CURRENT AND RESISTIVITY DEPENDENCE OF ELECTROMIGRATION FROM A STATISTICAL

ANALYSIS OF METALLIZATION FAILURE DATA

S. J. CHUA

Department of Electrical Engineering, University of Singapore, Kent Ridge Campus, Singapore 0511

(Received 28 April 1980)

Abstract-The time-to-failure among indentically produced linear metallization tracks deposited on a single wafer has been found to vary when they are subjected to current stress at a temperature of 270°C. This variation is accounted for by the statistical variation in resistivity and cross-sectional area and hence current density among the indentically produced tracks. The variation in cross-sectional area is principally caused by local over or under etching amd mask misalignment during exposure. Resistivity variation may be due to the random introduction of defects and vacancies during deposition. Statistical distributions are formulated for the variations in resistance and time-to-failure and these fit the experimental data well. From such distributions, the exponent of the current dependence has been found to be 2 for plain tracks and 2.14 for stepped tracks and for both cases linearly dependent on the resistivity.

INTJtODUCTlON

The metallization tracks used for interconnection in in- tegrated circuits are getting narrower to be compatible with the growing complexity and miniaturization of the circuits. Decrease in width of the tracks result in a reduction in cross-sectional area and a consequently higher current density. Generally at a current density above IO3 A/cm* a phenomenon known as electromigration [ 1,2] sets in whereby the atoms making up the metallization begins to move due to the presence of the electric field and by momentum exchange with the charge carriers[3]. Electromigration is generally charac- terised by the growth of hillocks on the metallization surface [4], whiskers approximately 1 pm in diameter from both the surface and sides and depletion of the metal from certain areas of the track[S]. This is undesir- able for integrated circuits for two reasons. The growth of whiskers may result in the shorting of adjacent metal- lization tracks while depletion of the metal results in an open circuit.

The extent of electromigration depends on a number of factors among which are the grain size[6], current density[7], alloy additions in the metal[8] and tem- perature and interfacial stress between the metal and the substrate[9]. In simple test tracks, it is found that the time-to-failure or the time taken for the tracks to become open circuit when they are current stressed at a high temperature is described by the Arrhenius model expressed in the formula given below:

where c is a constant and is a number varying between 1 and 4[2,6,7]. There is as yet no consensus on the value of m, largely because it is dependent on factors which have not been completely characterised. E is the activa- tion energy for electromigration and its value depends on the metallization system.

DEWRIPllON OF JXPERMENT

It has been found that identically produced aluminium metallization tracks from the same wafer fail at different times under current stress at a high temperature. This has also been observed by others[6]. To find out the cause for the spread in the time-to-failure experimental devices as shown in Fig. 1 were fabricated. These consist of two linear Al metallization tracks per chip of dimen- sion 0.8 p x 5 p x 500 p laid on a silicon oxide layer by electron beam deposition. The Al used was of six 9 purity and the vacuum achieved in the deposition unit prior to deposition was better than lo-‘torr. The oxide layer is 0.5 p thick except for a well 100 k x 200 ~1 where it is 0.2~ thick. One Al track is laid over this well to simulate the nonplanar tracks in integrated circuits. The other track is laid over an even thickness oxide layer. To distinguish between the two tracks, the former is refer- red to as a stepped track and the latter a plain track. At the ends of the tracks are two bonding pads each carry- ing two leads. A pair of leads, one from each end of the track is used for current flow and the remaining pair for monitoring the voltage drop across the track. Each chip is mounted on a header encapsulated in a dry nitrogen atmosphere with a humidity of less than 1OOppm.

In each experiment 100 such mounted chips were current stressed at 25 mA (a nominal current density of 6.25 X lo’ A/cm*) at a temperature of 270°C. A Solatron Data Logger periodically monitors the voltage drop across the 200 metallization tracks and these values are punched on paper tapes for analysis. The voltage drops across the tracks increase with time and generally many of the tracks fail catastrophically as observed by the abrupt increase in the voltage.

RESULTS AND ANALYSIS

Metallization tracks from one sample S12 (all from a single wafer) where the Al was deposited at a substrate temperature of 190°C and annealed at 470°C for 15 min were current stressed as described above. Before the

173

174 S. J. CHUA

Fig. 1. Diagram showing the Al metallization tracks on one chip. The upper is the stepped track and the lower a plain track.

experiment, the voltage drops across all the tracks were measured with the current through the tracks all main- tained at 25 mA. The variation in voltages can therefore be attributed to the variation in the factor p/A where p is the resistivity and A the cross-sectional area of the Al tracks. Variation in cross-sectional area may be due to uneven deposition of the metal over the wafer, local over or under etching and also misalignment of the mask during exposure. Resistivity on the other hand may vary due to random defects and vacancies formed during deposition. If the cross-sectional area A and the resis- tivity p are each assumed to be uniformly distributed within the limits A,, A2 and pl, pz respectively, then as shown in Appendix A, the cumulative distribution func- tion of Z = p/A is given by

F(z) = f 1 p(2) dz’ =

1 2(A~-A,NprpJX

A22z+T-2A2p,; FSz+ 2

2Az(p*-p,)-pz2-plz; -&z+ Z A2 !

2A~(p~-p,)+2p,A,-A,2z-~: gSz,z (1)

z is related to the initial voltage by the factor LI where L is the length of the Al track and I the current through the track. Since this factor is inconsequential to the calculation of cumulative distribution function, z is treated to be equal to the initial voltage drops. By plotting IF(Z) vs z, the straight line portion has a gradient equal to [l/l - (AI/A2)] as shown by eqn (Al 1). Figure 2 shows such a plot with a gradient 2.82 giving Al/At=0.645. This value can be further ascertained from the sample mean value of the initial voltage f by using eqn (A12) of Appendix A. With f= 0.187V and the y intercept of the straight line in Fig. 2 being -0.429, the value of Al/AZ is deduced to be 0.646. This agrees well with the value obtained from the gradient and confirms the consistency in the choice of the best straight line through the data point of Fig. 2. Thus the ratio of the extreme cross-sectional area of the Al tracks in this sample is 0.645.

The value of p,/pz is determined as described in Ap- pendix A and found to be 0.804. Figure 3 shows the fit of the theoretical distribution function F(z) from eqn I to the data points and is seen to be reasonably accurate

ilO -2 ,

I

24 t 22

20 1 18

16

14 1

;

N

12 1 10

B-

6-

4-

2r

I

.i .

s

:i- 1

.

;

Y a z C

i .

i x

_I 0’ .J ” ” ’ 1 ”

0.16 0.!8 0.20 0.22 0.24 0.26

VOLTAGE DROPUI

Fig. 2. Plot of zF(z) vs I where I is the initial voltage drops across the Al tracks at a current of 25 mA (nominal current density is 6.25 x IOI A/cm’). F(z) is the cumulative distribution of the voltage drops. The least-square linear fit (-) is drawn through the central portion. Coefficient of correlation is 0.994.

Gradient is 2.821 and y axis intercept is -0.429.

with p,lA,=O.l35V, pzlAz=0.168V, pllA,=0.210V and p2/Az = 0.261 V. Also shown dotted (for comparison) is the best Normal distribution fit for the data points obtained from Fig. 4. In Fig. 4, the cumulative dis- tribution of initiai voltage drops of the Al tracks are plotted on Normal probability paper and the least square linear fit is obtained. From a comparison of the coefficients of correlation of the least-square linear fit of Figs. 2 and 4, it is seen that the variation of voltage drops across the metallization tracks for the same current can be attributed to uniform distributions of the cross-sec- tional area and resistivity of identically produced sam- ples.

Study of electromigration from failure data

270°C may be simply written as

175

0.18 0.20 0.22 0.24 0.26

VOLTAGE DROP 121

Fig. 3. Theoretical cumulative function of eqns (1X-J fitted to the data points (x). The equations are (i) for 0.135Sz~0.168, 42.80062 +(0.7827/z)- 11.5664 (ii) for 0.1685-z 80.210, 2.8130- (0.4264/z) and (iii) for 0.210~2 ~0.261, 10.2674-(1.2080~. The Normal distribution fit (--_) to the data points is obtained from

Fig. 4.

Fig 4. The straight line shows the least-square linear fit to the cumulative distribution of initial voltages plotted on cumulative

Normal probability paper. Coefficient of correlation is 0.972.

When the same current of 25 mA is passed through the metallization tracks, the distribution in the current den- sity therefore assumes the same distribution as the vari- ation of the cross sectional area with the factor jJj2 = A ,/&. The variation in the activation energy among the tracks is considered small and its effect on the time-to- failure is reduced to a second order variation since the activation energy appears as an exponent. Thus the time-to-failure when the Al tracks are current stressed at

where K, n and m are constants, j the current density and p the resistivity.

It is shown in Appendix B that if j and p are uniformly distributed within the limit j,, jz and pl, p2 respectively, the probability function for the time-of-failure is given

by

p(r)=-L m+n

where

,,=K$; t2=Kg; f3=Kg Is II

and t4 = Kg 12

and tl c f2 < fS < l4 . t, and f, are the extreme time-to- failure of the Al tracks.

As shown in Appendix B, by plotting ln{(j2/j2- jl)- G(t)} vs In t, the central portion of the graph between f2 and tX can be fitted with, a straight line of gradient -(l/m). G(t) is the cumulative distribution of the time- to-failure. Figure 5 shows the plot of In {(j2/jz- j,) - G(r)} vs In r for the stepped (0) and the plain (X) tracks. The least-square linear fit are also drawn. For the step- ped tracks the slope is -0.467 giving a value m = 2.14 with a coefficient of correlation 0.99. For the plain tracks the slope is 0.498 giving a value m = 2.01 with a coefficient of correlation 0.97. The results show that the current exponent factor is slightly higher for the stepped than the plain tracks.

The ratio (f3/t2) = (pl/p2)n(j2/jdm and the values of t2 and t3 are obtained from Fig. 5 where the data points deviate from the linear fit in the upper and lower ends. For the plain track t3/t2 = 2.117 while for the stepped track f3/f2 = 1.86.

With the values pJp2 = 0.804, At/A2 = 0.645, m = 2.01 for plain tracks and m = 2.14 for stepped tracks as found earlier the values of n are calculated to be n = 1.023 for plain tracks and n = 1.091 for stepped tracks. It is there- fore seen that the time-to-failure of the metallization track is linearly dependent on resistivity.

176 S. J. CHUA

Fig. 5. Plot of In {(jz/jz- jr)- G(t)} vs In r where G(t) is the cumulative distribution of the time-of-failure t for plain (x) and stepped (0) tracks. (it/h) = (Al/A3 = 0.645. The lines show the least-square linear fit to the central portion of the data points

between t2 and t,.

CONCLUSIONS

The process steps involved in the making of metal- lization tracks for interconnection in integrated circuits introduce statistical variation in their cross-sectional area and resistivity even for tracks made on the same wafer. Because of the nature of this variation, the results are therefore treated statistically. It has been shown that their variations fit well to uniform distributions. The exponent in the current dependence of the time-to-failure has been found to be 2.01 for plain tracks and a slightly higher value of 2.14 for stepped tracks. On the other hand, the time-to-failure is linearly dependent on resis- tivity.

Acknowledgemenfs-The author would like to thank Dr. J. Froom and Mr. T. C. Denton for the experiments carried out at Standard Telecommunications laboratories, Harlow, England.

RgFUtENCJ?S I. J. D. Venables and R. G. Lye, 10th Annual Proc. Reliability

2. 3.

4.

5. 6.

7. 8. 9.

Phys., IEEE Cat. No. 72CHO628-8 PHY pp. 159-164 (1972). J. R. Black, IEEE Trans. Electron. Deu. 16, p. 338 (1%9). A. K. Das and R. Peierls, I. Phys. C: Solid-St. Phys. 8, p. 3348 (1975). L. J. Gauckier, S. Hofmann and F. Haessner, Actn Metall. 23, p. 1541 (1975). F. M. D’Heurle, Proc. IEEE 59, p. 4107 (1971). M. Saito and S. Hirota, Rev. Elect. Comm. Lab. 22, p. 678 (1974). G. L. Hofmann, IEEE Trans. Electron Deu. 10, p. 883 (1970). A. J. Learn, I. Electron Mat/s 3, pp. 531-552 (1974). A. K. Sinha and T. T. Sheng, Thin Solid Films 48, p. 117 (1978).

APPENDIX A

Theoretical model for the distribution of resistance in linear metaliization tracks

The two important parameters that affect the resistance of metallization tracks of fixed lengths are the resistivity (p) and the cross-sectional area (A). We assume auniform distribution function for the cross-sectional area and also for the resistivity. Their respective distribution functions are therefore given by

f(A) = -!-. AZ-A,’

A,<.A<Az

g@)=J-. P2-PIv

Pl<P<PZ.

Define L = p/A and the problem of finding the distribution function for the resistance is reduced to finding the distribution function for z. Depending on the relative magnitudes of p~/p, and A*/A , , the following inequalities apply:

and

~<&<~<&. tz<_ .42

AZ A2 A, A,’ PI AI (Al)

&<&<pr<&. A2 AI A2 A,’

f>S. 642)

It is envisaged that inequality (Al) applies to our samples where the variation in cross-sectional area is larger than the variation in resistivity. The distribution function of I for this case is derived below.

Define an auxiliary function w = p and since p and A are independent, the distribution function of z is given by

p(r) = /_=/A)s@)J($ dw.

Where J(p, A/z, w) is the Jacobian of the transformation from (p, A) to (z, w). Thus

p(z)= 5 I z (p~-pt~tq~-AJdW’

The limits for the integration are found from the convolutions of the separate distribution functions of p and A. Thus

i.e.

1 P(Z)=2(Az-A,ti-p~)X

(A3)

(A4)

(As)

Study of electromigration from failure data 177

The cumulative distribution function is given by

I

S F(z) =

1 -~p(z’)dz’=2(A~-A,)(p~-P,)X

I

A12z+&A2P,; &z~: z A2

ZA&2-P,)-P22; ZS~S~ 2

~A&Q-P,)+~P,A,-A,~~-$ ;,z,$

The mean value of z is given by

0)

&== f_

(A7)

(A@

(A9)

In the range (PJAz)6z s (PI/AI) (intermediate range of resis- tance)

By plotting zF(z) vs z we obtain a straight line of the form y=Bztc where

B-1 P2+Pl

l-2 and ‘=-2(Az-A,)’ (All)

Itisseenthat5=-cln2. (Al21

When the cumulative distribution function F(z) is plotted against l/z the following graphical form is obtained. The central portion of the graph is linear.

100%

G h

‘A!!

K, = P(z) dz = - 2(4)

(A131

p2 1 K2= p(z) dz = z.

2 2-l ( )

(A141

1 1K”” ‘(O= (j2-jl)(fi-pl)m I

$$. G’S)

--------

L Kl - --

;I I 1

Q2 QI Q2 QI

The ratio p2/pl is obtained from eqns (Al3) and (Al4) to give reasonable fit to the values of K, and Kz. By the choice of K, and K2, (p,/A,), and (P2/A2) are determined. The end values (p2IA I) and (pI/A2) are determined from these values and from the ratio of p and A.

APPENDIXB Theoretical model for the distribu:ion of time-to-failure due to resistioity and current density variation

When the same current is passed through a batch of metal- lization tracks with varying cross-sectional areas, they are in fact subjected to varying current densities. As the metallization tracks are deposited in a single run on one wafer it is envisaged that the activation energy for electromigration is the same for all the tracks. Also, if there is a slight variation the effect on the time-to-failure is likely to be small as it appears in the exponent. Thus the time to failure when the Al tracks are current-stressed at a high temperature is given by

t,KP’ .m J

where K, m and n are constants, p is the resistivity and j the current density. p and j are uniformly distributed and their distribution function are given by

fti,=&; jl<j>j2

&)=L. P?_PI’

PI' PC&.

For the case where

(B’)

W)

k)‘<(j)

we have the following inequalities

C<E?Z<<<~. jzm j2m JI II

Define an auxiliary function s = p. The Jacobian

(B3)

J h _ lKllrnSnirn 1 ( > t,s m t”m+l

The probability distribution function of I is given by

The limits of the integration are found from the convolution of the separate distribution functions as shown below:

(B6)

037)

(BE)

here

Fig. Al.

178

and,

I, and t, are the extreme time-to-failure. On simplication

p(t)=-L m+n

(,_b)‘d”_l)[(g-(jG)]; t1ststz

j2 I

t2 4 t s ta

S. J. CHUA

The cumulative distribution function is given by

t, s t c t2 (B12)

(B9)

@lo) t, s t =s t4 (B14)

(B11) From eqn (B13). by plotting In{(j2/j2-j,)-G(t)} vs In t, the gradient of the central portion of the graph is linear and has a slope - I/m.