Crosstalk-Affected Propagation Delay in Nanometer Technologies

Transcription

1 Int'l Journal of Electroncs, 2008 ABSTRACT Crosstalk-Affected Propagaton Delay n Nanometer Technologes Shahn Nazaran Massoud Pedram Unversty of Southern Calforna, EE Dept., Los Angeles, CA shahn@usc.edu pedram@usc.edu Ths paper presents a detaled analyss of the crosstalk-affected delay of coupled nterconnects consderng process varatons. We utlze a dstrbuted RC- model of the nterconnectons to accurately model process varatons. In partcular, we perform a detaled nvestgaton of varous crosstalk scenaros and study the mpact of dfferent parameters on crosstalk delay. Whle accountng for the effect of correlatons among parameters of the neghborng wre segments, statstcal propertes of the crosstalk-affected propagaton delays are characterzed and dscussed. Monte Carlo-based smulatons usng Spce demonstrate the effectveness of the proposed approach n accurately modelng the correlaton-aware process varatons and ther mpact on nterconnect delay n the presence of crosstalk. Keywords Correlaton, crosstalk-affected delay, process varatons, statstcal statc tmng analyss, skew, varaton sheldng 1. INTRODUCTION The ncrease n package densty as well as the clock frequency of the VLSI crcuts has made nose, such as the capactve couplng nose, one of the most challengng problems n the desgn and verfcaton of modern VLSI crcuts. Furthermore, the nterconnect lnes get thcker and narrower (and longer n case of global nterconnects), whch all result n the aggravaton of crosstalk nose ampltude and duraton, and the crcut faults caused by such nose sources. Therefore as the VLSI technology scales down the role of nterconnect parastc effects n the sgnal ntegrty becomes ncreasngly more pronounced. Another unwanted sde effect of CMOS process technology scalng s the ncrease n process varatons. Dfferences between dentcal features n a certan lthographc process are referred to as process varatons. Lthography steps generate more process varatons n smaller geometrc feature szes. Therefore, cell and nterconnect delay characterzaton methods should consder the ncreasng mpact of process varatons on crcut performance and relablty. In addton to IC manufacturng process varatons, envronmental varatons, and devce/nterconnect agng processes create a rather large devaton of key crcut parameters from ther desgned values. These phenomena n turn produce tmng uncertanty and demand hghly sophstcated and robust crosstalk-aware analyss and optmzaton tools. The conventonal corner-based technques, to handle varablty of parameters wll not be effectve n nanometer technologes due to ther hghly pessmstc (and sometmes optmstc) vews. Statstcal analyss s vewed as an essental methodology for nanometer process technologes, whch enables applcaton of the actual statstcs of the process technology parameters for the accurate calculaton of desgn characterstcs such as delay and nose [1]. Crosstalk effect has been analyzed usng lumped RC models to fnd smple close-form formulas for crosstalk-nduced pulse and slowdown n [2]-[5]. However, ths model s naccurate for global nterconnects, especally at hgh clock frequences. Usng a dstrbuted couplng capactance model produces more accurate and realstc results. Closed-form expressons by usng 2 and 4 confguratons (whch are based on lnear crcut models) have been developed n [6] and [7], respectvely. However, the qualty of analyss and optmzaton tools degrades when usng lnear equatons to model the nonlnear behavor of drvers. In [8] and [9] dstrbuted RC modelng has been used to estmate the pulse nduced by crosstalk effect. In [10], by usng dstrbuted RC model wth a crcut smulaton engne, a number of nterestng observatons have been reported for weak spot defects n the presence of crosstalk nose. Indeed, dervng accurate closed-form expressons based on dstrbuted RC modelng has been a dffcult, and so far unsatsfactory, undertakng. As a result, t s common practce to make smplfyng, yet realstc, assumptons about the crosstalk nose n order to mtgate the complexty of crosstalk-affected analyss and optmzaton. Although a great deal of research has been done on statstcal statc tmng analyss, only a few approaches exsts n lterature that nvestgate the mpact of process varatons on crosstalk and nherently crcut performance. The statstcal model of [11] uses a lumped RC model to explore crosstalk-nduced pulse (gltch) effect, where a sngle resstance s

2 extracted to capture the effect of total self resstance of nterconnect, regardless of ts length. The case for self and couplng capactances s smlar. Also the correlaton between the crcut parameters, such as nterconnect lne resstance and capactance s assumed to be zero. The statstcal model proposed n [12] s more sophstcated and uses a crcut model wth hgher number of nodes; however, stll a sngle capactance s extracted to model the total couplng effect, whch makes t napproprate for long nterconnect lnes. The authors of [13] apply specal exponental waveform shapes to analytcally study the statstcs of crosstalk effect. However the exponental type waveforms cannot accurately represent nose-affected sgnal waveforms. Addtonally, the above approaches are unable to consder the correlaton between neghborng wre segments. The goal of ths paper s to study the effect of process varatons on some exstng crosstalk analyss technques, resolve ther shortcomngs, and fnally propose an effcent model to statstcally calculate the crosstalk-affected delay of the nterconnect vctm lne. More precsely, frst a dstrbuted RC- model s used to accurately capture the statstcal varatons n the physcal dmensons of the nterconnect lnes and the correspondng electrcal parameters. We frst provde the results of our extensve smulaton of crosstalk-affected delay and a number of mportant propertes of the crosstalk that may be exploted n valdaton and optmzaton tools to ncrease the accuracy of crosstalk effect analyss and reduce the computatonal complexty of these tools. In addton, we study the senstvty of the delay and transton tme of the output a crosstalk ste to crcut parameters such as ts couplng, wre capactance, and resstance values, and the drver strength, as well as the tmng parameters, namely the nput skew and the nput transton tme. The local effects of process varatons on the coupled wre segments and the correlatons among varatons n neghborng segments are consdered n statstcal analyses. Ths nformaton s then used to evaluate the correctness of the exstng crosstalk analyss technques n the presence of process varatons usng extensve sets of Monte Carlo smulatons to calculate the actual statstcal dstrbuton of vctm lne tmng parameters. Fnally based on our observatons, we propose a set of heurstc solutons for each technque to mprove ther applcablty n statstcal analyss of crosstalk effects. The paper s a major extenson of our conference papers [14] and [15]. Capactve couplng between a par of nterconnect lnes can nduce spurous pulses and/or cause delay effects. We refer to such effects as crosstalk-nduced effects. The porton of the layout where the couplng occurs s referred to as a crosstalk ste. Crosstalk-nduced slowdown occurs when an aggressor lne, A, and a vctm lne, V, make sgnal transtons (state changes) n opposte drectons. The net effect, n theory, of the couplng between the two lnes s that the transton on the aggressor lne tends to slow-down the transton on the vctm lne, makng t appear to be delayed n tme. The amount of slow-down s the dfference between the tme the sgnal transton at the far-end of the vctm lne crosses 0.5V dd when the aggressor has made a transton n the opposte drecton, and that when the aggressor remans quet. Slowdown s dependent on the vctm and aggressor sgnal transton tmes, the skew between ther sgnal arrval tmes, and the parameter values that are reflected n the capactve and resstve model components. The uncertanty about the crosstalk-nduced delay ncrease/decrease may be due to varatons n any of the above parameters. Consder a par of coupled nterconnect lnes n some metal layer, whch les n between two delectrc plates (cf. Fgure 1(a) n whch one segment of the coupled nterconnect s depcted.) The two nterconnect lnes run n parallel and are capactvely coupled. Ether lne can be consdered as a vctm, whle the other may be treated as the aggressor. The goal s to statstcally analyze the slowdown of the vctm lne due to aggressor lne actvty. A dstrbuted RC- model (cf. Fgure 1(b)) s used to accurately model the abovementoned nterconnect lne confguraton. In ths crcut, each RC- stage represents an nterconnect segment of a predefned length, L seg, whch s an mportant factor when consderng spatal correlaton among physcal. The couplng between two nterconnect lnes along segment s captured by the couplng capactance C m. The self capactance and resstance of the vctm nterconnect n segment are denoted by C v and R v, respectvely. Note that although lengths of all wre segments are dentcal, due to process varatons, parameter values for each segment are dfferent from those for other segments. (a) (b) Fgure 1: Dstrbuted capactve modelng of coupled nterconnects.

3 The complexty of dstrbuted RC- crcut model sgnfcantly lmts ts applcaton n real world desgns where mllons of nterconnect lnes and hence crosstalk stes are present. Therefore, crcut desgners try to derve the electrcal behavor of ths complex crcut model by approxmatng ts transfer functon usng dfferent model order reducton technques [16]. Generally speakng, the exstng models for coupled nterconnects tend to be naccurate when sgnfcant process varatons exst. On the other hand, the complexty of model order reducton technques sgnfcantly ncreases when consderng process varatons [17]. The remander of ths paper s organzed as follows. Sectons 2 to 4 focus on the slowdown effect of the crosstalk. Secton 5 deals wth crosstalk speedup effect. In sectons 6 to 8 we study the drver strength, sde load, and the nteracton of crosstalk stes. In secton 9 the basc aspects of nterconnect characterzaton and modelng consderng process varatons, ther local effects and correlaton between parameters are revewed. Secton 10 explans the expermental setup used for smulaton of coupled nterconnects n the presence of process varatons. Sectons 11 summarze the conclusons. 2. CROSSTALK SENSITIVY ANALYSIS: DEPENDENCE ON INPUT SKEW Tmng analyss s an essental aspect of determnng whether a crosstalk event can create a faulty output n a crcut. In partcular, the sgnal arrval tmes and transton tmes (nverse of slew rates) n a crcut can change as a functon of the crosstalk nose that s present n the crcut. Therefore, the accuracy of tmng analyss tools strongly depends on the accuracy of arrval tme and transton tme calculatons n the presence of crosstalk nose. In ths paper, we adopt the standard defnton of arrval tme and transton tme that s commonly used n statc and statstcal tmng analyss and ATPG tools, meanng that the arrval tme of a sgnal transton s set to the tme nstance at whch sgnal waveform crosses the 0.5V dd voltage level whereas the transton tme of a sgnal transton s defned as the slope of a lne connectng two specfc ponts on the nosy nput: the ponts are when the sgnal waveform crosses the 0.1V dd and 0.9V dd voltage levels. The skew between two sgnal transtons s the dfference between ther arrval tmes. In ths paper, we adopt the standard defnton of arrval tme and transton tme that s commonly used n tmng tools, meanng that the arrval tme of a sgnal transton s set to the tme nstance at whch sgnal waveform crosses the 0.5V dd voltage level whereas the transton tme of a sgnal transton s defned as the slope of a lne connectng two specfc ponts on the nosy nput: the ponts are when the sgnal waveform crosses the 0.1V dd and 0.9V dd voltage levels. The skew between two sgnal transtons s the dfference between ther arrval tmes. All experments n ths secton through secton 8 use confguratons that are smlar to the one depcted n Fgure 2(a). In ths confguraton, the nverter 4INV x s fed by a long nterconnect lne whch s a potental crosstalk vctm (The sze of einv f s e tmes as bg as that of INV f, where e can be 1, 4, 16, and 64 and f can be x and y.) Aggressor and vctm lnes run parallel to one another. Every 100 m long and 0.200µm wde wre segment s modeled by a sngle stage of an RC- structure. For example, for a 1000 m wre par we use 10 RC- stages as depcted n Fgure 2. The couplng of each stage s modeled by C m ; clearly, for a 1000 m wre par the total couplng value s 10 tmes the C m value. We perform our crcut smulatons wth the TSMC 0.13µm technology and use standard cells from a 130nm, 1.2V producton cell lbrary. The sheet resstance of metal nterconnect n ths technology s /square. The unt lne capactance s 22.6pF/meters. Therefore for a 1000 m wre, the total lne resstance s 370 and the total self capactance (capactance to the ground) of each lne s 22.6fF. From now on, we wll refer to INV x and INV y as the lne drvers. Smlarly, 4INV x and 4INV y wll be called the lne recevers. We wll refer to out_x and out_y (n_u and n_v) as the near-end (far-end) of the lnes. Ether lne can be consdered as a vctm when the other s an aggressor. Fgure 2(b) shows the sde-load, whch s used n some of our experments. out_x n_u sub1_x sub2_x 4INV n_x R R R x sub_f INV sde 4INV sde C C C C C C INV x 16INV x 64INV x R s C INV m C m C m C C y 4INV R R R y 16INV y 64INV n_y y out_y C C C sub1_y C C sub2_y C n_v (b) f x or y, sub0_x out_x, and sub0_y out_y (a) The crosstalk model for long parallel lnes Fgure 2: (a) The confguraton used n our experments, (b): The load confguraton used n some of the experments as the sdeload, connected to the ntermedate pont of lne f.

4 We defne the (normalzed) senstvty of varable p to varable q as =(q. p)/(p. q). We say p s nsenstve to q, when 0; furthermore, p s weakly senstve to q exactly f 0< <0.1, and s moderately senstve f 0.1< <1; otherwse we say that p s hghly senstve to q. At tmes we wll refer to p as slowdown or speedup of p dependng on whether p s postve or negatve, respectvely. We wll also use the term crosstalk-affected delay and transton tme of node p to refer to the delay and transton tme of that node when the mpact of crosstalk capactances are consdered. The senstvty of crosstalk-nduced slowdown to tmng parameters, namely nput skew and nput transton tmes s studed n ths secton. It s shown that crosstalk-nduced delay s hghly senstve to the nput skew and weakly senstve to the nput transton tme. Senstvty analyss s useful for better understandng of the mpact of the crosstalk nose on crcut performance. Consderng the ncrease n process varatons n current process technologes, the results of the senstvty analyss may be used to create more accurate statstcal models based on the varablty of nput parameters. As stated earler, the accuracy of nosy sgnal arrval tme and transton tme calculaton s crucal n tmng analyss; therefore we report the dependence of delay as well as the transton tme of the output ( and ) on the crosstalk nose. 2.1 Output Slowdown as a functon of Input Skew To study the senstvty of the crosstalk-affected output delay to nput skew, the coupled lnes, x and y (shown n Fgure 2,) have been consdered. They run n parallel and each of them s 1000µm long. We create sgnal transtons wth opposte drectons at the nputs of lne drvers, namely n_x and n_y, hence the sgnal transtons of both lnes wll be slowed down. The arrval tme of a fallng transton at n_y s set to 1000ps and the arrval tme of a rsng transton at n_x s swept from to ps; therefore the nput skew between n_x and n_y changes from -1000ps to +1000ps. We also set ther transton tmes to 100ps. Both and exhbt a crosstalk-nduced slowdown n ths case. Fgure 3 shows the slowdown of (delay of w.r.t. n_x) and (delay of w.r.t. n_y.) Couplng capactance s (C m =3.) Inverter cells wth nearly equal fall and rse tme rato s used for all INV cells n the confguraton. We refer to such cells as balanced cells n ths paper. Snce the cells n ths experment are balanced cells the maxmum slowdown at and are very close, e.g., they are less than around 75ps dfferent n case of couplng capactance of. 8.5E-10 8E E-10 7E E-10 6E E-10 5E E-10 4E E E E E E E-10 n_v n_u 2.7E E Fgure 3: Delay from n_x (n_y) to () as a functon of nput skew between n_x and n_y. Fgure 4: Delay from n_x (n_y) to n_u (n_v) and from n_x (n_y) to () as a functon of nput skew. P1: Crosstalk-affected delay can be hghly senstve to the nput skew. Especally, for skew values that are close to the one generatng the worst-case delay, a small change n the skew can sgnfcantly change the delay. For example n Fgure 3, a 25ps nput skew change can result n more than 105ps of the delay change for. P1 hghlghts the mportance of accurately computng the sgnal arrval tme n the presence of crosstalk nose. P2: The worst-case crosstalk slowdown at the output of the vctm lne recever occurs at a certan skew, but a sgnfcant slowdown (e.g., more than 20% delay ncrease) occurs wth a large range of skews. Our experments confrm that even a zero-skew between transtons at the near-end of the lnes,.e., out_x and out_y, may not necessarly create the worst-case crosstalk-nduced slowdown. Note that the crosstalk couplng of the aggressor and vctm lnes s dstrbuted along the length of the lnes and the crosstalk effect at one pont of the vctm lne propagates and affects the subsequent ponts along the vctm lne. Therefore, the crosstalk effect at each pont of the vctm lne s the summaton of couplng effects of that pont plus the delayed effects propagated from the precedng ponts. As a result, the maxmum crosstalk slowdown occurs over a much wder wndow of tme than s usually assumed. One way to reduce the crosstalk effect of a ste s to delberately change the delay of crcut lnes drvng the correspondng vctm and/aggressor lnes (e.g. by usng buffers.) Ths can change the nput skew such that the slowdown created by that crosstalk ste cannot create any error. P2 shows that n order to sgnfcantly reduce the slowdown from ts worst-case level, the nput skew may have to be changed by a rather large amount.

5 P3: The maxmum crosstalk slowdown does not necessarly occur for zero nput skew condton even for completely symmetrc nterconnects. P3 provdes motvaton for establshng a framework for algnment of multple aggressors and the vctm lne such that the worst-case crosstalk effect s generated Slowdown effect at the far-end of the vctm lne. In Fgure 4 we report the slowdown of n_u (n_v) w.r.t. n_x (n_y) and compare ths slowdown wth results of Fgure 3 (slowdown of () w.r.t. n_x (n_y). P4: Delay at the output of the vctm lne recever, (), follows the shape of the delay at the far-end of the vctm lne, n_u (n_v). 2.2 Output Transton Tme as a functon of Input Skew A new experment smlar to the one descrbed n Secton 2.1 s set up. The only dfference s that now the transton tme change at the nterconnect output (/) due to the crosstalk effect s smulated. Fgure 5 shows the dependence of transton tme of and on the nput skew. The followng summarzes the observatons made from ths experment. P5: The output transton tme can be hghly senstve to the nput skew. Especally, for skew values that are close to the one generatng the worst-case ncrease n transton tme, a small change n the skew can sgnfcantly change the transton tme. For example n Fgure 5 less than 20ps change n skew can result n more than 200ps ncrease n the transton tme of. P6: The maxmum transton tme at the output of the vctm lne occurs for a certan nput skew, wth a sgnfcant ncrease n transton tme occurrng for a large range of nput skew values. P7: The maxmum transton tme at the output of the vctm lne recever does not occur for the zero nput skew even for completely symmetrc nterconnects. Moreover, the skew that results n the maxmum transton tme may not be the one that results n the maxmum slowdown. 6.5E-10 6E E-10 5E E-10 4E E-10 3E E-10 2E E E E-09 n_v 1.2E-09 n_u 1E-09 8E-10 6E-10 4E-10 2E Fgure 5: Transton tmes of and as a functon of nput skew. Fgure 6: Transton tmes of n_u (n_v) and () as a functon of nput skew Transton tme change at the far-end of the vctm lne In Fgure 6 we compare the transton tme of the sgnal transtons at the far-end of the vctm lne,.e., n_u (n_v) wth that of the output of the vctm lne recever,.e., (). In contrast to what we observed n Fgure 4 for slowdown, the transton tme comparson shows dfferent characterstcs. P8: Electrcal waveforms for the transtons at the far-end and the output of the vctm lne recever dffer sgnfcantly n terms of ther characterstcs. In general the transton at the nput of a gate tends to be smoothed out, and hence, the transton at the gate s output wll not change as drastcally as the change n the gate s nput transton. 3. CROSSTALK SENSITIVITY: DEPENDENCE ON TRANSITION TIME To study the effect of transton tme of sgnals at the nput of the vctm lne drver and/or the aggressor lne drver, we keep the skew between the transtons at n_x and n_y fxed at zero. We apply a fallng transton at n_y and a rsng transton at n_x wth dentcal arrval tmes so that both and wll experence crosstalk-nduced slowdown. We consder a reasonable range of transton tmes from 0 to 600ps.

6 We wll consder two scenaros for the transton tme change. In the frst scenaro, the transton tmes of both n_x and n_y are changed. In the scenaro, only one of the nput transton tmes s changed whle the other one s kept constant. A balanced nverter cell has been used for both INV cells n the confguraton of Fgure Both Input Transton Tmes Change The transton tme of n_x and n_y are dentcal and vary n lockstep from 0 to 600ps. Fgure 7 llustrates how the slowdown of the transtons at and change based on change of transton tmes of n_x and n_y. It s seen that a 600ps ncrease n nput transton tme of both n_x and n_y causes only a 145ps slowdown for (wth a couplng capactance value of.) Therefore, assumng equal transton tmes for the aggressor and vctm nputs, the slowdown at the output of the vctm lne recever s only weakly senstve to ts nput transton tme. Comparng P12 wth P1, we conclude that crosstalk-affected delay senstvty to the nput transton tme s much lower than that to the nput skew. Ths has the mplcaton that, as far as crosstalk s concerned, the accuracy of arrval tme computaton s more mportant than the accuracy of transton tme computaton. Fgure 8 llustrates how the transton tme of the transton at the output of the crosstalk ste,.e. / would change when transton tmes of both n_x and n_y change. From ths fgure, a 600ps ncrease n the nput transton tme of both n_x and n_y changes the transton tme at by only 10ps. 3.2 Only One Transton Tme Changes We smulate the crosstalk effect by keepng transton tme of the sgnal transton at n_y constant at 100ps and then changng transton tme of n_x from 0 to 600ps. Other parameters have been set smlar to those of the experment reported n Secton 3.1. Fgure 9 shows the effect of transton tme change at one nput (n_x) on the slowdown seen at the outputs. Consderng n_y and as the nput and output of the vctm lne, there wll be less slowdown at f the transton tme at the nput of the aggressor lne drver, n_x, ncreases. 9.E E-10 8.E-10 7.E E-10 6.E-10 5.E E E E-10 3.E-10 2.E E E E Fgure 7: Delay from n_x (n_y) to () as a functon of nput transton tme (both transton tmes change) for dfferent couplng capactance values. Fgure 8: Transton tmes of and as a functon of nput transton tme (both change) for dfferent couplng values. P9: Crosstalk-affected delay and transton tme of the output of the vctm lne are only weakly senstve to ts nput transton tme. P10: For a gven transton tme at nput of the vctm lne drver, faster aggressor causes larger worst-case slowdown. P11: The maxmum slowdown occurs when the vctm has the largest transton tme whereas the aggressor has the smallest transton tme. Table 1 lsts the slowdown for several nterestng transton tmes taken from Fgure 7 and Fgure 9. In the last row, the slowdown values for the last three columns (664, 718, and 720) substantate P14. Comparng the column entry (920) wth entres of columns 1, 3 and 4 (919, 805, and 785) substantates P15. In Fgure 10 we study a smlar effect to what was presented n Fgure 8. However, only transton tme at n_x s changed. P12: Slower aggressor creates slower transtons at the output of the vctm lne recever.

7 1.E E-10 9.E-10 8.E-10 7.E E E-10 6.E-10 5.E-10 4.E-10 3.E-10 2.E E Fgure 9: Delay from n_x (n_y) to () as a functon of transton tme of n_x for dfferent couplng values. 3.0E E E E E Fgure 10: Transton tme () as a functon of transton tme of n_x. 4. CROSSTALK SENSITIVITY: DEPENDENCE ON CIRCUIT PARAMETERS In ths secton we nvestgate the dependence of crosstalk-nduced slowdown on the couplng capactance value, wre capactance, and wre resstance. 1.20E E E E E-10 skew = A(n_x)-A(n_y) 2.00E E Total Couplng (ff) skew = skew =- skew =- skew = Fgure 11: Delay from n_x (n_y) to () as a functon of couplng value for three dfferent nput skews. 5.00E E E E E E E E-10 skew = A(n_x)-A(n_y) E Total Couplng (ff) Fgure 12: Transton tmes of and as a functon of couplng value for dfferent nput skews. 4.1 Dependence on the Couplng Capactance Value The couplng (capactance) value s the man factor n determnng the magntude of any crosstalk nose. Nose senstvty analyss wth respect to the couplng value are thus mportant to optmzaton algorthms such as wre spacng and buffer nserton, whch am at reducng the couplng value and subsequent mnmzaton of the crosstalk effect. We ran the experment descrbed n Secton 2.1 wth dfferent values of the couplng capactance. The couplng capactance value, C m, was swept from 0 to 5,.e., the total couplng between lnes x and y was changed from zero to 50. The nput skew was changed from - to +. However, for the sake of readablty of the plots, only the results for three nput skew values (.e., -, zero-skew, and +) are provded. Fgure 11 shows the slowdown of the output, (delay of w.r.t. n_x) and (delay of w.r.t. n_y.) Fgure 12 provdes the correspondng transton tmes. As expected the crosstalk-affected delay s hghly senstve to the couplng value. Both output delay and output transton tme are well approxmated by a lnear model. An mportant mplcaton s that statstcal analyss of crosstalk effect as a functon of varablty n C m can accurately be modeled by a frst order canoncal model [18],.e., there s no need for more complcated models such as the quadratc ones suggested n [19]. P13: Both slowdown and transton tme ncrease at the output of the vctm lne recever are hghly senstve to the couplng capactance value. Furthermore, both of these quanttes are well approxmated by assumng a lnear dependence on the couplng value. Now we provde results for the same experments as above but ths tme wth respect to the nput skew. For the sake of mproved readablty, results for only four couplng values are provded. Fgure 13 shows the slowdown at the outputs, and, whereas Fgure 14 shows the correspondng data of the transton tme ncrease. -

8 8.50E E-10 6E E-10 5E E E E-10 3E E E-10 2E E E Fgure 13: Delay from n_x (n_y) to () as a functon of nput skew between n_x and n_y for dfferent couplng values. Fgure 14: The transton tmes of and as a functon of nput skew for dfferent couplng values. P14: The nput skew values that cause the maxmum slowdown and largest transton tme at the output of the vctm lne recever are a strong functon of the couplng capactance value. However, ncrease n the couplng value, and hence crosstalk slowdown, does not necessarly result n an ncrease n the transton tme. Referrng to Fgure 14 and focusng on the skew range of about -200 to -100ps, we observe that the transton tme of for the case of 300f couplng s lower than that for 20, whch confrms the latter part of observaton P14. It s worthwhle to menton that although throughout ths paper we report the results of experments performed on 1000µm parallel lnes, we have confrmed that the results are equally vald for shorter lnes. As an example Fgure 15 shows the crosstalk-affected delay of the outputs versus the nput skew for two lnes whch are 300µm each and run n parallel to one another (modeled by three stages of the RC- structure.) It s seen that the results follow smlar patterns to those for the 1000µm-long parallel lnes. 3.4E-10 transton tme(n_x) transton tme(n_y) delay() delay() E E E-10 15fF fF Table 1: Crosstalk-affected delay senstvty to the nput transton tme (n ps.) 1.4E nput skew (ps) Fgure 15: Delay from n_x (n_y) to () as a functon of nput skew for two 300 m-long parallel lnes. 4.2 Dependence on the Wre Capactance We performed a smlar set of experments to the ones n Secton 2.1 n order to assess the senstvty of crosstalk-nduced output delay and transton tme to the (self) wre capactance value. The total wre capactance for both wres s swept from zero to 50. Other parameters have been set as explaned n Secton 2.1. Fgure 16 and Fgure 17 show the output delay and transton tme vs. the wre capactance value, respectvely. As before for readablty purposes, the curves for only one couplng () and three nput skew (-, 0, +) values are shown. Results for other couplng and nput skew values are smlar. As expected both the output delay and transton tme exhbt a lnear relatonshp wth the wre capactance.

9 1.30E E E E E E E E E E-10 skew = A(n_x)-A(n_y) E w relne capactance (ff) Fgure 16: Delay from n_x (n_y) to () as a functon of the wre capactance for three dfferent nput skew values E E E E E E E E-10 skew = A(n_x)-A(n_y) 2.20E Wrelne capactance (ff) Fgure 17: Transton tmes of and as a functon of the wre capactance for dfferent nput skew values. To evaluate the crosstalk effect when the wre capactance of only one of the coupled lnes vares, we swept the wre capactance of lne x from zero to 50 whle keepng that of lne y at a constant value of 22.6fF. Results are provded n Fgure 18 and Fgure 19. Based on a lumped RC model for crosstalk nose analyss of [3] and [4] the crosstalk nose s nversely proportonal to the wre capactances of both the aggressor lne and the vctm lne. However, our experments show that ths s not true. For example, n Fgure 18, as the wre capactance of lne x ncreases the delay of that lne also ncreases,.e., the crosstalknduced delay ncreases monotoncally wth the vctm lne wre capactance. However, the delay of lne y decreases for zero skew but ncreases for large skew values. Ths hghlghts the fact that the nverse proportonalty relatonshp of the crosstalknduced effect to the aggressor lne wre capactance s n general not vald. Smlar behavor s seen n Fgure 19 for the output transton tme. P15: The crosstalk-affected propagaton delay s moderately senstve to the wre capactance value. In addton, t monotoncally ncreases as the vctm wre capactance ncreases (vctm output: ); However, t does not show a monotone behavor wth respect to the aggressor wre capactance (vctm output:.) 1.30E E E E E E E E E E-10 skew = A(n_x)-A(n_y) 3.00E Wre capactance of lne x (ff) Fgure 18: Delay from n_x (n_y) to () as a functon of wre capactance of lne x for three dfferent nput skew values E E E E E E E E-10 skew = A(n_x)-A(n_y) E Wre capactance of lne x (ff) Fgure 19: Transton tmes of and as a functon of wre capactance of lne x for dfferent nput skew values. In Fgure 19, even the output transton tme of shows a non-monotone relatonshp wth respect to vctm lne capactance,.e., the monotone behavor assumpton of crosstalk-affected output transton tme wth respect to the wre capactance of the vctm or the aggressor lne s n general nvald. P16: The crosstalk-affected output transton tme s moderately senstve to the wre capactance value, however n general t does not exhbt a monotone relatonshp wth respect to vctm or aggressor wre capactance. The corner-based worst-case analyss used n conventonal STA or ATPG tools s generally assumed to be pessmstc. However, the above results show that t can also be optmstc. Due to lack of the monotone behavor property and the msconcepton about nverse proportonalty of the crosstalk effect on the wre capactances, the corner-based analyss can ndeed underestmate the magntude and severty of the crosstalk problem. For example, from Fgure 18 the lack of the monotonc behavor for the slowdown w.r.t. the wre capactance produces around 20ps delay underestmaton at for wre capactance of 5 (50% error consderng the offset delay of 40ps for delay of w.r.t. n_y.)

10 4.3 Dependence on the Wre Resstance We swept the wre resstance of both lnes from zero to 500 ohms and ran a smlar set of experments to the one descrbed n Secton 2.1. Fgure 20 and Fgure 21 show the results for a total couplng value of. The delay of both outputs monotoncally ncreases; however, the output transton tme may exhbt non-monotone behavor n some cases (cf. transton tme at zero skew n Fgure 21.) 8.40E-10 skew = A(n_x)-A(n_y) 3.40E E E E E E w relne resstance ( ) Fgure 20: Delay from n_x (n_y) to () as a functon of the wre resstance for three dfferent nput skew values E E E E-10 ns E-10 0ns 2.20E E Wrelne resstance ( ) Fgure 21: Transton tmes of and as a functon of the wre resstance for dfferent nput skew values. Fgure 22 and Fgure 23 contan the data for the case when the wre resstance of only lne x s changed whle keepng that of lne y at a constant value of 370. From Fgure 22 the crosstalk-affected delay of the vctm output () decreases when the wre resstance of the aggressor lne (lne x) ncreases. Also the delay of ncreases whch shows that the delay of the vctm lne ncreases wth the ncrease n the vctm lne wre resstance. It s seen that both crosstalk-affected delay and transton tme of the outputs can well be approxmated by usng lnear equatons. skew = A(n_x)-A(n_y) 8.00E E-10 0ns 6.00E E E E Wrelne resstance of lne x 3.60E E-10 ns 3.20E E E E E E E Wrelne resstance of lne x Fgure 22: Delay from n_x (n_y) to () as a functon of the wre resstance of lne x for 3 dfferent nput skew values. Fgure 23: Transton tmes of and as a functon of the wre resstance of lne x for dfferent nput skew values. P17: The crosstalk-affected output delay and transton tme are weakly senstve to the wre resstance value. In partcular, they monotoncally ncrease as the vctm wre resstance ncreases (vctm output: ) whle monotoncally decreasng as the aggressor wre resstance ncreases (vctm output:.) In both cases the effect can be well approxmated by lnear equatons. The reader s remnded that the range of the parameters has an mportant role n extractng the propertes of crosstalk nose and ts mpact on crcut performance; t s suffcent to consder realstc ranges of parameters to reduce the complexty of analyss and modelng. The results presented prevously are all qualfed to the range of parameters consdered durng the smulaton. These results can change f the parameter range s modfed. To make ths pont more clear, let s suppose that the wre resstance of lne x can range from 0 to 5k (notce that the upper range s too hgh for typcal nterconnects n VLSI crcuts.) The last two experments (as reported n Fgure 22 and Fgure 23) are repeated wth ths extended wre resstance range and the results are provded n Fgure 24 and Fgure 25. Notce that the monotone behavor, whch was observed for the reasonable range of wre resstances (0 to 500 ), does not exst for ths new range (0 to 5k.)

11 3.00E E E E E E-10 skew = A(n_x)-A(n_y) - 0ns E Wrelne resstance of lne x ( ) Fgure 24: Delay from n_x (n_y) to () as a functon of the wre resstance of lne x (rangng from 0 to 5k ) for three dfferent nput skew values. 1.70E E E E E E E E E Wrelne resstance of lne x ( ) 0ns - skew =- 0ns Fgure 25: Transton tmes of and as a functon of the wre resstance of lne x (rangng from 0 to 5k ) for dfferent nput skew values. 5. CROSSTALK-INDUCED SPEEDUP To study the crosstalk-nduced speedup effect, we consder transtons wth the same drecton of change at nputs, n_x and n_y. We set the arrval tme of a rsng transton at n_y to 1000 ps and sweep the arrval tme of a rsng transton at n_x from to ps; therefore the nput skew between n_x and n_y changes from -1000ps to +1000ps. We also set the nput transton tmes to 100ps. Fgure 26 llustrates the speedups that occur at outputs, and based on the nput skew change. Fgure 27 llustrates the output transton tme vs. the nput skew for the speedup case. Havng compared Fgure 26 wth Fgure 3, we fnd that the maxmum speedup at the vctm s output s 233ps whereas the maxmum slowdown was 390ps. Fgure 5 and Fgure 27 how that the maxmum decrease n transton tme at the vctm s output s 110ps whereas the maxmum ncrease n transton tme for the slowdown case was 390ps. Hence, the amount of speedup for the same confguraton s lower than the slowdown and the transton tme change n the speedup case s less than that n the slowdown case. Snce both lnes make transtons n the same drecton, the and curves are symmetrc to each other. Other observatons are smlar to those of the slowdown ones. Fgure 26 and Fgure 27 show that even for transtons n the same drecton the zero skew may not create the worst case slowdown or output transton tme. So P3 and P7 must be true even for completely balanced cells wth equal rse and fall tme transtons. 3.8E E E E E E E E E E E Fgure 26: Delay from n_x (n_y) to () as a functon of nput skew for dfferent couplng values (speedup case.) 1.3E E Fgure 27: Transton tme of and as a functon of nput skew for dfferent couplng values (speedup case.) 6. DRIVER STRENGTH Drver szng s consdered as one of the most effectve means of crosstalk reducton n optmzaton tools [20], [21]. In ths secton, we frst study the effect of unbalanced cells on crosstalk nose and then do senstvty analyss of ths nose wth respect to the drver strength. 6.1 Dependence on the Drver Strength To study the behavor of the crosstalk-affected output delay as a functon of the drver strength, the sze of INV y s kept constant, whle that of INV x s swept from 0.2 to 10 tmes the sze of INV y. Note that the sze of the recever 4INV x and

12 4INV y are kept constant at 4 tmes the sze INV y n ths experment; other parameters are set as explaned n Secton 2.1. Fgure 28(a) shows the output delay versus the rato of sze(inv x ) to sze(inv y ) for couplng capactance values of 0, 50, and. The drver szng technques usually attempt to take advantage of the trade-off whereby ncreasng the drver sze aggravates the crosstalk effect for coupled lnes. Ths trade-off s reasonable when a lumped RC modelng s appled n whch the drvers are modeled as lnear crcut elements. Consderng lne x as the vctm lne and lne y as the aggressor lne, Fgure 28 confrms that the crosstalk-nduced slowdown of wll decrease exponentally f the nput drver sze of the vctm lne s ncreased. It also shows that the crosstalk-nduced slowdown of s aggravated by an ncrease n the sze(inv x ). However, the amount of slowdown aggravaton at s much less than (and almost neglgble compared to) the slowdown decrease at. Ths s more remarkable when notcng that the two lnes are completely symmetrc n everythng but the sze of the nput drvers. 3.6E E E E E E E E sze(invx)/sze(invy) 5 8.9E E E E E E E E E sze(invx)/sze(invy) (a) (b) zoomed n the reasonable sze rato Fgure 28: Delay from n_x (n_y) to () as a functon of nput drver sze rato. Inspred by the typcal concept of fanout-of-4 (FO4) for gate delay estmaton, t s reasonable to assume that the drver sze of an nterconnect s at least ¼ th of the sze of ts recever, Ths mples that the mnmum rato of sze(inv x ) to sze(inv y ) s set to 1. Wth ths szng constrant, Fgure 28(b) shows that the so-called trade-off may not be always vald. Ths s because the ncrease n crosstalk nose of the coupled lne y s very small and qute neglgble compared to the crosstalk reducton obtaned n lne x (even for a large couplng value of.) It s also worth notcng that ncreasng the drver sze rato beyond a certan pont, say 4, can hardly change the crosstalk-affected delay. Fgure 29 shows smlar results for the transton tme versus the drver sze change. In ths case a non-monotone behavor exsts. P18: Crosstalk-affected delay (transton tme) at the output of the vctm lne recever can be hghly (moderately) senstve to the drver strength of the vctm lne, but weakly senstve to that of the aggressor lne. 4.9E E E E E E E E E sze(invx)/sze(invy) 5 3.4E-10 (a) (b) zoomed n for a reasonable szng rato Fgure 29: Transton tmes of and as a functon of nput drver sze rato. 6.2 Unbalanced Cells So far we reported experments on confguratons wth nverter cells wth nearly equal rse and fall tmes. We call these cells balanced. To see how dfferent rse and fall tmes may affect the results, we use drver and recever nverter cells wth dfferent pulldown and pullup strengths. We refer to these types of logc gates as unbalanced cells. Fgure 30 and Fgure 31 show the delay and transton tme change vs. nput skew smlar to confguraton of Fgure 3 and Fgure 5 respectvely, but wth unbalanced cells used as lne drvers and recevers. The fallng transton at n_y occurs at +2000ps whereas the rsng 2.9E E E E E sze(invx)/sze(invy) 5

13 transton at n_x occurs between 0 to +4000ps,.e., the nput skew changes from -2000ps to +2000ps. The delay value for very large negatve or postve skews actually captures the delay of the nterconnect output whch s not affected by any crosstalk. For example, the delay of for the skew of -2000ps s around 470ps and that for the skew of +2000ps s around 410ps. The dfference between the two delay values s the delay of an nterconnect lne that s nfluenced by the voltage level of the other nterconnect through the couplng capactance. P19: Crosstalk-affected delay and transton tme at the output of the vctm lne recever are hghly senstve to the rato of pull-up and pull-down strengths of the nverter cells. 1.05E E E E E E E E E E E E E E E E E Fgure 30: Delay from n_x (n_y) to () as a functon of nput skew usng an unbalanced cell. Fgure 31: Transton tmes of and as a functon of nput skew for an unbalanced cell. Fgure 30 ponts out the non-monotone property of the crosstalk effect. Assume that the arrval tme of n_x s sped up (e.g., as a result of the speedup effect of a crosstalk ste n the transtve fan-n of node n_x) such that the nput skew between n_x and n_y s reduced from 400 to 300ps. Ths skew reducton creates a 650ps ncrease n the crosstalk-nduced slowdown at. Now, lookng at the same scenaro n the opposte drecton, we can see that an nput skew ncrease from 300 to 400ps wll reduce the delay at by 650ps. So, n general, crcut scenaros can be found such that a speedup at the nput lne of a crosstalk ste can result n ether a speedup or a slowdown effect at the output of the ste. Smlarly, an nput slowdown may cause an output slowdown or output speedup. In Secton 7 we wll further explore the mpact of ths nonmonotone behavor when the crosstalk stes nteract wth one another. P20: Crosstalk effect exhbts a non-monotone behavor wth respect to the skew between the arrval tmes of the nputs of the aggressor and vctm lne drvers. 7. INTERACTION OF SITES The crosstalk nduced speedup or slowdown effects of two crosstalk stes, each smlar to the one shown n Fgure 1, may nteract wth each other f one s n the transtve fan-out of the other. In ths secton we show ths nteracton can generate a total delay effect that s more sgnfcant than the delay effects caused by each ste n solaton. 7.1 Interacton of Two Slowdown Effects Assume each has a total couplng value of. Assume both stes use the same moderately balanced cell that was used n Secton 2.1 for all INV cells n the model. Consder a fallng and a rsng transton at the nputs n_x and n_y of the frst crosstalk ste. Fgure 13 showed the slowdown effect vs. skew for ths ste. From Fgure 13 a 30ps decrease n the arrval tme of the transton at n_x of the frst crosstalk ste, whch s equvalent to a 30ps ncrease n ts nput skew, can result n a slowdown of 150ps at of ths ste. Let s consder a worst-case scenaro where all of ths slowdown effect wll reach the nput of the crosstalk ste. Referrng to Fgure 13, a 150ps change n nput skew can cause an output slowdown of up to 400ps. Therefore the total slowdown along the path s or 580ps. If we study the slowdown effect of the crosstalk stes one at a tme, then we wll ncorrectly conclude that a 30ps change n the nput skew of the ste wll create 150ps slowdown on each ste, and thus to the total slowdown of the path s or 330ps. Therefore, separate worstcase analyss of the two crosstalk stes would underestmate the total path slowdown by 93%. In addton, we should take nto account the transton tme change created at the output of the crosstalk stes. For example, n the case that one crosstalk ste drectly feeds nto the other, from Fgure 14, a 30ps change n the nput skew causes a 90ps transton tme change at the output of the frst ste and nput of the ste. Ths n turn creates around 30ps extra slowdown at the output of the ste. Ths means that the total path slowdown s actually or 610ps. P21: Crosstalk stes along a path may result n a sgnfcant ncrease n crcut delay, whch can be much hgher than the summaton of delay ncreases caused by each ste ndvdually.

14 7.2 Interacton of Slowdown and Speedup Effects Assume a frst ste wth total couplng value of 5 uses the same moderately balanced cell that was used n Secton 5 for all INV cells n the model. Assume rsng transtons at the nputs of n_x and n_y of the frst crosstalk ste. Fgure 26 showed the speedup effect vs. skew for ths ste. A 240ps decrease n the arrval tme of the transton at n_x of the frst ste, whch s equvalent to a 240ps ncrease n the nput skew, can n the worst case cause a 60ps speedup at the output of the ste. Ths speedup s n turn equvalent to a decrease n the nput skew of the ste that s n the transtve fanout of the frst one. The ste has a total couplng value of. It uses the unbalanced cell used n Secton 6.2. Therefore, from Fgure 30, a 60ps decrease n the nput skew can create up to 650ps ncrease n slowdown. Therefore the total slowdown along the path s or 830ps. Studyng the ste n solaton, a 240ps ncrease n the nput skew of the ste, whch from Fgure 13 means no slowdown at the output of ths ste, could be generated by the ste f the frst ste dd not exst. The total slowdown created by each ste n solaton s =220ps. Therefore the total slowdown caused by the nteracton of ste s more than 3.7 tmes as large as the summaton of crosstalk effects n solaton. Ths example hghlghts the non-monotone behavor of crosstalk ste descrbed n P12. The key to the synergstc nteractons dscussed n 7.1and 7.2 s that crosstalk-affected delay s hghly senstve to the nput skew (refer to propertes P3 and P7.) 8. SIDE-LOAD ROUTING Consder the sde-load of Fgure 1(b); Assume t has to be routed n connecton wth lne x. In the process t may be connected to any ntermedate pont along lne x,.e., out_x, sub1_x,, or n_u. The queston s whch pont gves the best performance n terms of crosstalk-affected delay. The followng experment was conducted to answer ths queston. We connected the sde-load to out_x and then swept the sze of the sde-load, INV sde from 0.2 to 4 tmes the sze of lne x drver,.e., INV x. We repeated ths experment for the sde-load connected to other ntermedate ponts. Fgure 32 and Fgure 33 llustrate the output delay and transton tme for both lnes vs. the nput skew for sze(inv sde ) equal to that of INV x for three dfferent couplng values (couplng values n the range 0 to 50 showed smlar behavor.) 7.9E E E E E E E E skew (psec) = A(n_x)-A(n_y) 5.9E E E E E E skew (psec) = A(n_x)-A(n_y) Fgure 32: Delay from n_x (n_y) to () as a functon of nput skew for dfferent sde-load connecton ponts for couplng values 0, 50, and. sze(inv sde ) = sze(inv x.) Fgure 33: Transton tmes of and as a functon of nput skew for dfferent sde-load connecton ponts for couplng values. Fgure 34 llustrates the delay for the case of sze(inv sde ) four tmes as large as that of INV x, (the transton tme fgure s not shown snce the results are smlar to Fgure 30.) Note that for a certan couplng value, the curves correspondng to dfferent connecton ponts almost fully overlap each other (actually hardly dstngushable from one another,) meanng that the connecton pont of the sde-load of nterconnect does not change the crosstalk-nduced output delay (and transton tme) of the nterconnect. P22: Crosstalk-affected output delay and transton tme have zero senstvty wth respect to the connecton pont of the sdeload.