Numerical modelling of stress-induced failure in sub-micron aluminium interconnects in VLSI systems

PERGAMON

Solid-State Electronics 43 (1999) 255±261

Numerical modelling of stress-induced failure in sub-micron aluminium interconnects in VLSI systems P.M. Igic *, P.A. Mawby University of Wales, Singleton Park, Swansea, SA2 8PP, U.K. Received 29 January 1998; received in revised form 29 April 1998

Abstract Stress-induced failure is investigated based on numerical modelling of the di€usional relaxation and groove growth in an aluminium line. A numerical model, based on surface and grain boundary di€usion, is improved and made to be very useful for time to failure estimation. # 1998 Elsevier Science Ltd. All rights reserved.

1. Introduction Stress migration is one of the principle mechanisms that cause failure of aluminium wire interconnects found in modern VLSI processes. The problem, which was ®rst identi®ed in 1984 by Klema et al. [1] and Curry et al. [2], is likely to become even more signi®cant as integration levels increase (width of the crosssectional area reduce below 1 mm and interconnects become longer). This failure mechanism is caused by the nucleation and growth of voids in the aluminium line under conditions of high stress at elevated temperatures. This process is not assisted by current ¯ow in the metal line as is the case with electromigration, where the electron ``wind'' provides a driving force which moves aluminium ions in the conducting line. The mechanical stress which drives stress migration is thermally induced during the line passivation, in which the metal lines are covered with a glass ®lm (borophosphosilicate or phosphosilicate), which is deposited at high temperature and subsequently cooled down to room temperature. Since the thermal expansion coecient of aluminium is much larger than that of the passivation ®lms (approximately 20 times larger), the cooling process generates large (of the order of hundreds MPa) thermal stresses. The existence of this stress is detrimental to the reliability of the aluminium

* Corresponding author.

lines as it enhances the failure mechanism, which leads ultimately to open-circuit line failure. Many studies have targeted this phenomenon, to which various methods have been applied to calculate the stress, model the failure mechanisms in the aluminium line and estimate the mean time to failure of the aluminium line. One approach adopted by several authors was to develop an analytical approach to understanding of the stress migration problem [3±5]. However these have some serious shortcomings as they use simpli®cations which are sometimes unrealistic [5]. To fully understand the failure mechanisms in the aluminium line numerical analysis must be employed. In this paper, we investigated stress-induced failure in sub-micron aluminium interconnections. It is well known that, under conditions of high temperature, the stress distribution drives the di€usion of atoms (or vacancies) along a grain boundary. For example, migration can result in the formation of a creep cavity (size less than a few microns), the size of that is large enough for a micron or sub-micron conductor to break down. Whereas the failure of larger components requires further multiple processes, such as initiation, propagation and coalescence of microcracks. It is obvious, that the di€erences in the failure mechanisms request new approach for the reliability assessment of the microdevices, since the conventional methods developed for large components do not work e€ectively for that purpose. In this paper, we model stress-induced migration failure of micron or sub-micron aluminium line in

0038-1101/98/$ - see front matter # 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 0 3 8 - 1 1 0 1 ( 9 8 ) 0 0 2 5 2 - 4

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terms of interaction between di€usion along the surface and grain boundary. We employ a numerical modelling strategy [6±8] for estimation of this phenomenon. In addition we improve the model so that it is now useful for time to failure predictions. 2. Mechanism of stress induced failure When aluminium is deposited on a surface followed by cooling, it forms as a polycrystalline structure, the size of the grains formed depends on growth conditions. Fig. 1(a) shows an ideal section of such a grain boundary±surface intersection, displayed as a right angle. This is however, not the true situation, as it is known that a groove will develop at the point between grains and the surface. The exact shape of the surface is determined by the balance of local thermodynamic forces (the two surfaces tensions and the grain boundary tension) requiring an equilibrium angle of b, as it is shown in Fig. 1(b). As well as the groove at the boundary, ridges are formed either side of the boundary. These ridges tend to ¯atten by surface di€usion, however this ¯attening process upsets the equilibrium angle and forces the groove to deepen. Here we are considering the situation prior to deposition of the passivation coating so that there is no residual stress along the wire, thus as a result there is no atomic ¯ux along the grain boundary. The groove developed in this way will initially grow very fast gradually slowing down to reach an equilibrium situation in a period of a few hours. This process is known as the thermal grooving phenomenon [9]. Thermal grooving alone, therefore, does not cause the failure in the aluminium line by itself, but it provides an early nucleus for grain boundary cracking when the wire is subjected to high stress levels. The stress perpendicular to the grain boundary activates the atomic ¯ow along the boundary which enhances

the grooving [10, 11], and ultimately results open-circuit failure.

3. Basic equations for thermal grooving due to surface di€usion and acceleration of groove growth due to stress induced grain boundary di€usion Since many studies have targeted these phenomena [8±13] and the model is well established in the literature, the main concepts are only brie¯y reviewed here. From di€usion theory it is known that di€erences in chemical potential give rise to a force which causes atoms to migrate from regions of high potential to regions of low potential. Thus the atomic ¯ux along a free surface is de®ned as Js ˆ ÿ

Ds ds @ m OkT @ s

…1†

where Js is the number of atoms per unit time crossing unit length on the surface, Ds and ds are the surface di€usion coecient and the thickness of the di€usion layer respectively, k is Boltzmann's constant, T the temperature, O the atomic volume, @/@s the derivative with respect to the arc surface length. The chemical potential (m), on the surface is determined by the potential of the bulk (m0), the curvature of surface (K), the free surface energy (gs), and the atomic volume as m ˆ m0 ÿ gs OK:

…2†

By considering the conservation of matter, one can derive from the surface ¯ux in Eq. (1) an expression for the normal velocity of the surface, vN: nN ˆ ÿ

Ds ds Ogs @ 2 K
2: @s kT

…3†

The equilibrium angle b at any instance in time is determined as:   gb b ˆ cosÿ1 …4† 2gs where gb represents the free grain boundary energy. The ¯ux along the grain boundary is determined in the same way as for the free surface, except that the surface coecients are replaced by those for the grain boundary, and is given by: Jb ˆ ÿ

Fig. 1. Sectional view of a grain boundary; a right angle is formed between the surface and the grain boundary (a), and the pro®le after thermal grooving (b).

Db db @ m OkT @ s

…5†

where Db and db are the grain boundary di€usion coef®cient and the thickness of the di€usion zone. The chemical potential is likewise given by: m ˆ m0 ÿ Os

…6†

P.M. Igic, P.A. Mawby / Solid-State Electronics 43 (1999) 255±261

where s is the stress. Grain boundary di€usion is driven by the stress gradient as: Db db O @ 2 s
2 ‡ d 0 ˆ 0: kT @s

…7†

Here d 0 represents the thickening at the grain boundary due to the addition of matter to the adjoining grains. Additionally, at the groove tip, to provide continuity of chemical potential and the ¯ux of atoms, the following equations must be satis®ed: s0 ˆ gs Ktip ,

…8†

ds0 2Ds ds gs @ K ˆ
: dx Db db @ s tip

…9†

4. Numerical simulation The structure modelled in the current work is the bamboo-like aluminium line. In this structure the width of the line is much thinner than a single grain, and thus the wire forms a sequence of grains de®ned by a periodic boundary which completely traverses the width of the wire (see Fig. 2). In the example here, a typical line width of 1 mm is employed (W = 0.5 mm), and a variable remote tensile stress is applied. The surface is discretised spatially (Fig. 2) by a set of N

257

equally spaced points (nodes) on a ®nite-di€erence mesh. The non-dimensional normal velocity at each point is calculated, and then the new nodal co-ordinates are determined. Thus, the surface geometry is described by the position of the nodes in the mesh [8]. This procedure progresses with time to map the evolution of the surface and hence the groove growth. Eventually, when groove tip reaches across the line, open circuit failure will occur. Numerical methods for solving the problem of simulation of stress-induced failure in aluminium lines, based on surface and grain boundary di€usion (outlined above) are well established in literature [7, 8]. If the half width of the aluminium line is assumed to be W, then the equations described above can be normalised with W, kTW 4/DsdsOgs and gs/W as normalisation factors for the length, time and stress dimensions. Application of these factors to the system of equations above yields the following set of scaled equations: n N ˆ

@ 2 K @ s2

cos b ˆ

gb 2gs

s 1  2 @ K  
@ s ÿ …1 ÿ a†F Ktip ˆ 1 ÿ a 3 tip

…10† …11†

…12†

Fig. 2. Schematic representation of the aluminium line and surface discretisation. Due to the symmetry, only one quadrant of the conductor surface is modelled.

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and

s 1 ˆ

s1 W gs

…16†

n N ˆ

nN kTW 3 Ds ds gs O

…17†

dition is reached after only 2±3 h and is due to surface di€usion [9]. This is then taken as the initial condition to which the numerical methods are applied, since it is not critical as the time to failure is expected to be much longer than this (a few days at least). In order to compute the mean-time-to-failure (MTTF) as a function of ambient temperature a temperature dependent di€usion constant is required. The surface di€usion and grain boundary di€usion coecients are estimated by [4]:   22 ÿ5 …19† Ds …T † ˆ 10 exp ÿ 0:001987T

…18†

 Db …T † ˆ 10ÿ5 exp ÿ

s ˆ s=W

…13†

K ˆ KW

…14†

a ˆ a=W

…15†

Ds ds gs Ot t ˆ : kTW 4

Here, F = Dbdb/Dsds is the so-called di€usion ratio and s1 is the remote stress. Initially the surface geometry is obtained by Mullins' theory [9], which describes the surface pro®le due to thermal grooving. Applying this theory gives a normalised value of the initial groove depth of 0.58 (corresponding approximately to 0.3 mm), this con-

 11 : 0:0019877T

…20†

The di€usion ratio depends not only on the material and the temperature, but also on the microstructure of grain boundary and surface. Thus, the di€usion ratio (F) becomes temperature dependent, and takes a value between 0.1 and 104 [7]. Also, the fact that the residual thermal stress is reduced as temperature increases is taken into account.

Fig. 3. Groove pro®les after thermal grooving (initial pro®le): solid circle, and ®nal pro®le (after grain boundary di€usion): open circle (gs = 1.5 J/m2, W = 0.5 mm, Dbdb = 2.64  10 ÿ 23 m3/s, O = 1.66  10 ÿ 29 m3, k = 1.38  10 ÿ 23 J/K, T = 473 K, F = 100, s = 50, gb/2gs = 0.16).

P.M. Igic, P.A. Mawby / Solid-State Electronics 43 (1999) 255±261

259

Fig. 4. Time to failure vs normalised stress.

The stress is given by:   TP ÿ T s1 …T † ˆ s1 …473†exp kT ln TP ÿ 473

5. Results and discussion …21†

where, s1(473) is the stress value at 473 K, kT is a ®tting parameter (since the stress is not a linear function of temperature), and TP is the deposition temperature of passivation layer (TP = 673 K).

The modelled groove growth is shown in Fig. 3, which shows the initial (solid circles) surface pro®les and the ®nal pro®le (open circles) at the point where a groove grows in the aluminium line. The ®nal pro®le here corresponds to the situation when the half width of the line is reached by the groove tip (corresponding

Fig. 5. Normalised groove depth vs normalised time (for three di€erent stress values: 40, 50 and 60).

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Fig. 6. Time to failure vs temperature. The same values are used as in Fig. 3. The values of F(T) have a certain amount of ambiguity, but the actual values do not appreciably alter the present numerical analysis.

to open circuit failure), and is when the grain boundary di€usion becomes dominant. The time to failure versus normalised stress characteristic is shown in Fig. 4. It is obvious that time to failure decreases as stress increases, as observed in experimental data. The next ®gure (Fig. 5) shows groove depth against time, for the three di€erent normalised stress values: 40, 50 and 60. The same depth will be reached in a shorter time if the applied stress has a higher value. The time to failure versus temperature is shown in Fig. 6, with the numerical values of the constants used in this work being given on the appropriate diagrams.

It can be seen that the time to failure decreases as the temperature increases, as a natural result of easier diffusion. Comparing these numerical results for time to failure vs temperature with the analytical ones [4], one can see that the both characteristics have similar shapes, but the analytical model estimates time to failure tend be about few seconds (an unrealistic and very short time in comparison with experimental results, where time to failure is typically about hundred hours). This anomaly in the analytical model, occurs because it assumes that grain boundary di€usion process is much faster than surface di€usion, under stress

Fig. 7. Grain boundary (Db) and surface (Ds) di€usion coecients vs temperature.

P.M. Igic, P.A. Mawby / Solid-State Electronics 43 (1999) 255±261

261

Fig. 8. T/Ds and T/Db functions vs temperature; tFAILURE1T/Db (analytical modelling [4]), tFAILURE1T/Ds (numerical modelling).

applied conditions, and therefore controls the time to failure (see Figs. 7 and 8). However, the numerical model predicts that two processes exist simultaneously, and the slower one controls time to failure. The two processes are coupled at the groove tip by conditions of continuity of chemical potential and continuity of atomic ¯ux. The controlling process depends on the relative magnitudes of the surface and grain boundary di€usion coecients, Ds and Db. Since the two processes operate in series, the slower one is dominant. Thus, when Db < Ds, the grain boundary process controls the rate, and when Ds < Db (our case), the surface process is rate controlling.

here, that modelled failure mechanism can be signi®cantly diminished, and even stopped, introducing barrier metal e€ect [14], the fact that the Ti, TiN or TiW layers which are present under Al line can stop groove growth and extremely improve the stress-migration reliability of aluminium interconnections.

6. Conclusions

[1] Klema J, Pyle R, Domangue E. Proc 22nd Rel Phys Symp IEEE, Las Vegas, 1984. p. 1. [2] Curry J, Fitzgibbon G, Guan Y, Muollo R, Nelson G, Thomas A. Proc. 22nd Rel Phys Symp IEEE, Las Vegas, 1984 p. 6. [3] Niwa H, Yagi H, Tsuchikawa H, Kato M. J Appl Phys 1990;68:328. [4] Kato M, Niwa H, Yagi H, Tsuchikawa H. J Appl Phys 1990;68:334. [5] Jones RE Jr, Proc. 25th Rel Phys Symp IEEE, San Diego, 1987 p. 9. [6] Bross P, Exner HE. Acta Metall 1979;27:1013. [7] Martinez L, Nix WD. Metall Trans A 1982;13A:427. [8] Kitamura T, Ohtani R, Yamanaka T. JSME Int J A 1993;36:146. [9] Mullins WW. J Appl Phys 1957;28:333. [10] Hull D, Rimmer DE. Philos Mag 1959;4:151. [11] Chuang T et al. Acta Metall 1979;27:265. [12] Flinn PA et al. MRS Bull 1993;18:75. [13] Kordic S, Collart EJH. MRS Spring Meeting, San Francisco, 1997. [14] Wada T. IEICE Trans Fundament 1994;E77-A:180.

A numerical model for di€usion relaxation of thermal stress in aluminium lines, based on surface and grain boundary di€usion at elevated temperatures is improved. The results shown in this paper give, for the ®rst time, the correct qualitative agreement with experimental observations. We have also presented a method based on the theory of Mullins [9] for a more physically correct initial condition for the surface geometry due to thermal grooving. In addition the behaviour is more realistic than that predicted by analytic calculations. An important element, in obtaining the correct qualitative failure behaviour with temperature, is the use of the correct temperature dependence of the surface and grain boundary di€usion coecients. The work presented here is principally aimed at aluminium wires that are sub-micron in cross-sectional dimension, which are important in understanding VLSI processing as widths decrease with evolving technology. Note

Acknowledgements This work is ®nanced by LG Semicon UK Ltd. References