Improved Methods for Lithography Model Calibration

Transcription

1 Improved Methods for Lthography Model Calbraton Chrs Mack Austn, Texas Abstract Lthography models, ncludng rgorous frst prncple models and fast approxmate models used for OPC, requre calbraton usng measured lnewdth data. For models that predct process wndow behavor, the basc calbraton data s lnewdth versus focus and exposure over a range of feature szes and types. The most common numercal method of fndng the best ft model parameters s standard least-squares regresson. Whle smple, ths approach suffers from a number of well known problems. Frst, least-squares regresson n not robust, meanng that even one bad data pont can make the ft meanngless. Thus, outler rejecton becomes an mportant part of ths approach. Both outler rejecton strateges and the use of robust fttng methods wll be dscussed. Second, standard least-squares may weght the data usng the uncertanty n the measured lnewdths, but uncertanty n the nput varables, focus and exposure, s gnored. Often, at the extremes of focus and dose, errors n focus and dose actually domnate the resultng uncertanty n the measured lnewdth. Ths can be accounted for usng total least-squares regresson. Whle often computatonally dffcult, n ths paper an extremely fast and smple method for total least-squares regresson wll be presented for focus-exposure lnewdth data. Fnally, uncertanty n nomnally fxed parameters, such as the lnewdths of the features on the photomask used n the calbraton, can lead to sgnfcant uncertanty n the resultng model parameters. The two standard approaches for dealng wth ths would be to leave these parameters fxed, or allow them to float and be adjusted for best ft. Nether approach s satsfyng. A better soluton s to use Bayesan fttng, where a pror estmates of the mask feature wdths and ther uncertantes are used n the fttng mert functon. Keywords: Calbraton, parameter estmaton, regresson, least-squares regresson, total least-squares, Bayesan parameter estmaton 1. Introducton to Parameter Estmaton Obvously, a prmary goal of the use of a lthography model s for the output of that model to be predctve of some actual lthography process. Unfortunately, t s extremely unlkely that all of the nput parameter settngs for such a lthography model wll be known a pror to an accuracy that allows the predctve capablty of the model to be at ts maxmum. Thus, before usng a lthography model, whether that model s based on frst prncples or s purely emprcal, t wll be necessary to estmate the nput parameters of the model through the use of measured data. Wth respect to parameter estmaton, all nput parameters fall nto two categores: parameters that can be drectly measured and those that can t (or aren t). Drectly measured parameters can nclude such nputs to a lthography model as wavelength, numercal aperture, lens aberratons, source shape parameters (partal coherence), resst thckness, resst absorpton coeffcent, mask lnewdth, etc. Other parameters, however, are more dffcult to measure drectly. As a result, ther values are estmated by comparng model predctons to expermental lthography Photomask and Next-Generaton Lthography Mask Technology XIV, edted by Hdehro Watanabe, Proc. of SPIE Vol. 6607, 66071D, (007) X/07/$18 do: / Proc. of SPIE Vol D-1

2 data n a process known as model calbraton. Typcally, model calbraton data sets are made up of resst crtcal dmenson (CD) as a functon of mask CD and ptch, and often as a functon of focus and exposure. Parameters that are often estmated through the process of model calbraton nclude lens aberratons, source shape parameters, mask lnewdth (or bas from desgned lnewdth), and many resst model parameters (acd dffusvty, development rate parameters, reacton rate constants, emprcal resst model parameters, etc.). [Note that there s sgnfcant overlap between the lst of parameters that can be drectly measured and those that are often estmated through ndrect calbraton. Not all parameters that can be measured are measured, often to the detrment of model predctablty. Also, t s mportant to remember that even drect measurement of parameters leads to parameter uncertanty snce all measurements nclude measurement error. Ths subject wll be address below n the dscusson of Bayesan parameter estmaton.] Model calbraton nvolves adjustng the unknown nput parameters of the model so that the predctons of the model are, n some way, a best ft to the expermental calbraton data. One of the keys to maxmzng the predctablty of the resultng model s n defnng an approprate metrc of best ft for ths calbraton process. One of the most common approaches, whch wll be revewed brefly here, s called least-squares regresson. The least-squares regresson method begns by defnng a cost functon that compares model to data: χ = N = 1 ) y y σ = N z = 1 (1) where χ s the cost functon (pronounced ch squared ), N s the total number of expermental data ponts, y s the th expermental data pont (wth an assocated set of nput condtons, such as the dose, focus, mask CD, and ptch used n the experment for that data pont), y ) s the model predcton based on the same condtons as the th expermental data pont, and σ s an a pror estmate of the measurement uncertanty for y. The ) term ε = y y s called a model resdual, and z s the reduced (normalzed) resdual. The goal of least-squares regresson s to fnd the set of model parameters that mnmzes ths cost functon. There s an obvous queston that arses: In what sense are the model parameters estmated n ths way the best parameters possble gven ths set of data? It can be readly shown that f the model resdual s a normally dstrbuted random varable wth a mean of zero (and all resduals are ndependent of each other), the least-squares regresson method produces the maxmum lkelhood parameters. Ths means that f dfferent sets of expermental calbraton data were to be collected under dentcal expermental condtons, the parameters generated by least-squares regresson of the frst calbraton set would do the best possble job of predctng all of the other data sets n the lmt of an nfnte number of data sets. The maxmum lkelhood nterpretaton of a least-squares regresson makes ths method of parameter estmaton extremely appealng. However, some very restrctve condtons must be met before ths nterpretaton s possble. In order for the model resduals to be normally dstrbuted wth a mean of zero, there must be no systematc expermental errors n the collecton of the y data ponts, and the model must be a perfect representaton of the expermental process (that s, wth no systematc model errors). Further, the measurement uncertanty σ must be known a pror. In realty, none of the condtons are ever met they can only be approached through very careful theoretcal and expermental effort. Expermental CD data over a wde range of condtons wll most certanly contan both random and systematc errors. And snce lthographc models are contnung to evolve and mprove over tme there s lttle lkelhood that they do not Proc. of SPIE Vol D-

3 contan systematc dfferences as compared to realty (ths s especally true of the smplfed models used for optcal proxmty correcton). Because of these problems, the least-squares regresson technque does not n general produce the set of model parameters that maxmzes the predctablty of the model. In the sectons below, a number of mprovements to the least-squares technque wll be descrbed. These mproved parameter estmaton methods are well known and well establshed wthn the mathematcal statstcs communty, and thus n one sense ths paper represents a farly straghtforward revew of the parameter estmaton lterature. However, t s ths author s experence that none of these technques are wdely practced (nor wdely known) wthn the lthography communty.. Regresson Robustness and the Expermental Outler As mentoned above, one assumpton of the least-squares regresson s that all expermental errors are random. In fact, systematc errors always exst when makng a measurement. Obvously, f these systematc errors were known they would be mmedately corrected for. Thus, t s the unknown systematc errors n an experment that are of concern. One nstance of a systematc error that s extremely troublesome s the data outler (or flyer). The classc bad data pont, an outler s best defned as an observaton that devates so much from other observatons as to arouse suspcons that t was generated by a dfferent mechansm. 1 As an example, consder the measurement of a focus exposure matrx usng a top down CD SEM. If, at a certan exposure and focus condton, the pattern to be measured happens to experence pattern collapse (where the tall resst lne mechancally fals and falls over on ts sde), the resultng CD value wll be far off compared to what would otherwse be expected. If the model beng calbrated wth ths data does not nclude pattern collapse mechansms, t certanly would not be expected to do a good job of predctng ths CD SEM measurement value. Snce data outlers are not at all uncommon n lthography model calbraton, ther nfluence on the resultng estmated parameters must be understood. Ideally, the parameter estmaton method should not be overly senstve to a few outlers. A lack of senstvty to outlers s called model calbraton robustness, whch can be defned n several dfferent ways. One smple descrpton of robustness s the breakdown pont, the mnmum fracton of outlers n the calbraton data set that can cause a change of arbtrary magntude n the estmate of a parameter. One extremely problematc aspect of least-squares regresson s that ts breakdown pont s zero: even one bad data pont can completely nvaldate the usefulness of the estmated parameter set. Ths stuaton arses from the squared nature of the cost functon the nfluence of a data pont on the ft goes as the square of ts dfference from ts expected value (.e., from the model). Snce data ponts wth purely random errors typcally devate from the best-ft model by one standard devaton, one data pont wth a 5σ error wll nfluence the ft as much 5 more typcal data ponts. The most common way to deal wth the lack of robustness n least-squares regresson s through outler rejecton removng data ponts suspected of beng outlers before usng the data for calbraton. As one mght magne, the challenge s n determnng whch data ponts are truly outlers, rather than vald data ponts that smply don t match our expectatons. There are many outler rejecton methods 3, a few of whch are lsted below, n ncreasng order of statstcal rgor (though some crteron are better under certan condtons, such as for small data sets): Manual removal Multple of sgma Multple of the nterquartle range Proc. of SPIE Vol D-3

4 Q-test Chauvenet s crteron Perce s crteron Grubb s test (extreme studentzed devate) All of these methods, even the more rgorous Grubb s test 4, make the assumpton that, other than the presence of a very small number of outlers, the resduals of the model ft are normally dstrbuted random varables wth a mean of zero. In other words, systematc errors n data or model are not consdered when determnng statstcally whch data ponts are outlers. Unfortunately, the applcaton of ths assumpton for outler removal tends to be self-renforcng: data ponts whch mght ndcate a sgnfcant problem n ether data collecton or the model used are systematcally rejected from the analyss. An alternatve to outler rejecton s the use of a robust parameter estmator rather than the leastsquares maxmum lkelhood estmator. The most robust estmator avalable s probably the least medan of squares (LMS) regresson 5, wth a breakdown pont that can approach the theoretcal maxmum of 0.5 (that s, nearly 50% of the data can be bad and a good ft can stll be obtaned). LMS regresson, however, has very low effcency (meanng that a large number of data ponts s requred to get a good ft, as compared to a least-squares regresson), and tends to be used only n extreme condtons of many expected outlers. By far the most popular robust estmator s the Huber s M-estmator 6. The cost functon uses the square of the reduced resdual, z, when that reduced resdual s small, then becomes lnear wth z for larger errors. Cost of data pont z = c( z c ) z z c, > c z ) y y = () σ The regresson term c can be adjusted to optmze the trade-off between effcency and robustness (hgher values ncrease effcency whle decreasng robustness), wth values near 1.3 beng common. The Huber s M-estmator has a contnuous frst dervatve (though a dscontnuous second dervatve), allowng for the use of reasonably effectve mnmzaton algorthms. The use of robust parameter estmaton methods s reasonably common n many dfferent felds, but I am unaware of ther use today for lthography model calbraton. 3. Total Least Squares Regresson In the conventonal least-squares regresson, the resdual between experment and data s weghted by one over the estmated uncertanty n the expermental data. Generally, ths uncertanty, expressed as the standard devaton of the data pont, s estmated based on a basc knowledge of the measurement method. For example, a CD SEM mght be characterzed pror to data collecton n order to determne ts statstcal precson as a functon of the range of measurand types beng used n the calbraton (for example, precson as a functon of feature sze, or as a functon of ptch). Ths precson s most often determned by repeated measurements of the same sample. It s reasonably easy to provde a much more comprehensve estmate of data pont uncertanty. Consder an experment to measure one output y (for example, CD) as a functon of K nput varables v (for example, focus and exposure, so that K = ). Uncertanty n the output s caused by measurement uncertanty, but t s also a functon of the uncertantes n settng the nput varables durng the experment. For example, Proc. of SPIE Vol D-4

5 f for one data pont the focus s set to -150 nm, the actual focus experenced by the resst feature to be measured wll dffer from ths amount due to random focus errors bult nto the expermental process. The mpact of nput varable uncertanty can be estmated usng a standard propagaton of uncertantes approach. Gven y = f(v 1, v,, v K ), and assumng the errors n the nput varables are small enough so that the functon f can be approxmated as lnear, the uncertanty n y can be estmated by K y σ y = σ y measurement + σ v (3) j= v j 1 j Thus, an uncertanty n an nput varable (σ v ) propagates to an uncertanty n the output based on the senstvty of the output to the nput (the partal dervatve term evaluated at the nomnal condtons of that data pont). If the nput varables are assumed to have no error, the uncertanty n the output s determned only by the uncertanty n the measurement of that output. If, however, the nput settngs have errors, then the output wll have an added uncertanty that s ndependent of output measurement uncertanty. If the varance n output as calculated from equaton (3) s used n the least-squares cost functon, the regresson s called a total least-squares regresson, or an error-n-varables regresson. The reduced resdual becomes z = σ ) y y K y + j= 1 v j Note that total least-squares regresson s completely compatble wth the use of robust estmators, such as the M-estmator descrbed n the prevous secton. Consder agan the focus-exposure example. For a data set as shown n Fgure 1, the senstvty of the output to focus and exposure wll be a mnmum at best focus and exposure. Out of focus or underexposed, the partal dervatves of CD wth respect to focus or dose wll be hgher, and often much hgher. Thus, gven estmates n the uncertanty n settng focus and exposure for each data pont n the experment, the out-offocus and underexposed data ponts wll have much larger uncertanty of the resultng CD value than the nomnal dose/focus condton. In fact, errors n focus and exposure, propagated to the output CD, often domnate the resultng CD uncertanty at the extremes of focus and exposure. As such, those extreme dose/focus data ponts wll be weghted less heavly n a model ft usng total least-squares regresson as compared to a conventonal least-squares ft. Whle conceptually smple, the total least-squares approach has one serous drawback t s dffcult to evaluate the partal dervatves of the output versus nputs functon durng the regresson. There s a soluton, however, that takes advantage of the fact that uncertanty estmates for each data pont (the weghts used n the regresson) do not need to be extremely accurate n order to get good estmated parameters from the model ft. Whle we often want our model to predct CD to wthn a percent or two, calbraton requres that the uncertanty estmates for each data pont need only be on target to wthn 10 0% at best. Thus, estmatng the data pont uncertantes can use a much less rgorous approach than fttng the model to the data based on those uncertantes. σ v j (4) Proc. of SPIE Vol D-5

6 80 Resst Lnewdth (nm) Focal Poston (nm) Fgure 1. Typcal focus-exposure matrx showng resst CD as a functon of focus for dfferent exposures (lnes connectng data ponts of the same exposure dose are for vsualzaton purposes only). The followng smple approach can be used to fnd the partal dervatves of the CD(focus, exposure) functon and thus the weghts to be used n the least-squares regresson. Frst, ft the measured focusexposure data set (usng standard least-squares regresson) to a smple but robust polynomal functon (my favorte s based on a physcal understandng of mage formaton and s commonly used for process-wndow calculatons usng expermental data 7 ). Once a reasonable ft s obtaned, the partal dervatves of the polynomal can be easly evaluated for each data pont, allowng a regresson weght to be calculated usng equaton (3). Gven ths new weght per data pont, standard least-squares regresson can be used to calbrate the lthography model. The result s a calbraton that s not overly senstve to hghly uncertan data ponts at the extremes of the process wndow. The applcaton of total least-squares regresson to focus-exposure data s reasonably straghtforward, but I am unaware of ts use today for lthography model calbraton. 4. Bayesan Parameter Estmaton In the dscusson above, better use of measurement error analyss results n better parameter estmaton when fttng a model to calbraton data. But as was mentoned n the Introducton, models also nclude parameters whch are measured drectly. And of course, all measurements nclude uncertanty, even the drect measurement of model parameters. How are estmates of the uncertantes n those drectly measured parameters to be used when calbratng a model and estmatng other parameters? In fact, there are some lthography model parameters whch are frequently measured (or assumed) and fxed, such as mask CD and partal coherence, but are also frequently used as adjustable parameters n the fttng to calbraton data. When s t most approprate to fx a parameter, versus lettng t float as an adjustable parameter? Is there a way to avod such an arbtrary dstncton between known and unknown parameter values? Bayesan statstcs attempts to make the best use of all a pror nformaton when fttng a model to a data set. It s named for the Reverend Thomas Bayes ( ), a mathematcan who frst put the dea Proc. of SPIE Vol D-6

7 of statstcal nducton on a frm mathematcal footng. In terms of parameter estmaton, the basc dea s ths: If a model nput parameter has an a pror best estmated value, wth an estmated uncertanty n that best estmated value, ths nformaton should be used n the model calbraton process. One very smple and straghtforward way to use ths nformaton s through an addtonal cost-functon term: N ) M ~ 1 y = y w p p j p j χ + N = 1 σ M j= 1 σ p j (5) Here, there are M parameters of the model that are beng adjusted to fnd the best ft, p 1, p,, p M. Each parameter has an a pror estmated value p j and an uncertanty n that a pror value of σ p j. The value of the overall parameter weght w p should be emprcally determned based on experence wth the problem doman, but can be set ntally to one. Lookng at equaton (5), one can see that the usual approaches to parameter fttng, where a parameter s ether fxed or allowed to float freely, are smply extreme cases of the more general Bayesan fttng approach. If the uncertanty n the a pror parameter estmate, σ p, goes to zero (meanng there s no uncertanty n our knowledge about the parameter), the cost functon wll always drve the parameter value to ts mean n other words, the parameter remans fxed at the best estmate of ts value. If the uncertanty n the a pror parameter estmate, σ p, s nfnte (meanng we have no pror knowledge about the parameter), ths term wll not contrbute to the cost and the parameter wll be allowed to float freely durng the fttng. Bayesan parameter estmaton smply allows a contnuum between these extreme cases, where our a pror knowledge about a parameter value s somethng between perfect and nonexstent. As an example of where Bayesan parameter estmaton can be used effectvely for lthography model calbraton, consder mask bas. Masks have a specfcaton of mean-to-target CD, whch when coupled wth mask measurement uncertanty leads to a reasonably good estmate of the uncertanty of the mask CD at any gven locaton on the mask. Snce mask CD s an nput to a lthography model, users are often faced wth the choce of fxng or floatng a bas between the actual mask CD and ts desgn value when calbratng a lthography model. Bayesan parameter estmaton, as used n equaton (5), removes ths false dchotomy and allows all of the nformaton possessed by the user to be appled to the calbraton problem. I beleve that Bayesan parameter estmaton for lthography model has the potental to greatly mprove the predctve accuracy of calbrated models, but further work should be done n assessng and optmzng ts use for lthography model calbraton. 5. A Note of Cauton and Some Random Ideas A smple output of a least-squares regresson s the root-mean-square (RMS) dfference between the best-ft model and the calbraton data. Due to ts convenence, many model users consder ths RMS calbraton error to be a smple measure of model goodness: gven two competng models, the model that produces the lowest RMS calbraton error s the better model. Whle such an nterpretaton s rfe wth problems for even standard least-squares regresson, the fnal calbraton RMS value s completely useless for any of the more advanced calbraton technques descrbed here. By defnton, standard least-squares wll produce the lowest possble calbraton RMS value. Thus, total least-squares, robust parameter estmaton, and Bayesan parameter estmaton technques wll always produce RMS calbraton values that are hgher than smple least-squares regresson. Or course, ths should never be thought of as a problem. The goal of Proc. of SPIE Vol D-7

8 calbraton s not to predct the calbraton data set, but to predct other data sets wth maxmal accuracy. Maxmal predctve accuracy s not acheved wth standard least-squares regresson, and so a model wth mnmum RMS dfference to the calbraton data wll not be the best model. Ths brngs up the nterestng subject of how to valdate a model: gven a model wth a set of calbrated parameters, what s ts predctve accuracy? The calbraton RMS value offers very lttle nformaton about predctve accuracy. And whle the subject of model valdaton s an extremely broad one (and thus can not be adequately covered here), t s worth mentonng that model valdaton must rely on outof-sample data, that s, data that was not used to calbrate the model. A very nterestng experment that could be performed s as follows. Consder a large data set collected for the purpose of model calbraton. Randomly select half of the data and calbrate the model usng the calbraton process of record. Usng ths calbrated parameter set, fnd the RMS dfference between the model and the second half of the data (the outof-sample data that was not used for calbraton). Now repeat these steps many tmes usng dfferent randomly selected halves of the data. The collecton of RMS valdaton values wll have a mean and a standard devaton that can be used to assess the true predctablty of the model (for a gven calbraton data set sze and a gven calbraton process). The model valdaton experment of the prevous paragraph can also be used to compare dfferent calbraton approaches. If, for example, the above experment was repeated for two dfferent sample szes, one could see how much the model predctablty would ncrease wth ncreasng calbraton sample sze. Addtonally, dfferent calbraton processes (standard least-squares versus total regresson, for example) can be compared usng the ultmate metrc of success, model predctablty, as the judge. Fnally, as a random note, resdual analyss s an underutlzed tool for judgng the qualty of the data, the model, and the calbraton process. Snce a major assumpton of standard least-squares regresson s that the resduals are normally dstrbuted, ths assumpton should be checked by examnng the actual resduals of the calbraton. For the large data sets that are commonly used for lthography model calbraton, testng the normalty of the resduals n not dffcult and can be very revealng (I have never seen a set of resduals from a lthography model calbraton that was even remotely close to normally dstrbuted). Further, the propertes of the ch-squared dstrbuton under the assumpton of normal resduals are well known (for example, the expectaton value of the reduced ch-squared, ~ χ = χ / N, s 1 and ts varance s /N). When the actual mnmzed ch-squared value dffers sgnfcantly from ts expected value, one or more of the assumptons of the ft are lkely wrong. If the ch-squared s too large, there are lkely to be systematc errors ether n the data or n the model. If the ch-squared s too small, the model probably has too many adjustable parameters and has begun to ft the nose. Of course, there s always the chance that the a pror estmates of the data pont uncertantes are sgnfcantly wrong (a not uncommon problem). 6. Conclusons Whle least-squares regresson s very commonly used for lthography model calbraton (for both rgorous and smplfed models), there are many mprovements that could and should be made to ths smple technque. Robust parameter estmators can sgnfcantly reduce the senstvty of the ft to outlers, whle also elmnatng the dangers nherent n outler rejecton schemes. Total least-squares regresson can mprove the accuracy of estmated parameters by takng nto account errors n the nput varables of the experment (a technque that s especally useful for focus-exposure matrx data sets). Fnally, Bayesan approaches have the potental to further ncrease model predctablty by usng nformaton about the uncertanty of nomnally fxed model parameters. The wdespread use of one or more of these smple extensons to current calbraton processes could sgnfcantly mprove the predctablty of the resultng calbrated models. And Proc. of SPIE Vol D-8

9 predctablty, after all, s the ultmate goal of the use of a lthography smulator. One of my gudng prncples s ths: Data s expensve, data analyss s cheap. It s ncumbent on the modeler to extract as much nformaton as possble from the data that s avalable. Acknowledgements Much of the research that underles ths paper was performed whle I was the vstng Melchor char professor at the Unversty of Notre Dame n the fall of 006. I am ndebted to Notre Dame, and my sponsor Dr. Gary Bernsten, for provdng me wth such a wonderful opportunty and envronment n whch to learn. 7. References 1 D. Hawkns, Identfcaton of Outlers, Chapman & Hall (1980). P. J. Rousseeuw and A. M. Leroy, Robust Regresson and Outler Detecton, John Wley & Sons (New York: 1987). 3 Bors Iglewcz and Davd C. Hoagln, How to Detect and Handle Outlers, ASQ Qualty Press (December, 1993). 4 F. E. Grubbs, Sample Crtera for Testng Outlyng Observatons, Annals of Mathematcal Statstcs, Vol. 1, No. 1 (Mar., 1950) pp P. J. Rousseeuw, Least medan of squares regresson, J. Amer. Statst. Assoc., Vol. 79 (1984), pp P. J. Huber, Robust Statstcs, John Wley& Sons (New York: 1981). 7 C. A. Mack and J. D. Byers, Improved Model for Focus-Exposure Data Analyss, Metrology, Inspecton and Process Control XVII, Proc., SPIE Vol (003) pp Proc. of SPIE Vol D-9