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88 L . K . J . Vandamme and A . J. van Kem enade

of change in mi c r o - s t r uc t u r e a r ound t he ho l es o r t o d i s cus s t he i n f l uence o f ma t e r i a l

pa r amet e r s . Her e , t h e m i n i mum i nc r ease in no i se and r e s i s tance i s ca l cu l a t ed t ha t can be

expec t ed due t o a l oca l inc r ease i n cur r en t dens i t y aRer degr ada t ion . Ca l cu l a t i ons a r e checked

by expe r i ment a l r e su l ts .

M o d e l

W e c o n s i d e re d l o n g n a r r o w f i lm r e si s to r s , d e g r a d e d b y h o l e s o r k i n k s , s h o w n

schemat i ca l ly i n F i g . 1 . T he gen e r a l equa t i ons f o r r e s i s tance R and l / f no i se i n r e s i s tance SR

due t o condu c t i v i t y f l uc t ua t ions a r e [ 1 ,2 ]

R = [ 1 / 1 2 ] f p j 2 d A ( l )

S R = [ 1 / I 4 ] / [ u p 2 / n f ] j 4 d A = [ 1 / I 4 f ] / C u s p 2 j 4 d A ( 2 )

wher e 10(~ "~) i s the shee t r e s i s t ance ; J ( A / m) t he t wo- d i m ens i ona l cu r r en t den s i t y ; dA ( m 2) an

a r ea e l ement ; n ( m 2) t he t wo- d i m ens i ona l fr ee cha r ge c a r r i e r concen t r a t i on ; c t a d i mens i on l es s

l / f n o i s e p a r a m e t e r a n d C , , ( m 2) t he ch a r ac t e r i s ti c l / f no i se f o r a un i t a r ea [ 3 ,4 ] . T h e p r ob l em

can a l so be so l ved i n t h r ee d i mens i ons w i t h p ( f ~ cm) , J ( A /m2) , and d A ( m 3 ) b e c o m e s a v o l u m e

elemen t , and n (m 3) c a r r i e r concen t r a t i on pe r un i t vo l um e [ 1 ] .

W e appr o x i ma t e the i nc r eased l oca l cu r r en t dens i t y a r ound a ho l e o r k i nk by J = I / W ,~ ove r a

l eng t h equa l t o t he w i d t h W wi t h W o~ = W - 2a , wh er e 2a i s t he d i ame t e r o f t he ho l e . T h e

cur r en t dens i ty i n t he undam aged pa r t s i s J = I / W . T he nu mber o f ho l e s o r k i nks i s k. T o

s i mpl i fy he ca l cu l a t i ons we a s sume a l l ho l e s o r k i nks have t he s ame d i men s i ons . T he r e s i s t ance

o f th e k d a m a g e d p a r t s t h e n b e c o m e s R ~ = k p W / W c f r. T h e t o t a l r e si s ta n c e o f t h e u n d a m a g e d

par t s i s R~ = p ( L - kW ) / W . T he r e s i s t ance o f t he en t i re s ampl e becom es

R = R u + R d = 1 + \ W e ft - 1 ( 3 )

h ,

' I '

hole kink

~ L " , ~ _ l

2 a t

F ig . 1 . G eom et r y o f a r e s i s t o r w i t h a ho l e and a k i nk .

B



Resistance noise measurement

Th e r e s i s t a n c e 1% o f t h e u n d a m a g e d sa m p le i s 1% = p L/W , h e n c e t h e r a t i o R /Ro b e c o m e s

R / R 0 = 1 + \ ~ - (4 )

Th e sa m e a p p ro x im a t io n s a p p l i ed t o e q . (2 ) l e a d s fo r t h e r e s i s t a n c e n o i se i n t h e k d e g ra d e d

p a r t s t o

= R d C u s / f k W W e fr (5)R d 2

a n d in t h e u n d a m a g e d p a r t s t o

2S Ru = R u C u s / f W ( L - k W ) (6 )

respec t ive ly . Resis tance noise in the d i f fe ren t par ts i s uncorre la ted hence the to ta l resis tance

noise densi ty SR is the su m ofe qs. (5) and (6) .

S R _ p 2 C u s [ kW L - kW q_ - - ~ + (7 )

f W;tt VVq J

2Re s i s t a n ce n o i se o f a fa i l u re f l e e sa m p le i s SRO = R o C u s / f W L . Th e r a t i o SR/SRo SHOWS

the increase in resis tance noise

S R / S R 0 = 1 L [ \W e f fJ - 1 ( 8 )

The noise increase in Equa t ion (8) can a lso be w ri t ten as a func t ion of resis tance increase wi th

R - 1% = A R and SR - SRO = A SR with eqs. (4) and (8) f ollo ws

A . I ( ,o ,SR0 [kWeffJ \W eft)

Eq u a t io n s (4 ) a n d (8 ) a re fu n c t io n s o f W/L , W /Wa r an d k , a n d t h e t h i rd p o w e r i n (8 ) SHo ws th e

st ronger increase in noise than in resis tance due to holes or k inks. Fro m eq . (9) i t i s a lso c lea r

tha t for W /W ar> 1 the increase in noise i s s t ronger than in the resis tance . For k i s la rge r than 1 ,

t h e r e s i s t o r c a n b e c o n s id e re d a s k r e s i s t o r s i n se r i es o f l e n g th L /k a n d d e g ra d e d b y o n e h o l e o r

k ink , each having the same R/1% and SR/SR0 as the to ta l resis tor . O nly when degrad a t ion i s du e

to a n i n c re ase i n t h e n u m b e r o f h o l es o r k in k s a l l ha v in g t h e sa m e W/W ,n wi l l we f i n d a l i n e a r

re la t ion be tween ASR/S~o and AR /Ro as g iven by eq . (9) and SR/S~o and R/1% as g iven by

S R / S R o = R / R o ~ - ~ - + 1 - ~ % ( io )

89



90 L. K. J. Vandamrne and A. J. van Kemenade

EXPERIMENTS AND RESULTS

The purpose of our experiments is not to study the degradation process in film

resistors, but the effect of a degradation type (holes and kinks) on the increase in resistance

and the noise.

To verify equations (4) and (8), experiments with decreasing W~ as well as experiments with

increasing k were performed. Carbon paper film resistors with various W/L ratios were used.

The result o f a degradation is realised by punching holes in the carbon paper resistors at the

same spot with increased diameter or by punching an increasing number of holes at regular

distances, all with the same diameter. Before degradation we started by measuring the

reference values R~ and SR0 of the undamaged resistors. First, experiments on samples

degraded by one hole or kink were performed. After the first damage was made, R and SR

were measured. This process was repeated after each subsequent enlargening of the hole or

kink by using punches with a larger diameter ranging We~W from 0.1 to 1.

The calculated and experimentally observed relative increase in noise versus a relative

increase in resistance for decreasing W~ is presented in Fig. 2. The full lines were calculated

for W/L values of 10/100 and 10/300 from eqs. (4) and (8) with k = 1. The squares represent

the experimental results obtained from 2 resistors with different W/L ratios with one hole with

increasing diameter. The triangles stem from a third sample degraded by an increasing kink.

Second, the experiment with increasing k was performed on a carbon film resistor with

a W/L ratio of 5/563. Again Ro and SR0 were measured first and this was repeated for each

additional hole all with the same diameter. Measurements were taken for one up to ten holes.

All holes had the same diameter and were distributed equidistantly across the length of the

resistor.

Fig. 3 shows the relative increase in noise versus the relative increase in resistance for

increasing numbers of holes k. The full line was calculated with eqs. (4) and (8), W/Wc~ being

constant.

DISCUSSION AND CONCLUSION

Although carbon filmq are of no interest in microelectronics they can be used to verify

the calculations. The correlation for the effect of current crowding between a simulation with a

carbon film and A1 filing is perfect.



3 0 - , [ , , ' -

1 0 - W / 0 1 3 0 , 0

5W / l = 1 0 / I 0 0 _

3

2I

~ R / R oI I I I I

Ii

1 1 . 2 1 . 4 1 . 6F i g . 2 . R e l a t i v e i n c r e a s e i n l / f n o i s e v e r s u s r e l a t iv e i n c r e a s e i n r e s i s t a n c e d u e t o o n e h o l e o r

k i n k . F u l l li n e s a r e c a l c u l a t e d f r o m e q s . ( 4 ) a n d ( 8 ) . E x p e r i n a e n t a l r e s u l t s w i t h a h o l e :

W / L = 1 0 / 1 0 0 U ; W / L = 1 0 / 3 0 0 ~ ; a n d w i t h a k i n k : W / L = 1 0 / 3 0 0 A .

2. 0

1. 8

T s / s 0

1. 5

1 . 3 I " / •

R/Ro4F-

1 0 ' " ' 'IIF

1 . 0 0 1 . 0 5 1 . 1 0 1 . 1 5

F i g . 3 . R e l a t i v e i n c r e a s e i n l / f n o i s e v e r s u s r e l a t i v e i n c r e a s e i n r e s i s t a n c e d u e t o a n

i n c r e a si n g n u m b e r o f h o l e s f o r W / L = 5 / 5 6 3 .

o.

t-



9 2 L . K .J . Va n d a m m e a n d A, J . v a n Ke m e n a d e

Ex p e r im e n ta l re su l t s a r e i n g o o d a g re e m e n t wi th m o d e l c a l c u la t io n s . A q u a n t i ta t i v e

e x p la na t io n sh o ws w h y re s i s ta n c e n o i se m e a su re m e n t s a re m o re se n s it i v e t h a n r e s i s ta n c e

measurem ents on ly . The re la t ive increase in 1 / f no ise i s a complem enta ry ra ther than a

c o m p e t i t i v e t e c h n iq u e c o m p a re d t o c l a ssi c fa i lu re a n a ly si s t e s t s a n d i s a p p l i c a b l e t o LSI AI

in te rconnec ts and th in-f i lm resis tors . As ea r ly as 1971, Vo ssen [5] pro po sed the sc reening o f

m e ta l f i lm d e fe c t s b y c u r re n t n o i se m e a su re m e n t s wi th o u t e x p l a in in g wh y n o i se i s m o re

se n s it i ve t h a n r e s i s t a n c e m e a su re m e n t s o n ly . F ro m th e p e r sp e c t i v e o f n o i se a s a d i a g n o s t i c t o o l

" l / f y noise" i s ex t remely usefu l in s tud ies on e lec t rom igra t ion ac t iva t ion energ ies [6 ,7] and re f .

i n [ 7 ] .

We m u s t s t r e ss , h o we v e r , t h a t e q . (8 ) o n ly t a k e s i n to a c c o u n t t h e i n c re a se in n o i se

in t en s i t y d u e t o c u r re n t d e n s i t y in c re a se a ro u n d th e h o l e s o r k in k s . I f t h e m ic ro s t ru c tu re a ro u n d

these dam ages i s a ffec ted , the noise in tensi ty wi l l be h igh er and eq . (8) w i l l no longer be v a l id .

A c a t a s t ro p h ic i n c re a se i n n o i se o v e r m o re t h a n a f e w d e c a d e s fo r o n ly a f e w p e rc e n t

a d d i t io n a l r e s i s t an c e i s d u e t o m ic ro sc o p ic c h a n g e s i n t h e s t ru c tu re a ro u n d h o l e s a n d k in k s a n d

n o t d u e t o a s im p le in c re a se o f c u r re n t d e n s i t y a ro u n d a h o l e i n t h e c o n d u c t in g p a th . Ou r

m o d e l g iv e s t h e m in im u m in c re a se in n o i se t h a t c a n b e e x p e c t e d o n t h e g ro u n d s o f i n c re a se i n

curren t c rowding .

A nois e measurem ent se t -up for s amples wi th a resis tance la rger than 10 f~ i s s tandard

[3]. O ne must rea l ize tha t i f the impedan ce of the sam ple i s a t least 10 C~, bu t the n umb er o f

f lee ca rr ie rs in the samp le i s la rger than roughly 1013, major d i f f icu l t ies wi l l be encoun te red in

o b se rv in g t h e n o i se d u e t o c o n d u c t io n f lu c tu a ti o n s a b o v e th e o m n ip re se n t t h e rm a l n o i se a n d

backg round nois e of the ampl if ie rs. This i s due to the fac t tha t the re la t ive noise i s inverse ly

prop ort iona l to th e num ber of e ffec t ive ca rr ie rs [2] . I f in te rconnec t resis tances a re in se r ies

wi th r e l a t iv e la rg e se r i e s r es i s t an c e s , t h e l / f n o i se o f t h e d e g ra d in g p a r t c a n a g a in b e h id d e n b y

the thermal no ise o f the s e r ies resis tance . The m ethod is no lon ger appl icable in such cases .

R E F E R E N C E S

[1] L .K.J . Va n d a m m e , On th e Ca l c u l a ti o n o f l / f No i se o f Co n ta c t s , Ap p l i e d Ph y s i c s, 1 1 ,

89-96 (1976) .

[2] F .N. Ho o g e , T .G.M. Kle in p en n in g a n d L .K.J . Va n d a m m e , Ex p e r im e n ta l s t u d i e s o n i / f

no ise , Reports on Progress in Physics , 44 , 479-522 (1981) .



Resistance noise measurement 93

[3] L.K.J. Vandam me, Criteria o f low -noise thick-fihn resistors, Elec troc om pon ent Science

and Technolog y, 4 171-177 (1977 )

[4 ] L K . J . Vandamme, C omm ents on "An exac t fo rmula fo r the e ffect s o f re s is to r

geom etry on current noise" , IEE E Transact ions on Electron Devices , ED -33, 1833-

1834 (1986).

[5] J .L. Vo ssen, Screening of metal film defec ts by current noise measurements, A ppl.

Phys. Lett. 23, 287 (1971).

[6] W. Ya ng and Z. Celik-Butter, A model for e lectromigration and low frequen cy noise in

thin metal fiim~, Solid State Electron. 34, 91 1-91 6 (1991).

[7 ] L . K~ J .Vandamme, No ise as a diagnos tic tool for qual i ty and re l iabi li ty o f e lec tron

devices , IEE E Tran s Electr. Dev. 41, 2176 -2187 (1994).

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