An analytical MOS Transistor Model Dedicated to Crosstalk Noise Evaluation

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1 An analycal MOS Transsor Modl Ddcad o rossalk Nos Ealuaon Parca Rnaul*, Prouz Bazargan-Sab*, Fabrc lpons *LP6 Unrsy o Pars 6, 4 Plac Jussu 755 Pars Franc ST-Mcrolcroncs - 85 Ru Jan Monn rolls Franc Absrac : n dp submcron chnologs, h nos nroducd on sgnals hrough h crossalk couplng s an mrgng problm. Ths papr prsns a nw analycal MOS ranssor modl ald n all rgons o opraon and ddcad o crossalk nos aluaon and rw som usul mhods and rlad work on crossalk analyss. Th sam MOS ranssor modl can b usd or h cm and aggrssors drrs. Th proposd modl s aldad by comparson o SPE smulaons. Ky-Words: MOS Transsor, Modl, SPE Smulaon, rossalk Nos. nroducon Mul-mllon ranssor crcus ar mad usng h las procsss. Th aurs o hs chnologs nclud an ncrasd numbr o mal lls, hnnr mal wdh, ncrasd wr hgh rsus wdh rao and smallr wr spacng. Ths nw aurs nroduc nw causs o alur. Ths s h rason why dsgnrs spnd up o 8 % o a dsgn on h rcaon sp. Thror, nw ools abl o prorm rcaons n a rasonabl dlay on complx crcus ar ndd o b dlopd. s wll known ha som up o laly nglcd physcal cs n submcron chnologs, such as crossalk, can sgncanly ac h bhaour o h crcu (mng and/or unconal alur. Nowadays, h dsgn mhodologs [] and ools, such as rour, ha o ak no accoun h crossalk phnomnon [2][3][4][5][6][7]. Thn, o cry h corrcnss o a dsgn mad by hos ools, som nw back-nd rcaon ools ha o b dlopd. n hs papr, w propos a smpl and sll accura modl o alua h mpac o h crossalk nos or ach sgnal o h crcu. Th nx scon dscrbs h crossalk phnomnon. Scon 3 rws som usul mhods and rlad work on crossalk analyss. Scon 4 dals h crossalk nos modl and hs mpromn. Som rsuls ar shown n scon 5. W compar h nos aluad rom h proposd modl agans a SPE smulaon and nally concludng rmarks and uur works ar dpcd n h las scon. Th crossalk phnomnon s du o h xsnc o a capacanc bwn wo nghborng wrs. Whnr a wr maks a ranson, a nos s producd hrough hs couplng capacanc on h ohr wr o h coupl. L s consdr wo sgnals A and drn by wo nrrs (Fg.. Whn h sgnal A maks a ranson, a nos s njcd on h sgnal. s n a sady sa, h nos has h orm o a spk and s absorbd by h s drr ar som dlay. On h conrary, s makng s ranson n h sam m, h crossalk nos lads o a shorr or a longr ranson dlay. Th sgnal A s calld h aggrssor and h cm. A Fg. : Two sgnals n crossalk couplng n a ral crcu, a gn sgnal may b coupld wh sral housand sgnals. Howr, a a gn m, all h aggrssors ar no makng a ranson. Th aggrssors whch ar n a sady sa and do no conrbu o h nos producd on h cm ar calld sln aggrssors. n urn, ach aggrssor may ha many ohr couplngs. Whn a gn sgnal s consdrd as a cm, all h ohr cms o s aggrssors ar calld scondary cms. n submcron procsss, many acors conrbu o h apparon o h crossalk nos: h shap o wrs, h rducon o h dsanc bwn wrs, h grar numbr o mal layrs, c. Ths acors nd hr o rduc h capacanc o h ground or, o ncras h couplng capacanc. Anohr paramr ha drmns h mporanc o h nos s h mpdanc o h aggrssor and h cm s drrs. Ths sudy s par o a Ph.D. hss currnly undrgong a Unrsy o Pars 6 and undd by ST-Mcrolcroncs.

2 2. Prous works Many modls ha bn proposd o sma h pak alu o h nos producd on a cm. Som o hs modls ak no accoun h rssanc-capacanc o h nrconnc, ohrs ocus on sudyng h nos producd by h smulanous ranson o sral aggrssors: Th P Modl [] ak no accoun coupld R nrconncs and gs a pak nos smaon on cm ln causd by on aggrssor. Th pak nos alu rpord or mulpl aggrssors s h suprposon o nos conrbuons []. Maxmum rror n h pak nos compuaon or an xampl hr-aggrssor cas s abou 3%. Th modl usng h aggrssor slop m [2] rplacs h aggrssors by h m whn h aggrssor n bgns o all and nds. aks no accoun o mulpl aggrssor cass bu no o R nrconncs. Th rror bwn an lcrcal smulaon and hs mhod s abou 3 %, n h cas o an aggrssor, and abou 7% n h cas o 3 aggrssors. [3] s a cas sudy o a mhod dlopd n [4], whch works wh a R r and mulpl aggrssors. Th or smaon o h pak nos s around 5 % n h cas o a crcu powrd by a.8 and wh a as rs m o ps. Th accuracy o hs modl dgrads wh rducon n rs m o h aggrssors. n a prous papr [9] w ha proposd a modl ha gnors h R nrconncs bu gs a syng smaon o h pak usng a smpl approach. n addon, hs modl aks no consdraon som scond ordr cs such as h xsnc o sln aggrssors or scondary cms. Hr w g an orw o hs modl: L s consdr h cas o gur 2. hr ranson n h urad mod. Thus, wo drn qualn rssors ar calculad or ach drr: a rs rssanc s usd whn h sgnal s consdrd as a cm and a scond on whn h sgnal s akn as an aggrssor. 2 A A sln A2 Fg. 3: Sgnal s drs rplacd by rssanc n h nx sp, or ach cm, h sln aggrssors ar rplacd by an qualn capacanc. Thn, or ach o h rmanng ac aggrssors, hr scondary cms ar rplacd by an qualn capacanc. qasln A + + q A 2 q2 Fg. 4: Sln aggrssors and ohr cms rplacd by capacanc Fnally, ac aggrssors ar rplacd by an qualn currn sourc whr h currn dcrass n an xponnal orm. Each currn sourc s calculad such as h pak producd on h cm occur a h sam m and ha h sam alu as h pak nducd by h corrspondng aggrssor. R A Fg. 5: Ac aggrssors rplacd by currn sourcs 2 A 2 A sln Fg. 2: omplx crcu Lk many ohr modls, n h proposd approach a rs approxmaon consss n rplacng sgnal s drrs by a smpl rssanc (Fg. 3. Howr, h cm and h aggrssors ar no approxmad n h sam mannr. Th ranssors o h cm s drr ar n hr lnar rgon whras h aggrssors mak h mos par o Thn, h c o h drn currn sourcs ar addd and h cm s waorm s xprssd: 2 k k ( = R ( k k= k Th xprnc has shown ha h rla rror o hs mhod compard o an lcrcal smulaon rmans lss han 2 %. Ths rrors com manly rom hr sourcs. Approxmang aggrssors by an qualn rssanc s rsponsbl o h major par o h rror. Anohr problm s ha rplacng aggrssors by an qualn currn sourc nglcs h nracon bwn h

3 aggrssors. En looks rasonabl, h rplacmn o h cm s drr by an qualn rssanc can lads o a sgncan rror. Acually, as long as h pak gnrad on h cm s blow h uraon olag o h cm s drr ranssor, h approxmaon bhas corrcly and h rror rmans rasonably low. Howr, h conguraon o h nos bcoms such ha h pak alu xcds h uraon olag, h smaon drgs rom h ral alu. Moror, snc h currn dlrd by h rssor s lnar whras h ranssor s urad, h pak s always undrsmad. Furhrmor, hs rror occurs or h hgh pak alus whr h rsk o unconal alur s mporan. n hs arcl, w propos an mpromn o h prous modl ha nds o rduc h rror nroducd by hs las approxmaon (Fg. 6. GS GS2 GS3 Fg. 6: Dran currn/olag or ranssor and h rssanc modl Ths smpl modl has bn usd o dlop sral rcaon ools nsd h Allanc AD Sysm ncludng a sac mng analyss and a rs rson o crossalk nos smaor. Howr, h dsconnuy bwn h lnar and h urad mods mak h aluaon o h pak alu dcul whn h pak xcds h uraon olag. Thus, w propos a und modl o h dran currn as a dcrasng xponnal uncon o ds : = ( whr s a paramr whch dpnds on h procss. Fgur 7 shows h proposd currn modl compard o a Spc smulaon or a.25µm chnology. Th drn paramrs o h modl ha bn s o mach h dran currn or h hgh alus o gs. ds(a E-3 9E-4 8E-4 7E-4 6E-4 5E-4 4E-4 3E-4 lcrcal smulaon modllng 2E-4 3. Modllng h cm s drr 3.. Transsor modl Th proposd approach s basd on a spcc modllng o h dran currn n shor channl ranssor (M. Hr w g a br orw o hs modl. A dald dscrpon o M can b ound n [8]. n M, a MOS ranssor and h currn can dr s characrzd by 3 rgons and h ollowng xprssons: Blockd mod ( gs < ds = ds Lnar mod ( ds < ds = 2 W A( gs Saurad mod( ds > ds= L ( + B( gs Sauraon olag = K( gs n h lnar mod, h currn s proporonal wh h ds olag. n h uraon rgon, h dran currn s consan and dos no dpnd on ds. n hs modl A and B ar wo consan paramrs ha dpnd on h procss and characrz h dran currn. K s a consan rprsnng h uraon acor. Ths paramrs ha o b s hrough a s o lcrcal smulaons o mak h M modl b d o a gn procss. E-4 E+,,5,,5 2, 2,5 ds( Fg. 7: Dran currn s. dran olag gn by lcrcal smulaon and h proposd modl 3.2. Dscharg o a capacanc Many currn orms may b usd o mach h currn drn by a ranssor. Nrhlss, h modl mus rspc som consrans. Th parcular orm o h dran currn has bn chosn n such a way allows o sol h drnal quaon dscrbng a capacanc dscharg hru a ranssor. L consdr h cas o Fgur 8 whr a capacanc s dschargd hrough a ranssor and whr h grd has a consan olag. Fg. 8: Dscharg o a capacanc n a ranssor Th ollowng rs ordr non lnar quaon characrzs h crcu: ( ' +. = Th rsoluon s oband by makng a arabl rplacmn:

4 ( =.log( ( ( +. ' = Th xprsson o ds can b sad gn h nal alu ds ( =. ( =.log( + (. Fgur 9 llusras a comparson bwn our xprsson and a Spc smulaon or h sam.25µm procss. ds ( 2,5 2,,5,,5, 4E- 8E-,2E-9,6E-9 m (s lcrcal smulaon modllng Fg. 9: Dran currn rsus dran olag or lcrcal smulaon and modl 3.3. urrn sourcs L now consdr h cas o h nos nducd on a cm by a s o currn sourcs. Each currn sourc rprsns a swchng aggrssor and has h ollowng orm: = /.... n Fg. : Mulpl currn sourcs Th crcu s dscrbd by h quaon sps ar ndd o g no a rsoluon. L s consdr h ollowng quaon whr has bn supprssd: ' + g = A soluon o hs quaon s gn by: (. (. g d = Back o h orgnal quaon a scond arabl rplacmn s proposd: = h ' h( + g + h' = h' = Thn ( can b xprssd: ( g d ( ( = d En g( can b asly ngrad, ( canno b xprssd bcaus o h scond prm. Thror, an approxmaon s ndd o oban h xprsson o (. L consdr h ollowng xprsson: = g ( p( d W propos o approxma p( by p q ( such as:. pq ( = k. = Th drn k paramrs ar dnd by orcng p q ( o rspc h ollowng condons: lm p( = lm p ( + + and j N, p( j. = pq ( j. Acually, lmng h numbr o paramrs o 4 lads o a syng prcson. Fgur shows h uncon p( and s approxmaon p q. ( q p pq ( + ' g( = 5 n whr / g( =. = Th sam arabl rplacmn chnqu can b usd o smply h quaon: =. log( ( + ' g = Unorunaly h rsulng quaon s no lnar and canno b sold n a smpl way. Thror, sral E+ 5E- E- 2E- 2E- 3E- 3E- 4E- 4E- Fg. : Approxmaon o h uncon p( by h uncon p q ( Th m consan s drmnd such as h slop o p( and p q ( b h sam a =. Thus, dp dp ( = d d q (

5 n... = = n = Thn, rplacng p( by s qualn gs h analyc xprsson o ( k d ( = *. = Thn, consdrng h nal condon: (=, w oban ( =.log( (. n j= k j ( j j. n = + = Thn, gn h xprsson o ( pak alu o h nos can b xprssd. Frs, h pak m s calculad rom: d ( pak = d An approxma soluon o hs mplc quaon can b oband usng an ra rsoluon chnqu (Nwon-Raphson. Thn, h pak olag s calculad rom: pak = ( pak Fgur 2 gs a comparson bwn h waorm oband rom an lcrcal smulaon and h analyc xprsson o ( whn h cm s aggrssd by 3 aggrssors.,8 Th bnch crcu s composd o a cm sgnal surroundd by 3 ac and 3 sln aggrssors. Each aggrssor s coupld o h sgnal and o 3 ohr scondary cms a a couplng capacanc (Fg. 3. Drn s o couplng and ground capacanc ha bn xprmnd o produc drn yps o pak on h cm. A s *3 Fg. 3: Bnch crcu Tabls o 3 show h pak alu and h corrspondng rla rror oband on h sgnal. Th comparson bwn h prous modl and h nhancd modl s prormd by rplacng h ac lmns o h crcu by an qualn rssor or by h proposd ranssor modl. Th wo rs lns concrn h rplacmn o h cm's and h ac aggrssors' drrs. Th hrd shows h combnd c o hs wo rplacmns. Fnally, n h wo las lns, scondary cms and sln aggrssors ar rplacd by an qualn capacanc as xpland n scon 3. Th rsuls ar gn or a.25µm procss, whr h supply olag s 2,25. n h rs suaon, h pak alu gn by a Spc smulaon s,38. Modllng Prous modl ( Enhancd modl ( *6,7,6 lcrcal smulaon modl cm,397-3,6,357 6,8 ds(,5,4,3,2 Agrssors,34,,36 5,8 c + Ag,357 6,8,33 3,5,, E+ 5E- E- 2E- 2E- 3E- 3E- 4E- 4E- 5E- 5E- m(s Fg. 2: Pak olag or lcrcal smulaon and modl 4. Rsuls Th mplmnaon o hs nhancd modl n a prooyp rossalk Nos Ealuaon ool s currnly undr dlopmn. Howr, a smpl bnch crcu can b usd o chck h rlanc o h proposd ranssor currn modl. Scondary cm,357 6,8,33 3,5 Sln Agrssors,362 5,5,295 23, Tabl : Prous s. Enhancd modl or a pak o,38 Th rror oband rom h nhancd modl s hghr bu rmans rasonabl xcpd h approxmaon o sln aggrssors. Ths rror s manly du o h ac ha, h proposd currn modl s no accura or low alus o. Bsds, h qualn capacanc s calculad usng h rss modl.

6 For h scond xampl, h pak alu s,6 and s clos o h uraon olag. n hs cas, h rror or h wo modls ar comparabl. Modllng Prous modl ( Enhancd modl ( cm,584 4,,594 2,3 Agrssors,495 8,6,555 8,8 c + Ag,489 9,5,533 2,3 Scondary cm,49 9,5,534 2,3 Sln Agrssors,54 7,,58 6,6 Tabl 2 : Prous s. Enhancd modl or a pak o,6 n h las cas, a hgh pak (,35 s producd on h cm. Modllng Prous modl ( Enhancd modl ( cm,25 9,7,36, Agrssors,96 5,4,36, c + Ag,888 2,8,35, Scondary cm,892 2,4,3,4 Sln Agrssors,99 2,,39 -,3 Tabl 3 : Prous s. Enhancd modl or a pak o,35 As xpcd, hs rs rsuls show ha h proposd mpromn dos no ac h rror or h low alus o h pak. Nrhlss, whn h nos s such ha h cm's drr nr n s uraon rgon, h rror nducd by h rss modl s largly rducd. 5. oncluson and uur works An orw o sral modls currnly usd n our crossalk nos analyss and an mpromn o our prous modl ha bn prsnd. Th prlmnary rsuls oband rom h applcaon o h proposd ranssor modl o h cm's drr sm o conrm h xpcd gan n h pak olag calculaon and push o xplor h applcaon o h modl n rplacmn o h aggrssors' drr. Th proposd ranssor dran currn lads o a mor complx pak xprsson and rqurs sral addonal compuaon sps. Howr, hs ncras o h complxy s compn by a sgncan rducon o h rror or h hgh paks. Thus, o lm h orhad o compuaon m nroducd by h nhancd modl, a mxd approach may rprsn a suabl rad-o. n hs conguraon, h nos rcaon ools opras n wo sps. n a rs sp, a rough smaon o h pak s prormd usng h rss modl. Thn, n h scond sp, h pak aluaon s mprod by usng h nhancd modl or hos sgnals ha show a hgh pak alu. Bsds, h crossalk modl has o b xndd o ak no accoun h nracon bwn aggrssors and h R o h nrconncs. Rrncs: [] R. Salh, D. Orhausr, S. Taylor, Full-hp o UM Dsgns, AD, EEE/AM, 998, pp [2] P. hn, K. Kuzr, Tru rossalk Nos Analyss, DA, EEE/AM, 999, pp. -6. [3] K.L. Shpard,. Narayanan, Nos n Dp Submcron Dgal Dsgn, AD, EEE/AM, 996, pp [4] B.N. Shhan, Prdcng oupld Nos n R rcus, DATE, EEE, 2, pp [5] B. ho, D.M. H. Walkr, Tmng Analyss o ombnaonal rcus ncludng apac ouplng and Sascal Procss araon, DATE, EEE, 2, pp [6] T. Xu, E.S. Kuh, D. Wang, Pos Global Roung rossalk Rsk Esmaon and rducon, AD, EEE/AM, 996, pp [7] H. Zhou, D.F. Wong, An Opmal Algorhm or Rr Roung wh rossalk onrans, AD, EEE/AM, 996, pp [8] A. Hajjar, R. Marbo, A. Grnr, P. Kan, TAS: An Accura Tmng Analysr or MOS LS, EDA, EEE, 99, pp [9] P. Bazargan-Sab, F. lpons, Modlng rossalk Nos or Dp Submcron rcaon Tools, DATE, 2. [] A.B. Kahng, S. Muddu, D. dhan, Nos and dlay uncranly suds or coupld R nrconncs, AS/SO, EEE, 999, pp. 3. [] A.B. Kahng, S. Muddu, N. Pol, D. dhan, Nos Modl or Mulpl Sgmnd oupld R nrconncs, SQED, 2, pp [2] P. hn, K. Kuzr, Towards Tru rossalk Nos Analyss, AD, 999. [3] P. B. Moron, W. Day, An Ecn Squnal Quadrac Programmng Formulaon o Opmal Wr Spacng or rossalk Nos Aodanc Roung, SPD, 999. [4] A. Dgan, Ecn oupld Nos Esmaon or On- hp nrconncs, AD, 997

t=0 t>0: + vr - i dvc Continuation

t=0 t>0: + vr - i dvc Continuation hapr Ga Dlay and rcus onnuaon s rcu Equaon >: S S Ths dffrnal quaon, oghr wh h nal condon, fully spcfs bhaor of crcu afr swch closs Our n challng: larn how o sol such quaons TUE/EE 57 nwrk analys 4/5 NdM