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Surface Science Letters

Current-induced non-linear dynamics of voids in metallic thin films: morphological transition and surface wave propagation M.Rauf Gungor, Dimitrios Maroudas * Department of Chemical Engineering, University of California, Santa Barbara, CA 93106-5080, USA Received 20 March 2000; accepted for publication 10 May 2000

Abstract A theoretical non-linear analysis based on self-consistent numerical simulations is presented of current-induced morphological evolution of void surfaces in metallic thin films. The analysis focuses on cases of low symmetry of surface diffusional anisotropy. Our simulations predict a surface morphological transition and the onset of oscillatory dynamics at a critical strength of the applied electric field. Voids migrate along the film at constant speed with surface morphologies that are either steady or time-periodic, characterized by wave propagation along the surface in the direction of the electric field, for electric fields weaker or stronger than critical, respectively. Both of these types of void surface morphology are stable and do not lead to failure of the metallic thin film. © 2000 Elsevier Science B.V. All rights reserved. Keywords: Aluminum; Models of non-equilibrium phenomena; Models of non-linear phenomena; Surface diffusion; Surface electrical transport (surface conductivity, surface recombination etc.); Surface structure, morphology, roughness, and topography; Surface waves

Surface electromigration has been generating tremendous research interest both as a fundamental atomic transport mechanism of surface morphological evolution [1–3] and as a mechanism of interconnect failure in integrated circuits [4]. For example, current-driven morphological evolution of transgranular void surfaces has been identified as a major source of failure in polycrystalline Al thin-film interconnects with bamboo grain structure [4,5]. Current-induced non-linear phenomena on metal surfaces and surfaces of voids in metallic films have attracted much theoretical attention, and motivated a substantial body of recent detailed * Corresponding author. Fax: +1 805 893 4731. E-mail address: [email protected] (D. Maroudas)

numerical simulation studies on a mesoscopic scale [6–17]. Formation of soliton-like features and void-mediated failure mechanisms are among the most intriguing of these surface dynamical phenomena. Schimschak and Krug reported simulations of surface electromigration-induced solitonlike features which followed curved trajectories indicating a decrease in drift velocity with increasing wavelength [6 ]. More recently, Bradley presented an asymptotic analysis that demonstrated current-induced soliton propagation on a metal surface in the direction of the electric field with velocity that decreases linearly with amplitude; the analysis assumed isotropic surface diffusion and was carried out in the limit of very strong electric fields [18]. We have demonstrated a very rich

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current-induced non-linear dynamical behavior of transgranular void surfaces in metallic thin films, including void faceting [9,11,13], steady void migration [9–11,13] with speed inversely proportional to the void size [11], facet selection and wedge shape formation [9,13], propagation of slitlike features from the void surface leading to film failure [9–11,13], as well as stress-induced formation of crack-like features at void surfaces [12,14]. Furthermore, in agreement with experimental observations [19], we have shown that soliton-like features may propagate on surfaces of large-size voids prior to the failure of the metallic thin film [10,13]. The purpose of this paper is to investigate the possibility of current-induced propagation of stable surface waves on void surfaces in metallic thin films, i.e., the generation of surface waves that can propagate due to surface electromigration without causing failure of the metallic film. For a finite-size void at the film’s edge, formation of such a surface wave pattern would result in stable temporal oscillations of the void’s surface. Our theoretical analysis predicts that such surface electromigration-driven wave propagation is indeed possible through a morphological transition of the void surface at a critical electric field strength. This transition is associated with the onset of oscillatory dynamics of the void surface, i.e., a time-periodic solution for the void surface morphological evolution; the corresponding frame of reference is one that translates at constant velocity equal to the void migration velocity. Our theoretical analysis was based on the continuum formalism of surface mass transport under the action of an electric field, which has been presented in detail in Refs. [10] and [13]. In summary, the total surface atomic flux is expressed through a Nernst–Einstein equation including contributions from curvature-driven surface diffusion and electromigration. Mass conservation gives the evolution of the void displacement normal to its surface as being proportional to the surface divergence of the total surface flux according to the continuity equation. The analysis is two-dimensional in the xy-plane of the metallic film of width w extending infinitely along x, where the void extends throughout the film thickness consistently

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with experimental observations. The morphological evolution of the void surface is coupled selfconsistently with the distribution of the electrostatic potential in the metallic film, governed by Laplace’s equation; the electric field is given by the gradient of the potential and the surface electromigration force at every surface location is proportional to the local electric field component tangent to the surface. Dimensional analysis of the evolution equations gives two dimensionless groups that govern void dynamics: the surface electrotransport number, C, which scales the electric field with capillarity and the dimensionless void size, L, which controls current crowding around the void surface. E q1w2 C¬ 2 s (cV) and L¬w /w, where E is the strength of the t 2 applied electric field, E =E xˆ, q1 is the surface 2 2 s effective charge [1–4], c is the surface free energy per unit area, V is the atomic volume, and w is t the initial extent of the void across the film; under typical electromigration testing conditions for an Al film of width w#1 mm, C≥50. An important element of our surface transport model is that surface adatom diffusivity, D , is taken to be s anisotropic and depend on the local void surface orientation; this is expressed by D (h)= s D [1+f(h)], where D is the minimum surs,min s,min face diffusivity, f(h)≥0 is an anisotropy function, and h is the angle formed by the tangent to the void surface and E . Specifically, we have adopted 2 the simple form f (h)=A cos2[m(h+w)], where A, m, and w are dimensionless parameters that determine the strength of the diffusional anisotropy, the grain symmetry, and the misorientation of a symmetry direction of fast surface diffusion with respect to E , respectively. On the other hand, c 2 is assumed to be isotropic as justified by the computed c(h) dependence in fcc metals, which is much weaker than that of D (h) [20]. The resulting s time scale is t¬

k Tw4 B , (D d cV) s,min s

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where k is Boltzmann’s constant, T is temperB ature, and d /V is the number of surface atoms s per unit area; for the material and conditions mentioned above, t is on the order of 104 h [10,13]. In our numerical simulations, the electrostatic potential was computed using a Galerkin boundary element method (BEM ) with linear trial functions to solve Laplace’s equation in the two-dimensional domain [10,13]. The electric field component tangent to the surface, E , was then computed using s the method of surface derivative calculation developed in Ref. [21]. Our BEM discretization was adaptive and employed several hundred nodes along the void surface. The surface flux divergence was computed through a centered finite-difference scheme and the continuity equation was integrated using an explicit Adams–Bashforth algorithm with variable time step size. Additional details of the computational methods employed in our simulations have been described in Ref. [13]. Consistently with the isotropic c assumption, the initial void shape in the two-dimensional domain was a circular arc. The void surface was taken to intersect the film’s edge at right angles. The effects of such a condition on the simulated surface dynamics are discussed below. The condition is very satisfactory for passivated films: in such films, the ‘edge’ of the film corresponds to a metal/dielectric interface, along which the mobility of metal atoms can be considered negligible compared to that on the free metal surface; experimental observations suggest that this condition also is satisfactory for unpassivated metallic films (see, e.g. Ref. [22]). As in Ref. [10], we followed a systematic parametric study for non-linear analysis of the current-induced morphological evolution of the void surface at constant void volume. Specifically, a parametric search was carried out of the fivedimensional parameter hyperspace defined by the five dimensionless parameters C, L, A, m, and w. Here, we focus on low, two-fold grain symmetry, m=1, for which we had observed current-driven wave propagation on void surfaces prior to failure in our previous studies [10,13]. We carried out a detailed parametric study in the neighborhood of parameter space, where surface wave propagation was observed at m=1: A=10 and w=p/(2m)=90° [10,13]. To prevent locally intensive current crowd-

ing on surfaces of large voids that can lead to void tip extension and failure [13], the size of the voids that we examined was limited to L<0.70. Here, we report results of a one-dimensional parametric study in C for a given void size, L=0.60. This study captures all the qualitatively interesting phenomena and, more importantly, the surface morphological transition under consideration. A more detailed parametric search and morphological stability analysis will be presented in a forthcoming publication [23]. Two qualitatively different types of void surface morphological response are shown in Fig. 1, as the strength of the applied electric field increases. Specifically, Fig. 1 demonstrates the currentinduced evolution of the void surface area, S(t), for two different values of C; S is the surface area 0 of the initial void configuration with semi-circular cross section on the xy-plane. The dashed curve of Fig. 1 corresponds to C=65: under such an applied electric field strength, the void surface morphology reaches a steady state characterized by a surface area that does not change over time after an initial transient; the duration of this transient period depends on the initial void morphology. The solid curve of Fig. 1 corresponds to C=85. This increase in C results in a time-periodic asymptotic state for the morphology of the void surface after the initial transient: instead of reaching a steady value, the void surface area fluctuates about a constant value; this fluctuation is characterized by a constant amplitude and period. The steady value of S and the oscillation characteristics of S(t) in the former and latter case, respectively, are determined completely by the values of the five parameters: C, L, m, A, and w. Most importantly, the results of Fig. 1 indicate that for given void size, L, there is a critical value of electric field that corresponds to a surface morphological transition from a steady to an oscillating void shape characterized by the propagation of stable surface waves. The corresponding evolution of the void surface morphology under the action of the two different electric field strengths, C=65 and C=85, is shown in Figs. 2a and b, respectively. In the former case, Fig. 2a, the initial semi-circular void morphology in the xy-plane of the metallic film evolves to a stable shape that does not change over time, while

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Fig. 1. Evolution of void surface area, S(t), from an initial configuration of area S with semi-circular cross section on the plane of 0 the film. The corresponding parameters are L=0.60, m=1, A=10, and w=90°. C=65 and 85 for the dashed and solid curve, respectively, indicating a surface morphological transition for 65

the void migrates along the film at constant speed. Note that this stable shape is not faceted, as a result of the low grain symmetry, m=1; at higher grain symmetries, such as six-fold (m=3), stable migrating voids are faceted [9,13]. Also, note the analogy between a solitary wave and the stable void of Fig. 2a migrating at constant speed. In the latter case, Fig. 2b, the void shape evolution is monitored at the time-periodic state over one period of S(t) that corresponds to the time interval between the two dotted vertical lines of Fig. 1. Specifically, seven configurations corresponding to equidistant points in time are shown over the period 1.350×10−3t≤t≤1.450×10−3t. The first of these configurations resembles the stable void shape of Fig. 2a, with a slightly sharper void-tip feature. This feature travels toward the cathode (right) end of the void, while the formation of a second tip is initiated to the left of the initial one. This part of the sequence can be viewed as a void tip splitting mechanism that reduces the void surface area; alternatively, such tip splitting can be considered as formation of a protrusion on the void surface. However, surface electromigration continues to drive the first void-tip feature to the cathode end with simultaneous decrease in its

amplitude, which results in its disappearance as it approaches the cathode end. At the same time, the second feature becomes extended and amplified; the seventh configuration is identical to the first one and the dynamical sequence keeps repeating itself periodically. Note that, contrary to the cases reported in Refs. [10] and [13], the second tip feature does not extend enough for current crowding to cause film failure over the region of parameter space examined here. Finally, it should be mentioned that in the asymptotic state where the void surface fluctuates in a time-periodic fashion, the void also migrates along the film at constant speed. We have found that the void migration speed provides a practical means of demonstrating the surface morphological transition; note that such a speed is analogous to the propagation speed of a solitary wave. The constant speed of migration, v , of a stable cylindrical void in an anisotropic m metallic film can be expressed as [11]: d q1E 1 D s,min s s 2 b(h(s); s). v = m l(L) k T B

(1)

In Eq. (1), l is the circumference of the void in

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Fig. 2. Evolution of initially semi-circular voids in a metallic film under an electric field directed from left to right. The corresponding parameters are L=0.60, m=1, A=10, and w=90°; C=65 and 85 in (a) and (b), respectively. In (a), the corresponding times for the different morphologies from left to right are: t=0, 0.280, 0.575, 0.898, 1.227, and 1.556×10−3t. In (b), seven configurations are shown corresponding to uniformly distributed times over one period of the time-periodic state indicated by the two dotted vertical lines in Fig. 1.

the plane of the film, i.e., S=lh for film thickness h, and b is a function that depends on current crowding around the void surface and surface diffusional anisotropy, f (h); s is the arc-length coordinate along the void surface in the plane of the film. Constant migration speed implies that b is constant; this condition provides the relationship between current distribution around the stable void surface and surface diffusional anisotropy for steady migration. Thus, in Eq. (1), b can be considered essentially as a shape constant. Therefore, different types of void morphologies would correspond to different shape constants, i.e., to different values of b. Considering that C3E , 2 by definition of C, and l3L, with an O(1) geometrical proportionality factor, implies that v 3(C/L)b. Consequently, plotting v versus C m m over the appropriate range of C under constant

void size L is expected to indicate clearly a void surface morphological transition. Fig. 3 shows the dependence of void migration speed on applied electric field, v versus C, over m the range 50≤C≤100. Two different regimes of surface morphology are evident with shape constants b and b , b ≠b , resulting in two straight 1 2 1 2 lines of different slopes that intersect at C=C . c The dashed parts of each straight line indicate ranges of C where the corresponding surface morphologies are unstable; specifically, surface morphologies represented by the steady configuration of Fig. 2a are stable for C

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Fig. 3. Dependence of the migration speed of stable voids, v , on the electric field strength expressed by C for parameter values: L= m 0.60, m=1, A=10, and w=90°. Two straight lines according to Eq. (1) are fitted to the computed data points shown by open squares and circles, respectively. The intersection of the two lines locates the critical point, C=C , for the surface morphological transition. The c dashed parts of the lines indicate the range of C that corresponds to unstable surface morphologies and the exchange of stability that occurs at C=C . The lower-right insert to the figure shows the dependence on C of the transient period, t , required to reach an asymptotic c o time-periodic state starting from a semi-circular void configuration: as CC (for C>C ), t 2. The upper-left insert to the figure c c o shows the evolution of the void migration displacement along the metallic film for C=65 (bottom trajectory) and C=85 (top trajectory).

Fig. 4. Evolution of an initially semi-circular transgranular void in a metallic film under an electric field directed from left to right and for parameters values C=150, L=0.60, m=1, A=10, and w=90°. The configurations correspond to uniformly distributed times over the indicated time period. Surface diffusion is considered both on the void surface and over the entire area of the film’s edge.

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mechanistic similarities regardless of the treatment of this surface junction, as well as differences both in the character of the asymptotic state and in the range of C for stability of the corresponding asymptotic states and, therefore, the value of C . c In general, if the constraint is released, stronger electric fields are required for the propagation of surface waves on the migrating voids, which correspond to soliton-like features on the free surface. A dynamical sequence completely analogous to that of Fig. 2b is shown in Fig. 4 for C=150 and with the whole edge of the film in contact with the void treated as a free boundary that evolves through surface diffusion and surface electromigration. Eleven configurations corresponding to equidistant points in time are shown over one period of the surface’s oscillatory response. The same void tip splitting mechanism and current-driven tip feature propagation along the void surface are evident. Note, however, the smoothness of the surface in the region where the void meets the edge of the film. A comprehensive analysis of the surface morphological response for such a freestanding film will be presented in a forthcoming publication [23]. In conclusion, we have presented a theoretical non-linear analysis based on self-consistent numerical simulations of current-induced surface wave propagation on void surfaces in metallic thin films. The analysis focused on a low (two-fold ) grain symmetry and relatively weak surface diffusional anisotropy, where surface wave propagation associated with void evolution toward film failure have been reported in our previous studies [10,13]. Our simulations revealed a surface morphological transition at a critical strength of the applied electric field. For electric fields weaker than critical, stable void shapes with steady surface areas migrate along the film at constant speed. For electric fields stronger than critical, surface waves propagate along the void in the direction of the electric field resulting in a stable time-periodic surface morphology that also migrates along the film at constant speed. Mathematically, this morphological transition corresponds to a Hopf bifurcation [24] in the current-induced surface morphology; the nature of the transition asymptotically close to the onset also will be addressed in

a forthcoming publication [23]. Additional nonlinear phenomena in the current-induced void dynamical response are currently under investigation and will be reported in future publications.

Acknowledgement This work was supported in part by the National Science Foundation (Award No. ECS-9501111) through a CAREER award to one of the authors (D.M.).

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