EE65R: Reliabiliy Physics of anoelecronic Devices Lecure 9: Feaures of Time-Dependen BTI Degradaion Dae: Sep. 9, 6 Classnoe Lufe Siddique Review Animesh Daa 9. Background/Review: BTI is observed when he source and he drain erminals of a p-mos are grounded and a negaive gae bias (w. r.. source and drain) is applied. I is aribued o he breaking of Si- bonds a he Si-SiO inerface and hereby creaing inerfacial raps (). ere, he applied ve gae bias acs as a sress. When he above menioned bias configuraion is mainained, he densiy of ( ) follows a emporal variaion as shown qualiaively in Fig.. I is essenially an power-law variaion. One needs o have a quaniaive perspecive of such variaion o esimae and improve he device lifeime. When he sress (he gae bias) is removed he bonds sar `healing and he rap densiy reduces. The overall variaion is shown qualiaively in Fig.. ln Threshold for proper circui operaion n eed o be on his curve o increase lifeime Lifeime of circui Fig. : The variaion of rap densiy, ln wih ime (solid line). G V G S D V G 3 n A + ξ + Fig.. Trap densiy variaion under sress and wihou sress.
9. Reacion-Diffusion Model for BTI: In his lecure we will deduce he variaion of rap densiy ( ) under sress. The dynamics of he variaion can be expressed by he reacion diffusion equaions: d () k f ( () ) kr ( ) ( x, ) () d d d ( x, ) d ( ) ( ) ( x, ) D + x, µ E x δ () d dx d where, is he oal Si- bond densiy a he inerface, is he densiy of free ydrogen, D and µ are he diffusion consan and he mobiliy, respecively, of he same. E is he elecric field and δ is he widh of he small region near he inerface where he bond-breaking reacion akes place. In all he analysis in his lecure, x denoes he Si-SiO inerface and posiive x-direcion direcs owards he oxide as shown in Fig 3. ( x) Region where he reacion akes place ydrogen generaion due o bond breaking Free ydrogen curren due o drif and diffusion x x δ x Si SiO Fig. 3: Coninuiy of free hydrogen The firs erm on he righ hand side of equaion () denoes he forward reacion (i.e. breaking of bonds o increase he rap densiy) has been se proporional o he densiy of he unbroken bond. The second erm denoes he reverse reacion (i.e. resoraion of he bond) has been se proporional o he rap densiy (broken bond densiy) and he available ydrogen a he inerface. Equaion () is essenially he coninuiy equaion for he free ydrogen a he inerface. The firs erm on he lef hand side of his eq. () denoes he generaion rae of free ydrogen due o he bond-breaking and he erm inside he parenhesis denoes he curren of ydrogen a he inerface (Fig. 3). The difference of hese wo erms is proporional o he rae of increase of ydrogen concenraion a he inerface, which is zero a seady sae.
Anoher convenien form of eq. () is is inegral form: f () () ) dx where, he upper limi of he inegraion is a funcion of ime and will paramerically depend on D and µ for he case of diffusion and drif respecively. This equaion merely saes ha he oal amoun of free ydrogen inside he oxide is equal o he rap densiy. The exac funcional form of f () will become clearer as we proceed furher ino he discussion. We will now consider he soluion of he reacion diffusion equaion under differen condiion. 9.3 Soluion of Reacion-Diffusion Equaion in he Sress Phase 9.3. Righ afer he sress has been applied: We can neglec in comparison o in eq. () and also neglec he second erm on he righ hand side of eq. () as much of he bonds remain inac and no much free ydrogen is produced. Wih hese approximaions we ge from eq. (): () k f (4) which is an exponenial variaion wih an exponen of. 9.3. Afer some ime has elapsed beyond he applicaion of sress: ow, is no ha small o be discarded in he way as we did before as considerable amoun of bonds, by now, are broken and here is significan amoun of free ydrogen a he inerface. I is sill small compared o as in he previous case. Bu, unlike he previous case we canno neglec he reverse reacion anymore. Moreover, he ne rae of increase of rap densiy becomes very small as he reverse reacion sars. We ge, wih all hese assumpions, from eq. (): k f () ( x, ) (5) k r We will now consider hree cases: (i) ydrogen flow due o diffusion only: Diffusion is he dominan process when he free ydrogen is no ionized. Assuming ha here is no any kind of recombinaion of ydrogen inside he oxide we ge a linear profile of free ydrogen densiy inside he oxide as shown in Fig. 4. A any ime he profile exends o a disance of D. This is he funcional form of f () in eq. (3). Subsiuing his value of f () in eq. () and considering a linear profile as shown in Fig. 4 we ge: D Dividing eq. (6) by eq. (5) and hen solving for () we ge: (3) ) dx D ( x, ) (6)
/ /4 /4 ( ) kf D A kr which is again an exponenial variaion wih a fracional exponen of /4. (7) ) ( x, ) D x Fig. 4: Free ydrogen densiy profile under diffusion (ii) ydrogen flow due o drif only: Drif becomes he dominan process when he ydrogen is ionized (proon). In his case he ydrogen densiy profile looks fla all he way upo he edge of he prifle a x µ E (Fig. 5). So, he funcional form of f () is µ E. Subsiuing i in eq. (3) and considering a fla profile as shown in Fig. 5 we ge: µ E ) dx µ E Dividing eq. (8) by eq. (5) and solving for () we ge: ( x, ) (8) k µ E k r which is an exponenial variaion wih a fracional exponen of /. f () / (9) ) ( x, ) µ E Fig. 5: Free ydrogen densiy profile under drif x
(iii) eural diffusion (wih +! ) Since he molecules are charge neural diffusion is he dominan process. So we ge: D ) dx D (, ) () x The reason for muliplying he inegral by is ha here are wo ydrogen aoms in a x x, ydrogen molecule. We also need a relaion beween (, ) and ( ) k { ( x ) } k ) (), Finally, from eqns. (5), () and () we ge: / 6 ~ : The summary of cases (A), (B) and (C) is shown in Fig. 6 where we see ha he slope of he variaion becomes fracional afer iniial phase. The exac value of he slope depends on he underlying mechanism of ydrogen flow inside oxide and he mechanism of reacion. () ln n Drif of + n 4 n 6 Diffusion of Diffusion of n Righ afer he applicaion of sress ln Fig. 6: Exponenial variaion of rap densiy wih ime as per he reacion-diffusion model 9.3.3 Sauraion of rap densiy a he end of reacion: The variaion of rap densiy wih ime a he end of he reacion, when all he bonds are broken, becomes sauraed. From eq. (), we can wrie for neural ydrogen: d () d ( x, ) ( x, ) D D (3) d dx D
A he end of he reacion when approaches we canno neglec rap inside he firs erm of righ hand side of eq. (). We are sill neglecing he ne rae of increase of rap densiy. Wih all hese consideraions we ge from eq. (): k f ( ) (, ) (4) () x kr Subsiuing eq. (4) ino eq. (3) and solving for we ge: + / ln k (5) which can be approximaed by (easily verified by second order expansion in Taylor series) 4 η e (6) which shows ha as,, i.e. he rap densiy smoohly sauraes o he value of Si- bond densiy. The oal variaion of rap densiy is shown in Fig. 7. ln Drif of + Diffusion of Sauraion Diffusion of Righ afer he applicaion of sress Fig. 7: Toal variaion of rap densiy as per he reacion-diffusion model ln 9.4 Conclusions: Today we discussed one feaure of he emporal degradaion of BTI, namely he origin of power-exponen in he sress phase. We will consider he oher wo aspecs of emporal degradaion relaxaion and frequency-independence in he nex class.