Forbidden area avoidance with spacing technique for layout optimization

Transcription

1 Forbdden area avodance wth spacng technque for layout optmzaton ShChang Sh *, lfred K. Wong, Tung-Sang Ng Dept. of EEE, The Unversty of Hong Kong, Pokfulam Road, Hong Kong BSTRCT The use of subresoluton assst features (SRFs) n the photo mask s one of resoluton enhancement technques n photolthography, whch can mnmze lnewdth varaton caused by proxmty effect. However, the process lattude wth SRFs through varous ptches s not unform. From the pont of vew of lthography, ptches wth low process lattude, called forbdden areas, should be avoded. These forbdden areas exst often n the layout after routng snce they are larger than the mnmum clearance requred n the desgn rules. In ths paper, a ptch optmzaton method appled n the post-routng phase s proposed to avod the forbdden areas. Expermental data of lthography technques and geometrc constrants from the layout are formulated nto a constraned quadratc optmzaton problem. By usng the spacng technque, wre segments n the affected area are adjusted to ther new locatons obtaned from solvng the optmzaton problem by quadratc programmng. Examples show that the proposed method can avod most forbdden areas n the layout after normal routng. Keywords: photolthography, resoluton enhancement technque, subresoluton assst features, layout optmzaton. INTRODUCTION Snce the md 99s, the optcal lthography technque has entered the era of subwavelength lthography. [] The mnmum dmenson of transstor s smaller than the wavelengths of the lght used to prnt them. Thus more notceable deformatons n mage such as lne shortenng, corner roundng, nonlnearty, and lnewdth varaton [] are ntroduced by optcal proxmty effects when layout patterns on a mask area transcrbed to a wafer. In order to enhance the mage qualty of mask, resoluton enhancement technques are wdely adopted, ncludng phase shft mask (PSM), optcal proxmty correcton (OPC), off-axs llumnaton optcs, and the use of subresoluton assst features (SRFs). SRFs are small patterns under resolutons. When used n conjuncton wth off-axs llumnaton, they show great mprovement on lthographc process wndow whle reducng proxmty effects. [3] By addng SRFs to both sdes of the sparse features to create a dense envronment, lnewdth varaton tolerance can be controlled. The number of SRF to be used, ther sze and placement are optmzed accordng to the ptch of man patterns. Fg. shows the expermental process lattude vs. ptch when applyng dfferent number of SRFs. [] It s obvous that wthout usng SRFs, the process lattude drops sharply when the ptch of man pattern ncreases. If one SRF s added, the mage qualty can be enhanced. However, t s not the case when the ptch keeps ncreasng. When the second SRF s added, the process lattude rses agan. Therefore, the use of SRFs through varous ptches s not straghtforward. Process wndow across ptch s not unform; there exsts sudden jumps n the fgure when nsertng addtonal SRF. From the optcal lthography standpont, n order to get hgh manufacture robustness, t s desrable to forbd the ptches wth lower process lattude to exst n the layout. These ptches wth low manufacturablty are called forbdden areas or forbdden ptches, [4][5] λ λ λ λ as the ptches n the ranges..4 and.7.8 n Fg.. N N N N The requrement of avodng forbdden areas n the layout challenges the physcal desgn automaton, especally n routng. Ths s because forbdden areas are larger than the mnmum clearance n the desgn rules; they exst often n the layout after routng. Modfyng the routng algorthm by ntroducng more rules or geometrc constrants to avod the *scsh@eee.hku.hk; phone ; fax Desgn and Process Integraton for Mcroelectronc Manufacturng II, edted by Lars W. Lebmann, Proc. of SPIE Vol (SPIE, Bellngham, W, 4) X/4/$5 do:.7/

2 wre segments formng forbdden areas s a reasonable approach, but t ncreases the complexty of routng algorthm and tends to nonoptmal result, such as larger routng area, more faled paths n routng. Ths paper presents an effcent soluton by modfyng the layout wth spacng technque durng post-routng or clean process so as to keep the wre segments from forbdden areas. The whole process can be dvded nto two phases: optmzaton and spacng. Durng the optmzaton phase, the process lattude of wre segments formng forbdden area and ther neghbors s formulated nto a quadratc objectve functon. Based on geometrc constrants n the layout, optmal locatons for wre segments n the affected area are obtaned by quadratc programmng method. In the spacng phase, these wre segments are adjusted to ther new optmal locatons by spacng technque. Spacng technque, smlar to layout compacton, s a method to adjust the dstances between wre segments so as to mprove the layout performance whle keepng the wre connecton and desgn rules ntact. However, t not only compacts but also stretches the wre segments. The rest of paper s organzed as follows: n Secton, the optmzaton phase of the proposed technque s ntroduced. The spacng phase s explaned n Secton 3. Secton 4 dscusses the detaled mplementaton n routng and fnally, concluson s gven n secton 5.. OPTIMIZTION PHSE In order to dmnsh or even clear away the forbdden areas exstng n the layout after normal routng, wre segments formng forbdden areas are adjusted to new locatons by spacng technque. However, due to the fact that the space adjustment for a specfc wre segment always nteracts wth nearby segments, the layout modfcaton should be consdered n all affected areas, not just the wre segments formng forbdden area only. Besdes, ths adjustment cannot sacrfce the orgnal mage qualty. The man objectve n optmzaton phase s to determne the new locatons of all wre segments n the affected areas wth the am to acheve the hghest process lattude for the layout. The key s to solve a constraned quadratc optmzaton problem, whch s formulated from expermental data of lthography technques and the geometrc constrants from the layout.. Quadratc curve fttng The expermental data, whch have the hghest process lattude under dfferent SRFs across the ptch, are presented n Fg.. To solve the optmzaton problem n a mathematc way, data are ftted by two quadratc curves wth the least-squares approxmaton. The two curves are shown n Fg., wth ptch range ( P, P ) and ( P, P 3 ) respectvely. Therefore, the expermental process lattude n Fg. can be approxmated by the followng equaton: ( p B ) + C, P p P ; f ( p) = () ( p B ) + C, P p P3. Fg.. Expermental process lattude across ptch wth the use of SRFs. [] In (), p s the ptch value;, B, C and, B, C are the coeffcents of the two quadratc curves; P normally equals to the mnmum clearance requred by desgn rules; P s the cross Fg.. Two quadratc curves ft the expermental data. 68 Proc. of SPIE Vol. 5379

3 pont of the two curves; P 3 s the upper boundary of curve II, and t has some freedom for the desgner to decde. It should be ponted out that the reason why the expermental data are ftted by two quadratc curves nstead of one hgh degree polynomal curve s based on two consderatons: one s to smplfy the fnal objectve functon and the other s due to the dstrbuton of the expermental data. If an acceptable lne s determned as n Fg., then ptch range ( a, b) and ( c, d) can be defned as forbdden range I and forbdden range II respectvely. It s desrable to keep the ptches out of these two ranges so that the mage qualty of the mask can be enhanced.. Objectve functon n example s frst presented n ths part to demonstrate the process n obtanng the objectve functon, and the formulas for the general case s gven at the end of Secton. Fg. 3(a) s a part layout of an 8-bt rpple-carry adder chp after normal routng. TSMC 5nm standard cells are adopted for ths chp and nets are routed over-the-cell (OTC). Patterns n metal ncludng wre segments and vas are gven n Fg. 3(b). Based on the expermental data n Fg., for the 5nm technology, the ranges of 45-55nm and 65-66nm can be defned as forbdden range I and forbdden range II respectvely. Then ptch I and ptch II n Fg. 3(b) are both forbdden areas. Wres segments formng ptch I and II should be adjusted so as to mprove the mage qualty. However, ptch III may be affected durng the adjustment process. Therefore, the optmzaton are done for all affected areas ncludng ptch I, II and III. Snce wre segments n Fg. 3(b) are only adjusted n the horzontal drecton, the layout can be smplfed as n Fg. 3(c) and four vertcal wre segments are labeled as,,, and 3 respectvely. In order to acheve the maxmum process lattude for the affected areas n the layout, the objectve functon for the example layout n Fg. 3 s gven as where [ ] T Max F( p) = Max [ F( p ) + F( p ) + F( p3 )] = Max [ α f ( p) + α f ( p ) + α 3 f ( p3 )]. () p = p, p, p3. Fg. 3. (a) Layout of the example. (b) The patterns n metal (two forbdden areas exst). (c) Vertcal patterns and ther geometrcal szes. In (), F ( p) s the objectve functon; p, p, p3 are szes of ptch I, II, III; f ( p), f ( p ), f ( p3 ) are the process lattude of ptch I, II, III accordng to (); α, α, α 3 are weghts for the three ptches and are proportonal to the overlappng area between the two wre segments formng the ptch. In our ptch optmzaton method, any wre segment par havng overlappng area and havng no drect connecton contrbutes to the objectve functon. However, ptches wth vas are not consdered n the objectve functon due to ther small geometrcal szes. For wre segment par whch overlaps and drectly connects to each other, they do not contrbute to the objectve functon, ether. They are both taken care of n the layout constrants dscussed n the next subsecton. Proc. of SPIE Vol

4 For any wre segment par and j whch overlaps wth each other wthout drect connecton, ts coordnaton n the layout are assumed as n Fg. 4. Then the ptch between the wre segment par and j s j = j + p. (3) Snce p j s n the ptch range ( P, P ) defned n Fg., the contrbuton of the wre segment par and j to the objectve functon usng () s: j + α j j + + F( ) = [ ( B ) C ] α ( x x ) B ( x x ) + B ]. (4) = j [ j + j + + C The constant part n (4) can be neglected n the objectve functon because t does not affect the fnal optmal soluton. Then (4) can be wrtten as F( + ) = α [ ( + ) B ( + )] j T j j + x + + = α j + j B. (5) j j j j T Fg. 4. Two vertcal wre segments and ther geometrcal sze. Denote and then (5) becomes H f +, j +, j = j j j j j B =, (7) j B T + + T + F( j + ) = H +, j + f +, j. (8) j j j (6) For the weght α j n (6) and (7), f the rato of the overlappng area s set to, then α. (9) j = y j+ y From (8), the contrbuton of any wre segment par and j to the objectve functon s n quadratc form and can be expressed by two sub-matrxes H +, j and f +, j whch relate to the varable [ x +, j ]. ll the elements n these two sub-matrxes can be determned by the geometrc nformaton of the wre segment par and j. Snce the objectve functon s the sum of contrbutons from all the wre segment pars, the elements n the fnal matrxes H and f whch relate to all varables n the objectve functon are the accumulaton of elements from all sub-matrxes. The objectve functon for the example n Fg. 3 s therefore T T Max F( x) = Max ( x Hx + f x). () where x = x, x, x, x, x, x, x, x ], and [ T 7 Proc. of SPIE Vol. 5379

5 H = , f B B B =. B 3 B 3 B Though the above process n obtanng the quadratc objectve functon s based on a smple example wth vertcal wre segments, the prncple to generate the objectve functon for the general case and horzontal wre segments s the same. t the end of ths secton, more precse descrpton for buldng the objectve functon for the general case s presented..3 Layout constrans In ths part, layout constrants for solvng the quadratc objectve functon n () are dscussed. They can be dvded nto: desgn rule constrants, nherted constrants, and compulsory constrants..3. Desgn rule constrants Snce the ptch adjustment should keep the layout desgn rules ntact, the soluton to the objectve functon n () s therefore constraned by desgn rules. For example, f the mnmum clearance requred n desgn rules for wre segment par and j n Fg. 4 s d, then the followng constrant mn x + () j d mn need to be mposed. Besdes, n order to smplfy the optmzaton problem and ensure the soluton converge, the boundares for the affected areas are suppled to lmt the optmzaton n a specfc area n one metal layer. Ths can be realzed by ntroducng new desgn rules of the boundares n the technology table. s a result, layout constrants related to the boundares are also desgn rule constrants..3. Inherted constrants Inherted constrants exst n each wre segment par. s n Fg., more than one quadratc curves are utlzed to approxmate the process lattude n the whole ptch range. Therefore, constrants should be added to ensure that the ptch stll les wthn the orgnal range after modfcaton. For example, f the ptch of the wre segment par and j n Fg. 4 s n the range P, ) defned n Fg., then ther coordnaton should be kept as follows: ( P x + () j P, x + j. (3) P Wdth constrant s another type of nherted constrant. For each wre segment, two varables are related n the objectve functon. For example, n Fg. 4, varables x and x + are used to express the left and rght border of wre segment. s only ptches are adjusted for the layout, wdth of segments should always be kept constant. If the wdth of wre segment equals to w, then the wdth constrant s gven as follows, + w. (4) =.3.3 Compulsory constrants Compulsory constrants, whch are mposed by the desgner to nfluence the soluton of the objectve functon, are used to deal wth cases that the outcome s worse off (worse case). Due to forbdden ranges only occupy a small fracton of the ptch range and the shft range for each wre segment n the layout s suffcently wde, most forbdden areas can be avoded by solvng the constraned quadratc optmzaton problem and applyng the spacng technque. However, n some worse cases where the shft range s narrow because of layout constrants, the process lattude of the forbdden area Proc. of SPIE Vol

6 may be sacrfced to enhance the whole process lattude of the affected area. Therefore, a check step s added to compare the process lattude of the forbdden area after each optmzaton. If the process lattude of the forbdden area becomes worse, compulsory constrants are added to the constrant matrxes and b and the optmzaton process s repeated. For example, f the ptch between the wre segment par and j n Fg. 4 s n the forbdden range ( a, b) defned n Fg., one of the followng compulsory constrants + b (5) j + j a (6) can be added n the constrant matrx to avod the forbdden area under the worse case..4 Formulaton for the general case Wthout loss of generalty, f a forbdden area formed by two vertcal wre segments exsts n the layout, and n vertcal wre segments n the boundary are affected. Then the ptch optmzaton for the affected area can be formulated to a constraned quadratc optmzaton problem as follows, subject to Where [ ] T x x, x,, ( n ), ( n ) + = L. T T Max F( x) = Max ( x Hx + f x) x b, (7) x =, (8) eq b eq x. In the above formulas, elements n matrx H and f are obtaned by accumulatng all the contrbutons of wre segment par whch overlaps wth each other wthout drect connecton. ll the layout constrants wth non-equalty are expressed wth matrxes and b n (7), and those wth equalty are expressed wth matrxes eq and b eq n (8). Snce constrants are consdered altogether, some constrants may be redundant and should be thrown away from the constrant matrxes. It s obvous that the objectve functon s quadratc, and the constrants are lnear. For such constraned quadratc optmzaton problem, soluton can be obtaned by quadratc programmng (QP). [6] In our ptch optmzaton method, the PI functons n the Matlab optmzaton toolbox [7] are called to solve ths optmzaton problem. More detals are dscussed n Secton SPCING PHSE fter the optmzaton phase, vertcal wre segments (stll take vertcal drecton as an example.) have ther new locatons. Then n the spacng phase, they are compacted or stretched n the horzontal drecton to ther new locatons. Durng the ptch adjustment process, wre connectvty and layout-rule correctness are kept. The ptch adjustment technque s named as spacng technque rather than layout compacton because wre segments are also stretched n the process. Spacng technque s developed from the plowng algorthm provded by the Magc layout edtor. [8] s an nteractve operaton, plowng can stretch and compact Manhattan VLSI layouts whle keepng connectvty and layout-rule correct accordng to the parameters suppled by desgners. These parameters nclude plow drecton, plow dstance and plow layers. The key of the plowng algorthm s fndng edges and movng them based on the rules n ts technology table. [9] Spacng technque modfes the man algorthm n plowng on three parts. The frst s to change the operaton from nteractve to automatc. s presented above, plowng only operates after desgners supply the plow drecton, plow dstance, and plow layers to t. In contrast, spacng operaton automatcally determnes the drecton, dstance and layer for wre segments by comparng ther orgnal geometrc sze wth ther optmal requrements from the optmzaton 7 Proc. of SPIE Vol. 5379

7 phase. The second s forbd the ntroducton of new jogs durng the modfcaton process. Ths s because plowng always nserts addtonal jogs when t compacts or stretches. The last s to amend the technology table of the plowng algorthm. Some new desgn rules related to boundary are appended. Due to ts smlarty wth the plowng algorthm, spacng technque s not dscussed n ths paper. Detals of the algorthm can be referred to n [9]. 4. IMPLEMENTTION We develop a lthography frendly router called FnRouter, whch mplements ptch optmzaton n ts post-routng phase. The router s a grdless multlayer router based on the corner sttchng data structure. [] It s developed on the open C source code of Magc and supports routng of a maxmum of fve metals. It accepts netlst and placement nformaton from other CD tools such as Cadence Encounter by LEF and DEF fles. Routng by FnRouter proceeds n a smlar manner to exstng routers. It comprses route, rp-up and reroute steps. When FnRouter fnshes routng, ptch optmzaton s executed. 4. Ptch optmzaton n the post-routng phase For a specfc lthography technque, ts expermental process lattude wth respect to the ptch wth the use of SRFs can be approxmated as (). Relevant parameters n () can be nputted to FnRouter n a technology fle. Thus, forbdden ranges can be obtaned after the desgner determnes the acceptable process lattude. In the post-routng phase, patterns n each metal layer are searched and locatons of forbdden area are recorded. For each forbdden area, frstly, the objectve functon and layout constrants n the affected area are produced, whch can be expressed by matrxes H, f,, b, eq, b eq and outputted as a Matlab M scrpt fle. C++ program whch utlzes Matlab PI optmzaton toolbox [7] s called by FnRouter to solve the constraned quadratc optmzaton problem based on the Matlab M scrpt fle. If the soluton converges, t produces the new optmal locatons for wre segments n the affected area. Otherwse, FnRouter narrows the boundary and generates the objectve functon and layout constrants for the forbdden area agan. fter each optmzaton, a check step s executed. If t s a worse case, the compulsory constrant s added and optmzaton s repeated. Lastly, spacng technque s appled to adjust the ptches n the affected area. The above procedure s repeated untl there s no forbdden area left or the preset maxmum number of teratons s reached. 4. Test results In ths part, examples are gven to llustrate the ptch optmzaton method n avodng forbdden areas and mprovng the mask qualty. Two comparsons are presented for each example. One s the comparson between the orgnal process lattude and that after the ptch optmzaton for each forbdden area. The process lattude can be obtaned from (). The other s the comparson between the orgnal mask qualty and that after the ptch optmzaton for each affected area. parameter n α f ( p ) γ = (9) = n = α f ( p ) s defned to represent the mprovement of the layout qualty where n ptches exst n the affected area. In (9), f ( p ) s the orgnal process lattude of ptch and f ( p ) s that after the ptch optmzaton. α s the weght of ptch, whch s proporton to the overlappng length of the two wre segments formng ptch. Proc. of SPIE Vol

8 Fg. 5. (a) Process lattude mprovement for forbdden areas n the -bt rpple-carry adder. (b) Layout mprovement n the affected areas n the -bt rpple-carry adder. (c) Process lattude mprovement for forbdden areas n the 8-bt -taps FIR flter. (d) Layout mprovement n the affected area n the 8-bt -taps FIR flter. Test results from two chp layouts, one s a -bt rpple-carry adder and the other s an 8-bt -taps FIR flter, are presented n Fg. 5. The acceptable lne s expressed as the dash lne n Fg. 5(a) and Fg. 5(c). It s obvous that the process lattudes of most forbdden areas can be ncreased after the ptch optmzaton. However, there are stll 9% and 4% of the ptches n forbdden ranges n Fg. 5(a) and Fg. 5(c) due to ther narrow shft range n the layout. Fg. 5(b) and Fg. 5(d) show the layout mprovement n each affected area. Snce more than one forbdden areas could exst n the same affected area, the number of data n Fg. 5(b) and Fg. 5(d) s smaller than those n Fg. 5(a) and Fg. 5(c). It s clear that mask qualty for most affected area can be mproved and γ can be more than 5%. However, for the affected areas where forbdden areas cannot be avoded due to ther layout constrants, γ s very small and could even be. 5. CONCLUSION The forbdden area problem dscussed n ths paper orgnates from usng SRFs n the mask layout to mprove the mage qualty. Because of the hgh cost n ntroducng new desgn rules and the lmtatons n the routng algorthm, applyng ptch optmzaton after normal routng s a compromse way to solve the forbdden area problem. ptch optmzaton method wth spacng technque has been developed and ncorporated nto a lthography frendly router,.e. FnRouter. Examples have shown that most forbdden areas can be avoded and the mask layout has been optmzed and the mage qualty has been mproved. 74 Proc. of SPIE Vol. 5379

9 REFERENCES. ndrew B. Kahng, Y. C. Pat, Sub-wavelength lthography and ts potental mpact on desgn and ED, Proceedngs of the 36th CM/IEEE conference on Desgn utomaton, pp , CM Press, New York, lfred Kwok-Kt Wong, Resoluton Enhancement Technques n Optcal Lthography, pp. 9-93, SPIE Press,. 3. Scott M. Mansfeld, Lars W. Lebmann, ntonette F. Molless, lfred K. Wong, Lthographc comparson of assst feature desgn strateges, Optcal Mcrolthography XIII, Chrstopher J. Progler, Proceedngs of SPIE, Vol. 4, pp ,. 4. Robert Socha, Mrcea Dusa, Lug Capodec, Jo Fnders, Fung Chen, Dons Flagello, Kevn Cummngs, Forbdden ptches for 3nm lthograph and below, Optcal Mcrolthography XIII, Chrstopher J. Progler, Proceedngs of SPIE, Vol. 4, pp. 4-55,. 5. F. M. Schellenberg, Olver Toublan, Lug Capodec, Bob Socha, dopton of OPC and the mpact on desgn and layout, Proceedngs of the 38th CM/IEEE conference on Desgn utomaton, pp. 89-9, CM Press, New York,. 6. Jorge Nocedal, Stephen J. Wrght, Numercal Optmzaton, Sprnger-Verlag, New York, MathWorks, Inc., Matlab Standard lgorthms (Optmzaton Toolbox). 8. John K. Ousterhout, Gordon T. Hamach, Robert N. Mayo, Walter SS. Scott, George S. Taylor, Magc: a VLSI layout system, Proceedngs of the st CM/IEEE conference on Desgn utomaton, pp. 5-59, IEEE Press, Pscataway, June Walter S. Scott, John K. Ousterhout, Plowng: nteractve stretchng and compactng n Magc, Proceedngs of the st CM/IEEE conference on Desgn utomaton, pp. 66-7, IEEE Press, Pscataway, June John K. Ousterhout, Corner sttchng: a data structurng technque for VLSI layout tools, IEEE Transactons on Computer-ded Desgn, Vol. CD-3, No., pp. 87-, January Mchael L. Reger, Jeffrey P. Mayhew, Srdhar Panchapakesan, Layout desgn methodologes for sub-wavelength manufacturng, Proceedngs of the 38th CM/IEEE conference on Desgn utomaton, pp , CM Press, New York,.. Lars W. Lebmann, Layout mpact of resoluton enhancement technques: mpedment or opportunty?, Proceedngs of the 3 nternatonal symposum on physcal desgn, pp. -7, prl 3. Proc. of SPIE Vol