Optical Proximity Correction (OPC)-Friendly Maze Routing

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1 . Optcal roxmty Correcton (OC)-Frendly Maze Routng L-Da Huang Computer Scences Department Unversty of Texas at Austn, Texas 787 Martn D. F. Wong ECE Department, Unversty of Illnos at Urbana-Champagn Urbana, Illnos 68 ABSTRACT As the technology mgrates nto the deep submcron manufacturng (DSM) era, the crtcal dmenson of the crcuts s gettng smaller than the lthographc wavelength. The unavodable lght dffracton phenomena n the sub-wavelength technologes have become one of the major factors n the yeld rate. Optcal proxmty correcton (OC) s one of the methods adopted to compensate for the lght dffracton effect as a post layout process. However, the process s tme-consumng and the results are stll lmted by the orgnal layout qualty. In ths paper, we propose a maze routng method that consders the optcal effect n the routng algorthm. By utlzng the symmetrcal property of the optcal system, the lght dffracton s effcently calculated and stored n tables. The costs that gude the router to mnmze the optcal nterferences are obtaned from these look-up tables. The problem s frst formulated as a constraned maze routng problem, then t s shown to be a multple constraned shortest path problem. Based on the Lagrangan relaxaton method, an effectve algorthm s desgned to solve the problem. Categores and Subject Descrptors: J.6: Computer- Aded Engneerng General Terms: Algorthms. Keywords: mcro-lthography, VLSI, maze routng, optcal system, manufacturng, OC.. INTRODUCTION Mcrolthography has become one of the key technques for the deep submcron technology. The mprovements of IC devce ntegraton and fabrcaton have been facltated by mprovng the stepper technology. Nowadays, ntegrated crcuts (IC) are manufactured wth features smaller than the lthographc wavelength []. The lthography technol- Ths work was partally supported by the Natonal Scence Foundaton under grant CCR-6 and CCR-6. L-Da Huang currently s workng at Texas Insturments. ermsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. DAC, June 7,, San Dego, Calforna, USA. Copyrght ACM //6...$5.. ogy wth shorter wavelengths s stll too costly and unstable. Any wavelength shorter than 85nm would be absorbed by the oxygen n the atmosphere. Furthermore, the photo resstance and development must react on a certan regon of the wavelength. Therefore, the.5um and.8um technologes are stll adoptng the DUV scanner (wavelength=8nm). The 9nm process adopts the 9nm wavelength optcal system, and major IC fabs have announced that the 65nm process wll adopt the 9nm wavelength optcal system to leverage the mass captal nvestment n the 9nm node. The sub-wavelength lthography ntroduces a huge burden n the manufacturng process because the dffracton of lght physcally lmts the crtcal dmenson (CD), such as the shortest length of the gate channel. The dstortons n optcal lthography nclude corner roundng, pullng back at the end of the narrow lne, and the wde varaton of lne wdth. To compensate for the dstorton or cancel out the nterference from the neghborng lght dffractons, two technques Optcal roxmty Correcton (OC) [5][6][] and hase Shft Maskng (SM) [][][9][7] have been demonstrated to show sgnfcant mprovements n sub-wavelength lthography technology. OC functons by addng or subtractng some fne features as serfs or lne segments; and SM operates by changng the phase of the transmtted lght through certan regons on the mask. However, a rule-based OC system that decdes the added OC features based on geometrcal rules cannot handle rules that are too complcated; a model-based OC system that requres many teratons of smulatons takes a long tme to choose the szes and postons of the added features; and SM ncreases the cost of the masks by ntroducng the phase shft mask. The major goal of OC s to keep the geometry prnted on the substrate as close to the layout as possble. If the layout contans crtcal paths or regons, OC cannot effectvely mprove the process wndow. Sometmes the stuaton becomes worse because the added feature narrows down the space between ts neghbors. The orgnal layout actually domnates the process varaton allowances such as the exposure lattude (EL) and depth of focus (DOF), despte the fact that the post processes lke OC and SM help reducng dstorton. Generally, the qualty of the aeral mage determnes the process wndow. If the aeral mage has sharp edges on each polygon, then the layout would have a wde process wndow [8]. Snce the desgn rules are the only gudes for routng, and these smple spacng rules cannot reflect the lght dffracton, the resultng layout often loses the yeld rate or the compactness. In modern technologes, such as the nodes beyond nm, 86

2 most of the metal layers need OC to control the lne wdth and length varatons []. An OC-frendly routng maxmzes the effects of the correctons and reduces the efforts that these fne features requre to be nserted. In ths paper, we propose a maze routng algorthm that s aware of the optcal effects. Snce the maze routng s a sequental routng algorthm, only one sgnal net s routed at a tme. The routed nets ntroduce some optcal effects all over the routng grds on the same layer. In addton, when routng a new net, t also affects the prevously routed nets. Based on the Lagrangan relaxaton technque, a robust and effectve algorthm that s able to be modfed to accommodate more complcated models can be derved. Secton II revews OC technologes and defnes the OCfrendly routng problem. In secton III, the OC cost functon derved from the models of optcal systems and the methodology of buldng the look-up tables are descrbed. In addton, an algorthm based on the Lagrangan relaxaton s proposed to solve the problem. Expermental results are presented n secton IV, and secton V concludes the paper. around the orgnal patterns, as shown n Fg.. Ths technque s called optcal proxmty correcton (OC). OC s not effectve nor effcent for an OC-unfrendly routng. Fg. s the routng for the same nets, but t has dfferent paths. Fg. 5 llustrates how the OC features are added nto Fg.. Obvously, less amount of OC teratons are needed n Fg. 5, and the process wndow s broader for the OC-frendly routng.. OC TECHNOLOGIES AND ROBLEM DEFINITION There are two major OC technologes for the subwavelength processes: rule-based and model-based. Rule-based OC extracts the geometrcal measurements from the orgnal layout. Accordng to the pre-bult OC rules, OC features are nserted to compensate for the dstorton from the lght dffracton. Model-based OC smulates the optcal system teratvely on many pre-determned ponts of the orgnal layout. If the lght ntensty and the constrast at the checkng pont are below certan requrements, a small OC feature would be added or subtracted from the orgnal layout. Fg. s an example of OC. Fgure : Optcal proxmty correcton. To smplfy the problem descrpton, we use a smple example to llustrate the OC-frendly routng. Fg. contans three nets on two conductor layers. All three nets are routed wth mnmum lengths. Snce the lne wdth and ptch are beyond the wavelength of the optcal system, all of the grds contan the lght dffracton from the routed nets. The extra lght constructvely or destructvely affects the routed patterns wth each other. These optcal proxmty effects can be compensated and corrected by addng fne features

3 Fgure : The same three nets wth dfferent paths. Fgure 6: A smple optcal system. Fgure 5: Fewer and less complcated OC features are needed n the layout. The routng s more frendly to the OC process, and the process wndow s wder. roblem. OC Constraned Maze Routng (OCCMR) Gven a grd routng graph wth m nets routed on K conductor layers, fnd the shortest route for net m, such that R j Cj for all = m, andj = K. Rj s the cost of net on layer j, andc j s the constrant for net on layer j. Instead of mnmzng the costs of all nets, the cost of each net s constraned and the total cost s balanced among the nets. Ths approach avods an OC crtcal net that narrows down the process wndow. OCCMR s a mult-constraned shortest path (MCS) problem. In the next secton, we propose an effcent method to calculate the OC cost. Based on the cost, a vectorweghted graph s constructed, and a robust algorthm s adopted to route the mult-constraned nets on ths graph.. OC COST CALCULATION AND OC- FRIENDLY MAZE ROUTING ALGORITHM. Optcal System Models Fg. 6 s a smple optcal system n the mcrolthography. The numercal aperture (NA) s defned as NA =snα. It represents the qualty of the lenses n the optcal system. The smallest representable sze n the optcal system s proportonal to λ. NA Let f(r), where r s a two dmensonal vector representng any poston on a plane, be the mask for a certan layer of a layout. The f(r) has a bnary output: zero means the lght s blocked, and one allows the lght to go through the mask. The ntensty of the output mage I(r) for an optcal system wth the ampltude-mpulse-response h(r) can be calculated by the followng three models: coherent llumnaton: I(r) = f(r) h(r) () ncoherent llumnaton: I(r) =f(r) h(r) () partally llumnaton: I(r) = n = β f(r) h (r) () where β s the scale factor. artally coherent systems can be approxmated as the sum of coherent systems as n Fg. 7. f(r) h (r) h (r)... h n (r) Fgure 7: artally coherent llumnaton model. The deal ampltude-mpulse-response functon h(r) sa snc-lke functon, as shown n Fg. 8. If the wavelength of the optcal system s λ, wth numercal aperture NA, the wdth W of the man lobes of h(r) wouldbe W = λ () NA λ Thewdthofthesdelobewouldbe. Snce the ampltude decays sharply beyond the frst sde lobe, we can NA thnk + I(r) 88

4 of the closest edges of the two adjacent patterns as the frst order factor n the cost functon. The rest of the edges on the patterns are second order, thrd order, and so on. If the edge falls beyond the frst sde lobe, ts effects would be gnored. The desgn rules bascally capture the frst order factor from the geometry. However, snce the CD s smaller than the wavelength nowadays, the second order or the thrd order edge would fall nto the effectve regon and cannot be gnored a (-,, 9) b (-,, 7) (,, ) - c (-, -, 6) d (, -, ) e (, -, 5) Fgure 8: h(r) of the coherent llumnaton system. λ = 8nm, NA =.5.. OC Cost Calculaton The optcal nterference s lmted wthn a regon of several wavelengths. To calculate the nterference on a certan edge from other routed patterns on the routng grd graph, only patterns wthn the effectve regon centered at the edge are necessary, as shown n Fg. 9. Note that the coordnates represent the center of each routng edge. All patterns wthn the effectve regon are marked wth coordnates of the left-most edge and the lengths of the patterns. The optcal nterference on the routng edge s the summaton of the nterference from all effectve patterns. As long as the relatve postons stay the same, all of the optcal effects would be equvalent. For example, to obtan the nterference from pattern b shown n Fg. 9, the pattern would be shfted and mrrored, as shown n Fg., Fg., and Fg.. Thus, a look-up table s bult as shown n Fg.. Let T j be the table for the length j. T j contans the results of two-dmensonal convoluton, f j(r) h(r). f j(r) s the bnary mask functon for a bar wth the length of j routng grds centered at the orgn, and h(r) s the ampltude-mpulse-response of the optcal system. For example, the lght dffracton from the pattern b to edge e as shown n Fg. s T 7(, ). Note that f the optcal system does not have the symmetrc property n certan axs, the mrror operaton would not be allowed and the sze of the table would be doubled. The optcal nterferences from all effectve patterns are looked up from the table. The sum of the values represents the total effect of the nterferences. The energy of the total effect s the weght of the edge representng the cost from all effectve patterns, wrtten as: 5 ( w(e, p )) where p represents the effectve patterns, and w(e, p )sthe optcal nterference obtaned from the look-up table T length(p ). For example, w(e, p b ) n Fg. 9 s obtaned from T 7(, ) as shown n Fg.. The other costs we need to obtan are 5 Fgure 9: Fve patterns are wthn the effectve wndow of the edge (, ). Each effectve pattern s denoted by the left most edge coordnate and ts length. For example, pattern a starts at (, ) wth length b (-,, 7) (, -, ) Fgure : attern b s shfted to the horzontal axs. the nterferences from the routng edge to the effectve patterns. Ths scenaro s more complcated because the net to be routed does not exst yet. We evaluate the cost of the routng edge to a certan effectve pattern as the maxmum energy of the nterference represented by: max(w(e, g)) g p For example, the cost of the routng edge to b n Fg. 9 s the maxmum nterference energy on the 7 edges of b from the routng edge centered at the orgn (the maxmum of T (, ),T (, ),,T (, ) ). The vector-weghted graph can be constructed as follows: The grd nodes and edges that are occuped by routed nets or obstacles are removed from the grd graph. Assgn the weght vector (v e,v e,,v e m)onedgee,wherev e,= (m ) are the cost nterferences from edge e to net f net m s routed on edge e, andv e m s the sum of the nterferences from all other nets on edge e. The entres of the vector are formulated as: v e =max g p (w(e, g)), = (m ) 89

5 b (-,, 7) (-, -, ) All lengths - - Fgure : attern b s shfted to be centered at the orgn (-, -, ) b (-,, 7) Fgure : The evaluated edge can be mrrored to the upper part of the effectve wndow f the optcal system s symmetrc on the horzontal axs. m vm e =( w(e, p )) =. OC-Frendly Maze Routng Algorthm Gven the vector-weghted graph, the OCCMR s a MCS problem on the graph. The Lagrangan relaxaton method solves the MCS by relaxng the constrants nto the objectve functon after gvng relatve weghts on the constrants. The Lagrangan relaxaton method s usually presented as two sub-problems: Lagrangan Sub-roblem (LS) and Lagrangan Multpler roblem (LM). The goal of the LS s to fnd the shortest path for a gven set of Lagrangan multpler; and the LM maxmzes the lower bound of MCS by adjustng the Lagrangan multplers. The LS of the OCCMR s defned n the followng problem. roblem. The Lagrangan Sub-roblem of OCCMR Gven the vector-weghted graph of OCCMR and a set of non-negatve constants u,j, = m, j (+ = K, fnd a K m path n the graph such that j= e j = u,jve j ) m = K j= u,j C,j s mnmzed, where ej s the edge on layer j and C,j s the constrant for net on layer j. Note that the second term, m = K j= u,j C,j, n the Fgure : The optcal nterference s smulated for all lengths of patterns centered at the orgn. The result s kept n a look-up table. Note that the routng grd sze s dfferent from the optcal smulaton grd sze. The mean value wthn the grd sze s recorded n the table. above equaton s a constant for fxed the Lagrangan multplers. By assgnng + m = u,j ve j as the weght on edge e j, the LS can be solved by usng Djkstra s shortest path algorthm []. Let L(u) be defned as the optmal soluton (.e., the shortest path) to the LS for a gven u,j,= m, j = K. Thats: { m K L(u) =mn + u,j( C,j)} e = j= e j The LM of the OCCMR s formulated as: roblem. The Lagrangan Multpler roblem of OCCMR Maxmze L(u) subject to u Snce u,j, t s obvous that the soluton to OCCMR s: : mn { e mn e j { e e j { mn e C,j,, j} m K u,j( + = j= C,j,, j} + m K = j= e j u,j( e j C,j) : C,j)} = L(u) (5) Note that the above relatonshp exsts for every u. Therefore, the optmal soluton to the LM s the lower bound of the soluton to OCCMR: : mn { e e j C,j,, j} max L(u) (6) u 9

6 Ideally, f a path meets all of the constrants (.e., e j ve j C,j,, j) and e =L(u), then would be the optmal soluton to OCCMR and u would be the optmal soluton to the correspondng LM. Snce the LM s a convex programmng problem, the sub-gradent method s an effectve approach to solve ths problem. The algorthm s descrbed as: Algorthm: Sub-gradent Method for the LM of OCCMR roblem. t =;u,j =,= m, j = K;. Solve LS by Djkstra s shortest path algorthm; Routable Nets OC frendly maze routng Shortest path maze routng. Check constrants; If satsfed, halt;. Else u,j =max{, θ t( e j ve j C,j)},, j; t = t + ; goto step. The above algorthm converges to the optmal soluton f t θ t and = θt as t []. Snce the number of teratons to get the optmal soluton s undefned, the maxmum number of the teratons s set to lmt the executon tme of each path routng. The feasble solutons are kept durng the teratons. If the sub-gradent method dd not generate the optmal soluton, the shortest path n the feasble solutons would be adopted.. EERIMENTAL RESULTS The OC-frendly maze routng based on the Lagrangan relaxaton s mplemented n C++ on the Sun Ultra workstaton. We perform the experments on an ndustral netlst that contans 5 nets. The parameters for the optcal system are: NA =.5 andλ = 9nm. We use the smple coherent optcal model to smulate the aeral mage and buld the look-up tables. The sze of the optcal grds s nm. The aeral mages are sampled at the optcal grds. Hgher precson of the smulaton can be obtaned by settng fner grds, but the smulaton tme would be longer. The lne wdth and space are both nm, whch are obtaned from the routng layers except the hghest layer n the ndustral 9nm process. The hghest layer has a much wder space and lne wdth for the global routng and specal nets. Both the lne wdth and space are more than one λ. Therefore, OC s not requred for the hghest layer. However, OC s necessary for the rest of the routng layers to compensate for the optcal dffractons. The sze of the routng grds s set as nm. We mplement the router on two routng layers (.e., H-V layers). The netlsts are frst routed to fnd the mnmum lengths and obtan the costs. Afterwards, we set the constrants as %, %, 8%, 6%, %, 6%, and 85% of the maxmum cost obtaned n the conventonal maze routng. The netlst s rerouted to meet the constrants by usng the OC-frendly maze router. The paths that volate the constrants are removed from the netlst. The amount of the routable nets n both routng results are shown n Fg.. 5. CONCLUSIONS The optcal proxmty costs are the energy of the lght dffractons from other patterns. The costs of all nets are OC Constrants (%) Fgure : The routng results of the OC-frendly maze router and the conventonal maze router. Routable nets are the nets that meet the constrants. constraned durng the sequental routng. The routng problem s shown to be a mult-constraned shortest path problem, and an effectve routng algorthm based on the Lagrangan relaxaton s proposed. As far as the authors knowledge extends, ths s the frst routng algorthm that s aware of the optcal effects of the sub-wavelength technologes. The mplementaton demonstrates consderable mprovement on the routng qualty REFERENCES [] R.K.Ahuja,T.L.Magnant,andJ.B.Orln. Network Flows: Theory, Algorthms, and Applcatons. rentce Hall, 99. [] N. Cobb and A. Zakhor. Large Area hase-shft Mask Desgn. SIE, 97:8 59, 99. [] N. Cobb, A. Zakhor, and E. Mloslavsky. Mathematcal and CAD Framework for roxmty Correcton. SIE, 76:8, 996. [] T. H. Cormen, C. E. Leserson, R. L. Rvest, and C. Sten. Introducton to Algorthms. MIT ress,. [5] C.-C.Fu,T.Yang,andD.R.Stone. Enhancementof lthography patterns by usng serf features. IEEE Trans. Electron Devces, 8():599 6, Dec. 99. [6] K. Harazak, Y. Hasegawa, Y. Shchjo, H. Tabuch, and K. Fuj. Hgh Accurate Optcal roxmty Correcton under the Influences of Lens Aberraton n.5 um Logc rocess. In Internatonal Mcroprocesses and Nanotechnology Conference, pages 5,. [7] M. D. Levenson, N. Vswanathan, and R. A. Smpson. Improvng Resoluton n hotolthography wth a hase-shftng Mask. IEEE Trans. Electron Devces, 9():88 86, Dec. 98. [8] C. A. Mack. Understandng focus effects n submcrometer optcal lthography: a revew. Optcal Engneerng, ():5 6, Oct. 99. [9] Y. at, Y.-T. Wang, J.-W. Lang, and T. Kalath. hase-shft Masks: Automated Desgn and Mask Requrements. SIE, 97: 7, 99. [] F. Schellenberg. Desgn for manufacturng n the semconductor ndustry: the ltho/ desgn workshop. th Internatoal Conference on VLSI Desgn, pages 9, Jan [] TSMC. In U.S. Technology Symposum,. [] K. Yamamoto, S. Kobayash, T. Uno, T. Kotan, S. Tanaka, S. Inoue, S. Watanabe, and H. Hgurash. Herarchcal optcal proxmty correcton on contact hole layers. In Internatonal Mcroprocesses and Nanotechnology Conference, pages,. 9