Mixture Model. A dissertation presented to. the faculty of. In partial fulfillment. of the requirements for the degree. Doctor of Philosophy.

Transcription

1 Sem-paramerc Bayesan Inference of Acceleraed Lfe Tes Usng Drchle Process Mure Model A dsseraon presened o he faculy of he Russ College of Engneerng and Technology of Oho Unversy In paral fulfllmen of he requremens for he degree Docor of Phlosophy X Lu December X Lu. All Rghs Reserved.

2 2 Ths dsseraon led Sem-paramerc Bayesan Inference of Acceleraed Lfe Tes Usng Drchle Process Mure Model by XI LIU has been approved for he Deparmen of Indusral and Sysems Engneerng and he Russ College of Engneerng and Technology by Tao Yuan Assocae Professor of Indusral and Sysems Engneerng Denns Irwn Dean Russ College of Engneerng and Technology

3 3 ABSTRACT LIU XI Ph.D. December 205 Mechancal and Sysems Engneerng Sem-paramerc Bayesan Inference of Acceleraed Lfe Tes Usng Drchle Process Mure Model 00 pp. Drecor of Dsseraon: Tao Yuan Acceleraed lfe esng ALT s commonly used o esmae he relably of hghly relable producs. Ths dsseraon develops sascal models o predc useful lfe of nano devces wh daa colleced under consan-sress ALT and sep-sress ALT. As an eample of nano devces nc-moo embedded ZrHfO hgh-k delecrc hn flm s suded wh respec o s physcal properes falure mechansms and long-erm sably. The devces used for ALT and relably predcon demonsraon have dencal srucure wh hs nc-moo embedded devce. Ths research develops a sem-paramerc Bayesan mehod o analyze ALT. The model assumes a log-lnear lfeme-sress relaonshp whou assumng any paramerc form of he falure-me dsrbuon. The Drchle Webull mure model s employed o model he falure-me dsrbuon under a gven sress level. The model s fed wh a smulaon-based algorhm whch mplemens Gbbs samplng o analyze ALT daa and predcs he falure-me dsrbuon a a normal sress level. Two praccal eamples relaed o he relably of nanoelecronc devces are presened for consan-sress ALT ncludng one rgh-censored daa and one complee daa se. One rgh-censored praccal eample s demonsraed for smple sep-sress

4 ALT. All hree eamples llusrae he capably of he proposed mehodology o provde accurae predcon of he falure-me dsrbuon a a normal sress level. 4

5 5 DEDICATION Ths dsseraon s dedcaed o my dear faher Janzhen Lu and o my moher Jnpng Deng.

6 6 ACKNOWLEDGMENTS Over he pas fve years of docoral ourney I have receved numerous encouragemen and suppor from a grea number of persons and nsuons. My work canno be compleed whou her suppor. Frs and foremos I would lke o epress my deepes apprecaon o my advsor Dr. Tao Yuan for hs connuous suppor and gudance hroughou my graduae sudes. I would never have been able o fnsh hs dsseraon whou hs perssen help and deep knowledge. I also would lke o hank Dr. Yue Kuo who has offered me he grea opporuny o do ess n Thn Flm Nano & Mcroelecroncs Research Laboraory Teas A&M Unversy for hs suppor on hs dsseraon and for gudng me o he semconducor devces feld. Ne I would lke o hank Dr. Andrew Snow Dr. Davd Koonce and Dr. Dana Schwercha for servng my commee members and offerng many nsghful commens and suggesons o my research. I am deeply graeful o Dr. Cha-Han Yang and Dr. Ch-Chou Ln for fabrcang samples eachng me he devce conceps and menorng me on esng and measuremens. Thanks Mnghao Zhu for he XPS and XRD analyzes. Thanks Dr. Saleem Z. Ramadan and Dr. Wen Luo for supplyng me daa ses as my praccal eamples. Lasly I would lke o hank my famly for her connuous suppor nspraon and love. Ths dsseraon was parally suppored by Naonal Scence Foundaon proec CMMI and CMMI

7 7 TABLE OF CONTENTS Page Absrac... 3 Dedcaon... 5 Acknowledgmens... 6 Ls of Tables... 9 Ls of Fgures... 0 Chaper : Inroducon.... Movaon and Obecve....2 Relably and Relaed Funcons Acceleraed Lfe Tesng Acceleraon Model Sep-Sress Acceleraed Lfe Tesng Commonly Used Esmaon Mehod Paramerc mehod Nonparamerc Bayesan mehod Drchle process mure model Nanocrysals Embedded Hgh-k Devce Dsseraon Overvew Chaper 2: Leraure Revew Inference of Consan-Sress Acceleraed Lfe Tesng Paramerc esmaon of consan-sress ALT Sem-paramerc esmaon of consan-sress ALT Inference of Sep-Sress Acceleraed Lfe Tesng Paramerc esmaon of sep-sress acceleraed lfe esng Sem-paramerc esmaon of sep-sress acceleraed lfe esng Drchle Process Mure Model... 4 Chaper 3: Problem Saemen Noaons Assumpons... 45

8 3.3 Problem Descrpon Invesgaon of meal ode nanodos-embedded ZrHfO hgh-k flm Developmen of smulaon-based algorhm Comparson beween paramerc Webull log-lnear ALT model and DP Webull mure ALT model Chaper 4: Memory Funcons of Molybdenum Ode Nanodos Embedded ZrHfO Hghk Fabrcaon of Nanocrysals Embedded ZrHfO Hgh-k Devce Physcal Properes of Nanocrysals Embedded ZrHfO hgh-k Devce Charge Trappng and Derappng Mechansms Charge Reenon Capably Conclusons Chaper 5: Bayesan Analyss For Acceleraed Lfe ess Usng Drchle Process Webull Mure Model Mehodologes Drchle process Webull mure ALT model Smulaon-based model fng Illusrave Eamples Complee daa se eample Rgh-censored daa se eample Conclusons Chaper 6: Bayesan Analyss For Smple Sep-Sress Acceleraed Lfe Tesng Mehodologes Illusrave Eamples... 8 Chaper 7: Conclusons Memory Funcons of MoO Nanodos Embedded ZrHfO Hgh-k Predcon of CDF of Falure-Tme Dsrbuon a Normal Sress Level Fuure Research References

9 9 LIST OF TABLES Page Table 5.: Eample : mes-o-breakdown of MOS capacors esed a four elecrcal feld sresses...70 Table 5.2: Eample 2: mes-o-breakdown of nanocrysals-embedded hgh-k memores esed a four volage sresses...75 Table 6.: Tmes-o-breakdown of nanocrysals-embedded hgh-k memores under smple SSALT...8

10 0 LIST OF FIGURES Page Fgure.: The relably funcon CDF and PDF of he Webull dsrbuon wh=.5 and= Fgure.2: Acceleraed lfe esng concep...5 Fgure.3: Sep-sress acceleraed lfe esng...8 Fgure.4: Cross-seconal vew of nanocrysals embedded ZrHfO capacor...28 Fgure 4.: a XPS O s peak and b XRD paern of he MoO embedded ZrHfO sample...49 Fgure 4.2: C-V hyseress curves of he nc-moo embedded ZrHfO capacor measured a MHz. The nse s he hyseress curve of he conrol sample...5 Fgure 4.3: J-V curve of nc-moo embedded capacor V g swep from -8 V o +8 V. The nse s he J-V curve of conrol sample...52 Fgure 4.4: Reenon propery of holes rapped n he nc-moo embedded capacor. The nse shows he erapolaon of he curve o 0 years proecon Fgure 5.: Eample : predced falure-me CDF a he normal sress 0 = 7. MV/cm...72 Fgure 5.2: Eample : esmaon of falure-me CDF a 7.9 MV/cm...74 Fgure 5.3: Eample 2: predced falure-me CDF a he normal sress 0 = 7. V...76 Fgure 6.: Trace plos of and...82 Fgure 6.2: Predced falure-me CDF a he normal sress 0 = 7.5 V...83

11 CHAPTER : INTRODUCTION. Movaon and Obecve For he purpose of ncreasng compeve advanages decreasng coss and sasfyng ncreasng cusomer epecaons manufacurers srve o desgn and produce hghly relable producs. Consequenly he sudes of relably daa analyss have been promoed. Quanave mehods o predc and assess produc relably are requred o mprove relably of esng producs and ensure he hgh relably of new producs []. The daa obaned from lfe ess are commonly suppled o sascal mehod for relably esmaon [2]. Convenonal lfeme daa analyss assumes he falure me followng ceran paramerc dsrbuons such as eponenal Webull or log-normal and esmaes he parameers of he dsrbuons [3]. However n some suaons may be dffcul o choose he approprae paramerc model. For eample modern comple producs may nvolve mulple falure modes and herefore smple lfeme dsrbuons are no adequae o descrbe her falure mechansms. For some new echnologes such as nanoechnology he falure mechansms have no been well undersood. Therefore more fleble nonparamerc mehods are needed. In recen years he developmen of Markov chan Mone Carlo MCMC mehods for smulaon-based mplemenaon and analyss assures he feasbly of nonparamerc daa analyss [4]. Ths dsseraon ams o assess he relably under normal sress levels based on ALT usng Bayesan models nvolvng Drchle process mure Models DPMM. Boh consan-sress ALT and sep-sress ALT are suded n hs work. The res of hs chaper

12 2 provdes a bref background nroducon of relably and relaed funcons ALT acceleraon models sep-sress ALT and commonly used esmaon mehods..2 Relably and Relaed Funcons The relably s he probably ha a sysem or componen can perform s funcon under operaon condons for a specfed perod of me. I can be mahemacally presened as: R = Pr{T }. where T s a random varable denong he me o falure of a non-reparable devce or he me o he frs falure of a reparable devce and R s called he relably funcon of he falure-me dsrbuon. Denoe F as he cumulave dsrbuon funcon CDF of he falure-me dsrbuon whch s he probably ha he lfeme s less han.e. F= -R = Pr{T <}..2 If he CDF s a connuous funcon of a hrd funcon defned by he dervave of F s used o descrbe he shape of he falure-me dsrbuon. I s called he probably densy funcon PDF and can be epressed as: df dr f.3 d d where f s a non-negave funcon. Takng Webull falure-me dsrbuon as an eample s PDF CDF and relably funcon can be wren as: f ep.4

13 3 / F ep d e 0.5 / and R F e..6 where and are he shape and scale parameers respecvely. Fgure. graphcally shows hese hree funcons when he me o falure follows he Webull dsrbuon wh he shape parameer =.5 and he scale parameer = R F f Tme o Falure Fgure.. The relably funcon CDF and PDF of he Webull dsrbuon wh=.5 and= 7. Anoher mporan funcon he falure rae or hazard rae funcon s used o provde an nsananeous falure rae whch can be denoed as: F F f h lm. 0 Pr{ T } R.7

14 4 When h s an ncreasng consan or decreasng funcon he falure rae can be descrbed as ncreasng IFR consan CFR or decreasng DFR respecvely [5]. The mean me o falure MTTF and he medan me o falure are commonly used as measures of he cener of a falure-me dsrbuon. The MTTF s he average lengh of me unl falure and can be defned as he epeced value of T.e. 0 0 MTTF E T f d R d..8 For a gven value of p[0 ] f p sasfes F p = p.9 hen p s called he ph percenle of he lfeme dsrbuon whch means 00p% of he falures occur before me p. If p=0.5 hen 0.5 s he medan me o falure. When he dsrbuon s hghly skewed he medan s preferenally used as he measure of he cener locaon of a dsrbuon. The medan 0.5 as well as he lower and upper 25% percenles 0.25 and 0.75 respecvely are mporan characerscs of a lfeme dsrbuon..3 Acceleraed Lfe Tesng Relably lfe esng s carred ou o oban falure nformaon for he purpose of quanfyng relably [5]. Hghly relable producs such as elecronc devces usually have long lfemes. Therefore only very few falures can be obaned whn reasonable me perod under acual operang condon and s dffcul o oban adequae falureme daa for sascal mehods. One approach o solve hs problem s o use ALT n whch he uns are placed under hgher han operaonal sress condons o speed up he falure occurrence [2]. Classcal sress ncludes volage curren humdy emperaure

15 5 pressure cyclng rae or load [6]. Then he falure-me daa colleced a elevaed sress levels are analyzed and erapolaed o predc he relably a he normal sress level hrough an acceleraon model. ALT s based on he fundamenal prncples ha he un under es wll have he same falure mechansms n a shor me a a hgh sress level as ehbs n a longer me a a lower sress level [7]. Fgure.2 shows he basc concep of an ALT esmaon whch uses lfeme daa colleced from a four-level sngle-sress es o esmae mean lfe under normal sress. The varable denoes he sress level wh 0 represenng he normal sress. Accordng o Dasgupa and Pech [8] here are four caegores of falure mechansms: sress-srengh damage-endurance challenge-response and olerance-requremen. The ALT s he mos approprae wh he damage-endurance falure and some cases of olerance-requremen falure [7]. Fgure.2. Acceleraed lfe esng concep. The consan-sress ALT and he sep-sress ALT are he wo ypcal ypes of acceleraed lfe esng. In consan-sress ALT each sngle un s placed only under one

16 6 hgher han normal sress level whle he sep-sress ALT SSALT allows several sress levels..4 Acceleraon Model One dffculy of ALT daa analyss s how o predc he relables of uns a he normal sress level from he falure-me daa under hgher sress levels. Bascally a funconal relaonshp called he acceleraon funcon s appled o descrbe he relaonshp beween he lfemes and he sress condons [6]. The log-lnear model whch assumes he log-lnear relaon for he lfeme s wdely used can be epressed as: ln a b... b m m.0 where denoes he sress relaed characersc and m denoe he sress facors or proper ransformaon of hem. The log-lnear model s used because s smple and can be ransformed from many oher relaonshps. For eample when he emperaure prmarly conrbues o he falures he Arrhenus model s commonly used: E ep a r A KT. where r s he reacon or process rae A s consan E a represens he acvaon energy n elecron vols T s he absolue emperaure K and K s he Bolzmann s consan a known physcal consan equals o ev/ K [9]. Ths equaon can be ransformed o he log-lnear model as lnr = lna-e a /K /T wh =/T.

17 7 The generalzed Eyrng model s applcable for falures relaed o wo ypes of sresses one hermal and one nonhermal sress. The smples form of Eyrng model can be presened as: r AT E ep a C ep B s KT T.2 where r s he process rae A B and C are consans E a s he acvaon energy T s he absolue emperaure K K s he Bolzmann s consan and s s he second sress [0]. When =0 and C=0 hs equaon can be ransformed o he log-lnear model as ln r ln A Ea / K/ T Bs wh =/T and 2 =s. The E-model can be used o sudy he me-dependen delecrc breakdown of gae delecrc hn flms where he sress s he elecrcal feld. The E-model s epressed as: L ep G E E.3 where L and G L are unknown emperaure-dependen consans E s he appled elecrcal feld and E bd s he feld above whch breakdown occurs mmedaely []. The E-model can also be epressed as a log-lnear funcon ln=ln L + GL Ebd E. In hs sudy he log-lnear acceleraon model wll be used and he deals wll be nroduced n secon Sep-Sress Acceleraed Lfe Tesng Sep-sress acceleraed lfe esng SSALT can furher ensure enough falures whn reasonable me perod by esng he uns hrough more han one level of sress. In he SSALT he sress levels change usually ncrease a pre-specfed mes me-sep L bd

18 8 sress ALT or afer pre-specfed numbers of falures falure-sep sress ALT. A es wh only one change of sress s called a smple sep-sress ALT whle a es wh several sress changes s called a mulple sep-sress ALT [6]. The number of falures n he mesep sress ALT s random a each level of sress. The me duraon of each level of sress n he falure-sep sress ALT s also random. If he es ends a a pre-specfed me s called ype-i censorng; f he es ends when a pre-specfed number of falures has acheved s called ype-ii censorng. Fgure.3 shows a mul-ssalt wh four sresses and ype-i censorng. The es ends a a pre-specfed me c. All uns ha have no faled by c are censored. The s and s are he sress levels and sress changng mes respecvely. In order o analyze SSALT daa a model descrbng he effec of changng sress s needed. Nelson [2] proposed he cumulave eposure model whch assumes ha he remanng lfe of specmens depends only on he curren cumulave fracon faled and curren sress- regardless of how he fracon accumulaed. Sress Level Tme Fgure.3. Sep-sress acceleraed lfe esng. c

19 Denoe F as he CDF of he falure-me dsrbuon under sress F under a sep-sress paern epressed by he cumulave eposure model can be wren as: 9 F F2 u F3 2 u2 F F m m where τ s he me of changng he sress from he h sress level o he +h sress level F s he CDF under he h sress level and u s he soluon of F + u =F τ - τ - + u -. Ths dsseraon sudes he smple SSALT wh wo sress levels and assumes he falure-me dsrbuon under each sress level followng a Drchle process mure model wh Webull kernel. The deals of nference wll be gven n Chaper 6..6 Commonly Used Esmaon Mehod The commonly used daa esmaon mehod for ALT analyss can be classfed as paramercal and non-paramercal dependng on f here are parameers assumed n he model. Two wdely appled paramerc esmaon groups are Mamum Lkelhood Esmaon and paramercal Bayesan esmaon. Whle commonly used nonparamerc esmaon mehods for ALT analyss are emprcal mehod and nonparamercal Bayesan Inference. In addon he Drchle Process Mure Model s a popular nonparamerc Bayesan esmaon mehod frequenly ced n leraure.

20 20.6. Paramerc mehod The paramerc mehod s one approach o performng ALT model nference. The paramerc nference assumes he lfeme dsrbuon under each sress level comes from he same paramerc famly and s preassgned wh a heorecal dsrbuon such as Webull eponenal or log-normal dsrbuon. Afer ha an acceleraon model s chosen and he parameers are esmaed. The Mamum Lkelhood Esmaon MLE s one of he mos wdely used paramerc esmaon mehod o esmae he model parameers gven he sample daa. Gven n falure-me daa = n colleced from he es he pon esmaor of parameers are obaned by mamzng he lkelhood funcon L : n L Θ L Θ.5 where L s he lkelhood conrbuon of he h observaon and s he parameer vecor o be esmaed. For he complee daa se L = f and for rghly censored daa L = R where s he censored me. The logarhm of lkelhood funcon loglkelhood funcon nsead of he lkelhood funcon s usually mamzed for compuaonal convenence. In general he mamum lkelhood esmaor s obaned by solvng he followng ses of equaons: ln L 0 ln L k 0 k.6 where k s he number of parameers and s he esmaed parameers.

21 2 The MLE mehod assumes unknown parameers as fed. Therefore n order o oban precse esmaon a large sample sze and accurae model assumpons are requred. Anoher paramerc approach o conducng ALT model nference Bayesan nference has been appled. Bayesan nference assumes he parameers are random and descrbes he unceranes by a on pror dsrbuon. The pror dsrbuon s formulaed before daa collecon and s based on he hsorcal daa or epers opnons. The man advanage of Bayesan nference s he ably of combnng he colleced daa wh any relaed nformaon avalable for relably analyss whch can rela he sample sze lmaon. The Bayesan daa analyss esmaes parameers usng he poseror dsrbuon f whch s obaned by ncorporang s pror dsrbuon f and he lkelhood funcon of daa L. The pror dsrbuon s updaed afer he daa s colleced accordng o he Bayes heorem: L Θ f Θ f Θ.7 f where f L Θ f Θ.8 whch s called he preposeror margnal dsrbuon of. In many praccal problems whch are comple and nvolve more han one parameer mulple levels of negraon are necessary n Bayesan nference. Mosly hese negraons are analycally nracable and herefore numercal mehods are used nsead. For eample he Markov chan Mone Carlo MCMC smulaon has been wdely used for numercal negraons. Generally smulaes a Markov chan n such a

22 way ha he saonary dsrbuon of he chan s he poseror dsrbuon of he parameers and hen uses he smulaed daa o compue Bayes esmaon [3]. The Gbbs samplng whch s usually appled when s dffcul o drecly sample from mulvarae probably dsrbuon s a ype of MCMC smulaon ha s parcularly useful n hgh dmensonal problems. For eample when samples from f 2 are needed and s dffcul o sample drecly from her margnal dsrbuons f and f 2 her condonal dsrbuons f and f are sampled nsead n 2 2 each eraon. When he number of eraons s suffcenly large he samples obaned from condonal dsrbuons can be regarded as smulaed observaons sampled from her margnal dsrbuons. In hs sudy he Gbbs samplng wll be appled n he semparamerc Bayesan mehods for smulang poseror dsrbuons. Generally he paramerc relably analyss s performed by fng he lfeme daa wh a suable paramerc model. Usually a falure dsrbuon wh parameers s assgned and hen s esmaed based on he observed daa. Some commonly used falure-me dsrbuons are Webull eponenal or log-normal probably dsrbuon. In he acceleraed lfe esng an acceleraon relaonshp s also seleced and hen he parameers are esmaed..6.2 Nonparamerc Bayesan mehod The accuracy of convenonal paramerc esmaon s based on he paramerc assumpons ha are assgned o he daa ha s he parcular paramerc famly of dsrbuons assumed. However he falure mechansm of some producs may be unknown and may nvolve mulple modes or seps whch are mpossble o model usng 22

23 23 a smple lfeme dsrbuon. Therefore more fleble mehods-nonparamerc mehods have been developed. One class of nonparamerc mehods s emprcal whch have no resrcve assumpons on he lfeme dsrbuons and derve he relably properes such as PDF and CDF drecly from he daa. Some commonly used emprcal mehods nclude Kaplan-Meer esmaor and Medan rank whch are shown n equaons.9 and.20 respecvely. R ˆ.9 n : where s he ordered falure mes and n s he number remanng a rsk us pror o he h falure. ˆ 0.3 F.20 n 0.4 where s he h ordered falure and n s he sample sze. The nonparamerc Bayesan nference s anoher group of nonparamerc mehods whch has been proposed o esmae a probably dsrbuon. The nonparamerc Bayesan mehods n he praccal use are acually probably models wh nfnely many parameers on funcon spaces [4]. In ALT analyss he falure-me daa under each sress level s no suggesed by any sandard model. Therefore s dsrbuon-free. Generally a pror dsrbuon on he class of all dsrbuon funcons s placed and he poseror dsrbuon on he class of all dsrbuon funcons s obaned from daa. The pror dsrbuons for he underlyng dsrbuon funcons consue a sochasc process.

24 24 There are many nonparamerc Bayesan mehods for dfferen applcaons ncludng Gaussan process GP splne models and DPMM ec. [4]. These mehods are wdely used n sascal nference problems such as densy esmaon regresson and cluserng. In hs sudy he DPMM s proposed o be used o esmae he ALT daa. Alhough he nonparamerc mehods are more fleble when boh paramerc and nonparamerc mehods are applcable for a problem he paramerc mehod s preferred because of s effcency and compuaonal convenence [3]..6.3 Drchle process mure model The Drchle Process DP s by far he mos popular nonparamerc model n he leraure [4]. The DP pror whch was formally developed by Ferguson [5] s he frs pror defned for spaces of dsrbuon funcons [6]. I fulflled wo desrable properes of pror dsrbuon for nonparamerc problems: large suppor of he pror dsrbuon and analycally manageable poseror dsrbuons [5]. The foundaon of Drchle process s he Drchle dsrbuon. Defnng he probables of n dscree space = {... n } are Θ= {θ θ n }.e. px= = θ. Then he PDF of Drchle dsrbuon can be defned as: n y p y... ym.2 B y where By s he normalzng consan epressed n erms of gamma funcon: n y B y n..22 y

25 Denoe y as he concenraon parameer of and m = {m m n }= / as he base measure he Drchle dsrbuon can be epressed as: 25 p M n m. n.23 m The concenraon parameer shows how much he probably would be concenraed around m. When n=2 he Drchle dsrbuon reduces o a Bea dsrbuon. The Drchle process can be regarded as he eenson of he Drchle dsrbuon o connuous spaces whch ncludes wo parameers: a posve scalar parameer M and a probably base measure G 0. The base dsrbuon G 0 s where he nonparamerc dsrbuons are cenered and usually represens he pror belef [7]. The concenraon parameer M ndcaes he degree of concenraon of he dsrbuon around G 0. The greaer he M he more samples from Drchle process are concenraed around G 0 [8]. Therefore he dsrbuon funcon G wh a DP pror can be wren as [9]: G ~ DPG I s worh nong ha alhough he Drchle process s defned over a connuous space s sll dscree as consss of counably nfne pon probably mass. Undersandng he forma of a mure model s a necessary sep before undersandng DPMM. Gven he observaon of n ndependen random varables v v 2 v n generaed from a populaon wh unknown PDF kv a paramerc mure model wh k componens can be wren as [20]:

26 k 26 f v k v θ.25 where kv s a paramerc kernel wh parameer vecor and he mng proporons 0< < sasfy. Defne a laen allocaon varable z as he group o whch he observaon v belongs and z s supposed o be drawn ndependenly from he dsrbuon [2]: p z... k..26 The herarchcal form of he paramerc mure model can be wren as: v z ~ k θ = 2 3 n v z θ ~ G 0 = 2 3 k.27 z ~ Mulnomal = 2 3 n ~ Drchle /k /k where s concenraon parameer of Drchle dsrbuon. In hs model all θ s are assumed o come from a common dsrbuon G 0 and he mng proporon s assgned as a Drchle pror. When k he Drchle dsrbuon becomes he Drchle process and he paramerc mure model becomes he Drchle process mure model gven by: v θ ~ kv θ = 2 3 n θ ~ G = 2 3 n.28 G ~ DPG 0. In he DPMM he parameer vecor 'sare assumed o be from a common dsrbuon funcon G wh a DP pror. The unknown PDF f modeled by he DPMM can be epressed as:

27 27 f v k v θ dg θ..29 The behavor of he DPMM s affeced by he choce of he kernel dsrbuon k. Many common dsrbuons have been appled n he DPMM such as DP Gaussan mure model DPGMM DP eponenal mure model DPEMM and DP Webull mure model DPWMM. In hs sudy he DPWMM s appled for ALT analyss..7 Nanocrysals Embedded Hgh-k Devce The ALT daa whch are analyzed n hs sudy are he falure mes of nanocrysals embedded hgh-k devces colleced a Thn Flm Nano & Mcroelecroncs Research Laboraory Teas A&M Unversy. When he meal ode semconducor feld-effec ranssor MOSFET s scaled down o he nano scale o sasfy he requremens of echnology developmen he hckness of he slcon dode SO 2 gae delecrc layer has o be reduced drascally. Ths degrades he devce performance and relably [22]. One effecve soluon s o use a hgh delecrc consan hgh-k flm o replace SO 2. The hgh-k flms have also been used n memory devces [23]. The convenonal poly-s floang-gae nonvolale memory NVM devce conans a connuous poly-s layer n he SO 2 gae delecrc as he charge-rappng medum. Therefore a sngle leakage pah n he unnelng ode may quckly dran all he charges. The nanocrysals embedded delecrc srucure can solve hs problem because he nanodos are solaed from each oher by he surroundng delecrc maerals. Therefore a sngle leakage pah can only dran charges sored n one or a few dos locally [24]. Varous conducve and semconducve maerals such as S ITO ZnO and MoO have been prepared no he nanocrysallne form and embedded n

28 28 hgh-k flms as elecron- or hole-rappng meda [23] [25] [27]. Fgure.4 shows a cross-seconal vew of a general sngle-layer nanocrysals embedded ZrHfO capacor. The relably of hs knd of devce has no been well suded whch s mporan o he praccal applcaons. The me-dependen delecrc breakdown refers o damageendurance falures [7] and herefore ALT can be appled for obanng he falure daa of delecrc devces. In hs sudy he falure-me daa of hs knd of devce are appled n he DPMM for relably analyss. nanocrysals Al Gae ZrHfO hgh-k Inerface Layer p-s Fgure.4. Cross-seconal vew of nanocrysals embedded ZrHfO capacor..8 Dsseraon Overvew The balance of hs dsseraon s organzed as he followng: Chaper 2 revews prevous research on nference of acceleraed lfe esng and Drchle process mure model; Chaper 3 descrbes he noaons and assumpons as well as he problem solved n hs sudy; Chaper 4 nroduces he nano-crysals embedded sample nvesgaed n he research; Chaper 5 sudes he consan-sress ALT; Chaper 6 nvesgaes he sep-sress ALT; and fnally Chaper 7 concludes he dsseraon.

29 29 CHAPTER 2: LITERATURE REVIEW Ths chaper revews prevous research on ALT nference ncludng consansress ALT and sep-sress ALT. The revew of Drchle process mure model s also summarzed. 2. Inference of Consan-Sress Acceleraed Lfe Tesng A radonal paramerc approach o develop he nference on consan-sress ALT requres wo assumpons. Frsly he falure-me dsrbuon under each sress level s assumed o come from he same paramerc dsrbuon famly. For eample s ypcal o assume ha a falure-me dsrbuon s from a locaon-scale famly such as eponenal Webull log-normal or normal. Secondly an acceleraon relaonshp called he me ransformaon funcon such as he Arrhenus and Eyrng law s assumed o relae he parameers of he dsrbuons under varous sress levels. A sem-paramerc ALT analyss usually relaes one of hese wo assumpons and has been appled wdely n leraure. 2.. Paramerc esmaon of consan-sress ALT The ALT assumng varous me ransformaon funcons and falure-me dsrbuons have been analyzed wh dfferen paramerc esmaon mehods n leraure. Sngpurwalla e al. [28] Kahn [29] and Barbosa and Louzada-Neo [30] used Leas Squares Esmaon for he ALT. Sngpurwalla e al. [28] handled he censored daa usng Eryng model wh normal falure-me dsrbuon. The auhors defned a lnear model for he parameers n Eryng model and esmaed he parameers usng Leas

30 30 Square Esmaon. Kahn [29] followed he approach of Sngpurwalla e al. [28] and used he nverse power law wh an eponenal raher han normal falure-me dsrbuon. Barbosa and Louzada-Neo [30] esmaed he mean lfeme of he uns under workng condons. The censored daa was handled. The auhors assumed a Webull dsrbuon for lfeme and a log-lnear relaonshp as he me ransformaon funcon. The pon esmaon of parameers was obaned wh Ieravely Reweghed Leas Squares Algorhm and he nerval esmaon was obaned wh MLE. Whman [3] Abdel-Ghaly e al. [32] Wakns [33] [34] Glaser [35] Hrose [36] and Newby [37] appled ML mehod o esmae he parameers. Whman [3] assumed he falure-me dsrbuon o be log-normal and used he Arrhenus model for he medan me o falure as he me ransformaon funcon. The auhor esmaed he parameers n he Arrhenus model and he medan me o falure a a ceran sress and also provded her confdence nervals. Boh complee daa se and daa wh censorng have been modeled. Abdel-Ghaly e al. [32] esmaed he daa wh ype-ii censorng assumng a 3-parameer Pareo falure-me dsrbuon and nverse power law as he me ransformaon funcon. The auhors predced he value of he shape parameer as well as he relably funcon a a msson me under operang condon. Wakns [33] [34] fed Webull dsrbuon o ALT daa. Wakns [33] assumed power-law model and deal wh a complee daa se. The common shape parameer of Webull dsrbuon and parameers of power-law model were esmaed. The ML esmaors were obaned by Newon-Raphson erave mehod. Wakns [34] specfed a log-lnear relaonshp o descrbe he scale parameer of Webull dsrbuon. The auhor fed daa wh

31 3 censorng and esmaed parameers wh Newon s mehod. Glaser [35] esmaed a Webull ALT model. Boh scale and shape parameers were epressed as funcons of he esng envronmen whle n mos cases he shape parameer was assumed o be consan. Besdes complee daa and censored daa hs paper also handled grouped daa by nroducng a saus ndcaor. Hrose [36] developed a Webull nverse power law ncludng a hreshold sress consderng ype-i censorng. The auhors consdered boh scenaros of common shape parameer and sress-relaed shape parameers. Newby [37] nroduced a generalzed reamen of AFT model usng general shape scale and locaon parameer famles of dsrbuons. The Arrhenus model was assumed n he paper. Mazzuch and Soyer [38] and Mazzuch [39] dd Bayesan nference for he ALT model wh power-law ransformaon funcon. Mazzuch and Soyer [38] assumed an eponenal lfeme dsrbuon. The auhor rewroe he power-law funcon by akng logarhm and hen se up a lnear model. The Lnear Bayesan Approach was used o solve he model. Mazzuch [39] presened a Bayesan procedure for nference assumng Webull falure-me dsrbuon. The procedure was based on he General Lnear Model se up by lnearzng he me ransformaon funcon and hen employed Lnear Bayesan Approach o produce compuable resuls. Ba and Chung [40] proposed boh consan-sress and progressve-sress ALT based on Webull lfeme dsrbuon and nverse power law. Boh MLE and Bayesan mehods were used o esmae he parameers. A Mone Carlo sudy was carred ou o nvesgae he behavor of esmaors.

32 32 Berna e al. [4] dropped he assumpon ha he falure-me dsrbuon a dfferen sress levels was from he same famly dsrbuon. Insead he auhors assumed he falure-me dsrbuon mgh be governed by dfferen forms of dsrbuon. The auhors fed each CDF of falure-me dsrbuon wh emprcal CDF and compued quanles of each CDF. They hen performed regresson on each quanles based on he nverse power law. The esmaed parameers of nverse power law funcon were obaned from regresson and he CDF n use condon was a sep funcon based on he compued quanles. Ths mehod can only produce a dscree CDF. Km and Ba [42] AL-Hussan and Abdel-Hamd [43] [44] developed mure models for daa wh more han one falure mode. Km and Ba [42] analyzed ALT daa under wo falure modes by assumng he log lfeme followed mure of wo locaonscale dsrbuons and each locaon parameer had a lnear relaon wh he sress. The ML esmaes of he dsrbuon parameers and he mng proporons were obaned by he epecaon and mamzaon algorhm. AL-Hussan and Abdel-Hamd [43] [44] assumed ha he ype-ii censorng lfemes under varous falure modes were dsrbued accordng o a fne mure model n whch each falure mode was represened by a nonnegave and connuous funcon. In boh papers he power-law relaonshp appled o he mure of wo Webull componens was presened. AL-Hussan and Abdel-Hamd [43] esmaed he parameers relably and hazard rae funcons usng he Bayesan mehod whle AL-Hussan and Abdel-Hamd [44] appled he MLE mehod. Moreover AL-Hussan and Abdel-Hamd [44] used mures of wo eponenals Raylegh and Webull componens models as llusrave eamples.

33 Sem-paramerc esmaon of consan-sress ALT A commonly used sem-paramerc esmaon for ALT n leraures drops he frs assumpon of paramerc form of lfeme dsrbuon and reans a paramerc acceleraon funcon. For eample Shaked e al. [45] Shaked and Sngpurwalla [46] Ba and Lee [47] and Basu and Ebrahm [48] all followed hs approach. Shaked e al. [45] assumed a paramerc relaonshp beween any wo sress levels and esmaed he MTTF under normal sress based on he emprcal dsrbuon funcon. Shaked and Sngpurwalla [46] assumed nverse power law as me ransformaon funcon and also esmaed he MTTF and CDF under use condon based on he emprcal CDF. Moreover he auhors esed wheher he esmaed CDF followed a member of a specfed famly of CDF s usng Kolmogorov-Smronov sasc and obaned he confdence bounds for CDF based on he es resul. Ba and Lee [47] assumed nverse power law and used emprcal esmaor o esmae he ALT under nermen nspecon n whch he es uns were only nspeced a specfed pons of me. Snce he emprcal dsrbuon funcons are dscree may be dffcul o handle censorng daa wh emprcal esmaors. Basu and Ebrahm [48] eended he work of Shaked and Sngpurwalla[46] o nclude he censorng daa by nroducng he scale model. Some leraures drop he second assumpon- paramerc acceleraon model and rean he assumpon ha he falure-me dsrbuon a each condon s from a famly of dsrbuon. Dorp and Mazzuch [49] [50] proposed a model for general ALT ncludng regular ALT consan-sress ALT sep-sress ALT and profle-sress ALT. The lfeme dsrbuon a each sress level was assumed o be eponenal n Dorp and

34 34 Mazzuch [49] and Webull n Dorp and Mazzuch [50]. The auhors dd no assume any paramerc me ransformaon funcon. Insead he mulvarae Ordered Drchle dsrbuon was used as pror nformaon o defne a mulvarae pror dsrbuon for he scale parameer a varous sress levels and he common shape parameer. Ths approach requred nformaon of a specfed quanle on he msson me relably a he operang sress level o nfer he use sress lfe parameers esmaes. Lous [5] assumed he Webull famly and developed a scale-change model by nroducng a scale change parameer o parameerze he dfference beween wo survval dsrbuons. Then he effcen score sasc was used o esmae he scale change parameer. Schmoyer [52] analyzed ALT wh wo-level sngle-sress ncludng an acceleraed sress and a normal sress. The auhor proposed a general model of he form Pr; = Fgh where Pr; denoed he probably of falure by me a sress level F was a CDF on [0 g and h were nonnegave and nondecreasng g had S-shaped curvaure and g0=0. Eher F or h was assumed o be known. If F was known and Fu= - e -u hen hs was a proporonal hazards model; f h was known and h= was a acceleraed falure me model. Based on hese assumpons he auhor developed confdence bounds for low-sress long-me probables and quanles. Because no acceleraon funcon was assumed neher approach developed by Lous [5] nor Schmoyer [52] can predc he falure-me dsrbuon under use condon. Proschan and Sngpurwalla [53] and Maceewsk [54] relaed boh assumpons. Proschan and Sngpurwalla [53] dvded he es me o nervals and assumed he probably p of falure of a un n me nerval under sress was Bea dsrbued

35 35 and esmaed he falure rae wh weghed average falure rae. The esmaed p was obaned wh Bayesan esmaon and he wegh parameers were obaned wh Leas Square Esmaon. Maceewsk [54] frs f he CDF a each sress level o a paramerc famly of funcons wh emprcal CDF hen derved quanles of he levels and her dspersons for each CDF. The auhor defned a funcon whch could be derved from each quanle level and used hs funcon o calculae he esmae of each quanle level n use condon. Then f he CDF wh quanles n use condon. Some more recen leraures assumed a lnear or log-lnear model beween falure me and covarae resulng n a more generc acceleraon model. No paramerc dsrbuon funcon was assumed and regresson was carred ou. The regresson model was defned as lny = X + = n where Y was he falure me X was a p vecor of covaraes was a p vecor of unknown regresson parameers and s were s-ndependen and had an unspecfed dsrbuon funcon. To esmae he regresson coeffcen Ln and Geyer [55] developed compuaonal mehods o mplemen rank regresson procedures usng smulaed annealng. The unknown error funcon was esmaed usng he Kaplan-Meer esmaor. Komarek and Lesaffre [56] developed a Bayesan lnear regresson model wh pared doubly nerval-censored daa. The bvarae error dsrbuon was assumed as a fne mure of bvarae normal denses. The Bayesan approach wh he Markov chan Mone Carlo MCMC mehodology was used for nference. Komarek and Lesaffre [57] analyzed he mulvarae doubly-nerval-censored daa consderng cluserng. The unvarae denses of random errors and random effecs were modeled as penalzed Gaussan

36 36 mure wh an overspecfed number of mure componens. The Bayesan approach wh MCMC samplng was used o esmae he model parameers. Argeno e al. [58] eamned he acceleraed falure-me model for unvarae daa wh rgh censorng. The error dsrbuon was represened as a nonparamerc herarchcal mure of Webull dsrbuon on boh shape and rae parameers and he mng measure was a pror dsrbued as he generalzed gamma measure. Oher lnear or log-lnear regresson model nvolved wh Drchle process mures wll be revewed n secon Inference of Sep-Sress Acceleraed Lfe Tesng Besdes he wo radonal assumpons n consan-sress ALT a classcal approach o analyze a SSALT requres an addonal assumpon o relae he dsrbuon under sep sresses o he dsrbuon under consan sress. Maor models used n leraures nclude he Tampered Random Varable TRV model [59] he Cumulave Eposure CE model [2] and he Tampered Falure Rae TFR model [60] Paramerc esmaon of sep-sress acceleraed lfe esng The Cumulave Eposure model was frs proposed by Nelson [2] and has been wdely acceped and used n SSALT analyss. The CE model assumes he remanng lfe of he uns depends only on he cumulave eposure he uns have eperenced whou memory on how hs eposure was accumulaed. Nelson [2] presened he nference usng ML esmaors for Webull falure daa under he nverse power law. The daa of me o breakdown of elecrcal nsulaons was fed as llusraon. Xong [6] presened he nference of parameers n an smple SSALT wh ype-ii censorng assumng CE model and eponenal lfeme dsrbuon wh a mean ha was a log-lnear funcon of

37 37 sress. The confdence nervals were consruced usng a pvoal quany. Xong and J [62] suded a smlar problem wh ype-i censorng. Xong and Mllken [63] consruced he model wh he same assumpons o esmae he parameers n log-lnear funcon and furher predced he lfeme under desgn sress as well as ha durng a fuure SSALT. The falure-sep sress ALT was conduced n hs paper. Zhang and Geng [64] consruced he analyss for boh consan-sress and sep-sress ALT by applyng a Webull lfeme dsrbuon wh a lnear Arrhenus lfeme-sress relaonshp. The Leas Square Mehod was used o esmae he Webull parameers and a self-desgned sofware was programmed o predc he lfe under use condon. Abdel-Hamd and AL- Hussan [65] appled an eponenaed dsrbuon wh a scale parameer whch was a log-lnear funcon of he sress and hold he CE model. Specal aenon was pad o an eponenaed eponenal dsrbuon. The ML esmaes of parameers under consderaon were obaned based on ype-i censorng. Wang [66] derved he confdence nervals for he eponenal SSALT model under progressve ype-ii censorng whch employed he removal of survvng uns a me of falure. The mean lfe was assumed o be a log-lnear funcon of sress and he MLE was appled. Yn and Sheng [67] derved he lfeme dsrbuon under progressve sress ALT n whch he sress was proporonal o me. The lfeme dsrbuon was assumed o follow an eponenal or a Webull dsrbuon wh nverse power law and he parameers were esmaed usng MLE. Gouno [68] and Lee and Pan [69] [7] all assumed eponenal lfeme dsrbuon a each ndvdual sress level and used falure rae o descrbe PDF and relably under sep-sress. Ths resuled n he same form wh ha derved from CE

38 38 model. Gouno [68] presened a praccal mehod o analyze emperaure SSALT daa wh ype-ii censorng consderng an Arrhenus model. Boh Leas Squares and MLE were used o esmae he parameers of an Arrhenus model and falure rae under use condon. Lee and Pan [69] [70] presened Bayesan nference model for SSALT wh ype-ii censorng. The mean lfeme a each sress was assumed o be a log-lnear funcon of sress level. Lee and Pan [69] consruced he model for smple SSALT and derved he Bayesan nference wh conugae pror. Lee and Pan [70] consruced he model for mul-ssalt and used MCMC echnque o deal wh nonconugae pror. Lee and Pan [7] analyzed mul-sress ALT wh rgh censorng assumng a Generalzed Lnear Model GLM and log-lnear relaonshp. Boh he MLE and Bayesan esmaon were used o esmae he GLM parameers. Tang e al. [72] modfed he CE model o analyze ALT wh falure-free lfe FFL whch s he age of a produc below whch no falure should occur. The FFL was characerzed by a locaon parameer n he dsrbuon. The auhors proposed a Lnear Cumulave Eposure Model LCEM whch assumed he fraconal eposure was lnearly accumulaed. The 3-parameer Webull dsrbuon wh locaon and scale parameers epressed as nverse power law relaonshp wh sress was used o llusrae he esmaon procedure and he MLE was used. Khams and Hggns [73] proposed he KH model as an alernave o he Webull CE model n SSALT. The proposed model was based on a me ransformaon of he eponenal CE model. The new model was as fleble as he Webull CE model for fng daa whle easer o oban he ML esmaes of he parameers.

39 39 When an SSALT only has wo seps.e. smple SSALT he me ransformaon funcon s no necessary and many researchers use orgnal parameers a each sress level drecly. For eample Balakrshnan e al.[74] [75] Kaer and Balakrshnan [76] Balakrshnan and Xe [77] Balakrshnan and Han[78] and Han and Balakrshnan [79] all used orgnal parameers of dfferen assumed dsrbuons a each sress level as esmaes and appled he MLE mehod. Moreover Balakrshnan and Xe [77] consdered ype-ii hybrd censorng scheme n whch he es was ermnaed a me T f he rh falure occurred before me T; oherwse he es was ermnaed as soon as he rh falure occurred. Ths ype of censorng scheme ensures he es can oban a leas r falures. Balakrshnan and Han [78] and Han and Balakrshnan [79] consdered he smple SSALT wh wo faal causes for he falure and assumed dfferen rsk facors were ndependen and eponenally dsrbued. Balakrshnan and Han [78] analyzed ype-ii censored daa and Han and Balakrshnan [79] analyzed ype-i censored daa. Balakrshnan e al. [80] developed he model for mul-ssalt whou assumng any me ransformaon funcon. The auhors assumed an eponenal lfeme dsrbuon and developed he order resrced MLE under rgh censored samplng suaons. DeGroo and Groel[59] nroduced he Parally Acceleraed Lfe Tes PALT n whch f a un survved o a specfed me a desgn sress was swched o a hgher level of sress. The auhors modeled he effec of swchng he sress by mulplyng he remanng lfeme of he un by some unknown facor called he amperng coeffcen and he model was called Tampered Random Varable TRV model. DeGroo and Groel [59] assumed he lfeme under use condons s eponenal and appled Bayesan

40 40 esmaon. Mad [8] Abdel-Ghaly e al. [82] and Wang e al. [83] appled TRV model o analyze PALT consderng dfferen censorng schemes. Mad [8]proposed an emprcal Bayes approach o pool daa from several groups of uns ha were esed a dfferen nsances o esmae he parameers. Abdel-Ghaly e al. [82] and Wang e al. [83] assumed a Webull lfeme dsrbuon and used MLE o esmae he dsrbuon parameers and amperng coeffcen. Abdel-Ghaly e al. [82] consdered ype-i and ype- II censorng and Wang e al. [83] consdered mulply censored daa. Baacharyya and Soeoe [60] modfed he TRV model by assumng ha he effec of changng he sress was o mulply he falure rae funcon over he remanng lfe and he modfed model was called Tampered Falure Rae TFR model. The auhors assumed a Webull lfeme dsrbuon and esmaed he parameers wh MLE. An eenson o fully SSALT was derved wh he applcaon of log-lnear lfe-sress funcon. Wang and Fe [84] appled he TFR model o he progressve sress ALT assumng a Webull-me falure dsrbuon wh he scale parameer sasfyng nverse power law. The parameers were esmaed usng MLE. Zhao and Elsayed [85] proposed a general SSALT model based on he acceleraon models and produced some commonly used lfeme-sress relaonshp and her acceleraon facors. The MLE mehod was ulzed o solve for he Webull and lognormal lfeme dsrbuons Sem-paramerc esmaon of sep-sress acceleraed lfe esng Mos sem-paramerc esmaon nference for SSALT s obaned by droppng he lfeme dsrbuon assumpon whle holdng he paramerc me ransformaon

41 4 funcon and he CE model. Ln and Fe[86] Ba and Chun [87] and Tyoskn and Krvolapov [88] all used hs approach. The lfeme properes under use condons were obaned based on ransformed falure mes and some emprcal esmaors. Shaked and Sngpurwalla [89] developed a model whch assumed he dsrbuon of he lfeme was a funcon of he oal accumulaed V where V was he sress and was an unknown consan. The new model unfed and generalzed he TRV and CE models. Hu e al. [90] eended he work of Schmoyer [52] o a smple SSALT and obaned he upper confdence bounds for cumulave falure probably under use condons. Dorp e al. [9] assumed he falure mes a each sress level were eponenally dsrbued and dropped he me-ransformaon funcon. The model was developed for SSALT consderng lnear rampng sress. The mulvarae Ordered Drchle dsrbuon was used as pror nformaon o defne a mulvarae pror dsrbuon for he falure raes a varous sress levels. Bayes pon esmaes as well as probably saemen lfeme parameers under use condons were developed. 2.3 Drchle Process Mure Model The DP was frs formally developed by Ferguson [5] as a random probably model o defne prors for spaces of dsrbuon funcons. I consss of counably nfne pon probably masses. In order o rela he resrcon of dscreeness he DPMM was developed and has been wdely appled n he area of nonparamerc Bayesan daa analyss. The DP Gaussan mure model usng a normal kernel has been eensvely appled e.g. n densy esmaon [92] [95] curve fng [96] and regresson [97]

42 42 [00]. Some oher classcal kernels have also been appled. Koas [0] proposed a DPMM wh a Bea dsrbuon o esmae densy and nensy. Wes e al. [02] nroduced he DP Gamma mure model for herarchcal lnear regresson and densy esmaon. Mukhopadhyay and Gelfand [03] developed he Generalzed Lnear Models wh DP mure of bnomal and Posson kernels. Caroa and Parmgan [04] developed he DP Posson mure model for regresson problems n whch he response varable s a coun. Some researchers have appled DPMM n survval daa analyss. Koas and Gelfand [97] nroduced boh sem-paramerc and nonparamerc Bayesan modellng approaches for he error dsrbuons of medan regresson. Boh models were based on DP normal mure models. The censored survval daa was handled n he paper. Gelfand and Koas [98] demonsraed a compuaonal approach o oban he enre poseror dsrbuon for nonparamerc Bayesan nference wh DPMM. The applcaon of comparson of survval mes from dfferen populaons under farly heavy censorng was llusraed. Gelfand and Koas[99] eended he nonparamerc Bayesan modelng approach wh DPMM n Gelfand and Koas [97] o medan resdual lfe dsrbuon. The DP normal mure model was appled o survval daa. Koas [9] appled he DP Webull mure model o censored survval daa. The mes were appled over boh he shape and scale parameers of he Webull kernel. Kuo and Mallck [8] Hanson [05] and Ghosh and Ghosal [06] developed a DP mure model wh log-lnear acceleraon relaonshp for consan-sress ALT analyss. Kuo and Mallck [8] proposed wo herarchcal models o esmae error

43 43 dsrbuon of log-lnear model whch was shown n equaon.4 as ln T W. The frs model appled he DP mure model for he dsrbuon of V = epw and he oher modeled he dsrbuon of W. The DP mure model wh normal and log-normal kernel was used. Hanson [05] modeled he dsrbuon of V by he DP mure model wh gamma kernel and Ghosh and Ghosal [06] appled he DP Webull mure wh a fed shape parameer.

44 44 CHAPTER 3: PROBLEM STATEMENT Ths chaper frs descrbes he noaons and basc assumpons appled n he dsseraon. Then he prmary obecve as well as s sub-asks s presened o llusrae he problem ha wll be solved. 3. Noaons : sress or a proper ransformaon of he sress : shape and scale parameer of Webull dsrbuon : regresson coeffcen n he log-lnear lfeme-sress relaonshp : falure me c : censorng me : censorng ndcaor : precson parameer of he Drchle process : mng proporons of he paramerc mure process : parameer vecor of he paramerc kernel F f : cumulave dsrbuon funcon CDF and probably densy funcon PDF G: random dsrbuon funcon for G 0 : base dsrbuon of he Drchle process R: relably funcon of falure me R=-F K k : CDF and PDF of he paramerc kernel v: v= ep : sress changng me n a smple SSALT

45 Assumpons Ths dsseraon makes he followng basc assumpons: n dencal and ndependen uns are placed n he es. 2 The relaonshp of falure me under dfferen sress levels s log-lnear whch can be epressed as log = - + w = 2...n where w s error erm. 3 Each v s from an ndependen Webull kernel k. v 4 A cumulave eposure model s assumed for smple SSALT. 5 For consan-sress ALT hree acceleraed sresses are used n he es. The breakdown me was observed by he ump of leakage curren. Each un under es was esed ndvdually and broke down ndependenly. A sress lower han hese hree acceleraed sress s assumed o be normal sress level. 6 For smple SSALT a un s frs esed under he lower sress level L. If he un has no faled by a pre-specfed me he sress level s ncreased o H a he changng me and he es s connued unl falure or he censorng me c.e. ype-i censorng. The lower sress L s assumed o be normal sress level. 3.3 Problem Descrpon In hs dsseraon he sample sze n he esng sress levels and sress changng me for smple SSALT are pre-specfed. The prmary obecve s o predc CDF of lfeme dsrbuon under normal sress level from epermenal daa. Three epermenal daases ncludng one complee daase and wo rgh censored daases are used o llusrae he applcably of he proposed mehodology. In order o fulfll hs obecve he followng sub-asks are performed.

46 Invesgaon of meal ode nanodos-embedded ZrHfO hgh-k flm The epermenal daases used n hs dsseraon were colleced a he Thn Flm Nano & Mcroelecroncs Research Laboraory a Teas A&M Unversy. In order o furher undersand he devce hs dsseraon nvesgaes he fabrcaon process physcal properes and memory funcons of he devce esed for lfe me predcon Developmen of smulaon-based algorhm The DP Webull mure ALT model s a hgh dmensonal problem wh mulple parameers. Gbbs samplng s a popular algorhm o f DP mure model. Ths dsseraon develops Gbbs samplng algorhm o formulae poseror nference on parameers as well as falure-me CDF Comparson beween paramerc Webull log-lnear ALT model and DP Webull mure ALT model A paramerc ALT analyss s also performed for he purpose of comparson. I s assumed he falure me a a gven sress level comes from a Webull dsrbuon and he relaonshp beween lfeme and sress level s log-lnear. Ths Webull log-lnear ALT model s esmaed wh sandard MLE mehod and s mplemened usng Mnab. Then he CDF of falure-me dsrbuon under normal sress level predced wh paramerc ALT model and he proposed DP mure model as well as emprcal CDF are ploed and vsual compared.

47 47 CHAPTER 4: MEMORY FUNCTIONS OF MOLYBDENUM OXIDE NANODOTS EMBEDDED ZRHFO HIGH-K Ths chaper nroduces he devce esed n he dsseraon ncludng he fabrcaon process s physcal characerscs charge rappng and derappng mechansms as well as memory properes. 4. Fabrcaon of Nanocrysals Embedded ZrHfO Hgh-k Devce The ZrHfO unnel ode/ MoO /ZrHfO conrol ode gae delecrc sack was deposed on he HF precleaned p-ype 0 5 cm -3 S 00 wafer n one pump down whou breakng he vacuum. The unnel and conrol ZrHfO layers were spuered from a compose Zr/Hf 2/88 w % arge n an Ar/O 2 : mure a 5 mtorr and 60 W for 2 and 0 mn separaely. The MoO flm was spuered from he Mo arge n Ar/O 2 : a 5 mtorr and 00 W for 5 s. Afer he gae delecrc sack was accomplshed he pos deposon annealng PDA sep was carred ou by rapd hermal annealng RTA a 800 o C n he pure N 2 amben for mn. An alumnum Al flm was spuer deposed lhography paerned and we eched no gae elecrodes. The Al flm was also deposed on he backsde of he wafer for ohmc conac. The fnal MOS capacor was annealed a 300 C under H 2 /N 2 0/90 for 5 mn. The conrol sample.e. conanng only he ZrHfO gae delecrc whou he embedded nc-moo layer was also prepared and characerzed for comparson. The capacor s capacance-volage C-V and curren-volage I-V characerscs were measured wh an Aglen 4284A LCR meer Reprned wh permsson from X Lu Cha-Han Yang Yue Kuo and Tao Yuan Elecrochemcal and Sold-Sae Leers 202 vol. 5 ssue 6 H Copyrgh 202 The Elecrochemcal Socey.

48 48 and an Aglen 455C semconducor parameer analyzer respecvely. The hgh-k sack was analyzed wh X-ray phooelecron specroscopy XPS for chemcal bond saes and X-ray dffracon XRD for he crysallny. 4.2 Physcal Properes of Nanocrysals Embedded ZrHfO hgh-k Devce Fgure 4.a shows he XPS Os peak of he MoO embedded ZrHfO hgh-k sack prepare n hs sudy. I has been deconvolued no 4 sub peaks. The peak wh he bndng energy BE of ev s MoO 3. Peaks wh BE ev and 53.3 ev are relaed o HfO 2 and ZrO 2. The peak wh BE ev s probably conrbued by he Al 2 O 3 from he gae elecrode. A small Mo 3d 5/2 peak a ev BE was deeced whch s he Mo δ+ elemen [07]. The nanodos are crysallne MoO 3 deeced by X-ray dffracon XRD as shown n Fg. 4.b. The crysal sze s abou 28 nm deermned from he peak locaon and full wdh a half mamum usng he Scherrer equaon [08]. Prevously was demonsraed ha under he same process condon dscree nc-ito nanodos were formed n he ZrHfO flm[09]. However s no clear f he embedded nc-moo flm was composed of dscree dos. In addon he densy of he nanodos n he delecrc layer s mporan o he charge reenon and relably. In order o oban hese daa he sample has o be analyzed wh he hgh resoluon TEM whch s under sudy now. The equvalen ode hckness of he conrol sample and he nc-moo embedded sample are 7.8 nm and 8.5 nm respecvely calculaed from he C-V curve.

49 49 Fgure 4.. a XPS O s peak and b XRD paern of he MoO embedded ZrHfO sample 4.3 Charge Trappng and Derappng Mechansms The charge rappng characerscs of he capacor can be eamned from he C-V hyseress curves. Fgure 4.2 shows he C-V hyseress curves of he capacor wh he nc-moo embedded ZrHfO delecrc measured a MHz. The gae was sressed wh a volage V g from negave o posve.e. he forward drecon and hen back o negave.e. backward drecon n hree ranges.e. -3V o 3V o -3V ±3V -6V o 6V o -6V ±6V and -8V o 8V o -8V ±8V separaely. The memory wndow can be defned as fla band volage dfference V FB beween he V FB of he forward curve and ha of he backward curve. The conrol sample was prepared and compared wh he nc- MoO embedded sample for defec formaon consderaon. The charge rappng densy Q of he capacor can be esmaed from he followng equaon [0]:

50 50 Q CFB VFB 4. q where C FB s he flaband capacance and q s he elecron charge. For he conrol sample snce he C-V hyseress s very small.e. V FB = 0.06 V he ZrHfO flm has neglgble charge rappng capably. For he nc-moo embedded sample he C-V hyseress phenomenon s more pronounced. The charge rappng denses Q o s are cm cm -2 and cm -2 for he V g sweep ranges of ±3V ±6V and ±8V separaely. Therefore he charges are rapped o he nc- MoO se. The charge rappng capably s relaed o he amoun of charges suppled o he hgh-k sack. I s worh o noe ha he magnude of he V FB of he forward C-V curve s hghly dependen on he sarng V g.e V V and -.5 V for sarng V g = -3V -6V and -8V separaely. The V FB of he forward C-V curve of he fresh nc-moo embedded sample s V. Compared wh he conrol sample holes are rapped o he nc-moo sack are a densy of cm cm -2 and cm -2 wh he sarng V g of -3V -6V and -8V separaely. Therefore nc-moo s effecve n rappng holes whch s smlar o he nc-ruo or nc-ito case [] [3]. On he oher hand he V FB of he backward curve s more negave wh he ncrease of he sweepng volages.e V a ±3 V V a ±6 V and V a ±8 V. Therefore holes rapped n he forward sweepng are no compleely erased n he back sweepng drecon.

51 5 Normalzed Capacance V o 3V o -3V -6V o 6V o -6V -8V o 8V o -8V Normalzed Capacance Conrol sample -8V o +8V o -8V Gae Volage V Gae Volage V Fgure 4.2 C-V hyseress curves of he nc-moo embedded ZrHfO capacor measured a MHz. The nse s he hyseress curve of he conrol sample. To furher sudy he hole-rappng mechansm he leakage curren densy-volage J-V curves of he nc-moo embedded sample and he conrol sample were measured from -8 V o +8 V as shown n Fgure 4.3. The polary of he leakage curren s defned as posve when he curren flows oward he subsrae and negave when he curren flows oward he gae. Compared wh he conrol sample he nc-moo embedded sample has a larger leakage curren and he J-V curve s less smooh. There are several bumps n he J-V curve of nc-moo embedded sample. Frs a very small bump appears a pon A whch s near he V FB of he correspondng C-V curve n Fg.4.2 a V g = -.5 V. The curren changes s polary a hs pon because of he release of loosely rapped holes whch are probably locaed a he nc-moo /ZrHfO nerface[] [3]. Second a

52 52 slghly more obvous ump of he curren s observed a pon B near V g = 0 V whch s due o he furher release of a large number of remanng rapped holes from he change of he V g polary [] [3]. Thrd here are wo obvous peaks a pons C and D near V g = V. These are he negave dfferenal ressance NDR peaks commonly observed n floang gae memory rappng devces [25] [4]. Ths s caused by he Coulomb blockade effec.e. he nc-moo se s sauraed wh he rapped charges [23] [5] [6].When he V g s furher ncreased e.g. beyond 2 V an nverson layer s fully esablshed and he leakage curren ncreases drascally wh he ncrease of V g. Curren Densy A/cm 2 2.0E-07.5E-07.0E E E-2-5.0E E E E-07-8V o +8V A Curren Densy A/cm2 B 3.0E E-08.0E C Gae Volage V D -8V o +8V 0.0E+00 Conrol sample -.0E Gae Volage V Fgure 4.3 J-V curve of nc-moo embedded capacor Vg swep from -8 V o +8 V. The nse s he J-V curve of conrol sample.

53 Charge Reenon Capably The charge reenon capably of he capacor could be deermned by measurng he percenage of charge remaned n he devce afer releasng he sress volage for a perod of me. The followng equaon was used o calculae he percenage of charge from he V FB shf over a perod of me [3]: Charge VFB VFB fresh remanng % 4.2 V sress V fresh FB FB where V FB sress s he V FB mmedaely afer he V g sress V FB fresh s he V FB before he V g sress and V FB s he V FB afer releasng he sress for me. Afer releasng he wre sress he C-V curve was measured over a small V g range.e. -2 V o + V o deermne he V FB. Fgure 4.4 shows he charge reenon curve of he nc-moo embedded capacor afer beng sress a V g = -8 V for 0 seconds. The oal measuremen me was 0 hrs and he V FB was deermned every 800 s. Frs a quck loss of nearly 20% of rapped holes occurred whn 800 s afer releasng he sress V g whch s due o he derap of he loosely-rapped holes from he nc-moo /ZrHfO se [] [3]. Then he srongly-rapped holes were gradually released e.g. oally 9% loss from 800s o 36000s. The phenomenon of he 2-sep release of rapped holes have been observed n he nc-ruo and nc-ito embedded capacors [] [3]. The eac locaon of he charge rappng se can be clarfed usng he frequency dsperson mehod [26] [7]. The nse of fgure 4.4 shows he erapolaon of he curve o 0 year perod. Abou 54% of rapped holes remaned n he nc-moo embedded sample afer releasng he sress V g for 0 years whch s he desrable characersc for he nonvolale memory devce.

54 54 Charge Remanng 00% 80% 60% 40% Charge Remanng Afer -8V 0s sress 00% 0 yrs 80% 54% 60% 40%.0E+03.0E+05.0E+07.0E+09 Reenon Tme sec Reenon Tme sec Fgure 4.4 Reenon propery of holes rapped n he nc-moo embedded capacor. The nse shows he erapolaon of he curve o 0 years proecon. 4.5 Conclusons Memory funcons of he nc-moo embedded ZrHfO hgh-k MOS capacor have been suded. The C-V hyseress and J-V daa show ha holes were rapped o he nc- MoO se of he hgh-k sack. The charge rappng capably s affeced by he supply of he charges from he wafer subsrae. Holes rapped from he negave gae volage sress could no be compleely erased by a posve gae volage wh he same magnude. The J-V curve of he nc-moo embedded sample confrmed ha hose loosely rapped holes were released easly. The Coulomb blockade effec was observed under he elecron rappng condon. The charge reenon sudy shows ha abou 20% of he rapped holes were loosely rappng and more han half of hose orgnally rapped holes remaned n he devce afer 0 years. In prncple he nc-moo embedded ZrHfO hgh-k capacor s

55 55 a unque memory devce of whch he operaon s based on hole rappng and derappng. For he gga level applcaon n addon o he enlargemen of he sorage capacy hrough he opmzaon of he srucure and operaon parameers dealed charge rappng se and relably ssues need o be nvesgaed.

56 56 CHAPTER 5: BAYESIAN ANALYSIS FOR ACCELERATED LIFE TESTS USING DIRICHLET PROCESS WEIBULL MIXTURE MODEL 2 Ths chaper develops he Drchle proess Webull mure model for consansress ALT. The model s employed o predc falure-me dsrbuon a a gven sress level. A smulaon-based model fng algorhm ha mplemens Gbbs samplng s developed o analyze complee and rgh-censored ALT daa and o predc he falureme dsrbuon a he normal sress level. Two praccal eamples relaed o he relably of nanoelecronc devces are presened. The resuls have demonsraed ha he proposed mehodology s capable of provdng accurae predcon of he falure-me dsrbuon a he normal sress level whou assumng any resrcve paramerc falureme dsrbuon. 5. Mehodologes 5.. Drchle process Webull mure ALT model Secon.6.3 nroduced he general form of Drchle process mure model. In hs secon a Drchle process mure model wh he Webull kernel s developed o model falure-me dsrbuon a a gven sress level and a log-lnear regresson model s assumed o descrbe he relaonshp of falure me under varous sress levels. Assume n ems are esed n ALT and le d = { = 2 n} denoe he ALT daa where he h un s esed a he sress. Noe ha may be a ransformaon observaon and equals zero when s a rgh-censored observaon. Kuo and 204 IEEE. Reprned wh permsson from Tao Yuan X Lu Saleem Z. Ramadan and Yue Kuo Bayesan Analyss for Acceleraed Lfe Tess Usng a Drchle Process Webull Mure Model IEEE Transacons on Relably vol. 63 No. March 204

57 Mallck [8] and Ghos and Ghosal [06] consdered he followng semparamerc lnear regresson model for ALT 57 ln w = 2 n. 5. where s he h sress level s he coeffcen of sress level and w s he error erm If he error erms w = 2 n are assumed o be s-ndependen and dencally dsrbued..d. from he smalles ereme value dsrbuon or equvalenly f v = ep w = 2 n are..d. Webull random varables he model gven by Eq. 5. s he wdely used Webull ALT model wh a log-lnear lfeme-sress relaonshp [] [8]. Kuo and Mallck [8] modeled he dsrbuon of v by he DP mure model wh he normal kernel and he lognormal kernel. Ghos and Ghosal [06] modeled he dsrbuon of v by he DP mure model wh he Webull kernel wh a fed shape parameer. Ths sudy eends he work of Kuo and Mallck [8]and Ghos and Ghosal [06] and uses he DP mure model wh he Webull kernel where he shape parameer s no fed o model he dsrbuon of v. Koas [9] poned ou ha usng Webull kernel has he compuaonal advanage over he normal and lognormal kernels when dealng wh censorng because he Webull kernel has a closed form CDF. In addon mng on boh he shape and scale parameers of he Webull kernel can resul n a fleble mure ha can model a wde range of dsrbuonal shapes [9]. Assumng ha v = 2 n are d from he followng PDF f v G k v G d d 5.2 where k s he PDF of he Webull kernel. The PDF and CDF of he Webull v kernel are gven by

58 58 k v v ep v 5.3 and K v ep v 5.4 respecvely where > 0 s he shape parameer and > 0 s he scale parameer. From f v G gven by Eq. 5.2 and accordng o he log-lnear relaonshp funcon v ep he PDF of can be derved hrough he ransformaon of random varables: f dv G f v G ep k ep G d d. 5.5 d The CDF of hen can be derved as: F G 0 0 K ep k s ep G d d ds ep k s ep dsg d d ep G d d. 5.6 The base dsrbuon G 0 can be consdered as pror guess on and and n hs dsseraon he followng G 0 s adoped: G Unform 0 nverse Gamma d Ths base dsrbuon can offer boh compuaonal convenence and modelng flebly [9]. The nverse-gamma dsrbuon s he condonal conugae pror for he Webull scale parameer when he shape parameer s known. There s no naure conugae pror for he Webull shape parameer. d =2 s se so ha he nverse-gamma dsrbuon has an nfne varance whch can convey he lack of pror knowledge.

59 In summary he DP Webull mure ALT model can be wren n he followng herarchcal form: 59 G ~ ~ ~ ~ ~ ~ ~ ep k ep K ep G... n DP G f a f a f a f a b b 0 b b f f n 5.8 where f a b f a b f a b and f a b denoe he pror dsrbuons for and respecvely. denoes he ndcaor funcon. = ndcaes eac falure me and =0 ndcaes he censored observaon. The followng pror dsrbuons are assumed: ~ ~ ~ ~ Normal a Pareo a Gamma a Gamma a b b b b. The Pareo dsrbuon and Gamma dsrbuon are he conugae pror dsrbuons for and respecvely. Usng he Gamma pror for he precson parameer of he DP pror can resul n a very aracve compuaonal convenence [93]. A normal pror s assumed for because may be negave especally when s a ransformaon of he sress. The proposed DP Webull mure ALT model handles censorng dfferenly from he work of Kuo and Mallck [8] and Ghos and Ghosal [06]. The prevous wo sudes

60 60 used an mpuaon mehod o replace he censored observaons wh smulaed falure mes. Ths dsseraon uses he relably funcon of f s a censored observaon n he frs sage of he herarchcal model 5.8. Ths s nuvely appealng because s conssen wh he general approach o deal wh censorng n paramerc daa analyss n whch he lkelhood conrbuon from a rgh-censored observaon s he relably funcon Smulaon-based model fng Ths secon presens a smulaon-based algorhm for fng he DP Webull mure ALT model. Gbbs samplng has become he sandard algorhm for fng he DP mure models as a useful ool for hgh dmensonal problems wh many parameers. Each eraon of he Gbbs samplng cycles hrough he unknown parameers samplng a value of one parameer condonng on he laes values of all he oher parameers. When he number of eraons s large enough he sample drawn on one parameer can be regarded as smulaed values from s margnal poseror dsrbuon. Sample sascs can hen be used o formulae poseror nference on ha parameer [9]. A key feaure of he DP mure models s he dscreeness of G under he DP assumpon nducng cluserng of s [5] [93]. Denoe n as he number of clusers n = 2... n and = 2.. n as he dsnc clusers. The vecor of ndcaors c = {c c 2 c n } s nroduced o ndcae he cluserng confguraon. c = when = ndcang ha he h observaon belongs o he h cluser. Le n denoe he number of members n he h cluser.

61 6 Ths dsseraon develops a Gbbs samplng algorhm o f he model 5.8. Each eraon of he algorhm consss of he followng seps: a Draw from he condonal poseror dsrbuon denoed by: f { ' ' c' ' } dfor... n; b Adus he cluser locaons by samplng from condonal poseror dsrbuon: f c dfor... n ; c Updae and by samplng from her condonal poseror dsrbuons; d Sample from s condonal poseror dsrbuon; e Sample F G 0 he falure-me CDF a he normal sress level 0. Deals of hese fve seps wll be descrbed n he remander of hs secon. A. Sep a: Updae Sep a draws new values of and updaes he cluserng locaon ndcaor c for each observaon. The new values of can eher be one of he ' or could be new values drawn from G 0. Denoe n - as he number of clusers n { ' }.e. he number of clusers when s removed from { 2 2 n n } and le = n - denoe he dsnc clusers n { ' }. Also denoe n as he number of members n he cluser for = n -. If s an eac falure-me observaon.e. f = he condonal poseror dsrbuon n Sep a has he followng mure form:

62 62 } ' { 0 0 ' ' ' n o o n o o o q n q q n h q c f 5.9 where denoes a pon mass a and ep ep ep o k k q 5.0 and ] ep [ ] ep [ ep ] ep [ ep ] ep [ ep ep ep d d d d o d d d d d d d G k d d G k q 5. whch can be easly compued numercally. The superscrp o s used o ndcae ha s an eac falure me observaon. Therefore equals wh he probably of o o o q n q q n 0 and wh he probably of 0 0 n o o o q n q q are new values drawn from o h. Heren I ep ep ep } 0 { 0 d d o d k G k h 5.2 where {} I s an ndcaor funcon. o h can be epressed as:

63 63 o f f h where I ] ep [ ] ep [ } 0 { 0 d o d h f 5.3 and. ] ep [ Gamma ~ nverse ] ep [ ep 2 d d f 5.4 In order o draw new values of from o h frs a value for s sampled from f gven by equaon 5.3 by dscrezaon and hen a value for s drawn from he nverse-gamma dsrbuon gven by equaon 5.4. If s a rgh-censored observaon.e. f = 0 he condonal poseror dsrbuon } ' { ' ' ' c f can be derved n a smlar way by replacng ep k wh ep K. Because he Webull kernel has he closed form CDF censorng can be easly handled n he compuaon as } ' { 0 0 ' ' ' n c c n c c c q n q q n h q c f 5.5

64 64 where he superscrp c s used o ndcae ha s a rgh-censored observaon ep c K q ] ep [ ] ep [ d d c d d d G K q 5.6 and. ] ep [ 0 o G K h Epress c h as: c f f h can be shown ha I ] ep [ } 0 { d f 5.7 and. ] ep [ Gamma ~ nverse d f 5.8 Afer are sampled from he condonal poseror dsrbuon gven by equaon 5.9 or 5.5 he confguraon ndcaor c and he cluser locaons are updaed accordngly. B. Sep b: updae Once Sep a s compleed for all n observaons he cluserng confguraon vecor c and he cluser locaons = 2 n have been updaed. Sep b

65 65 aduss he cluserng locaons condonng on c and d. For each cluser locaon le o and c denoe respecvely he se of eac falure me observaons and he se of rgh censored observaons n he h cluser.e. } and : { o c and } 0 and : { c c. In addon le } : { c c o denoe all he observaons belongng o he h cluser. The condonal poseror dsrbuon for s gven by: c o K k G f d c ]. ep [ ep To draw he values for frs draw a value for from d c f and hen draw a value for from d c f. I can be shown ha ep Gamma ~ nverse ep ep. ep ep ep ep ep o c o o n d d n d f d c 5.20 where o n denoes he number of eac falure me observaons n he h cluser and

66 66 o o n f d c ep ep ep } 0 { 5.2 whch s a non-sandard densy funcon. The daa augmenaon mehod dscussed by Damen e al. [20] can be mplemened here o sample a value from d c f. Inroduce aulary varables } : { o u u and } : { 2 2 u u so ha he on densy can be epressed as:. ep ep 0 ep 0 } 0 { 2 2 o o u u n f d c u u 5.22 The Gbbs samplng can be eended o frs draw u unformly on he nerval ep 0 and u 2 unformly on he nerval ep 0 ep and hen draw from 2 n o f d c u u 5.23 resrced o he nerval deermned by } ep ep { } ep { 0 2 o u u. Ths can be easly done usng he nverse-cdf mehod. C. Sepc: updae and Sep c draw values for and from her condonal poseror dsrbuons. Escobar and Wes [93] developed an augmenaon mehod for samplng precson parameer from s condonal poseror dsrbuon when he Gamma pror s assumed

67 67 for. An aulary varable u s nroduced such ha u c d ~ Bea + n. Then he new value for s sampled from a mure Gamma poseror dsrbuon epressed as pgammaa + n b logu+ pgammaa + n b logu 5.24 where p = a + n / n b logu+ a + n. The pror dsrbuon for s pareo dsrbuon herefore accordng o Bayes Law he condonal poseror dsrbuon for can be wren as: } ma{ ma ~ Pareo } { } { } { } { n b n a b a f b a f b n a n b n a a 5.25 Smlarly gven he pror of gamma dsrbuon he condonal poseror dsrbuon for s:. ~ Gamma ep ep ep n n d n a n d d a a b d n a b d a b b f b a f 5.26 D. Sep d: updae Sep d samples a new value for from s condonal poseror dsrbuon n b a f K k b a f n f 0. ep ep ep ] ep [ ep ep } 2... { d 5.27

68 68 The daa augmenaon mehod can be appled agan o smulae a value from f { 2... n} d. For each {: =} draw an aulary varable u unformly on he nerval 0 ep. For each =2 n draw an aulary varable u 2 unformly on he nerval 0 ep ep. Then a value for s sampled from f a b resrced o he nerval ma {log u / } mn { log log u2 }. { : } { 2... n} I s roune o sample from he runcaed normal dsrbuon. E. Sep e: updae he falure-me dsrbuon One of he obecves of ALT analyss s o predc he falure-me dsrbuon a he normal sress level 0. Kuo and Mallck [8] proposed wo mehods for predcng he falure-me CDF a 0. The frs mehod smulaes a sample of falure mes a he normal sress level 0 from he predcve densy and consrucs he falure-me dsrbuon a 0 from he smulaed falure mes usng some emprcal esmaors such as he Kaplan- Meer esmaor. The second mehod uses he funconal form of he kernel for evaluaon and needs o combne draws from mulple chans of he Gbbs samplng. In hs dsseraon he mehod proposed by Koas [9] s eended. Once Seps a-d have compleed n one eraon of Gbbs samplng Sep e draws values for he falureme CDF a 0 condonal on he values of { = 2 n} and obaned a ha eraon. Accordng o equaon 5.6 he falure-me CDF a he normal sress level 0 s gven by: F G K ep G d d

69 69 The CDF gven by equaon 5.28 can be appromaed by a fne mure wh a large L m 0 l m 0 l l l number of mng componens.e. F G K ep ' ' for a prespecfed grd of values m m = 2 N over he suppor of F G where 0 L s he number of mng componens and ' ' l = 2 L are..d. draws from l l n he mure n [ G ]. The wegh coeffcens l l = 0 { } 2 L are smulaed as follows. Inroducng an aulary varable z l frs smulae z l..d. from Bea +n and hen compue l accordng o l z L l zl z for 2... s s l L and L. The above procedure s based on he consrucve defnon of DP whch was dscussed by Sehuraman [2]. L = 2000 s used. A sensvy analyss has proven ha 2000 s a relable and conservave choce for L. 5.2 Illusrave Eamples Ths secon uses wo praccal eamples o llusrae he proposed ALT model and algorhm. Boh eamples use epermenal daa colleced a he Thn Flm Nano and Mcroelecroncs Research Laboraory a Teas A&M Unversy College Saon. The frs eample sudes he relably of a new med odes hgh-k delecrc maeral for nanoelecronc applcaons []. The second eample evaluaes he relably of a novel nanocrysals-embedded hgh-k nonvolale memory devce [3]. Meal-Ode- Semconducor MOS capacors wh he hgh-k delecrc flm were subeced o acceleraed elecrcal sresses.

70 Complee daa se eample Table 5. lss he falure-me observaons of hgh-k delecrc esed a four sress levels. Ths s a complee daa se whou censorng. Alhough 7. MV/cm s no he normal sress level we analyze he falure-me daa colleced a and 7.9 MV/cm o predc he falure-me dsrbuon a 7. MV/cm. The predced falure-me dsrbuon a 0 = 7. MV/cm can be compared wh he epermenal daa colleced a 7. MV/cm. Table 5.. Eample : mes-o-breakdown of MOS capacors esed a four elecrcal feld sresses [28]. Sress MV/cm Tme-o-breakdown seconds

71 7 For he purpose of comparson a paramerc ALT analyss s also performed. I s generally acceped ha he breakdown of gae delecrcs belongs o he class of weakeslnk problem of ereme value sascs and hus s assumed o have he Webull falureme dsrbuon a a gven sress level [22]. As nroduced n secon.4 E-model s a wdely used physcal lfeme-sress relaonshp whch relaes he me-o-breakdown o he elecrcal feld sress. The E-model s acually a log-lnear lfeme-sress relaonshp. Therefore he Webull dsrbuon wh he log-lnear lfeme-sress relaonshp s frequenly used as he paramerc ALT model for sudyng he me-dependen delecrc breakdown. The sandard MLE mehod s used o f he Webull log-lnear ALT model and o predc he falure-me CDF a 0 = 7. MV/cm. Because of he absence of any pror knowledge we choose nonnformave prors o reflec our absence of pror knowledge. Apply he Gamma0.00 pror for and and he Normal0 0 6 pror for. Boh Gamma0.00 and Normal0 0 6 are wdely used dffuse prors n he sense of no favorng any value [23]. For Pareo pror s used whch has an nfne varance o reflec large pror uncerany due o absence of pror knowledge. The algorhm s coded n Malab. The Gbbs samplng algorhm runs for 0000 eraons and he frs 5000 eraons are dscarded before daa anlayss. Convergence s verfed by runnng mulple chans from dverse sarng pons eamnng he race plos and monorng he Gelman-Rubn sascs [9]. Poseror predcon on he falure-me CDF a 0 s based on he poseror medan. Fgure 5. shows he falure-me CDF a he normal sress level predced by he proposed DP Webull mure ALT model and he Webull log-lnear ALT model. Usng

72 72 he epermenal daa colleced a 0 = 7. MV/cm an emprcal CDF s compued from he medan rank esmaor.e. ˆ 0.3 F = 2 n n 0.4 where = 2 n denoe he ordered falure mes. The proposed DP Webull mure ALT model provdes much beer predcve resul han he Webull log-lnear ALT model. The Webull log-lnear ALT model fals o accuraely predc he falureme dsrbuon a he normal sress level. Ths may be caused by he nvolvemen of mulple falure modes. Fgure 5.. Eample : predced falure-me CDF a he normal sress 0 = 7. MV/cm.

73 73 Fgure 5.2 shows he Webull f o he daa colleced a 7.9 MV/cm and he emprcal falure-me CDF a 7.9 MV/cm. The Webull dsrbuon does no f he daa adequaely. Degraeve e al. [24] proposed a bmodal breakdown model wh he followng falure-me PDF f p p[ f R f R p f a e a e a e where he subscrps e and denoe respecvely ernsc breakdown and nrnsc breakdown p s he fracon of devces wh defecs and f and R are he Webull PDF and he Webull relably funcon respecvely. For defec-free devces he falures occur nrnscally. For devces wh defecs he ernsc and nrnsc falure modes are n compeon wh each oher. Fgure 5.2 ndcaes ha he bmodal breakdown model fs he daa beer han he Webull model. Because he Webull dsrbuon s no adequae o descrbe he falure-me dsrbuon a a gven sress level he Webull log-lnear ALT model fals o accuraely predc he falure-me dsrbuon a he normal sress level. On he oher hand he DP Webull mure ALT model descrbes he falure-me dsrbuon nonparamercally usng he DP Webull mure model. Ths modelng flebly leads o an mproved predcon of he falure-me dsrbuon a he normal sress level.

74 74 Fgure 5.2. Eample : esmaon of falure-me CDF a 7.9 MV/cm Rgh-censored daa se eample Table 5.2 lss he mes-o-breakdown of memores esed a four gae volage sress levels. Ths daa se consss of 0 8 and 37 rgh-censored observaons a and 7. V respecvely. Agan we assume he lowes sress level.e. 7. V used n he epermen as he normal sress level and use he daa colleced a and 7.5 V o predc he falure-me CDF a 0 = 7. V. Fgure 5.3 shows he falure-me CDF a he normal sress level predced by he DP Webull mure ALT model and he Webull log-lnear ALT model and compares hem wh he emprcal CDF compued by he medan rank esmaor. Because of he censorng he emprcal CDF s runcaed a he censorng me of 600 seconds. For hs

75 75 eample a vsual comparson of he falure-me dsrbuons predced by he Webull log-lnear ALT model and he DP Webull mure ALT model shows ha he wo models provde very close resuls and her predced falure-me CDFs agree very well wh he epermenal daa colleced a 0 = 7. V. Table 5.2: Eample 2: mes-o-breakdown of nanocrysals-embedded hgh-k memores esed a four volage sresses Sress vols Tme-o-breakdown seconds censored censored censored

76 76 Fgure 5.3: Eample 2: predced falure-me CDF a he normal sress 0 = 7. V. 5.3 Conclusons Ths sudy proposed he DP Webull mure ALT model and developed he Gbbs samplng algorhm for model fng. Ths ALT model descrbes he falure-me dsrbuon a a gven sress level usng he nonparamerc DP mure model wh he Webull kernel whch offers grea modelng flebly. Two praccal eamples relaed o he relably of nanoelecroncs have demonsraed ha he proposed mehodology s capable of provdng accurae predcaon of he falure-me dsrbuon a he normal sress level whou assumng any resrcve paramerc falure-me dsrbuon. The Gbbs samplng algorhm presened n Secon 5..2 alhough cumbersome n appearance s easy o mplemen. All he condonal poseror dsrbuons nvolved n he algorhm are roune o sample. Of course he proposed model and algorhm are

77 77 compuaonally more nense han he ML esmaon of he Webull log-lnear ALT model. Anoher dsadvanage of he nonparamerc Bayesan analyss s ha may requre larger sample szes han he paramerc analyss especally when nonnformave pror dsrbuons are used. Ths sudy consdered only one sress varable. I s sraghforward o eend he proposed mehodology o consder mulple sress varables by allowng and n Eq. 5. o be vecor-valued. Ths sudy consdered rgh censorng. Oher censorng mechansms can be easly consdered due o he fac ha he Webull kernel has he closed-form CDF and he flebly and generaly of he Gbbs samplng algorhm.

78 78 CHAPTER 6: BAYESIAN ANALYSIS FOR SIMPLE STEP-STRESS ACCELERATED LIFE TESTING Ths chaper develops Drchle process Webull mure model for smple sepsress ALT. The cumulave eposure model s appled o descrbe he effec of changng sress. The conssen model fng algorhm wh consan-sress ALT analyss s ulzed. The resuls of praccal eample show hs mehodology s capable of accuraely predcng he falure-me dsrbuon a he normal sress level. 6. Mehodologes The SSALT model nroduces cumulave eposure model o he consan-sress ALT model. The CDF and PDF of he falure-me dsrbuon n a smple SSALT wh log-lnear acceleraon model can be respecvely epressed as: and F F F2[ ep L H ] 0 f f f 2 [ ep L 0 H ] where s sress changng me and L and H are lower and hgher sress levels respecvely. Then he herarchcal form of DP Webull mure model for SSALT can be wren as:

79 79. ~ ~... ~ 0 ]ep ep [ and ]ep ep [ ep 0 ep ep ~ 0 b a f b a f b a f b a f G DP G n G K k k H L H L 6. Agan assgn normal pror for Pareo pror for Gamma pror for and as followng.. Gamma ~ Gamma ~ Pareo ~ Normal ~ b a b a b a b a The smulaon algorhm of SSALT nference s smlar o ha of consan-sress ALT as followng. a Draw from s condonal poseror dsrbuon and updae cluserng ndcaor for each. For all > replace wh ' where ; ep ' H L b Updae cluser locaons from... for n M f d c replace wh ' for all >; c Updae and based on her condonal poseror dsrbuons; d Sample value of from s condonal poseror dsrbuon usng slce samplng.

80 80 For each eraon samples a new value for from s condonal poseror dsrbuon ep ep ep ep ' ep ' ep ep ep ] 'ep [ 'ep ep ep ep } 2... { n c c c c c T b a f b a f K k k b a f n f d 6.2 where. ep 0 H L T Agan apply daa augmenaon o smulae a value for from } 2... { d n f. For each {: =} draw an aulary varable u unformly on he nerval 0 ep. For each =2 n draw an aulary varable u 2 unformly on he nerval 0 ep ep T. Then a value for s sampled from he runcaed normal dsrbuon b a f wh resrc o he nerval } log log { mn } / {log ma 2 } 2... { } : { n u T u. a Updae he falure-me CDF a he normal use sress level.

81 8 6.2 Illusrave Eamples The praccal eample used n hs secon also use epermenal daa colleced a he Thn Flm Nano & Mcroelecroncs Research Laboraory a Teas A&M Unversy College Saon. Table 6. presens he mes-o-breakdown observaons of devce esed a a smple SSALT. The low and hgh sresses appled n he es are 7.5V and 8.3 V respecvely. The sress changed a = 480 s and he ess ended a c = 600 s. The daa se consss of 52 falure observaons and 8 rgh-censored observaons. Agan he lowes sress level.e. 7.5 V used n he epermen s assumed as he normal sress level and he daa colleced n he smple SSALT s used o predc he falure-me dsrbuon a 7.5 V. Ths ype of sep-sress ALT s known as sep-sress parally ALT.e. a un s frs esed under normal sress f he un does no fal by me ncrease he sress level and es he un unl fals or censors. The predced dsrbuon can be compared wh he emprcal CDF compued from he medan rank esmaor. Table 6.: Tmes-o-breakdown of nanocrysals-embedded hgh-k memores under smple SSALT. Tme-o-breakdown seconds censored

82 82 Nonnformave prors are chosen o reflec absence of any pror knowledge. Assume Gamma 0.00 pror for and Pareo pror for and Normal0 0 6 pror for. Ulze Malab o code he algorhm and run Gbbs samplng algorhm for 0000 eraons wh he frs 5000 eraons dscarded before daa analyss. Fgure 6. shows he race plos for parameers and wh hree dfferen sar pons. I s reasonable o conclude ha he convergence has acheved snce he hree chans appear o m well afer 5000 eraons. Fgure 6. Trace plos of and

83 83 The poseror predcon on he falure-me CDF a 7.5V s based on he poseror medan. The 95% poseror nervals are also ploed o demonsrae he accuracy of predcon. Fgure 6.2 shows he falure-me CDF a 7.5 V predced by DP Webull mure SSALT model he paramerc Webull log-lnear SSALT model and emprcal CDF compued from he medan rank esmaor. The emprcal CDF s runcaed a he censorng me of 600 seconds. DP Webull mure SSALT model and paramerc model provde very smlar resuls. In addon he 95% poseror nerval of DP Webull mure model covers all epermenal daa. Fgure 6.2 Predced falure-me CDF a he use sress 0 = 7.5 V

84 84 CHAPTER 7: CONCLUSIONS Ths dsseraon presens a sem-paramerc Bayesan nference for acceleraed lfe esng usng Drchle process Webull mure model. A novel srucure of nanoelecroncs s nvesgaed. Then he epermenal daases colleced from smlar srucure of devces are appled n he sem-paramerc ALT model proposed n hs dsseraon. The resuls demonsrae he capably of our model. In general compare o paramerc acceleraed lfe esng model he DPWM model s more applcable for emergng producs.e. here es unceranes on he falure-me dsrbuon of producs and very lle pror knowledge on he model parameers are avalable. 7. Memory Funcons of MoO Nanodos Embedded ZrHfO Hgh-k The memory properes of nc-moo embedded ZrHfO hgh-k capacor are shown o be conrbued by hole rappng and derappng. The C-V hyseress and J-V curves deec ha holes are rapped a he nc-moo se. Those wo curves also confrm ha some loosely rapped holes are easly released. The reenon sudy furher demonsraes hose loosely rapped hole appromae 20% of oal rapped holes release quckly. More han 50% of rapped holes reman n he devce afer 0 years. Ths s desrable for nonvolale memory devces. 7.2 Predcon of CDF of Falure-Tme Dsrbuon a Normal Sress Level The consan-sress ALT model proposed n he dsseraon assumes a log-lnear relaonshp beween lfeme and sress levels. The nonparamerc DP Webull mure model s used o descrbe he falure-me dsrbuon a a gven sress o rela he lmaon of lack of dsrbuon nformaon. The sep-sress ALT model nroduced he

85 85 CE model o descrbe he effec of changng sress. Malab s used o mplemen he smulaon algorhm and plo he predced CDF curve of falure-me dsrbuon under use condon. The smulaon has proved o be converged. The resul s compared wh he paramerc Webull log-lnear ALT model. In complee daase eample DPWM model yelds closer curve o emprcal CDF. For rgh censored daase boh semparamrc and paramerc models yeld close curves. 7.3 Fuure Research Ths dsseraon develops a semparamerc mehod for ALT wh one sress varable and nvolves complee and ype-i censored daase. I can be generalzed o nclude mulple sress varables and ype-ii censored daa due o he flebly and generaly of Gbbs samplng algorhm. In addon oher dsrbuons can be appled o Drchle process mure model such as eponenal normal or lognormal dsrbuon.e. DPEM DPNM or DPLNM respecvely. Anoher fuure research wh DPWM can be degradaon analyss. Insead of hard falure he falures for some elecroncs are defned as sof falure. For eample falure me of lgh dsplay devces s defned a he me when a devce lumnosy drops below 50% of s nal lumnosy [25]. The falure of some laser producs are defned as more han 20% oupu power loss. The analyss of performance degradaon pah boh under normal condon and acceleraed sress condon can furher reduce lfees me and mprove relably nference. Degradaon analyss can be performed as a nonparamerc regresson wh DMWM model.

86 86 The cumulave eposure model s appled o deal wh smple SSALT n hs dsseraon. Oher models for eample he Tampered Random Varable TRV model and he Tampered Falure Rae TFR can be appled and combned o DMWM model. Furhermore smple SSALT modellng only deals wh wo sress levels. The algorhm nvolvng mulple sress levels can be deduced by generalzng he cumulave eposure model wh a log-lnear funcon o mulple sresses.

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