ln(i G ) 26.1 Review 26.2 Statistics of multiple breakdowns M Rows HBD SBD N Atoms Time

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1 EE650R: Reliability Physics of Naoelectroic Devices Lecture 26: TDDB: Statistics of Multiple Breadows Date: Nov 17, 2006 ClassNotes: Jaydeep P. Kulari Review: Pradeep R. Nair 26.1 Review I the last class we discussed the criteria of hard breadow (HBD) ad soft breadow evets (SBD). Gate oxides is ot destroyed after SBD ad the chage i the gate leaage curret per SBD is almost quatized (i.e. gate curret icreases i costat steps as the SBD evets happe (see Fig. 1) because the variatio i percolatio resistace is just 4-6 times. If R proc is very small, it results i hard breadow. I this lecture we will see the statistics of the multiple soft breadow evets (before the last hard breadow) Statistics of multiple breadows l(i G ) HBD SBD M Rows N Atoms Time Fig. 1(left): HBD (blue) ad SBD (dashed blac) evets. (right) A oxides fails whe a colum of defects shorts the isulators from substrate to gate. It is easy to see from right-had part of Fig. 1 that the probability that a colum of fails (p) = q M where, q = probability that oe trap is beig geerated i the oxide (~at a ) M = Number of defects eeded to short a colum

2 Probability that colums are shorted (P ) is give by biomial distributio as: N! P = p (1 p)!( N )! N Followig the percolatio model used earlier (referece lecture otes#22) we get, P χ = e. ;! t χ = where = αm ad t is the time η Thus the failure probability that the trasistor fails at th faulty colum formatio is give by (1-F ): 1 χ 1 F = P e = = 0 = 1! HBD (=1) N=1, HBD W ß W ß N=2 ß N=3 1/12 l(t) l(t) 1 st trasistor with 1 st b/d 2 d trasistor with 1 st b/d 1 st trasistor with 2 d b/d 1 st trasistor with 3 rd b/d For HBD coditio, trasistor fails at 1 st faulty colum formatio, i.e. =1 Substitutig this value i above equatio, we get 1 F1 = e W = l( l(1 F)) = l( χ) = l( t) l( η) 1 1

3 For SBD coditio, trasistor fails at th faulty colum formatio, therefore the Weibull fuctio is give by: 1 χ W = l( l(1 F )) = l l 1 e = 0! F ca be expressed as Taylor s series as (Icomplete Gamma Fuctio): 1 2 χ χ 1 χ χ F = 1 e = +... = 0! Γ ( ) + 1 ( + 2).2! For small values of F, l(1 F = 2 χ χ ) l( l(1 F )) = l( ( F )) = l( F) W = l( χ) l(!) = l( t) C The term ß gives the Weibull slope. As see i figure o previous page, as icreases the time of first failure icreases cosiderably (X axis is i log scale). That meas the time at which the first product would fail will icrease cosiderably compared to 1 st breadow coditio. Mathematically it is derived as follows: χ F =! χ = (!. F ) t t η 1/ = (!. F ) = η(!. F ) t η 1/ 1/ 1 2 1/ 1/ 1 1/ = (!) ( F ) = ( F ) e 2 π 1 2 * t 1 F 1/ = ( ) t * 1 e 2 π F1

4 The term (F * /F * 1 ) 1/ gives the amout of improvemet if th breadow is required for trasistor failure compared to 1 st breadow. If we reduce the operatig voltage slightly ad help the trasistor to survive from =1 to =3 rd breadow, it will give approximately 1000 fold improvemet i the lifetime! 26.3 Marov Chai Process for Soft Breadow: R 0 (No Shorts) K=1 R 1 (1 Short) K=1+? R 2 (2 Shorts) K= 1+2? A differet way of doig the statistics of SBD ivolves usig the geeralized Marov chai. We ca bi the trasistors accordig to the umber of faulty colum formed. Iitially all the trasistors would be i bi R 0, where there are o shorts preset. R 1 groups all the trasistors with 1 short. R 2 groups trasistors with 2 shorts. The term? gives the correlatio betwee the local trap geeratios (ehacemet factor). At ay poit i time, the sum of trasistors i every bi should be same as the total umber of trasistors. R N = N 0

5 The differece equatio ca be writte at a bi boudary as: dp = p p dχ 1 1 First term correspods to the trasistors movig from the previous bi ad secod term correspods to trasistor movig to the ext bi. At first bi boudary, dp0 dχ = p = e 0 0 At secod bi boudary ' ' p1 = e χ e χ + χ dχ' = χe χ 0 I geeral the differece equatio at th bi boudary is give as: ( χ') p+ 1= p 0 1χ' e d χ' With this Marov Chai approach, we would get the same result as we obtaied usig the simple probability theory (Sec. 26.2) Summary I this lecture we discussed the statistics of multiple breadow evets related to TDDB. We saw that for thi oxides at low operatig voltages, oxide defect formatios are statistically ucorrelated ad hece the device lifetime is may orders of magitude more tha that associated with hard breadow. I the ext class we will see how the Marov Chai approach is used i I DDQ measuremet for determiig the Weibull slope very efficietly.