Copyright 2003 by the Society of Photo-Optical Instrumentation Engineers.

Transcription

1 Copyrght 003 by the Socety of Photo-Optcal Instrumentaton Engneers. Ths paper as publshed n the proceedngs of Metrology, Inspecton, and Process Control for Mcrolthography XVII, SPIE Vol. 5038, pp It s made avalable as an electronc reprnt th permsson of SPIE. One prnt or electronc copy may be made for personal use only. Systematc or multple reproducton, dstrbuton to multple locatons va electronc or other means, duplcaton of any materal n ths paper for a fee or for commercal purposes, or modfcaton of the content of the paper are prohbted.

2 Improved Model for Focus-Exposure Data Analyss Chrs A. Mack and Jeffrey D. Byers KLA-Tencor, FINLE Dvson 8834 N. Captal of Texas Hghay, Sute 301, Austn, TX USA e-mal: Abstract The paper ntroduces an mproved, physcs-based functon for fttng lthographc data from focusexposure matrces. Unlke smple polynomal functons, the coeffcents of ths equaton offer physcal nsght nto the meanng and nature of the data. Dervaton of ths equaton from frst prncples of the physcs of lthographc magng s presented. Examples based on typcal expermental data are shon and the advantages of usng a physcs-based fttng functon s descrbed based on mproved fttng and nose flterng. Keyords: Focus-exposure matrx, process ndo, data analyss, ProDATA I. Introducton Systematc analyss of focus-exposure matrx data s vtal to the accurate determnaton of process ndos and the calculaton of depth of focus and best focus [1-3]. Ths analyss s generally accomplshed by frst fttng the data to a mathematcal functon, then usng ths functon for process ndo determnaton. The advantage of ths approach s that the goodness of ft can be used as an objectve means of data fler removal, and the natural smoothness of the fttng functon can reduce the mpact of expermental nose n the data on process ndo determnaton. Improper selecton of the fttng functon, hoever, can lead to other problems. A functon th to fe parameters could elmnate real and sgnfcant patterns n the data. A functon th to many parameters can produce artfacts that do not actually exst n the data. Choosng the correct functon, th the correct number and type of fttng coeffcents, s crtcal to proper process ndo determnaton. The paper ntroduces an mproved, physcs-based functon for fttng lthographc data from focus-exposure matrces. Unlke smpler polynomal functons, the coeffcents of ths equaton offer physcal nsght nto the meanng and nature of the data. Dervaton of ths equaton from frst prncples of the physcs of lthographc magng s presented. Numerous examples based on typcal and unusual expermental data ll be shon and the advantages of usng a physcs-based fttng functon s descrbed. II. Polynomal Focus-Exposure Matrx Data Analyss Snce the effect of focus s dependent on exposure, the only ay to judge the response of the process to focus s to smultaneously vary both focus and exposure n hat s knon as a focus-exposure matrx. Fgure 1 shos a typcal example of the output of a focus-exposure matrx usng lnedth as the response (sdeall angle and resst loss can also be plotted n the same ay) n hat s called a Bossung plot [4]. As one can see, the shapes of the Bossung curves are qute complcated. As a result, most efforts to ft ths data to an equaton has nvolved the use of polynomals n focus (F) and exposure (E) [1-3]. One very general expresson s

3 CD j0 a j E F j (1) Although ths functon has 0 adjustable coeffcents, for most data sets good fts are obtaned hen a 03, a, a 14, a 3, a 4, a 33, and a 34 are fxed and set to zero. CD (nm) mj/cm mj/cm 00.0 mj/cm 0.0 mj/cm 40.0 mj/cm 60.0 mj/cm 80.0 mj/cm mj/cm 30.0 mj/cm Focus (m) Fgure 1. Example of the effect of focus and exposure on the resultng resst lnedth (symbols) and the best ft of ths data (lnes) to equaton (1). III. Curve Fttng and Statstcal Analyss Gven a set of measured ponts (x,y ) th x the measurement poston vector (.e., nput parameter values) and y the measured value (output) for each measurement, let F(x) be the functon to be ftted. The most common ay to determne the coeffcents s to calculate those coeffcents that optmze the mert functon y F( ) () x Summng the squares of the dstances at each data pont, (ch-squared) measures the agreement of the fttng functon and the data. If the coeffcents are chosen such that ch-squared s mnmzed, a functon th the best average approxmaton for each data pont s found. A standard statstcal method to handle data th large measurement errors ( data flyers ) n a curve ft s to perform a second ft after removng those data ponts that exceed a certan devaton from the frstly

4 obtaned functon. In other ords, an algorthm to optmze s used to tmes: frst, t calculates the coeffcents as mentoned above, usng all data ponts n the analyss ranges. Next, those data ponts hose devaton from the ftted functon exceed a specfed tolerance are removed and the algorthm s used agan to calculate the fnal coeffcents. A good choce for the devaton tolerance s usually to tmes the standard devaton from the frst ft, here the standard devaton s defned as (3) N 1 and here N = number of data ponts. Hoever, another multple of or the drect selecton of a devaton tolerance can be used. Some data sets have a center n hch the measured values have more mportance than values at the edges of the data range. Focus-Exposure matrces especally are measured around an estmated best focus and best exposure and the data closest to the center of the range s most mportant. A ay to represent ths n the curve ft s to assgn to each data pont an ndvdual eght. By optmzng the eghted ch-square, y F( x ) (4) nstead of ch-squared, the obtaned functon ll tend to ft data ponts th more eght more closely than data ponts th less eght. One approach to data eghtng s eght each data pont n nverse proporton to the uncertanty n the data. If repeat measurements are made, ether on a sngle expermental observaton or on repeat experments, the statstcs of the measurements ll produce a standard devaton hch can be used as an estmate of the uncertanty of the data pont. By eghtng each pont as one over the standard devaton of the measurement, the most certan ponts ll have the greatest nfluence on the ft. In practce, the out of focus features, th ther poor resst profles and hgher senstvty to process varatons, ll have greater uncertanty and thus ll, n general, be eghted less. IV. Improved Physcs-Based Fttng Functon Usng a smple polynomal to ft expermental focus-exposure CD data can have certan problems. Usng the polynomal functon th to fe parameters could elmnate real and sgnfcant patterns n the data. Usng the functon th to many parameters can produce artfacts that do not actually exst n the data. It s often dffcult, over a de range of data sets, to determne the best number of coeffcents to use n the ft. A physcally based fttng functon can ad n ths decson, th the added beneft of physcal meanng for the coeffcents. Addtonally, a ell desgned physcally based fttng functon should allo the best ft th the feest number of adjustable parameters, thus ncreasng the confdence n data fler removal and n preservng the ntegrty of true data patterns. Consder frst the aeral mage of a smple pattern of lnes and spaces of ptch p. In one dmenson (x), the aeral mage can alays be expressed as a Fourer seres. ( 0 1 I x) cos(x / p) cos(4x / p)... (5)

5 As the smplest example, consder equal lnes and spaces of dth (= p/). Note that the dervaton that follos s conceptually smpler for the case of equal lnes and spaces, but the results are not lmted to ths case. It ll be convenent to replace the x poston th a coordnate that s the devaton of x from the nomnal lne edge. x (6) Wrtng equaton (5) n terms of ths ne coordnate gves I ( x) 0 1 sn cos... (7) Eventually e ll use ths equaton to predct the behavor of feature sze th focus and exposure. Snce the features of nterest ll be near the nomnal sze, e ll be most nterested n equaton (7) for the case of small /, th values beteen 0. and 0. of greatest nterest. Thus, t ll be reasonable to expand the sne and cosne terms of ths equaton n a Taylor seres, keepng only the frst fe terms. I ( x) (8) 3 here the coeffcents are just lnear combnatons of the terms (for example, 0 = 0 -, etc.). Although the exact values of the and terms are a functon of the ptch, avelength, numercal aperture, and partal coherence, for the smple coherent llumnaton case and equal lnes and spaces 0 = 0.5, 1 = 1, = 1, 3 = 0.411, etc. Gven an aeral mage one can estmate the change n crtcal dmenson (CD) as a functon of exposure dose, E. Usng the smple approxmaton of a thn, nfnte contrast photoresst, the photoresst ll be removed henever the dose exceeds some threshold dose, E th. In other ords, EI( CD ) E th (9) Keepng only through the second order terms n equaton (8), one can solve for the resultng CD as a functon of dose. CD E E th (10) Agan, for small / the argument of the square root must necessarly take the form of one mnus a small number. Notng that the dose to sze, E s, must be equal to E th / 0 and takng the Taylor expanson of the square root,

6 CD 0 E 0 1 s E s E E 1 1 (11) Puttng equaton (11) nto a more general form and keepng only the frst N terms n the seres, CD N n1 c n Es n1 (1) E In general, keepng three terms or less n ths seres gves very good fts to all data sets that e have observed. For many data sets, keepng only the frst term provdes adequate results. As an nterestng asde, equaton (10) smplfes to the follong expresson for the case of coherent llumnaton of small equal lnes and spaces: CD 1 1 Es E (13) Before contnung th the dervaton and addng a focus dependence to ths equaton, t s nstructve to consder the lthographc sgnfcance of the coeffcents c n n equaton (1). The coeffcent c 1 represents the slope, on a log-log scale, of the CD versus dose curve (and s thus the nverse of exposure lattude). c lncd ln E EEs NILS (14) here NILS s the normalzed mage log-slope (the proportonalty to NILS s n fact an equalty n the lmt of an nfnte contrast photoresst). The second order coeffcent c represents the curvature of the log-cd versus log-dose curve, hch s the change n exposure lattude th exposure dose. By notng the relatonshp beteen c 1 and NILS, t becomes possble to ncorporate the mpact of focus errors on CD. It s ell knon that NILS falls off th ncreasng defocus. For a small lne/space pattern th coherent llumnaton, the behavor of NILS th defocus dstance s just NILS NILS n focus cos( / p ) (15) Keepng th our theme, e ll agan expand the cosne term th a Taylor seres. Thus, the frst coeffcent of equaton (1) ll become a functon of focus as ln CD c 1... (16) ln E n focus Although n the deal case equaton (16) has only even poers of defocus dstance, real lthographc results do exhbt some asymmetry th defocus. Thus, a generalzaton of equaton (16) ould become

7 c n M b nm m0 F F * m (17) here F s the focal poston, F* s best focus and the defocus dstance s F-F*. In general, M = 4 s suffcent to descrbe the vast majorty of data sets and often the odd terms are very small or neglgble. Combnng equatons (1) and (17) and smplfyng, CD M N m0 n0 a nm n E s m 1 F (18) E Equaton (18) represents a physcally based fttng functon that has mproved fttng performance over the orgnal polynomal formulaton of equaton (1). As others have noted [5,6], a CD varaton th one over exposure dose s more physcally accurate than assumng that CD s proportonal to poers of dose. V. Applyng the Improved Fttng Functon One of the goals of usng the fttng functon descrbed above rather than the orgnal polynomal expresson s to provde better fts of expermental data th feer terms (and thus feer adjustable fttng coeffcents). Dong so should lead to mproved tolerance to statstcal nose n the data and better data flyer removal decsons. In order to test out these attrbutes of a data fttng functon an deal nose free data set as generated usng smulaton. A typcal 48nm chemcally amplfed resst process as used to create 130nm dense lne/space focus-exposure matrx data sets. Fgure shos example fts of to data sets usng the smple polynomal and the ne physcally based fttng functon. The to sets are dentcal except that the second data set used a der range of focus. In each case the number of adjustable parameters n the to fttng functons as kept the same. For the lmted focus range case, the orgnal polynomal th sx terms ft the data th a one sgma goodness of ft of 3.49nm. The physcally based fttng functon shoed a goodness of ft of 1.47nm. For the extended focus range case, usng a greater number of terms n each expresson to capture the more nterestng focus behavor, the physcally based functon agan resulted n a much smaller goodness of ft, 1.90nm versus 4.6nm for the orgnal polynomal. To test the robustness of each expresson th respect to nose n the data, random nose of varous amounts as added to the frst data set and fts usng both fttng expressons ere repeated. To make the nose as realstc as possble, a metrology nose as added to the CD values tself, but nose as also added to the focus and exposure values n the data set. Snce a random number generator as used to generate the added nose, several nosy data sets ere generated for each nomnal magntude of added nose and then the actual RMS nose amount as measured from the result. Fgure 3 shos example fts hen 4.1nm RMS of random Gaussan nose as added to the data set. The goodness of fts ere 4.53nm and 4.4nm for the orgnal and mproved fttng functons, respectvely. Note that the advantage that the mproved functon enjoyed n terms of goodness of ft seems to be ashed aay by the nose n the data. The effect of nose on the goodness of ft s explored n more detal n Fgure 4, here the RMS magntude of added nose s vared and the goodness of ft of each expresson s plotted. One can see that the sx term physcally based functon out performs the sx term smple polynomal at lo nose levels, and n fact s comparable to a telve term polynomal ft. Hoever, at hgh nose levels all functons gve approxmately the same goodness of ft.

8 (a) (b) (c) (d) Fgure. Comparson of the orgnal (a and c) to the ne (b and d) fttng functon usng the same number of terms n each expresson and nose free data generated by smulaton.

9 (a) (b) Fgure 3. Comparson of the orgnal polynomal (a) to the physcally based fttng functon (b) to data th 4.1nm RMS of added nose Goodness of Ft (nm) Poly 6Term Mack 6Term Poly 1Term RMS Nose Level (nm) Fgure 4. Comparson of goodness of fts for the varous fttng functons n the presence of nose added to the data.

10 But s the goodness of ft to nosy data the best metrc of the approprateness of a functon to the task of data fttng? A goal of usng a fttng functon to descrbe expermental data s to flter out nose and extract the true, core behavor present n the data. In general one does not kno the true behavor of expermental data n the absence of nose. In our case, hoever, the data has been generated by addng a set amount of nose to an deal nose free data set. Thus, a more approprate metrc for ho ell the fttng functon flters out random nose n our experment ould be to use an RMS model error to judge the result: RMS Model Error (Model Nose Free Data) (18) N Note that the model s frst ft to the nosy data n the standard ay, by mnmzng the goodness of ft to the nosy data. Then, the effectveness of the model s measured usng the RMS model error of equaton (18). Fgure 5 shos the results. As can be seen the physcally based model does a much better job of flterng out nose and keepng the RMS model error lo n the presence of large amounts of nose. In fact, the sx term physcally based functon does a slghtly better job of preservng the orgnal nose free behavor than even the telve term polynomal. Addng more polynomal terms results n the fttng of nose, gvng a goodness of ft that s belo the amount of added RMS nose. As a result, the hgher term polynomal may n fact have a orse RMS model error than the sx term polynomal hen a hgh nose level s present. 8 RMS Model Error (nm) Poly 6Term Poly 1Term Mack 6Term RMS Nose (nm) Fgure 5. Behavor of the varous fttng functons n the presence of added nose usng the RMS model error as the metrc to judge the effectveness of each model at flterng nose.

11 VI. Conclusons Data analyss s an mportant part of the photolthography engneer s job. As lnedth control becomes more crtcal and process ndos become smaller and smaller, accurate analyss of lthography process data becomes essental. Automated, statstcally sound technques for analyzng data, removng bad data ponts, and extractng relevant lthographc nformaton can dramatcally mprove one s ablty to montor, characterze, and optmze a process. Ths paper ntroduces an mproved physcs-based functon for fttng lthographc data from focusexposure matrces. Unlke smple polynomal functons, the coeffcents of ths equaton offer physcal nsght nto the meanng and nature of the data, based on ts dervaton from frst prncples of the physcs of lthographc magng. The advantages of usng a physcs-based fttng functon as shon based on mproved fttng and nose flterng. The mproved physcs based fttng functon presented here has been ncorporated nto the softare tool ProDATA TM. References 1. C. A. Mack, D. A. Legband, S. Jug, Data Analyss for Photolthography Mcro- and Nano- Engneerng 98, Proc., (1998) pp C. A. Mack, S. Jug, D. A. Legband, Data Analyss for Photolthography, Metrology, Inspecton, and Process Control for Mcrolthography XIII, Proc., SPIE Vol (1999) pp S. Jug, R. Huang, J. Byers, C. A. Mack, Automatc Calbraton of Lthography Smulaton Parameters, Lthography for Semconductor Manufacturng, Proc., SPIE Vol (001). 4. J. W. Bossung, Projecton Prntng Characterzaton, Developments n Semconductor Mcrolthography II, Proc., SPIE Vol. 100, pp (1977). 5. C. P. Ausschntt, Rapd Optmzaton of the Lthographc Process Wndo, Optcal/Laser Mcrolthography II, Proc., SPIE Vol (1989) pp D. Fuard, M. Besacer, P. Schavone, Assessment of Dfferent Smplfed Resst Models, Optcal Mcrolthography XV, Proc., SPIE Vol (00).