributio of process variatios o the sythesized mask patters. Keywords: optical lithography, robust iverse lithography, process variatios, distributio of process variatios, covolutio-variatio separatio (CVS) method.. ITRODUCTIO Iverse lithography techology (ILT) is cosidered as a effective ad ecoomically viable way to meet various challeges i curret ad future techology odes [, ]. Its objective is to sythesize a iput mask to deliver a desired output patter o the wafer. Curretly, as critical dimesio (CD) shriks, patter desity of itegrated circuits gets much deser ad lithographic process variatios, such as les-wafer defocus ad exposure dose variatio, become more proouced. It therefore requires through process widow compesated approaches to sythesize masks for the high volume productio. To this ed, the average wafer performace with respect to process fluctuatios is optimized via miimizig the expectatio of the differece betwee the output patter ad the desired patter [3, 4]. This approach takes ito accout process variatios explicitly, ad is well uderstood ad easily accomplished. It is a commoly used robust ILT, ad oe of its drawbacks is that it eeds to calculate a large umber of optical images uder differet process variatios durig its optimizig process. The imagig formatio i optical lithography, especially adopted i ILT, is usually formulated i a biliear form [5] accordig to Hopkis imagig theory [6], i which the trasmissio cross coefficiet (TCC) matrix is itroduced as a combiatio of the illumiatio source ad pupil, ad the optical image ca be calculated by a equatio with a four-fold itegral. I order to simplify the itegral operatio, the cocept of sum of coheret systems [7-9], which approximates the partially coheret system by the superpositio of coheret systems, was itroduced ad it achieved a sigificat speedup by eigeaalyzig the TCC matrix ad eglectig small eigevalues. evertheless, the TCC kerels are usually obtaied uder the omial (best) process coditio, ad a imagig simulator may have to repeat the costly TCC decompositio ad mask-kerel covolutios whe process parameters vary. Most recetly, we proposed a covolutio-variatio separatio (CVS) method for modelig of optical lithography [], which separates process variables from the coordiate system ad hece eables fast computatio of optical images * Correspodig author: shyliu@mail.hust.edu.c; phoe: ; webpage: Optical Microlithography XXVII, edited by Kafai Lai, Adreas Erdma, Proc. of SPIE Vol. 95, 95C 4 SPIE CCC code: X/4/$8 doi:.7/.463 Proc. of SPIE Vol C- Dowloaded From: o 4//4 Terms of Use:

2 through a wide rage of process variatios. The fudametal ratioale behid the CVS method is that a physical quatity beig depedet o both a spatial coordiate ad other parameters ca be represeted by a sum of multiple series expasio terms, with each term cosistig of oe fuctio depedet oly upo the spatial coordiate ad aother fuctio depedet oly upo the other parameters. I this work, we provide detailed formulatio of robust iverse lithography makig use of the CVS method. I additio, we ivestigate the impacts of the statistical process variatios distributio o the sythesized mask patters i robust iverse lithography.. METHODLOGY. The lithography imagig model with process variatios I this work, we cosider two mai process variatios i optical lithography, i.e., les-wafer defocus ad exposure dose variatio. So the lithography imagig model is decomposed ito two parts, amely the defocused optical image formatio ad the resist developmet with a exposure dose variatio. The defocused optical image I(r; h) geerated by a partially coheret imagig system ca be expressed by a biliear trasform i the spatial domai I(; r h ) = D( r r, r r ; h) M( r ) M ( r )drd r, () where r is the two-dimesioal (D) spatial coordiate (x, y), h is the defocus, M(r) is the mask trasmittace fuctio, deotes complex cojugatio, ad D( r, r ; h) = J( r r ) H( r; h) H ( r ; h) () is called double-impulse respose (DIR) fuctio [5] whose Fourier trasform is the familiar TCC [6]. Here, J(r, r ) is the mutual itesity fuctio of the illumiatio source, ad H(r; h) is the defocused poit spread fuctio (PSF) [4] of the optical system as A H(; r h) = IFT H % ()exp f jπ h f, (3) λ where f is the ormalized pupil plae coordiate, j =, A is the umerical aperture, λ is the icidet light wavelegth, H % () f = circ( f ) is the pupil fuctio at the best focus without ay aberratios, ad IFT{ } is the iverse Fourier trasform. Itetioally, we expad the expoetial fuctio portio i Eq. (3) ito Taylor series with the trucatio order as A A h A j π h j h j! π λ λ! π = = = = λ exp f f f. (4) Substitutig Eq. (4) ito Eq. (3), we ca reformulate the defocused PSF H(r; h) as H(; r h) = h ϕ (), r (5) = A ϕ () r = IFT H % () f jπ f. (6)! λ It is oted that the process variable defocus h ad the coordiate r i defocused PSF H(r; h) are separated. Subsequetly, puttig Eq. (5) ito Eq. (), we obtai a variable-separated DIR fuctio with the trucatio order as D( r, r ; h) = h D ( r, r ), (7) = Proc. of SPIE Vol C- Dowloaded From: o 4//4 Terms of Use:

3 D( r, r) = J( r r) ϕi( r) ϕ i( r ). (8) i= It is oted that D (r, r ) is Hermitia ad ca be sigular-value decomposed as K = k k k k = D ( r, r ) μ φ ( r ) φ ( r ), (9) where ϕ k (r) is the k-th kerels with K kerels i total, ad μ k is its correspodig eigevalue. Substitutig Eq. (9) ito Eq. (7) ad the ito Eq. (), the defocused optical image I(r; h) is rephrased as I(; r h) = h I (), r () = K μk φk k = I () r = () r M(). r () Here, deotes the D covolutio. It is oted that the basis image I (r) is idepedet of defocus which thus may be calculated ad stored i advace. So the defocused optical images ca be calculated efficietly ad accurately by summig pre-computed ad stored partial images I (r) weighted by the coefficiets h. The sigificace of Eq. () lies i that the fixed mask-kerel covolutios are fully separated from the process variable, defocus, ad ca be precomputed, so that optical images uder a large rage of defocus ca be quickly calculated. This is the fudametal ratioale behid the CVS method. The defocused optical image I(r; h) goes through the resist developmet to form the prited image o the wafer. The resist effect ca be approximated by a costat threshold resist model usig the followig Sigmoid fuctio, sig [ I( r; h); t] =. () + exp{ ai [ ( r; h) t]} where a is the steepess of the Sigmoid fuctio ad t is the threshold level of the resist. Specifically, exposure dose variatio ca be accouted for by varyig the threshold t. I this work, we use Z(r) to deote the output patter o the wafer of the iput mask M(r) uder the defocus h ad dose variatio t. Combiig Eqs. () ad (), we ca formulate the lithography imagig equatio as Z(; r ht,) = sig [ I(; r h); t] = sig hi (); r t. (3) =. Formulatio of robust iverse lithography We formulate the robust iverse lithography problem as [ ] * M G M κr M M = arg mi ( )+ ( ), (4) P Q δx δy * GM ( ) = α( hp) β( tq) Z( r; hp, tq) Z ( r ). (5) p q L Here, R(M) is regularizatio term with the correspodig weight κ to promote mask maufacturability, G(M) is called statistical edge distace error (EDE) [, ] to evaluate the differece betwee the output patter of iput mask M(r) ad the desired patter Z * (r) o the wafer, δ x ad δ y are the legths of the discretized mask grid alog the x ad y directios, respectively, L is the perimeter of the desired patter cotour, α( ) is the desity of a prescribed distributio of defocus, β( ) is the desity of a prescribed distributio of exposure dose, Z(r; h p, t q ) represets the output patter of the defocused image I(r; h p ) o the wafer uder the exposure dose level t q, ad is the L orm. Equatio (4) may be solved by a optimizatio algorithm [8], i which the forward lithography imagig simulatio is iterated may times ad each iteratio requires calculatig a total of P defocused images. Proc. of SPIE Vol C-3 Dowloaded From: o 4//4 Terms of Use:

4 3. SIMULATIOS Simulatios were performed o a partially coheret imagig system with a quasar source illumiatio (σ out /σ i /degree =.9/.6/45º). The wavelegth i the simulatios was set at 93 m, ad the umerical aperture (A) was.35. The resist effect was approximated by a Sigmoid fuctio with a = ad t =.4. All the simulatios were carried out with ihouse Matlab codes o a HPZ8 Workstatio. 3. Computatioal complexity aalysis of covolutio-variatio separatio method From the defiitio of robust iverse lithography problem i Sec.., umerous optical images uder differet defocus are required to be calculated ad repeated i each iteratio. I covetioal approaches, to calculate P discrete samplig poits of defocus, they eed to geerate ad use P sets of process models. Let T deote the computatioal complexity for simulatig oe samplig poit of defocus, ad let T(P) be the total computatioal complexity for simulatig P samplig poits of defocus; the the followig formula holds true for all covetioal approaches: T( P) = T P. (6) I the preseted CVS method, let T deote the computatioal complexity of calculatig all basis images I (r) i Eq. (), ad let δt deote the extra complexity for summig up the series i Eq. (). It is usually the case that δt T ad T T ( + ), where is the trucatio order of Eqs. (5) ad (7). Therefore, the computatioal complexity for simulatig oe defocus samplig poit is T CVS () = T + δt, whereas the cost for simulatig P samplig poits of defocus would be T ( P) = T + δt P. (7) CVS As demostrated i Fig., the advatage of usig CVS method become apparet as the defocus umber P > ( + ), ad gets larger. - 7» ti 7- Tcvs()+(P- ) X 87' Defocus Samplig Poits P Figure. Computatioal complexity of the covetioal approaches ad the CVS method for simulatig P defocus samplig poits. T deotes the computatioal complexity of the covetioal approaches for simulatig oe defocus samplig poit. Here, we set the trucatio order = Results of robust iverse lithography Figure shows a desired output patter o the wafer with CD of 3 m. The patter has a size of 4 4 pixels with a pixel size of m. The sythesized iput mask patters uder differet defocus ad exposure dose distributios are show i Fig. 3. As expected, differet distributios result i differet mask patters. Moreover, it is particularly iterestig to observe that the sub-resolutio assist features (SRAFs) i Figs. 3(a) ad 3(e) are dramatically differet. That meas differet SRAFs patters correspod to differet process distributios. [, ] Figures 3(d) ad 3(h) depict the exposure-defocus (E-D) trees of the sythesized mask patters by settig the EDE Proc. of SPIE Vol C-4 Dowloaded From: o 4//4 Terms of Use:

5 4 Y Positio (m) X Positio (m) Figure. Desired patter o the wafer. to withi ±% of the CD target. To clearly show the process distributio, we draw the E-D widow alog with its probability distributio. It demostrates that the proposed robust iverse lithography is sesitive to the differece of the process distributio, ad it has the capacity of sythesizig a mask patter to match the distributio. 4. COCLUSIOS I this work, we formulate the robust iverse lithography i a statistical maer, ad employ a CVS method to calculate umerous defocused optical images. Beefitig greatly from the CVS method, we ca take umerous defocus samplig poits to exactly model its real distributio without proportioally icreasig the computatioal complexity. Compared to covetioal approaches, the superior scalability of the CVS method is apparet whe the umber of defocus gets larger. I additio, we observe that differet process distributios result i mask patters with dramatically differet SRAFs i robust iverse lithography. Therefore, it has practical sigificace to cosider the real process distributio i Y Positio (m) 4 (a) X Positio (m) ormalized Defocus Distributio.5 (c) Defocus (m) Y Positio (m) 4 (e) X Positio (m) ormalized Defocus Distributio.5 (g) Defocus (m) ormalized Exposure.4..8 (b) (c).6.5 ormalized Exposure Distributio ormalized Exposure.4..8 (d) Defocus (m) ormalized Exposure.4..8 (f).6.5 ormalized Exposure Distributio ormalized Exposure.4..8 (h) Defocus (m) Figure 3. (a) The sythesized mask patter uder the exposure distributio (b) ad defocus distributio (c), (d) is its E-D widow. (e) The sythesized mask patter uder the exposure distributio (f) ad defocus distributio (g), (h) is its E-D widow. Proc. of SPIE Vol C-5 Dowloaded From: o 4//4 Terms of Use:

6 robust iverse lithography to raise yield. ACKOWLEDGMET This work was fuded by the atioal atural Sciece Foudatio of Chia (Grat o. 933, 559, 5), the Specialized Research Fud for the Doctoral Program of Higher Educatio of Chia (Grat o. 49), the atioal Sciece ad Techology Major Project of Chia (Grat o. ZX7), ad the atioal Istrumet Developmet Specific Project of Chia (Grat o. YQ6). REFERECES [] A. K. Wog, Resolutio Ehacemet Techiques i Optical Lithography (SPIE Press, ). [] D. O. S. Melville, A. E. Rosebluth, A. Waechter, M. Millstoe, J. Tirapu-Azpiroz, K. Tia, K. Lai, T. Ioue, M. Sakamoto, K. Adam, ad A. Tritchkov, Computatioal lithography: Exhaustig the resolutio limits of 93-m projectio lithography systems, J. Vac. Sci. Techol. B 9, 6FH4 (). [3]. Jia ad E. Y. Lam, Machie learig for iverse lithography: usig stochastic gradiet descet for robust photomask sythesis, J. Opt., 456 (). [4] Y. She,. Jia,. Wog, ad E. Y. Lam, Robust level-set-based iverse lithography, Opt. Express 9, (). [5] B. E. A. Saleh, Optical biliear trasformatios: Geeral properties, Opt. Acta 6, (979). [6] H. H. Hopkis, O the diffractio theory of optical images, Proc. R. Soc. A 7, (953). [7]. B. Cobb, Fast optical ad process proximity correctio algorithms for itegrated circuit maufacturig, Ph.D. Dissertatio (Uiversity of Califoria at Berkeley, 998). [8] P. Gog, S. Y. Liu, W. Lv, ad X. J. Zhou, Fast aerial image simulatios for partially coheret systems by trasmissio cross coefficiet decompositio with aalytical kerels, J. Vac. Sci. Techol. B 3, 6FG3 (). [9] W. Lv, S. Y. Liu, Q. Xia, X. F. Wu, Y. She, ad E. Y. Lam, Level-set-based iverse lithography for mask sythesis usig the cojugate gradiet ad a optimal time step, J. Vac. Sci. Techol. B 3, 465 (3). [] S. Y. Liu, X. J. Zhou, W. Lv, S. Xu, ad H. Q. Wei, Covolutio-variatio separatio method for efficiet modelig of optical lithography, Opt. Letter 38, 68-7 (3). [] W. Lv, Q. Xia, ad S. Y. Liu, Pixel-based iverse lithography usig a mask filterig techique, Proc. SPIE 8683, (3). [] W. Lv, Q. Xia, ad S. Y. Liu, Mask-filterig-based iverse lithography, J. Micro/aolith. MEMS MOEMS, 433 (3). Proc. of SPIE Vol C-6 Dowloaded From: o 4//4 Terms of Use: