Avlle onlne www.scencedrec.com Engneerng Frcure Mecncs 75 (8) 64 8 www.elsever.com/loce/engfrcmec Frcure of nodc-onded slcon-n flm glss-slcon rple scks L.Y. Sng, Z.L. Zng *, B. Skllerud Deprmen of Srucurl Engneerng, Norwegn Unversy of Scence nd Tecnology (NTNU), Trondem, Norwy Receved Ocoer 6; receved n revsed form Aprl 7; cceped 6 Aprl 7 Avlle onlne 7 My 7 Asrc In s sudy we focus on e frcure evor of wo ypes slcon-n flm glss-slcon (S-Glss-S) rple scks specmens w srp corner. We deermne e noc sress nensy fcor K n for o specmens usng comnon of e Wllms egenfuncon expnson meod, Sro s sexc formlsm, fne elemen nlyss, nd e p-ndependen H- negrl. Emprcl soluons of dmensonless sress nensy fcors re proposed for wo ypcl specmens, nd e dependence of geomery s nlyzed. Furermore, e effec of glss ckness on sress nensy s explored for nodc-onded S-Glss-S rple scks. We dscuss e fesly of usng crcl vlue of K n o correle e flure resuls for o specmens w vrous ond re nd glss ckness. Ó 7 Elsever Ld. All rgs reserved. Keywords: MEMS relly; Inerfce frcure; Wfer onds; Noc mecncs; P-ndependen negrl. Inroducon Wfer-o-wfer ondng s crcl n e producon of mcro-elecro-mecncl sysems (MEMS). Among e dfferen ypes of wfer ondng, e nodc ondng plys n mporn role especlly for slcon wfers. Anodc ondng (lso clled feld-sssed erml ondng, elecrosc ondng, ec), ecnque for selng glss nd slcon wfers, ws presened y Wlls nd Pomernz []. Te dvnge of nodc ondng for MEMS s e low emperure provdes melzon lyer does no degrde due o emperure effecs. Anodc ondng s commerclly vlle ecnque. However, n-flm nodc ondng, nvened y Brooks nd Donovn [], s no ye ndusrlzed. S-Glss-S rple scks w free edges nd corners re lso common n wfer-level vcuum pckgng of mcroelecronc devces nd mcrosensors. A n lyer of glss s eer elecron em evpored or spuered ono slcon wfer. Anoer slcon wfer s en onded ono e glss flm. Deled ecnques cn e found n Ref. []. All pulsed resuls ndce cern mnmum crcl ckness of e glss lyer s requred o ensure good ondng, oerwse, wll * Correspondng uor. E-ml ddress: Zlng.zng@nnu.no (Z.L. Zng). -7944/$ - see fron mer Ó 7 Elsever Ld. All rgs reserved. do:.6/j.engfrcmec.7.4.
L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8 65 Nomenclure lf ond lengs of MESA specmen, Sro egenvecors lf ond lengs of FRAME specmen B mrx C jk merl elsc sffness C merl elsc sffness w Vog s noon D mrx E elsc modulus f rrry funcons w rgumens z f j ngulr vron of e sress felds f w normlzed funcon of w for e MESA specmen f w normlzed funcon of w for e FRAME specmen g normlzed funcon of for e MESA specmen g normlzed funcon of for e FRAME specmen g ngulr vron of e dsplcemen felds e ecng dep (e eg of ond pd) K crck sress nensy fcor K n noc sress nensy fcor K C crcl sress nensy fcor K sffness mrx m deformon mode, m = I, II M merl ID n j e un ouwrd norml o C N,N,N mrces n convenonl sx-dmensonl lner egenvlue prolem p egenvlues ( =,,...,6) q, complex consns q M, M mrx Q, R nd T mrces reled o merl elsc consns r, polr coordnes glss cknesses ckness of e specmens n x drecon e rcon ny merl pon (r,) u dsplcemen funcon w lf wd of e MESA or FRAME specmen x,x,x Cresn coordnes Y non-dmensonl sress nensy fcor Y ref reference quny n e emprcl soluons due o normlzon Y m H-negrl non-dmensonl sress nensy fcor oned from H-negrl pproc Y m u-sympoc non-dmensonl sress nensy fcor from FEM sed on dsplcemens Y m s sympoc non-dmensonl sress nensy fcor from FEM sed on openng sress z complex vrle z = x + p x r j, u cul sresses nd dsplcemens r j ; u complemenry sresses nd dsplcemens r remoe ensle nomnl sress r f nomnl flure sreng r N ne-secon flure sreng d l Kronecker del k sress sngulry
66 L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8 k * u k e complemenry egenvlue sress funcon componens / sress funcon n () n () = cos + p sn g egenvecors c noc ngle C closed conour round e noc p m Posson s ro decrese e ond sreng or even resul n flure. Te mnmum ckness of e glss flm depends on e concenron of lkl ons n e glss, nd e ypcl mnmum ckness s lm [4]. In conrs o spuerng, evporon cn conn mny more sodum ons n e glss lyer nd e nodc ondng cn e performed lower emperure nd volge n e ondng process [5]. An dvnge w n-flm nodc ondng, s compred o nodc ondng, s prolems cused y dfferen erml expnson coeffcens nd Young s modulus of e glss nd e slcon re reduced. Addonlly, wo gly perned slcon wfers cn e onded ogeer w n-flm nodc ondng. Te dsdvnge w e ecnque s e process s no een suded nd opmzed o e sme degree s e well known nodc ondng process. In lner elsc frcure mecncs, e use of crcl sress nensy fcor K C o predc rle frcure of crcked solds s wdely cceped. Te fesly of usng e sngle prmeer K (or funcon of wo prmeers n mxed-mode cses) s due o e unversl nure of e sngulr sress feld surroundng crck p s sown y Wllms [6]. In numer of MEMS componens, srp re-enrn corners or noces re nroduced, usully o fcle frcon. Over e ls yers, noc mecncs nd nlyss of n nerfce eween wo elsclly dssmlr merls ve een n cve reserc feld. As we know, e ner-p sngulr sress feld n noced ody s crcerzed y e form r j ¼ K n I rki fj IðÞþKn II rkii fj IIðÞ. Te sress sngulry k for vrous sngle nd mulple-pse noc geomeres n o soropc [7 ] nd nsoropc [ ] med s een suded. For mulmerl med, e suon ecomes complced s n mxed-mode deformon e sympoc elsc felds depend on rdl poson, elsc msmc nd nerfce corner geomery. Accordngly, e mode I nd mode II felds re usully no symmerc w e noc secor s ey re for e soropc omogeneous cse. An effcen compuonl procedure o on sress nensy fcors round mulmerl nerfce corners cn e relzed y e p-ndependen H-negrl. Te neresed reders cn confer e revew of Hucnson nd Suo [], nd Lossere nd Dunn [] for more dels. Despe some exensve sudes reled o nsoropc merls nd merl med, few sudes focused on e ree-merl nerfce noc prolems. Ts gves movon of our work. In s pper, we exend e H-negrl pproc o compue sress nensy fcors nerfce corners of n-flm nodc-onded S-Glss-S rple scks. Two ypes of specmens, nmed MESA nd FRAME, w dfferen ond re nd vrous glss ckness re consdered. We pply s pproc o e mxed-mode I nd II lodng. Te sress nensy fcors oned from e H-negrl pproc re n good greemen w ose oned y mcng e sympoc soluons w deled fne elemen nlyss. Emprcl soluons of dmensonless sress nensy fcors re eslsed o fcle engneerng pplcon. Furermore, e effec of glss ckness on e sress nensy s suded for rple scks. Te mgnude of crcl sress nensy fcor for n-flm S-Glss-S rple scks s deermned from ess. Fnlly, mes effec nd some uncernes re dscussed.. Asympoc nlyss of nerfce noc p felds w nsoropc elsc merls Te sympoc nlyss of nsoropc elsc felds e merl nerfce corner s refly presened. Te sympoc nlyss solves wo egenvlue prolems oned y Sro s sexc formlsm [ 4] nd e egenfuncon expnson meod of Wllms [6].
.. Frs egenvlue prolem Sro s sexc formlsm Accordng o Sro, wen wo-dmensonl deformons depend only on x nd x, e dsplcemen u nd sress funcon / of n nsoropc elsc sold n fxed recngulr coordne sysem cn e generlly expressed s u ¼ X ½ f ðz Þþ f þ ðz ÞŠ ð:þ ¼ / ¼ X ¼ ½ f ðz Þþ f þ ðz ÞŠ L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8 67 Here f re rrry funcons of er rgumens z dependng on e geomery nd lodng, z = x + p x s e complex vrle. Te sx complex egenvlues p ssfy p þ ¼ p nd re e soluons of e qudrc egenvlue prolem (.4). In ddon, nd re e Sro egenvecors nd ssfy þ ¼ nd þ ¼ reled roug e mrx Q, R nd T descred ler. p, nd depend only on e elsc sffnesses C jkl. Wou loss of generly, e mgnry pr of p s ken o e posve. Overr of p, z,, denoes e complex conjuge. In erms of e sress srn lws r j = C jkl u k,l nd usng Eq. (.), e equlrum equons C jkl u k,lj = cn e wren C jkl ðd l þ d l pþðd j þ d j pþ k ¼ ð:þ n wc comm denoes dfferenon nd d l s e Kronecker del. Te ove equon cn e wren n mrx form s fq þ pðr þ R T Þþp Tg ¼ ð:4þ were Q k = C k, R k = C k nd T k = C k. For non-rvl soluon of, we mus ve dejq + p(r + R T )+p Tj =, wc resuls n sx roos for e egenvlue p. In mrx forms nd w Vog s noon C C 6 C 5 C 66 C 6 C 46 C 6 C C 4 6 Q ¼ 4 C 6 C 66 7 C 56 5; 6 T ¼ 4 C 6 C 7 C 4 5 nd 6 R ¼ 4 C 66 C 6 7 C 46 5 ð:5þ ð:þ C 5 C 56 C 55 C 46 C 4 C 44 C 56 C 5 C 45 C þ pc 6 þ p C 66 C 6 þ pðc þ C 66 Þþp C 6 C 5 þ pðc 4 þ C 56 Þþp C 46 6 C 66 þ pc 6 þ p C C 56 þ pðc 46 þ C 5 Þþp 7 4 C 4 5 ¼ ð:6þ symmerc C 55 þ pc 45 þ p C 44 Dfferenng Eq. (.) nd en nserng no consuve equon, we cn on r j r ¼ðQ k þ pr k Þ k f ðzþ ð:7þ r ¼ðR k þ pt k Þ k f ðzþ n wc e prme denoes dfferenon w e rgumen z. Mkng use of Eq. (.), e relon eween nd cn e wren ¼ðR T þ ptþ ¼ ðq þ prþ p ð:8þ Te ove qudrc egenvlue prolem (ree dmensonl) cn e recs s convenonl sx-dmensonl lner egenvlue prolem N N N N T ¼ p ) Ng ¼ pg; g ¼ ð:9þ
68 L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8 were N = T R T, N = T nd N = Q + RT R T. Snce p cnno e rel f e srn energy s o e posve [5], we ve ree prs of complex conjuges for p s well s for g.ifp nd g ( =,,...,6) re e egenvlues nd egenvecors, we le p þ ¼ p ; Imp > ¼ ; ; ð:þ g þ ¼ g were Im denoes e mgnry pr. Te Sro egenvecors re deermned up o n rrry consn. Tey re normlzed s ¼ ^ qffffffffffffffffffffff nd ¼ ^ qffffffffffffffffffffff ^ T^ ^ T^ ð:þ were ^ nd ^ re e non-normlzed egenvecors,.e. ose would e produced y sndrd egensolver; nd represen e drecon of e dsplcemen u nd rcon, respecvely... Second egenvlue prolem noc p sress sngulry A sress sngulry exss srp noces/crck ps. Te degree of sress sngulry s oned from solvng e second egenvlue prolem. Te rcon ny merl pon (r,) long e rdl lne from e noc p cn e wren ¼ d dr / or r k ¼ ou k ; r k ¼ ou k ð:þ ox ox were / s e sress funcon gven y Eq. (.) nd r s e rdl dsnce mesured from e noc p. By coosng f(z ) [9,,4] s f ðz Þ¼ k zk q nd f þ ðz Þ¼ k zk ð:þ were q nd re e unknown complex consns nd wll e deermned y Eq. (.9) once k s oned. Usng e expresson z = x + p x = rn ()=r(cos + p sn), e dsplcemen nd rcon funcons cn e wren u M ¼ X rk k ¼ M ¼ k X / ¼ rk r ¼ n M ðþk q þ n M ðþk n M ðþk q þ n M ðþk ð:4þ were superscrp M ndces merl A or B. Te second egenvlue prolem cn e solved y usng e oundry condons for e nerfce noc prolem, see Fg.. Tere re four ses of oundry condons: A ðþ ¼; X ¼ n A ðþk A qa þ X ¼ n A ðþk A A ¼ B ð Þ ¼; A ðþ ¼ B ðþ; X ¼ X ¼ n B ð Þk B qb þ X ¼ n A ðþk A qa þ X ¼ n B ð Þk B B ¼ n A X ðþk A A n B ðþk B X qb n B ðþk B B ¼ ¼ ¼ ð:5þ u A ðþ ¼u B ðþ; X ¼ n A ðþk A qa þ X ¼ n A ðþk A A X ¼ n B ðþk B qb X ¼ n B ðþk B B ¼
L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8 69 x γ α r β θ Merl A x Merl B Fg.. Confguron of merl nerfce corner. Te equons ove cn e rewren s B A q A þ B A A ¼ B B q B þ B B B ¼ A q A þ A A B q B B B ¼ A q A þ A A B q B B B ¼ ð:6þ were B A ¼ n A ðþk A ; na ðþk A ; na ðþk A ; B B ¼ n B ð Þk B ; nb ð Þk B ; nb ð Þk B ; M ¼ M ; M ; M ; M ¼½ M ; M ; M Š; qm ¼½q M ; qm ; qm ŠT ; M ¼½ M ; M ; M ŠT ðm ¼ A; BÞ From e frs wo equons, we ve A ¼ ðb A Þ B A q A q B ¼ ðb B Þ B B B By susung Eq. (.7) no e ls wo equons of Eq. (.6), we ge " A ðb A Þ A ðb A Þ ð B ðb B Þ #( ) B ðb B Þ Þ B A q A ¼ A ðb A Þ A ðb A Þ ð B ðb B Þ B ðb B Þ Þ B B B ð:7þ ð:8þ resuls n 6 smulneous egenvlue equons KðkÞD ¼ ð:9þ were D ¼½B A q A ; B B B Š T. For e sngle crysl slcon nd glss merl consdered n s sudy, K, cn e proned no nplne nd nplne deformons. K ¼ K IP ð:þ K AP In order o on non-rvl soluon, we ge e crcersc equon for k de½kðkþš ¼ ð:þ
7 L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8. H-negrl pproc A p-ndependen H-negrl pproc for clculon of e sress nensy fcor srp noces s refly presened ere. Te mn dvnge of s meod s e sress nensy fcor cn e oned y conour negrl round e noc p w only rcons nd dsplcemens requred. Consderng e confguron of merl nerfce corner sown n Fg., e upper nd lower noc fces re = nd = nd e noc ngle s c, were ngle s mesured from e nerfce. For nsoropc merls, er elsc sffnesses cn e rnsformed w respec o e x x xes [6]. Te noc fces re rcon-free nd e noced ody s loded remoe oundres y rcons or dsplcemens. In s pper we sudy only e nplne felds nd rel egenvlues k I nd k II solved y (.). W normlzons of e mode I felds y r ð ¼ Þ ¼K n I rk I nd of e mode II felds y r ð ¼ Þ ¼K n II rk II, e nplne sympoc sngulr felds ner e noc p cn e expressed s r M j ðr; Þ ¼Kn I rk I f IM j u M ðr; Þ ¼K n I rk I g IM ðþþk n II rk II f IIM j ðþ ðþþk n II rk II g IIM ðþ ð:þ nd efore normlzon f IM f IM ðþ ¼ X ðþ ¼X ¼ ¼ g IM ðþ ¼ k I X ¼ f IIM ðþ ¼ X ¼ f IIM ðþ ¼ X ¼ g IIM ðþ ¼ k II X M p M M nm ðþk I q IM þ p M þ M þ nm þ ðþk I IM nm ðþk I q IM þ M þ nm þ ðþk I IM M nm ðþk I q IM þ M þ nm þ ðþk I IM p M M nm ðþk II q IIM þ p M þ M þ nm þ ðþk II IIM M nm ðþk II q IIM þ M þ nm þ ðþk II IIM ¼ M nm ðþk II q IIM þ M þ nm þ ðþk II IIM ð:þ Here e superscrp M ndces merl A or B, K n m re noc sress nensy fcors were m corresponds o deformon mode I or II re nlogous o e openng nd sldng modes n omogeneous noced sold, k n re e sress sngulres, nd fj mm nd g mm re funcons dependng on ngle nd merl sffnesses. In Eq. (.), only e sress nensy fcors K n I nd Kn II cnno e deermned from e sympoc nlyss. Tey depend on e geomery of e noced ody, merl elsc consns, nd e fr-feld lodng. Te p ndependen H-negrl s sed on e pplcon of Be s recprocl work eorem wc suppose wo ses of elsc felds: e cul nd e complemenry. If k s roo of Eq. (.), sosk * = k [7]. Hence, e cosen complemenry soluon s gven y Eqs. (.) nd (.4). f ðz Þ¼ k zk q fþ ðz Þ¼ k zk r M j ðr; Þ ¼K n u M ðr; Þ ¼K n I r ki fj IM I r k I g IM ðþþk n ðþþk n II r kii fj IIM II r k II g IIM ðþ ðþ ð:þ ð:4þ
In e sence of ody forces nd sngulres, e H-negrl kes e form: Z H ¼ r j u r j u n j ds ¼ C were C s closed conour round e noc p, n j s e un ouwrd norml o C, r j, u nd r j, u re e cul nd complemenry sresses nd dsplcemens ssfy e equlrum nd consuve relons, respecvely. Tkng couner-clockwse conour round e noc p from e lower fce o upper fce, n polr coordnes, e ove equon s expressed s H ¼ Z r rr u r þ r ru r rr u r r r u r d ð:6þ nd n Cresn coordnes, s gven y Z Z H ¼ r x u x þ r xyu y r x u x r xy u y dy þ y x r y u y þ r xyu x r y u y r xy u x dx Tkng dvnge of Eq. (.5), we on n explc expresson for e mode I sclng fcor K n I H ¼ H I ¼ K n I nd smlrly e mode II sclng fcor Kn II y seng H ¼ H II ¼ K n II. Dmensonless consderons llusre e sress nensy fcor K n kes e form: K n ¼ r ðlengþ k Y ðgeomeryþ were r s nomnl sress nd Y s non-dmensonl funcon of geomery. 4. Applcon o S-Glss-S rple scks L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8 7 4.. Specfcon of S-Glss-S rple scks w mxed-mode I nd II lodng ð:5þ ð:7þ y seng In s work, rple scks re composed of S-Glss-S mulmerls. Two ypes of specmens, MESA nd FRAME, Fg., ve een suded. Te overll wd of e specmen (w) s 5 lm, e ecng dep (e eg of ond pd, ) s8lm. Fve ond lengs (), 7,, 44, 7 nd lm, nd four glss cknesses (),,.6, 5 nd lm, ve een consdered for e MESA specmens. For e FRAME specmens, fve ond lengs (), 4, 8,, 6 nd 9 lm, nd fve glss cknesses,,.6, 5, nd 8 lm, ve een nlyzed. A remoe ensle nomnl sress r = F/w s ppled o ll e specmens, were s e ckness of e specmen n x drecon. Brle frcure ned e nerfce corner. I sould e ð:8þ SI x SI x GLASS SI x 54.74º GLASS SI x 54.74º w w Fg.. 54.74 Noced () MESA nd () FRAME rple-scks specmens.
7 L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8 noed even oug e specmens ve remoe enson, ey re sujeced o mxed-mode I nd II lodng noc p. Te nerfce corner s c = 54.74 wc s resul of ecng process. Slcon s n elsc rle merl nd s elsc sffness mrx correspondng o e x x xes n Fg. s gven y Eq. (4.). Cornng #774 PYREX glss s used ere w elsc modulus E = 6 GP nd Posson s ro m =.. Mkng use of symmery, we only ke ccoun of lf-model n e plne srn fne elemen nlyss for o specmens. ABAQUS ws used n e nlyss. 94:6 6:9 5:4 6:9 65:7 6:9 5:4 6:9 94:6 C j ¼ GP ð4:þ 79:56 6 4 5:9 7 5 79:56 For e merls consdered, usng e sympoc sngulry nlyss descred n Secon, we on e egenvlues k I =.578, k II =.699. Te ngulr vron of e sress nd dsplcemen felds fj IðÞ, gi ðþ, fj I ðþ nd gi ðþ re sown n Fg., nd fj II ðþ, gii ðþ, fj II ðþ nd g II ðþ re sown n Fg. were,j = r,. In order o sudy e effec of glss ckness nd ond re on e frcure evor, dmensonl consderons of K n m yeld e followng form: K n m ¼ r w km Y n m w ; w ; ð4:þ were Y n m ; ; w w ðm ¼ I; IIÞ s dmensonless clron funcon nd s specfc for e geomery, e ckness of glss lyer, oundry condons, nd elsc consns. In s sudy, we only ke fxed vlue of ¼ :; consequenly, Y n w m ; ; w w reduces o Y n m ; w. Furermore, we cn wre Y n m w ; ¼ Y ref f g w for MESA Y n m w ; ð4:þ ¼ Y ref f g w for FRAME were f f ¼ Y n m ; ¼ : w ; g w Y ¼ Y n m ¼ :86; w ; Y ref ¼ Y n m ref Y ref w ¼ :86; ¼ : ¼ Y n m ; ¼ : w ; g w Y ¼ Y n m ¼ :4; w ; Y ref ¼ Y n m ref Y ref w ¼ :4; ¼ : 4.. Resuls 4... Vldon of e H-negrl pproc Before cn e convenenly ppled o sudy e frcure prolem, e ccurcy of e H-negrl sould e qunfed for e S-Glss-S rple sck prolem. I s found for e MESA nd FRAME specmens, e sress nensy fcors clculed from e H-negrl gree very well w sympoc soluons oned drecly y fne elemen compuons of dsplcemens long e noc flnks or of e sress long e nerfce. Te vrons re wn 5% for ll clculons. Tles 4 (Y ref n oldfce) sow e dmensonless sress nensy fcors Y n I nd Y n II oned from H- negrl nd sympoc nlyss w dfferen ond leng nd glss ckness for o MESA nd FRAME specmens.
f I j (θ) L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8 7 f I* j (θ).5.5.5 -.5 - f I θθ -.5 - -5 5 5 θ (degrees) - - - f I* θθ f I* rθ f I rr f I rθ f I* rr -4 - -5 5 5 θ (degrees) g I (θ) (θ) g I* 8 x -5 6 4 - -4-6 g I r -8 - -5 5 5 θ (degrees) 8 x -5 6 4 - -4-6 -8 g I* θ - -5 5 5 θ (degrees) g I θ g I* r f II j (θ) (θ) f II* j.5.5.5 f II θθ -.5 - f II rθ -.5 - f II rr -.5 - -5 5 5 θ (degrees) 5 4 - - - -4-5 -6 f II* rθ f II* θθ f II* rr - -5 5 5 θ (degrees) g II (Θ) (θ) g II* x -5 g II - r - - -4-5 g II θ -6-7 - -5 5 5 θ (degrees) 8 x -5 6 4 - -4-6 -8 g II* θ g II* r - -5 5 5 θ (degrees) Fg.. Angulr vron of e sress nd dsplcemen felds () fj IðÞ, gi I ðþ, fj ðþ, gi ðþ nd () fj II ðþ, gii ðþ, fj II ðþ, g II ðþ for S-Glss-S rple scks w noc ngle c = 54.74.
74 L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8 Tle Y n I nd Y n II oned from H-negrl nd sympoc nlyss w ¼ :86 for MESA specmens w / Y I H-negrl Y I u-sympoc Y I s-sympoc H-negrl u-sympoc.67.477.46.488.986.94.97..479.464.488.978.97.966.78.48.469.489.964.98.955.556.49.479.49.95.9.9 s-sympoc Tle Y n I nd Y n II oned from H-negrl nd sympoc nlyss w ¼ : for MESA specmens /w Y I H-negrl Y I u-sympoc Y I s-sympoc H-negrl u-sympoc..579.56.589.77.8.6.86.479.464.488.978.97.966.44.94.8.4.86.776.797.486.5.9.57.79.695.7.57...7.64.6.6 s-sympoc Tle Y n I nd Y n II oned from H-negrl nd sympoc nlyss w ¼ :4 for FRAME specmens w / Y I H-negrl Y I u-sympoc Y I s-sympoc H-negrl u-sympoc.67.494.478.56..98...497.48.56.8.979.5.78.5.488.59.7.97.998.556.58.56.5.99.97.986..56.54.56.98.96.98 s-sympoc Tle 4 Y n I nd Y n II oned from H-negrl nd sympoc nlyss w ¼ : for FRAME specmens /w Y I H-negrl Y I u-sympoc Y I s-sympoc H-negrl u-sympoc.4.497.48.56.8.979.5.9.5.5.4.69.668.685.4.75.66.8.57.546.564.457.46.8.5.5.49.55.54.4.7.8.486.466.48 s-sympoc 4... Emprcl soluons of Y n I nd Y n II for MESA nd FRAME specmens To fcle engneerng pplcon, we eslsed emprcl soluons of Y n I for MESA nd FRAME specmens. Te followng geomercl rnges re consdered: e soluons re pplcle over e rnge : 6 6 :57; :67 6 6 :556 for MESA specmens, nd over e rnge :4 6 6 :54; :67 6 w w 6 for FRAME specmens, respecvely. Accordng o Eq. (4.), e normlzed resuls oned from H- negrl pproc re ccurely fed y e power funcon f (geomery) = j(geomery) k nd g (geomery) = l(geomery) s, =,. Te es-f vlues of e prmeers j,k,l nd s re sown n Fgs. 4 nd 5 for MESA nd FRAME, respecvely. From Eqs. (4.) nd (4.), e dmensonless sress nensy fcor Y n I oned from e H-negrl cn e fed y e followng funcons: Y I H-negrl w ; ¼ Y I ref :9848 :5755 :55 :87 w ð4:4þ
L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8 75..9 f I (/w)=.9848*(/w) -.5755.. f I(/w).8.7.6 g I(/).. g I (/)=.87*(/).55.5....4.5.6.7 /w.99....4.5.6 / Fg. 4. f I nd g I for MESA specmens w : 6 6 :57; :67 6 6 :556 w...8 f I(/w).9.8.7.6.5 f I (/w)=.949*(/w) -.5695 g I (/).6.4.. g I (/)=.7547*(/).456.4....4.5.6 /w.98..4.6.8. / Fg. 5. f I nd g I for FRAME specmens w :4 6 6 :54; :67 6 6 w. were Y I ref ¼ Y I w ¼:86; ¼: ð Þ ¼ :479 nd : 6 w 6 :57; :67 6 6 :556: for e MESA specmens, nd Y I H-negrl w ; ¼ Y I ref :949 :5695 :7547 :456 w were ð4:5þ Y I ref ¼ Y I w ¼:4; ¼:Þ nd :4 6 w 6 :54; :67 6 6 : for e FRAME specmens. Fg. 6 compres e Y I H-negrl nd H-negrl. I cn e oserved from Fg. 6 o Y I H-negrl nd H-negrl decrese w ncresng ond re. 4... Effec of glss ckness In order o undersnd e effec of glss ckness on e sress nensy fcor, we ve rfclly exended glss ckness o = 75 lm. Fg. 7 compres e n-flm MESA specmen w glss ckness = lm nd e exreme merl MESA specmen w glss ckness = 75 lm. I cn e oserved e cker e glss lyer, e ger e Y I H-negrl. I lso s een found from Fg. 8 e Y n I ncreses w glss ckness wle sclly decreses for mode II lodng s e glss ckness ncreses. Furermore, for n-flm MESA nd FRAME specmens, Fg. 9, cn e oserved vryng ckness of glss does no ffec Y n I nd Y n II sgnfcnly. Nevereless,
76 L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8 Y H-negrl ( /=. )..8.6.4 Y I H-negrl H-negrl...4.6 /w Y H-negrl ( /=. ).8.6.4. Y I H-negrl H-negrl..4.6 /w Fg. 6. Y I H-negrl nd H-negrl for () MESA specmens nd () FRAME specmens..8.7 -negrl.6.5 I Y H-negrl ( =75μm,merl) Y H I.4. I Y H-negrl ( =μm,rple scks )....4.5.6 /w Fg. 7. Te dmensonless sress nensy fcors for wo glss cknesses. Y I H-negrl.9.8.7.6.5 Y I H-negrl ( /w=.86 ) Y I H-negrl ( /w=.44 ) Y I H-negrl ( /w=.57 ) H-negrl..9.8.7 H-negrl ( /w=.86 ) H-negrl ( /w=.44 ) H-negrl ( /w=.57 ).4.6. 5 5 Glss ckness (μm).5 5 5 Glss ckness (μm) Fg. 8. () Y I H-negrl nd () H-negrl for MESA specmens w glss ckness over e rnge 75 lm. for e oer MESA specmens w cker glss lyer, e sress nensy fcors dffer grely. Hence, cn e concluded e glss ckness s n mporn geomercl quny ffecng e sress nensy fcors for rple scks w ck glss lyer.
L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8 77 Y H-negrl ( /w=.86 )..8.6 Y I H-negrl H-negrl Y H-negrl ( /w=.4 )..8.6 Y I H-negrl H-negrl.4.4 4 6 8 Glss ckness (μm) 5 5 Glss ckness (μm) Fg. 9. Y I H-negrl nd H-negrl for n-flm () MESA nd () FRAME rple scks. 4..4. Flure creron Frcure ressnce of srucurl componens w sress concenron srp noces cn e evlued y dfferen flure crer. As dscussed erler, e noc p n o MESA nd FRAME specmens s sujeced o mxed-mode lodng. I s neresng o nvesge e flure creron. In order o ssess weer K n IC cn e used o predc frcure non, vrous MESA nd FRAME specmens were desgned for pull ess []. Ten o weny dfferen es specmens were used for ec specfc ond re nd e expermenl flure sreng ws fed o Weull dsruons. Cps were glued ono grnded exgonl ed cp screws w n lyer of Mcro Bond TM III CTCA- or Loce 4. Te dmensons of e es specmen re summrzed n Tle 5. One wfer n ec wfer couple for n-flm nodc ondng expermen ws covered w spuered lyer of PYREX, Cornng#774 glss (.6 ±. lm ck) on op of n delecrc lyer. A n delecrc lyer ws lwys grown or deposed efore spuerng of e glss lyer so s o reduce e possly of elecrcl rekdown durng e ondng process. Te n delecrc lyer ws removed from e cksde of e wfers efore ondng for elecrcl concng. I s well known rdonl yeld creron s no pplcle o correle w flure ecuse ypclly e flure lod mesured from ess depends on specmen geomery, sze nd ype of remoe lodng. Insed, some sudes ve sown e crcl sress nensy fcor K n C cn e used s sngle prmeer o correle frcure non srp noces, wc s possly ndependen of some geomery prmeers [,8,,8]. Te corner n e presen es specmens nvolves mxed-mode deformon. From e sympoc nlyss, e sress sngulry for mode I s srong, u s weker for mode II. Ts s dscussed y Dunn e l. [8]. In s cse, e crcl sress nensy K n C ¼ Kn IC ; Kn IIC s mnly crcerzed y K n IC. Te crcl noc sress nensy fcor, K n IC, cn e compued from Eq. (4.) usng r = r f, e nomnl flure sreng, wc s clculed usng e mesured flure force dvded y e overll srucure cross-secon re n e pull es. Accordng o verge frcure sreng r f [], e crcl sress nensy fcors K n IC sed on e es specmens re sown n Fg.. For e MESA specmen, e crcl sress nensy fcor K n IC oned for e verged flure sreng for e ree specmens of dfferen ond leng re.,.6 nd Tle 5 Dmensons of ond res for es specmens for pull ess [] Geomery nme Ouer dmensons (lm lm) Inner dmensons (lm lm) MESA MESA 44 44 MESA FRAME 7 7 FRAME 7 7 9 9 FRAME 7 7 Ec process 5 g KOH: l DIW, 8 C, 8 lm
78 L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8.5 FRAME MESA K IC.5.5..4.6 /w or /w Fg.. Te crcl sress nensy fcor for MESA nd FRAME specmens. Tle 6 Comprson of e expermenl nd predced nomnl flure sreng r f Specmen Weull verge expermenl r f (MP) Predced r f (MP) Specmen Weull verge expermenl r f (MP) MESA.6.6 FRAME.674.545 MESA.44.587 FRAME.497.775 MESA 4.86.7 FRAME 5.57 5.9 Predced r f (MP) 5 4 Expermenl nomnl flure sreng K IC n =.4MP mm.49 σ f (MP) S S S...4.5.6 /w 6 5.5 5 Expermenl nomnl flure sreng K IC n =.67MP mm.49 σ f (MP) 4.5 4.5 S.5 S..4.6 /w Fg.. Te expermenl nd predced nomnl flure sreng r f for () MESA nd () FRAME.
L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8 79.76 MP mm.49, respecvely; wle for e FRAME, e correspondng crcl sress nensy fcors re.75,.54 nd.7 MP m.49. Te verge vlue of K n IC s.4 MP mm.49 for MESA specmens nd.67 MP mm.49 for FRAME specmens, respecvely. Te verged expermenl nomnl flure srengs S S S S SS w w Fg.. Scemc defnon of ne-secon ond sreng r N for () MESA nd () FRAME. 5 Expermenl σ for MESA N Expermenl σ N for FRAME σ N (MP) 5 5....4.5.6 /w or /w Fg.. Ne-secon ond sreng r N for MESA nd FRAME specmens. Inerfce sress (MP) 5 4 S θθ (Fne Elemen Anlyss) S θθ (H-negrl) S rθ (Fne Elemen Anlyss) S rθ (H-negrl) - - - - r (μm) Fg. 4. Inerfce sress ner e noc p w ¼ :9 nd ¼ : for FRAME specmens. w
8 L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8 nd e srengs clculed usng e verged K n IC for e dfferen ond re re sown n Tle 6 nd Fg.. I cn e seen e verged K n IC cn e used o predc rle frcure of e nodc-onded FRAME specmens. However, for e MESA specmens e greemen s no very ssfcory. Te cuse of devons for MESA specmens wll e suded susequenly n more del (Secon 5.). I sould e noed e flure sreng employed n e ndusry s defned s e mesured flure force dvded y e overll onded re,.e., e ne-secon ond sreng r N (see Fg. ). Te ond sreng for MESA nd FRAME specmens re sown n Fg.. I cn e seen from Fg. dsnc geomery dependence s no oserved for MESA specmens, u for FRAME specmens sgnfcn geomery effec cn e seen. W respec o e ond re, lrge ond sreng of e nnes FRAME ws oserved, u ncresng e ond re for e FRAME specmen does no necessrly led o proporonlly ger ressnce gns n exernlly ppled force. Te flure creron ppled ere s sed on e ssumpon K-domned nnulus exss round e nerfce corner. Te elsc felds cn neer e very close o noc p wc s dsured y merl nonlneres nd geomerc rregulres nor e fr wy from e nerfce corner wc s ffeced y fr-feld lodng nd oundres. To qulfy s K-domned nnulus, fne elemen nlyss s performed flure lods. Te ccurcy of e flure creron s furer depced n Fg. 4 wc sows logrmc dgrm of e nerfce norml nd ser sresses ner e noc p w ¼ :9 nd ¼ :. Te norml sress r w oned from wo meods grees o 5% wn dsnce of.5 lm wle e ser sress r r grees o % wn dsnce of.7 lm. Smlr resuls re oned for oer specmens. 5. Dscusson 5.. Mes effec Te mporn feure of H-negrl pproc s relvely corse mes cn e used for e fne elemen nlyss [,9]. For exmple, for MESA specmen w ¼ :86 nd ¼ :78, fne nd corse mes w ner e noc p re depced n Fg. 5. Troug e glss lyer, only fve elemens re creed for e corse mes model wle 9 elemens re used for e fne mes model. Te smlles elemen sze for e fne nd corse mes s. nd lm, respecvely. I sould e poned ou e smlles mes sze s resrced y geomerc condons round e noc p, suc s glss ckness, ecng dep nd so on. Furermore, four negrl conours round e noc p re esed w e corse meses nd e vlues of mode I sress nensy fcors re.54,.7,.55, nd.45 MP mm.49, respecvely. As resul, e devon of sress nensy from.7 MP mm.49 oned from fne mes s less n %. 5.. Uncernes In s secon, we furer nerpre e resons resulng n devon for MESA specmens nd dscuss e pplcly of crcl sress nensy fcor. Fg. 5. () Fne mes nd () corse mes close look ner e noc p For MESA Specmen w w ¼ :86 nd ¼ :78.
L.Y. Sng e l. / Engneerng Frcure Mecncs 75 (8) 64 8 8 σ f (MP) 5 4 Expermenl nomnl flure sreng σ f D=μm D=μm D=μm D=5μm D...4.5.6 /w Fg. 6. Te effec of lodng mslgnmen on e predced nomnl flure sreng r f. For e MESA specmens, e predced nomnl flure sreng dd no mc very well w expermenl vlues. We eleve s s mnly due o e dffcul conrol of e lodng lgnmen. Te effec of mslgnmen s een furer suded, Fg. 6. Severl mslgnmen vlues ve een red. I s een found lodng mslgnmen of D = lm wll resul n muc eer greemen. Te FRAME specmens re less sensve o lodng mslgnmen due o lrge redundncy. An mporn ssue for noc mecncs s o undersnd e pplcly of e crcl sress nensy fcor K n IC for nerfce corners. Acully, Kn IC cn no longer e smply reed s sze-ndependen merl prmeer lke n e lner elsc frcure mecncs. I depends no only on nerfce corner geomery nd merl elsc consns, u lso on nerfce sreng resulng n dverse flure modes. For nsnce, Dunn e l. [8] nd Lossere e l. [] crred ou smlr ess w weker nerfce nd oned smller crcl sress nensy fcor n w srong nerfce. For e es specmens employed eren, we ve srongly onded nerfce nd ws lso oserved frcure ned from e noc p nd propged no e glss or prly long e onded nerfce. In ddon, resdul sress cn possly nfluence e resuls nd deserves enon n e fuure work. I my e one of e resons for e scer of flure sreng n e ess. Furermore, s oserved from fne elemen nlyses e ckness of S-wfers lso ffecs e dmensonless sress nensy fcor. For n S-wfers, for exmple,.5.75 mm ck, e vlues of mode I sress nensy cn pprenly vry w ckness of S-wfers. However, wen S-wfers re ck enoug, for nsnce, lrger n.75 mm, e ckness dependence wll dspper. Ts effec wll e nvesged more n e fuure. Fnlly, e crcl sress nensy fcor K n IC derved ere dd no yeld e sme vlue for o MESA nd FRAME specmens w e sme noc ngle. Te exc resons re no ppren. Apr from e dscussons ove, e possle resons ledng o e dfferences re s follows: e smplfed use of sngle prmeer K n IC n mxed-mode lodng, e conrol of lodng lgnmen, dfferen flure modes nd e effec of resdul sress. 6. Conclusons For e rple scks nerfce corner prolem consdered n s pper, e sress nensy fcors oned from e H-negrl pproc sow excellen greemen w ose oned from e sympoc soluons y fne elemen clculons of dsplcemens long e noc flnks or of sresses long e nerfce. Te devon of e dsplcemen pproc nd e sress pproc from e H-negrl pproc s less n 5%. Te effec of glss ckness on e sress nensy fcor s een suded. I s een found e mode I sress nensy fcor ncreses w glss ckness wle sclly decreses for mode II lodng s e glss ckness ncreses. Moreover, e sress nensy fcors vry sgnfcnly w cker glss lyer u re no ffeced grely for n-flm nodc-onded rple scks. Hence, urns ou e glss ckness s n mporn geomercl quny ffecng e sress nensy fcors for rple scks w ck glss lyer.
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