EE650R: Reliability Physics of Nanoelectronic Devices Lecture 5: Statistical Projection Date:

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1 EE0R: Relably Physcs of Naoelecroc Devces Lecure : Sascal Projeco Dae: Sep 0, 00 ClassNoes: Jg L Revew: Jaydeep P. Kulkar. Revew/Backgroud I dscussg he BRW problem, we meoed ha vable relably ess mus o ake oo log ( < days ad ca o use oo may samples (~0-0. Based o hese shor me daa from few samples oe mus be able o deerme he log-me (~0 years relably of he hudreds of mllos of producs, so ha produc falure does o exceed 00 pars per mllo (00 ppm, for example. We have already dscussed he eld accelerao mehodology ha allows oe o calculae average falure mes (T av a operag codos based o daa from shor me acceleraed ess -- oday we wll dscuss he ssue of sascal (or percele projeco whch daa from few samples are used o esmae properes of very large populao. There are hree pars o hs dscusso: rs, we dscuss how o hadle he expermeal daa well: he sample sze s small, so we mus rea he daa carefully so ha we do o lose ay formao; Secod, we wll show how o f he expermeal daa wh emprcal probably dsrbuo fucos so ha we ca predc he falure raes for very large populao; Thrd, ad fally, we wll dscuss, he ex lecure, he cocep of goodess-of-f mercs oher words, whle may emprcal dsrbuos ca f he lmed daase, some f he daa beer ha ohers ad we wll see how o dffereae hem.. Noparamerc Aalyss of Expermeal Daa: The o-paramerc aalyss of expermeal daa s a powerful ool whe he umber of samples s small. There are several mehods for coducg a oparamerc aalyss, cludg he Mome Esmaes, Kapla-Meer ad so o... Mome Esmaes Gve a se of falure mes, (e.g. fsh fallg dow he waerfall, oe ca readly calculae he average falure me for he daase, T av where s he umber of samples. Smlarly, oher momes ca be compued as k δ T K ( T av k + k

2 where k meas he k h mome. Noe he use of (-k+ he domaor raher ha hs s because he degrees of freedom of he formao coe of he daa reduces by oe each me a parameer s exraced from he daase... Geeralzed Haze formula for (Ucesored Daa Nex oe would lke o use he expermeal falure mew o cosruc, exp ( he Cumulave Dsrbuo uco (CD. Ths s ofe doe by Geeralzed Haze formula. ( + The value of mus be chose carefully for ubased esmaes of. I has bee suggesed ha 0 works bes for relavely small samples, whle 0. s releva for larger sample sze. To udersad he procedure clearly, cosder a problem wh samples ( falg a mes,.. Wh (, we fd ; (blue le he gure below ; + 0. ; + 0. Kapla-Meer 0 0 Oe Sample s mssg. Kapla-Meer Mehod for Cesored Daa If few of he samples s cesored, ha s, removed from he experme a me before he had me o fal, he queso s: How would he exp ( be modfed? Ths may happe BRW problem f a fsh des before arrvg a he waerfall, or could arse

3 semcoducor devce esg f he probe p ges dscoeced from pad before compleo of he es, ec. We could jus hrow he sample ou from our cosderao ( reduces from o ad keep usg he Geeralzed Haze formula,.e., ; (red le Remarkably he frs wo pos are ow beg affeced by rd daapo (fuure affecg he pas, whch does o make sese uvely. The mpac of such cesorg ca be sgfca for small sample sze. So we roduce aoher mehod, amely Kapla- Meer Mehod, o solve hs problem. The Kapla-Meer mehod s a oparamerc (acuaral echque for esmag merelaed eves (he survvorshp fuco. I s used o predc he sascal dsrbuo whe he sample formao s los durg he experme ad s gve by f + s ( s or smplcy, we assume 0, so ha ( smplfes o f s + s +. Le us frs covce ourselves ha Kapla-Meer Mehod ca ge he same resul as Haze s formula f he sample space s o chagg (for ucesored daa. The umber of survvg sample ( s afer each me po ( s ls he able below: s before s afer 0 where s s he umber of survvg samples afer me ; ; ; ; The resuls cocde wh Geeralzed Haze formula (Blue le, as expeced. Now f he sample space s chagg due o cesorg of daa, he he resuls are que dffere. or example, assume ha a me, oe sample s ake ou of he expermes. rs wo pos are uaffeced by rd sesor daa. ; 7 ; ; ; 9 9

4 These hree cases dscussed above s ls hs able: Mehod T T T T T Haze s ormula Haze s ormula wh oe sample mssg Mssg Sample Kapla-Meer Mehod Mssg Sample Gve exp (, oe ca compue R exp ( as well as f exp ( based o he equaos dscussed Lecure. The reame of exp ( s very mpora, because subseque fg of emprcal probably dsrbuo, heory ( would amplfy ay error exp (.. Emprcal Sasc Dsrbuos Alhough dsrbuo fucos arse from varous physcal processes (.e., gae delecrc breakdow s ofe descrbed by Webull dsrbuo,, Elecromgrao s descrbed by log-ormal dsrbuo, ec., for he dscusso o follow, we wll assume ha he dsrbuos are already gve ad ca be used o f expermeal daa. or he me beg, we wll cofe our dscussos o hree dsrbuos ha arse frequely relably aalyss: Normal, Log-Normal, ad Webull. Some of he characerscs of he dsrbuos are dscussed below: Probably Desy uco (PD Normal Log Normal Webull e πσ ( µ σ e σ π (l l µ ( σ e f( σ f( Asymmerc f( PD µ

5 Cumulave Dsrbuo uco (CD µ + erf σ l lµ + erf σ e ( ( ( l(-l(-( CD Mome ( s µ Mome ( d µ e σ σ σ ( e σ µ e + Γ ( + Esmag he Parameers of he Theorecal Dsrbuos I geeral, here are hree ways o f he observed dsrbuo (Sec.. o a heorecal dsrbuo dscussed above o exrac he parameers of he heorecal model (e.g. a, b for he Webull model. These mehods are: Mehod of Momes: Ths mehod ca be employed o deerme parameer esmaes for a dsrbuo. The mehod of machg momes ses he dsrbuo momes (heory equal o he daa momes (experme ad solves o oba esmaes for he dsrbuo parameers. or example, for a dsrbuo wh wo parameers, he frs wo momes of he dsrbuo (he mea µ ad varace σ of he dsrbuo, respecvely would be se equal o he frs wo momes of he daa (he sample mea µ ad varace σ, respecvely ad solved for he parameer esmaes. Mehod of Leas Square: Leas square s a mahemacal opmzao echque whch, whe gve a seres of measured daa, aemps o fd a fuco whch closely approxmaes he daa (a "bes f". I aemps o mmze he sum of he squares of he ordae dffereces (called resduals bewee pos geeraed by he fuco ad correspodg pos he daa. Suppose ha he daa se,exp cosss of he pos (x,y wh,,,. We wa o fd a fuco,heroy such ha,exp, heroy. Ths s doe by mmmzg he error bewee observed ad heorecal dsrbuos,.e.

6 Error uco: E (,exp, heroy or example, he Webull parameers a ad b ca be obaed by mmzg he error wh respec o hose parameers: de.e. 0 d ad de d 0 ad solvg he wo equaos he process. Maxmum lkelhood Esmaor (MLE: Ths mehod s creded o sher (90 ad s wdely used for robus fg of expermeal daa. Maxmum lkelhood esmao begs wh wrg a mahemacal expresso kow as he Lkelhood uco of he sample daa. I oher words, he lkelhood of a se of daa s he probably of obag ha parcular se of daa, gve he chose probably dsrbuo model. Ths expresso coas he ukow model parameers. The values of hese parameers ha maxmze he sample lkelhood are kow as he Maxmum Lkelhood Esmaes or MLE's. Le us go hrough oe example o ge beer udersadg of hs mehod. Le us cosder Webull dsrbuo as he heorecal PD o f he expermeal daa. The ukow model parameers are ad, respecvely. Hece, he Lkelhood uco of he sample daa s assume o be L f(,, f Maxmum Lkelhood esmaor L Po eed o fd Maxmum Lkelhood esmaor, Sce produc fuco s dffcul o evaluae, oe ofe use he log of he prevous expresso,.e. l L l f(,, wh he reasoable assumpo ha same parameers would maxmze log(l ad L.

7 To fd he values of ad so ha he Lkelhood uco reaches s maxmum value, dl L we se 0 d ad dl L 0 d or Webull dsrbuo, oe ca readly show ha he above codos produce wo equaos wh wo ukows l( l(. Ths gves value of. You should ry o oba smlar equaos by MLE for Normal ad Log-Normal dsrbuos. Obvously, oe ca f varous heorecal dsrbuos o he same se of expermeal daa ad oba releva parameers. There s ohg he mahemacal procedure Mehod of Momes, Leas Square mehods, or MLE o o sugges f oe model s beer ha he oher. However, remember ha we are eresed he als of he dsrbuo (e.g % relably ad varous heorecal dsrbuos would predc dffere values of he al of he dsrbuos. Therefore he queso s: Whch oe of hese dsrbuos fs he daa he bes so ha he al of he dsrbuo s relable? We wll pck up hs queso of Goodess of Lecure.