Spatial Analysis of Ring Oscillator Devices

Transcription

1 6.780 Term Project Spatal Analyss of Rng Oscllator Devces A. Soman, L. W. Teo, and D. S. Bonng, Member, IEEE Abstract Spatal dependences tend to ntroduce correlatons among parameter values obtaned from test structures. These spatal correlatons obscure the parameter correlatons caused by common underlyng varables and make process dagnoss more dffcult. In ths work, lnear regresson method has been used to generate the spatal models for both wafer and chp level to nvestgate ther sgnfcance. It s found that the wafer level regresson model does not have a strong wafer level dependence on the rng oscllator devces. On the other hand, at the chp level, goodness of ft calculated for certan devce structures exhbt strong spatal dependence on the lnear regresson model. Ths proves that t s essental to consder dfferent levels of spatal dependence before makng concluson on the varatons caused by underlyng varables and parameter. Index Terms Spatal analyss, lnear regresson, least square method I. INTRODUCTION INCE the brth of the ntegrated crcut nearly four S decades ago, the semconductor ndustry has dstngushed tself from the other ndustres by ts rapd pace of mprovement. Most of the mprovements on ts products have resulted essentally from the ndustry s ablty to exponentally decrease the mnmum feature szes used to fabrcate ntegrated crcuts. As devce dmensons reduce and wafer szes ncrease, process unformty s becomng an ncreasngly mportant and dffcult task. A sound understandng of varatons, partcularly spatal varaton, s essental to both control the process and to desgn manufacturable crcuts []- [4]. Some process varatons, such as lne-wdth changes of poly or nterconnect, can sgnfcantly affect crcut performance. Ths requres the development of new technques to measure and extract varaton n a gven process and lnk t to crcut performance. Test crcuts are normally desgned to add ntentonal varatons n devce parameters. These varatons are carefully controlled n terms of ther magntude as well as ther behavor Manuscrpt receved May 4, Ths work was supported by NSF/SRC Engneerng Research Center for Envronmentally Bengn Semconductor Manufacturng and Sngapore-MIT Allance research grant. A. Soman s wth the Mcrosystems Technology Laboratores, Department of Materal Scence and Engneerng, Massachusetts Insttute of Technology, Cambrdge, MA 0239, USA. L. W. Teo s wth the Advance Materals Mcro- and Nano- Systems programme, Sngapore-MIT Allance, 4 Engneerng Drve 3, Sngapore D. S. Bonng s wth the Mcrosystems Technology Laboratores, Department of Electrcal Engneerng and Computer Scence, Massachusetts Insttute of Technology, Cambrdge, MA 0239, USA. towards expected process varatons. However, dfferent degree of spatal dependence on ether the wafer level or chp level may dstort the results drawn from these devces measurements. Therefore, n order to nfer any relable models or draw any conclusons, any possble trends of spatal correlaton wth the measured results have to be deconvoluted. Examples of common spatal dependences for unrelated process parameters could be radcal dependence of crystal pont defects, on mplantaton ncdence angle, and photoresst thckness. Snce varaton s manfested n several forms, the man am of ths paper s to dentfy any wafer level or chp level spatal dependence that would affect the conclusons drawn from any devce measurements. II. PROBLEM FORMULATION The present work focuses on establshng a relaton between spatal varance wth observed parameters such as frequency of rng oscllators n chp. Rng Oscllators (ROs) are standard test structures to determne the delay n dfferent process. The chp archtecture used n ths paper s based on the desgn by Panganban [5] and the testng methodology used to generate the raw data s clearly explaned by Gonzalez [6]. The chps were manufactured by TSMC usng 0.25 µm MOSIS [7]. Fg. shows the chp locatons wthn the wafer that the data are analyzed. Note that all the chps are obtaned from the top half of the wafer and a majorty of them were from the top left hand sde of the wafer. Y Fg.. Spatal locaton of wafer [6] X

2 6.780 Term Project 2 Typcal test structures found n each chps are normally dvded nto two types: Front End Of the Lne (FEOL) and Back End Of the Lne (BEOL) structures. FEOL structures are desgned to capture varatons n the devces that are part of a crcut. FEOL structures are tested by carefully layng out sets of rng oscllators (ROs) whose nverters have been carefully lad out to enhance a specfc source of varaton. Any varatons of these nature would be reflected n the overall output frequency of the RO. BEOL test structures on the other hand are structures that smulate common scenaros of nterconnectons wthn a chp. These structures must smulate parastc capactances such as frngng, couplng and plane capactance. To do so, rng oscllators are carefully lad out to enhance all these varatons, but exclusvely one at a tme n order to detect how ths specfc varaton s affectng the output frequency of the crcut. Fg. 2 shows the representaton of the entre chp. Note that each chp conssts of 54 rows of rng oscllator and 6 rows of hgher polyslcon densty, for a total of 60 rows. The 54 rows of normal polyslcon densty have 43 tles each. There are 30 tles among all 6 hgh polyslcon rows, for a total of 2,352 tles per chp. A complete summary of the devce postons can be found n Gonzalez [6]. Fg. 2. Chp Layout [6] In order to drawn any conclusons from the comparson between the parameter varatons for the dfferent devces, t s essental to nvestgate f there s any parameter correlatons caused by common underlyng varables due to some spatal dependence of a certan process. Wafer level and chp level spatal modelng are used to dentfy the trends. The detals of the mplementaton of the spatal modelng are dscussed n the followng secton. III. REGRESSION FUNDAMENTALS Ths paper prmarly focuses on standard analytcal technque of lnear regresson. In the lnear regresson model, the dependent varable s assumed to be a lnear functon of one or more ndependent varables plus an error ntroduced to account for all other factors: y = β x β K xk + u () In the above regresson equaton, y s the dependent varable, x,..., x K are the ndependent or explanatory varables, and u s the dsturbance or error term. The goal of regresson analyss s to obtan estmates of the unknown parameters β,..., β K whch ndcate how a change n one of the ndependent varables affects the values taken by the dependent varable. The am of regresson s to make some predctve models, whch can be used n future to estmate the dependent varable. The usual method of estmaton for the regresson model s ordnary least squares (OLS) [8]. Let b,..., b K denote the OLS estmates of β,..., β K. The predcted value of y s: y ˆ = b x bk xk (2) The error n the OLS predcton of y, called the resdual, s: e = y yˆ (3) The basc dea of ordnary least squares estmaton s to choose estmates β,..., β K to mnmze the sum of squared resduals: n e 2 It can be shown that: b = ( X X ) X y (5) where X s an n * k matrx wth (,k) th element x K, y s an n * k vector wth typcal element y, and b s a k * vector wth typcal element b K. The least squares fttng procedure descrbed below can be used for data analyss as a purely descrptve technque. However, the procedure has strong theoretcal justfcaton f a few assumptons are made about how the data are generated. One set of such assumptons, known as the Gauss-Markov assumptons that are suffcent to guarantee that ordnary regresson estmates wll have good propertes are summarzed [9]. The frst assumpton s that the errors u have an expected value of zero: E(u ) = 0 Ths means that on average the errors balance out. Second assumpton s that the ndependent varables are non-random. In an experment, the values of the ndependent varable would be fxed by the expermenter and repeated samples could be drawn wth the ndependent varables fxed at the same values n each sample. As a consequence of ths assumpton, the ndenpendent varables wll n fact be ndependent of the dsturbance. For nonexpermental work, ths wll need to be assumed drectly along wth the assumpton that the ndependent varables have fnte varances. Thrd, t assumes that the ndependent varables are lnearly ndependent. That s, no ndependent varable can be expressed as a (non-zero) lnear combnaton of the remanng ndependent varables. The falure of ths assumpton, known (4)

3 6.780 Term Project 3 as multcollnearty, clearly makes t nfeasble to dsentangle the effects of the supposedly ndependent varables. The fourth assumpton s that the dsturbances u are homoscedastc: E(u 2 )=σ 2 (6) Ths means that the varance of the dsturbance s the same for each observaton. Ffthly, t assumes that the dsturbances are not autocorrelated: E(u, u j )=0 (7) Ths means dsturbances assocated wth dfferent observatons are uncorrelated. Regresson models are ftted to the RO frequences for dfferent devces to test for wthn wafer level and wthn chp level spatal dependence. The center pont for the x-y coordnates n the case of wthn wafer level regresson models s defned to be at the center of the wafer. However, n the case of chp level lnear regresson models, the center pont for the x-y coordnates are taken to be at the center of each ndvdual chp. Goodness of ft, R 2, s calculated to nvestgate f the regresson models are of a good ft. Fg. 3. Chp Mean Frequences by Spatal Locaton IV. RESULTS Regresson models are ftted to the RO frequences for dfferent devces to test for wthn wafer level and wthn chp level spatal dependence. The center pont for the x-y coordnates n the case of wthn wafer level regresson models s defned to be at the center of the wafer. However, n the case of chp level lnear regresson models, the center pont for the x-y coordnates are taken to be at the center of each ndvdual chp. Goodness of ft, R 2, s calculated to nvestgate f the regresson models are of a good ft. A. Wafer level spatal dependence analyss The wafer level spatal dependence analyss s carred out n two parts. In the frst part of the analyss, the average RO frequences for all the devce structures wthn each chp are averaged and data obtaned for all the chps are analyzed usng the regresson model. In the second part, the average RO frequences for all the devces of dentcal devce structure wthn each chp are then calculated. The chp mean frequences by spatal locaton and the mean frequency for a best regresson result for a devce structure are plotted n Fg. 3 and 4 respectvely. It s observed that there are no clear spatal wafer level trends n both Fgs. and 2. Regresson model that uses parameters of x, y, xy, x 2 and y 2 yelds R 2 value of 0.54 for the chp mean frequences by spatal locaton. Regresson models for all the 45 devce structures also yeld low R 2 value rangng from 0.04 to Table and 2 summarzes the model coeffcents and the coeffcents sgnfcance at 95% confdence nterval for the case of chp mean as well as the best regresson ft for a certan devce structure. From the two tables, t can be seen that the all model coeffcents are not very sgnfcant at 95% confdence. Fg. 4. Mean Frequences for Plane Cap for ILD by Spatal Locaton Table : Chp Mean Wafer Level Spatal modelng Intercept X Y XY X Y R Square 0.5 Table 2: Mean for Sngle Devce Structure Wafer Level Spatal modelng Intercept X Y XY X Y R Square 0.22

4 6.780 Term Project 4 There can be several reasons for ths observaton; frstly there mght not be any spatal correlaton. Secondly, there can be a spatal correlaton but not lnear t mght be non lnear. Thrdly the data set mght not be a good sample as t s mssng values from other sdes of wafer as shown n Fg.. B. Chp level spatal dependence analyss After wafer level spatal analyss, chp level spatal dependence was carred out. As mentoned earler, n the case of chp level lnear regresson models, the center pont for the x-y coordnates are taken to be at the center of each ndvdual chp. RO frequences of ndvdual devce structure were regressed over x, y, xy, x 2 and y 2 to yeld models. There are 45 devce and 35 chps so t wll yeld (45x35=) 575 models. Table 3: R 2 value for dfferent types of FEOL structure tle_ro tle_ro_vert tle_ro3 tle_ro3_vert chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp chp Ths work focuses on some nterestng trends and devces, as t s not possble report such a large number of models. One of these nterestng trends s observed n vertcal and horzontal FEOL test structures. Horzontal and vertcal test structures are desgned to nvestgate the orentaton dependence of structure. They prmarly have the same archtecture. There can be dfference n ther frequency attrbuted to mask scan bas or on mplantaton effect. Table 3 summarzes the R 2 value for two types of FEOL structure, whch have both vertcal and horzontal structure for all chps. The two FEOL structures are Canoncal FEOL and 3x spacng FEOL. From ths table t s evdent that vertcal structures have better ft as compared to horzontal. Note that there are also some chps (e chp,2, ) that do not have any good R 2 value for all devce structures measured. Fg. 5 shows the spatal dstrbuton for vertcal Canoncal FEOL for partcular chp 9 and t can be observed that there s certanly a pattern to t. In gven dataset, Chp 9 has the replcaton, whch makes t more relable and sutable canddate for spatal plots. R 2 value for ths chp and devce s 0.86 and coeffcents are lsted n Table 4. Smlarly, Fg. 6 shows the spatal dstrbuton for horzontal Canoncal FEOL for Chp 9. R 2 value for ths regresson was around 0.49 and ts coeffcents are also lsted n Table 5. Fg. 5. RO_vertcal by Spatal Locaton Table 4: Chp 9 RO_Vertcal Chp Level Spatal Modelng Intercept X Y XY X Y R Square 0.86

5 6.780 Term Project 5 as FEOL horzontal. The understandng of the nterplay between the dfferent levels of spatal dependence wth the observed parameters s crtcal n order to establsh the actual lnk between the varatons n observed parameters to ther varous devce parameters. The fnal objectve s to solate the varaton n observed parameters due to varous devce parameters takng nto account the generated spatal models Fg. 6. RO_Horzontal by Spatal Locaton Table 5: Chp 9 RO_Horzontal Chp Level Spatal Modelng Intercept X Y XY X Y R Square 0.49 From Table 4, s noted that for the case of the RO vertcal devce structure, the coeffcents of x, y and y 2 n the regresson model are very sgnfcant. The hgh R 2 value for the spatal model correlates well wth the Fg. 5 that shows a clear trend that the measured devce frequency ncreases dagonally from the bottom rght to the top left hand corner of the chp. From Table 5, s noted that for the case of the RO horzontal devce structure, the coeffcents of x, y, x 2 and y 2 n the regresson model are sgnfcant. The relatve low R 2 value for the spatal model correlates well wth the Fg. 6 that shows that there exsts only a slght trend that the measured devce frequency ncreases dagonally from the bottom to the top of the chp. ACKNOWLEDGMENT The authors would lke to thank Karen Gonzalez for the useful dscussons and provdng the dataset for the analyss. The provson of a research grant from NSF/SRC Engneerng Research Center for Envronmentally Bengn Semconductor Manufacturng and a research scholarshp (L. W. Teo) and fnancal assstance to ths project by the Sngapore-MIT Allance are gratefully acknowledged. REFERENCES [] D. Bartelnk, Statstcal metrology-at the root of manufacturng control, J. Vac. Sc. Technol. B, vol. 2, pp , July/Aug. 994 [2] D. Bonng and J. Chung, Statstcal metrology: Understandng spatal varaton n semconductor manufacturng, n Mcroelectronc Manufacturng Yeld, Relablty, and Falure Analyss II: SPIE 996 Symp. On Mcroelectronc Manufacturng, Oct. 996, Austn, TX. [3] J. Kbaran and A. Strojwas, Usng the spatal nformaton to analyze correlatons between test structure data, IEEE Trans. Semcond. Manufact., vol. 4, pp , Aug. 99 [4] B. E. Stne, D. Bonng and E.C. Chung, Analyss and decomposton of spatal varaton n ntegrated crcut Processes and Devces, IEEE Trans. Semcond. Manufact., vol.0. no., Fed, pp. 24-4, Feb [5] J. S. Panganban, A rng oscllator based varaton test chp, Master dssertaton, Dept. Elect. Eng. And Comp. Sc., Massachusetts Insttute of Technology, Cambrdge, MA, [6] K. M. Gonzalez, Extracton of varaton sources due to layout practces, Master dssertaton, Dept. Elect. Eng. And Comp. Sc., Massachusetts Insttute of Technology, Cambrdge, MA, [7] Avalable: [8] G. E. P. Box, W. G. Hunter, J. S. Hunter, Statstcs for Expermenters, Wley Seres, New York, 978. [9] Avalable: V. CONCLUSION AND FUTURE WORK Wafer level spatal analyss shows that RO frequences obtaned from ths dataset don t have lnear spatal dependence on wafer level, as all R 2 values were low. It needs further nvestgaton n terms of non-lnear regresson modelng to verfy f there s any spatal model that exsts n the system. On the chp level, some nterestng trends were observed such as some chps lke Chp, 2, 3, whch shows really low R 2 value for all devce structures. Ths observaton needs to be further nvestgated f there exsts some potental process ssues wth these chps. On the other hand some devce structures does have good R 2 value (>.0.6) for few chps such