Triple/Quadruple Patterning Layout Decomposition via Novel Linear Programming and Iterative Rounding

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1 Triple/Qudruple Ptterning Lyout Deomposition vi Novel Liner Progrmming nd Itertive Rounding Yio Lin, Xioqing Xu, Bei Yu, Ross Bldik nd Dvid Z. Pn ECE Dept., University of Texs t Austin, Austin, TX USA CSE Dept., The Chinese University of Hong Kong, NT, Hong Kong Emil : {yiolin, xioqingxu}@er.utexs.edu, {ldik, dpn}@ee.utexs.edu Emil : yu@se.uhk.edu.hk ABSTRACT As feture size of the semiondutor tehnology sles down to 0nm nd eyond, multiple ptterning lithogrphy (MPL) hs eome one of the most prtil ndidtes for lithogrphy, long with other emerging tehnologies suh s extreme ultrviolet lithogrphy (EUVL), e-em lithogrphy (EBL) nd direted self ssemly (DSA). Due to the dely of EUVL nd EBL, triple nd even qudruple ptterning re onsidered to e used for lower metl nd ontt lyers with tight pithes. In the proess of MPL, lyout deomposition is the key design stge, where lyout is split into vrious prts nd eh prt is mnuftured through seprte msk. For metl lyers, stithing my e llowed to resolve onflits, while it is foridden for ontt nd vi lyers. In this pper, we fous on the pplition of lyout deomposition where stithing is not llowed suh s for ontt nd vi lyers. We propose liner progrmming nd itertive rounding (LPIR) solving tehnique to redue the numer of non-integers in the LP relxtion prolem. Experimentl results show tht the proposed lgorithms n provide high qulity deomposition solutions effiiently while introduing s few onflits s possile.. INTRODUCTION Triple ptterning lithogrphy (TPL) nd qudruple ptterning lithogrphy (QPL) re promising tehniques to enhne lithogrphy resolution when the feture size of semiondutor tehnology sles down to 0nm nd eyond. While it is true tht there re other tehniques suh s extreme ultrviolet lithogrphy (EUVL), e-em lithogrphy (EBL) nd direted self ssemly (DSA), the merits nd demerits of these tehniques result in vrious hoies ording to different pplitions. A typil proess of TPL onsists of litho-eth-litho-eth-litho-eth (LELELE) steps whih need three exposure/ething steps. In order to mnufture with TPL tehnology, it is neessry to split single lyer into three msks so tht the fetures on eh msk re fr wy enough to meet the resolution requirement of the optil system. This proess is lled lyout deomposition. Similrly for QPL, one lyer is deomposed into four msks for mnufturing. The ondition for splitting fetures is usully relted to the distne etween them. For ny feture pir, if two fetures re very lose to eh other, they should e split into different msks; otherwise, onflit is introdued nd it is not possile to mnufture them. In lyout deomposition, eh feture n e viewed s vertex in grph. If two fetures re too lose to e mnuftured in the sme msk, onflit edge is introdued to onnet orresponding verties. Verties tht shre the sme onflit edge must e ssigned to different msks, or in other words, leled with different olors. Then the lyout deomposition prolem n e formulted into grph oloring prolem s shown in Fig.. Tht is, TPL n e formulted to -oloring nd QPL n e formulted to -oloring. The minimum distne to insert onflit edge etween two fetures is defined s oloring distne. Grph oloring is known s n NP-omplete prolem for olor numer lrger thn two. A lyout my inlude su-grphs suh s -lique (K) struture tht is not -olorle. In Fig. (), verties,, nd d form K struture, so t lest four msks re needed. The min ojetive of lyout deomposition is to minimize the numer of onflits. For metl lyers, stithing my e llowed to resolve onflits; i.e. feture n e split into two prts with different olors. But stithes re foridden for ontt nd vi lyers. Even for metl lyers, Design-Proess-Tehnology Co-optimiztion for Mnufturility X, edited y Luigi Cpodiei nd Json P. Cin, Pro. of SPIE Vol. 978, 9780M 06 SPIE CCC ode: X/6/$8 doi: 0.7/.868 Pro. of SPIE Vol M-

2 d d e f e () () () (d) st msk nd msk rd msk th msk Figure. Multiple ptterning lyout deomposition: () metl lyer onflit grph, () orresponding deomposition, () ontt lyer onflit grph, nd (d) orresponding deomposition. some fs do not llow stith insertion s it degrdes the mnufturing yield. While lyout deomposition is still different from trditionl grph oloring, the NP-ompleteness still holds. Vrious lgorithms hve een proposed for the MPL lyout deomposition prolem, inluding integer liner progrmming (ILP), semidefinite progrmming (SDP) nd other heuristi pprohes. 9 Although ILP n solve the prolem optimlly, it suffers from exponentil runtime. SDP nd other heuristi pprohes re introdued to speedup the deomposition proess with trde-offs etween runtime nd solution qulity. In order to improve the feture uniformity on eh msk, density lne is lso introdued s seondry optimiztion trget during deomposition. 0, For row sed lyout strutures, even fster pprohes hve een proposed with the gurntee of optimlity. For single stndrd ell, Yu et l. proposed serh sed methods to enumerte ll possile oloring solutions. 6 Besides, Yu et l. developed n inrementl frmework to elerte onventionl lyout deomposition flow. 7 Reently, Guo et l. studied the lithogrphy impt of different deomposition solutions sed on lithogrphy models. 8 In this pper, we fous on the pplition of lyout deomposition where stithing is not llowed suh s for ontt nd vi lyers. Given lyout with either retngles or polygons where stith insertion is foridden, our gol is to provide high qulity deomposition results effiiently while introduing s few onflits s possile. Our mjor ontriutions re listed s follows.. We develop novel liner progrmming (LP) nd itertive rounding sheme to solve lyout deomposition effiiently with high performne.. We propose n odd yle sed tehnique to prune ntive non-integer solutions in the fesile set of LP, whih n effetively redue non-integer solutions in LP.. Experimentl results show tht our lgorithm gets.8x speedup for TPL nd.6x speedup for QPL ompred with the stte-of-the-rt lyout deomposer with less % degrdtion in finl onflit numers. The rest of the pper is orgnized s follows. In Setion, we disuss the preliminry nd prolem formultion. In Setion, we explin the lgorithms suh s integer liner progrmming formultion, liner progrmming relxtion nd our itertive rounding sheme. The experimentl result is presented in Setion nd we onlude this pper in Setion.. Preliminries. PRELIMINARIES AND PROBLEM FORMULATION Given lyout with retngulr or polygon shpes, we onstrut deomposition grph where eh feture is denoted y vertex in the grph. An edge is inserted if two orresponding fetures hve distne smller Pro. of SPIE Vol M-

3 g e f d h g e f d h g e f d h () () () st msk nd msk rd msk Figure. Exmple of () onflit nd () unlned deomposition () lned deomposition in triple ptterning lyout deomposition. thn minimum oloring distne. In the deomposition grph, if two verties onneted y onflit edge re ssigned to sme olor, onflit is generted. The min trget during lyout deomposition is to minimize the numer of onflits. Fig. () shows n exmple of onflit where node nd d re ssigned to the sme msk, whih results in onflit highlighted y red line. In ddition, the density of fetures t eh msk should lso e onsidered to redue lithogrphy hotspots nd improve CD uniformity. Fig. () nd Fig. () ompre two oloring solutions. The former one is unlned euse most nodes re ssigned to the first nd seond msk nd the third msk hs only one node. The ltter hs more uniform distriution of nodes on eh msk. Therefore, during the deomposition proess, it is neessry to mintin density uniformity in eh msk s well s the density rtios long different olors.. Prolem Formultion Given lyout with fetures of retngulr or polygon shpes, deomposition grph is onstruted. The lyout deomposition prolem ssigns fetures to different olors suh tht the totl numer of onflits is minimized nd menwhile the oloring density t eh msk is lned. The oloring density lning is defined s the differene etween the most frequently used olor nd the lest frequently used olor.. ALGORITHMS In this setion, we will go over the overll flow of our lgorithm. Then we introdue our ILP formultion of lyout deomposition whih is slightly different from onventionl deomposition nd explin its orresponding LP relxtion with the itertive rounding sheme in detils.. Overll Flow The overll flow of our frmework is shown in the left prt of Fig.. It first onstruts onflit grph from the lyout nd performs grph simplifition whih will generte set of simplified omponents. Eh simplified omponent will e fed to kernel oloring lgorithm sed on liner progrmming nd itertive rounding (LPIR). As some verties re removed from the originl grph to generte simplified omponents during grph simplifition, it is neessry to reover them during post olor ssignment. In the end, the finl oloring results re produed. The detiled flow for LPIR will e explined in Setion... ILP Formultion Given onflit grph G(V, E), it is neessry to introdue two inry vriles for eh node to represent three/four olors in the TPL/QPL lyout deomposition prolem. The ILP formultion is shown in Formultion (). For eh onflit edge in the edge set E, the sitution of identil olors on oth verties is foridden y Constrints () to (f); e.g. Constrint () requires tht x i, x i, x j nd x j nnot e zero t the sme time, whih would otherwise result in onflit t edge e ij, nd so forth for the other onstrints. Constrint () is only used to eliminte the fourth olor in TPL lyout deomposition, so it is not needed in the QPL deomposition prolem. Different from the ILP formultion in, dditionl stith edge vriles re not introdued sine stith insertion is not llowed. There re no dditionl onflit vriles either euse we hndle the Pro. of SPIE Vol M-

4 Input Lyout Simplified Component ILP with ojetive = 0 Construt Conflit Grph Grph Simplifition Generte Simplified Components Kernel Coloring - LPIR LP Relxtion Add dditionl onstrints nd ojetive ising Solving LP Binding Constrint Anlysis Vertex Color Reovery Finl Coloring Results Detet non-integer its Non-integer redued? N Colored Component Figure. Overll flow for our oloring frmework. Y minimiztion of onflits in the LP relxtion. Insted of minimizing the totl ost from onflits, the trget of our ILP formultion is to seek fesile olor ssignment to the vriles while optimizing the hngele ojetive funtion.. LPIR Flow min oj () s.t. x i + x i, () x i + x i + x j + x j, e ij E, () x i + x i + x j + x j, e ij E, (d) x i + x i + x j + x j, e ij E, (e) x i + x i + x j + x j, e ij E, (f) x i = x i, i V, (g) x i = x i, i V, (h) x i, x i {0, }, i V. (i) Although the ILP formultion is le to find optiml olor ssignment when there exists onflit free solution, it hs exponentil runtime nd is not ple of minimizing totl onflits if there is t lest one onflit in the optiml solution due to the infesiility. With LP relxtion of the prolem, it is possile to void the infesiility issue nd find solution with few onflits. The relxtion from ILP to LP will result in non-integer solutions, so it is ritil to find proper rounding sheme for qulity gurntee. The LPIR flow for our oloring frmework is demonstrted in the right prt of Fig.. The frmework strts with the LP with n ojetive of zero. To del with non-integers in the solution from LP, we identify some ntive non-integer solutions in the fesile set resulted from odd yles. Then dditionl onstrints re introdued to prune these ntive non-integer solutions. During eh itertion, ojetive funtion is hnged from the originl LP formultion to push non-integer to integers. In prtiulr, these dditionl onstrints nd ojetive funtion hnge will not rek the fesiility of possile oloring ssignment. We ontinue the LPIR itertions until no further improvement. Pro. of SPIE Vol M-

5 . Liner Progrmming nd Itertive Rounding The ILP formultion in () is relxed to LP y repling Constrint (i) with 0 x i nd 0 x i. The ritil issue from LP relxtion is tht it my introdue mny non-integers in the solution, euse typil LP lgorithm like simplex wlks through the oundries of the polyhedron spe of fesile set nd serh for est solutions. It is very likely tht the solutions t the oundries of the polyhedron spe ontin non-integers. For instne, with the onstrint (g), trivil fesile solution is x i = 0., x i = 0., i V. As shown in Fig., the fesile spe for the LP is denoted s the light green region. The dsh red line denotes the ojetive funtion with optiml vlue. The grids onsisting of dshed lk lines re possile solutions with integer its. We n see tht the optiml solution from LP is (0., 0.). Effiient tehniques re needed to push the LP solution to those lue dots in the fesile region with integer solutions while eing lose to the optiml point. (0.,0.) Figure. The polyhedron for fesile liner progrmming solutions... Odd Cyle Constrints It is known tht n odd yle in grph needs t lest three olors. For the odd yle exmple shown in Fig., if the first its of the verties re equl, e.g. x i = x j = x k = x l = x m = 0, it is not possile to otin solution without onflits y djusting x i, x j, x k, x l, x m. The LP relxtion will produe ll 0. solutions for x i, x j, x k, x l, x m to stisfy Constrints () to (f). These solutions re ntive non-integer solutions in the fesile set whih should e pruned. To void suh kind of situtions, the first its of the verties should not e equl. Just like Fig. () shows, s long s x i, x j, x k, x l, x m re not ll equl, it is very esy to find oloring solution without ny onflit. We n void the sitution of equlity of the first its y dding following onstrints, whih forids the ses of ll zeros nd ll ones: x i + x j + x k + x l + x m, ( x i ) + ( x j ) + ( x k ) + ( x l ) + ( x m ). () It must e noted tht lthough Constrints () nd () disllow the first its to e identil, it helps to resolve the potentil onflits nd non-integers for the seond its. Similr tehnique n e pplied to the seond its. For generl odd yle C, we hve the following onstrints. x i, i C () () ( x i ), () i C x i, i C () ( x i ), (d) i C Pro. of SPIE Vol M-

6 where Constrint () nd () forid the possiility of the first its to e ll zeros or ll ones; Constrint () nd (d) forid the sme thing to the seond its. These onstrints prune invlid solutions without losing the fesiility of the LP prolem. (0,0) i (0,0) m (0,) l (0,0) (0,) m (0,0) l j k (0,) (0,0) () j k (,) (0,) Figure. One possile odd yle in the onflit grph () oloring onflit from due to identil first it vlues of nodes () resolved onflit... Ojetive Funtion Bising To eliminte the non-integer results in n LP solution, one heuristi is to push the orresponding vriles to 0 or y djusting the ojetive funtion. For exmple, if x i turns out to e 0.6, it indites tht x i hs the tendeny to ; hene, we dd ( x i ) to the ojetive funtion so tht x i tends to e pushed to during the next itertion. It n e generlized to following rules: () oj = { oj + ( xi ), if x i > 0., oj + (x i ), if x i < 0.. ().. Binding Constrints Anlysis One drwk for the ojetive funtion ising tehnique nnot hndle non-integers like 0.. Therefore, we propose method to round those vertex with oloring solution (x i, x i ) = (0., 0.) pirwisely y nlyzing the relted inding onstrints. For onstrint in LP, if the inequlity turns out to e equlity ording to the LP solution, we ll it inding nd this onstrint is lled inding onstrint. Fig. 6 shows n exmple of onstrints for vertex whose solution is (0., 0.). Let S i e the set of onstrints only relted to x i, S i e the set of onstrints only relted to x i nd the set of shred onstrints is denoted y S i. For simpliity during illustrtion, ssume eh onstrint is formtted in suh wy tht ll vriles re on the left side of the inequlity opertor nd only onstnts re on the right side. At the sme time, the oeffiients for x i should remin positive. If ll onstrints in S i shre the sme kind of opertors, suh s, then these onstrints will not e violted if x i is pushed from 0. to 0; similrly, if ll opertors re, then x i n e pushed from 0. to. The ondition lso holds for x i y heking ll onstrints in S i. With the nlysis ove, we n generte ndidte rounded solution for (x i, x i ). The solution will not e epted unless the rounded solution lso stisfies ll onstrints in S i. For the exmple in Fig. 6, we n generte ndidte rounded solution (0, ) nd then hek if onstrints in S i re stisfied s well. If true, (x i, x i ) re rounded to (0, ). This tehnique will not ffet the fesiility of the LP... Anhoring Highest Degree Vertex During olor ssignment, one oloring solution n tully rotte to generte other oloring solutions. To redue the solution spe, it will not do hrm to the optimlity if one vertex of the grph is pre-olored. The degree of verties in the grph vries from vertex to vertex, nd the seletion of pre-olored vertex leds to different oloring results. As high-degree vertex hs lrge set of neighors, the solution spe will e lrgely redued if its olor is pre-determined. Therefore, when onstruting the mthemti formultion, we nhor the olor of the vertex with highest degree. Pro. of SPIE Vol M-6

7 S i S i S i x i =0. x i =0. i... + x i +... pple... + x i +... pple... + x i +... pple... + x i +... pple... + x i x i x i x i x i + x i +... pple x i x i x i + x i +... Figure 6. An exmple of inding onstrints nlysis.. Grph Simplifition Sine the onflit grph onstruted from initil lyout is very lrge, we perform grph simplifition to redue the prolem sizes. As the onflit grph is usully not strongly onneted, independent omponents re extrted. Then for eh independent omponent, two more steps re further dopted to simplify it: itertive vertex removl (IVR) nd ionneted omponent extrtion (BCE). It shll e noted tht if simplifition tehnique modifies the originl grph, it needs orresponding pproh to reover the olors verties t proper time. Sine the two simplifition methods we dopt will either remove verties from the grph or divide grphs into omponents, we will lso explin its reovery pproh. In our experiment, IVR is performed efore BCE, so during the reovery proess, the reovery lgorithm for IVR is exeuted fter tht of BCE... Itertive Vertex Removl nd Density Awre Reovery In the grph oloring prolem, if the degree of vertex is smller thn the numer of olors n, we n lwys remove it temporrily nd ssign olor lter, euse its neighoring verties in the grph will tke t most n olors. There will lwys e ville olors left for this vertex. When vertex is removed, some other verties my turn out to e removle, so this proedure n e performed itertively, whih is shown in Fig. 7. In eh itertion, the removed verties re pushed into stk whih is used to mintin the proper order during the vertex olor reovery. Figure 7. An exmple of itertive vertex removl in TPL. When ssigning olor to removed verties during vertex olor reovery, it is neessry to keep the popping order of the stk. For the exmple of Fig. 7, we will ssign olor in order of,,,,. During the reovery proess, vertex my hve multiple ville olors; e.g. if vertex is ssigned to olor, then vertex will hve two ndidte olors, i.e. nd, in TPL. The hoies of olors in the reovery stge ply n importnt role in olor density lning. Therefore, we design n lgorithm to onsider olor density during reovery, shown in Alg.. The si ide is to ompute the priority for eh ndidte olor of vertex v sed on the distne d,v with losest vertex of the sme olor. The lrger d,v is, the higher priority the olor hs. Eventully, the ndidte olor with lrgest d,v will e hosen. The min loop from line to line 7 in Alg. itertes though the verties with the order defined y the stk. For eh ndidte olor of vertex, the distne d,v is omputed in line 7. The est olor is omputed from line 0 to line... Bionneted Component Extrtion nd Color Reovery In grph theory, ionneted omponent is defined s mximl ionneted sugrph. In TPL nd QPL, we n divide grph into ionneted omponents so tht eh omponent n e solved independently. Fig. 8 shows n exmple of ionneted omponent extrtion. In the figure, the grph is split into three omponents, Pro. of SPIE Vol M-7

8 Algorithm Density Awre IVR Vertex Color Reovery Require: A stk S ontining unolored verties. Ensure: Assign olors with lned density. : Define ville olor set C for TPL/QPL nd ville olor set C v for vertex v; : Define distne d,v for vertex v s the distne with the losest vertex with olor ; : Define est olor v for vertex v nd d v s the orresponding distne; : while S do : v S.pop(); 6: for C do 7: Compute d,v ; 8: end for 9: Compute C v for vertex v; 0: v, d v ; : for C v do : if d,v > d v then : d v d,v, v ; : end if : end for 6: Assign olor v to vertex v; 7: end while, nd. Vertex is shred y omponent nd ; vertex is shred y omponent nd. These omponents n e olored independently nd reunited lter st msk nd msk rd msk Figure 8. An exmple of ionneted omponent extrtion nd olor reovery. Sine eh omponent is proessed independently, it is likely to result in the ondition tht the olors of shred verties in different omponents re different, like vertex in omponent nd. Therefore, olor rottion is neessry during the proess of reovery. The olor ssignments of omponent should e rotted in suh wy tht vertex in omponent hs identil olor to vertex in omponent. As the oloring solution of omponent is hnged, vertex nd vertex no longer remin the sme olor, so omponent should follow omponent nd rotte its oloring solution s well. For more generl proedure of olor rottion during the reovery, we n onstrut n undireted yli grph (UAG) in whih eh ionneted omponent is vertex nd two verties re onneted if the orresponding omponents shre vertex in the originl grph. The olor rottion for ionneted omponents n e solved y pplying depth first serh (DFS) to the UAG. Pro. of SPIE Vol M-8

9 . EXPERIMENTAL RESULTS Our lgorithms were implemented in C++ nd tested on n 8-Core.0 GHz Linux server with GB RAM. The sme enhmrks from Ref. re used. Guroi 9 is used s the ILP nd LP solver. The minimum oloring distne for TPL is set to 0nm nd tht for QPL is set to 60nm. We ompre our lgorithm with the stte-of-the-rt ILP nd SDP lgorithms from Ref. in Tle. Tle. Comprison on Runtime nd Performne Ciruit ILP for TPL SDP for TPL LPIR for TPL ILP for QPL 9 SDP for QPL 9 LPIR for QPL n# CPU(s) n# CPU(s) n# CPU(s) n# CPU(s) n# CPU(s) n# CPU(s) C C C C C C C C C C S S S N/A > S N/A > S N/A > vg N/A > In Tle, our lgorithm is shown s LPIR. Conflit numer is denoted y n# nd runtime is denoted y CPU in seonds. As stith insertion is not llowed, the stith numer is lwys zero nd thus not shown in the tle. We n see tht the LPIR lgorithm is le to hieve minimum onflits for ll enhmrks in TPL with 6x speedup to ILP nd.8x speedup to SDP. In QPL, the speedup from LPIR is even more impressive, whih hieves 60x speedup to ILP nd.8x speedup to SDP, while it only produes round % more onflits thn SDP. The smll degrdtion in onflits is resonle euse designers need to mnully fix onflits y modifying the lyout nywy. We lso study the density lning of the experimentl results, whih is hndled during the vertex reovery of IVR in Setion... We dopt the metri of density vrition in to evlute our density uniformity s in following eqution: σ = d mx d min, () where d mx is the mximum olor density of ll olors, nd d min is the minimum olor density. In idel se, σ should pproh zero; i.e. d mx is equl to d min. Tle shows the density vrition of LPIR for TPL nd QPL. We n see tht in TPL, the verge density vrition is only 0.% nd most enhmrks hve vrition pprohing zero. For QPL, the verge density vrition is %, whih is eptle. The outlier of the vrition in QPL lies in enhmrk C whih only ontins 09 verties. Although there re only more verties ssigned to the first msk thn tht to the fourth msk, the vrition is lrge due to the smll vlue of totl verties, euse smll enhmrks re more sensitive to smll density vrition.. CONCLUSION In this pper, we propose new nd effetive lgorithm for lyout deomposition prolem for TPL nd QPL. By utilizing the fetures in the polyhedron spe of the fesile set, we pproximte the ILP formultion with itertive rounding to the LP relxtion. Severl tehniques re proposed to shrink the polyhedron spe without losing the fesiility, suh s odd yle onstrints, inding onstrint nlysis nd vertex nhoring. Experiment results show the effetiveness nd effiieny of the lgorithm ompred with oth ILP nd SDP. Pro. of SPIE Vol M-9

10 Tle. Density Vrition Ciruit LPIR for TPL LPIR for QPL C C C C C C C C C C S S S S S vg Aknowledgment This work is supported in prt y Ntionl Siene Foundtion (NSF), Semiondutor Reserh Corportion (SRC), nd Chinese University of Hong Kong (CUHK) Diret Grnt for Reserh. REFERENCES [] Pn, D. Z., Yu, B., nd Go, J.-R., Design for mnufturing with emerging nnolithogrphy, IEEE Trnstions on Computer-Aided Design of Integrted Ciruits nd Systems (TCAD) (0), 7 (0). [] Krp, R. M., [Reduiility mong omintoril prolems], Springer (97). [] Yu, B., Yun, K., Ding, D., nd Pn, D. Z., Lyout deomposition for triple ptterning lithogrphy, IEEE Trnstions on Computer-Aided Design of Integrted Ciruits nd Systems (TCAD), 6 (Mrh 0). [] Ghid, R. S., Agrwl, K. B., Liemnn, L. W., Nssif, S. R., nd Gupt, P., A novel methodology for triple/multiple-ptterning lyout deomposition, in [Proeedings of SPIE], 87 (0). [] Fng, S.-Y., Chng, Y.-W., nd Chen, W.-Y., A novel lyout deomposition lgorithm for triple ptterning lithogrphy, IEEE Trnstions on Computer-Aided Design of Integrted Ciruits nd Systems (TCAD), (Mrh 0). [6] Lus, K., Cork, C., Yu, B., Luk-Pt, G., Pinter, B., nd Pn, D. Z., Implitions of triple ptterning for nm node design nd ptterning, in [Proeedings of SPIE], 87 (0). [7] Kung, J. nd Young, E. F., An effiient lyout deomposition pproh for triple ptterning lithogrphy, in [ACM/IEEE Design Automtion Conferene (DAC)], 69: 69:6 (0). [8] Zhng, Y., Luk, W.-S., Zhou, H., Yn, C., nd Zeng, X., Lyout deomposition with pirwise oloring for multiple ptterning lithogrphy, in [IEEE/ACM Interntionl Conferene on Computer-Aided Design (ICCAD)], (0). [9] Yu, B. nd Pn, D. Z., Lyout deomposition for qudruple ptterning lithogrphy nd eyond, in [ACM/IEEE Design Automtion Conferene (DAC)], : :6 (0). [0] Chen, Z., Yo, H., nd Ci, Y., SUALD: Sping uniformity-wre lyout deomposition in triple ptterning lithogrphy., in [IEEE Interntionl Symposium on Qulity Eletroni Design (ISQED)], 66 7 (0). [] Yu, B., Lin, Y.-H., Luk-Pt, G., Ding, D., Lus, K., nd Pn, D. Z., A high-performne triple ptterning lyout deomposer with lned density, in [IEEE/ACM Interntionl Conferene on Computer-Aided Design (ICCAD)], 6 69 (0). Pro. of SPIE Vol M-0

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