Effect of applied stress tensor anisotropy on the electromechanically driven
complex dynamics of void surfaces in metallic thin films
Georgios I. Sfyris, M. Rauf Gungor, and Dimitrios Maroudas
a)
Department of Chemical Engineering, University of Massachusetts, Amherst, Massachusetts,
01003-3110, USA
(Received 26 May 2011; accepted 2 August 2011; published online 19 September 2011)
We present a systematic computational analysis of the complex, electromechanically driven surface
dynamics of voids in thin films of face-centered cubic metals for h100i-oriented film planes charac-
terized by four-fold symmetry of surface diffusional anisotropy. The voids are located at an edge of
the metallic thin film, and the film is subjected simultaneously to an external electric field and an ani-
sotropic biaxial tensile stress. Our analysis is based on self-consistent dynamical simulations of
driven void surface morphological response according to a well-validated, two-dimensional, and fully
nonlinear model. We examine thoroughly the effects of the anisotropic mechanical loading on the
morphological evolution of the electromigration-driven void surface and the resulting asymptotic
states of the surface morphological response. We have found supercritical Hopf-bifurcation transi-
tions from stable steady to stable time-periodic states. For such films and over the range of electrome-
chanical conditions examined, the only possible stable asymptotic states are either time-periodic
states characterized by a single period of oscillation or steady states without any change in the void
shape. We have determined the stability domain boundaries of the various asymptotic states and their
dependence on the anisotropy of the applied stress tensor. The loading anisotropy has significant
effects on the stability domain boundaries, but it does not introduce any more complex void dynamics
in h100i-oriented films than that under isotropic mechanical loading.
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2011 American Institute of
Physics. [doi:10.1063/1.3638070]
I. INTRODUCTION
Electromigration-induced failure mediated by the driven
dynamics of voids in metallic thin-film interconnects is one of
the most challenging materials reliability problems in micro-
electronics. One mode of failure is the formation and propaga-
tion of voids in the metal, which may evolve to cause sudden
increase in electrical resistance or even sever the interconnect
lines and open the circuit. The literature in the area is very
rich with numerous theoretical studies that have analyzed the
electromigration-driven surface morphological evolution of
metallic surfaces.
1–33
The formation and propagation of vari-
ous surface wave patterns is of particular interest; they range
from solitary waves and nonlinear wave trains on surfaces of
bulk metals,
14,19
to soliton-like features that travel on large-
size void surfaces preceding the failure of metallic thin
films,
9,27
to stable wave propagation on smaller-size void
surfaces in films driven by a stronger-than-critical electric
field.
16,28
Electromigration-induced complex shape evolution
also has been demonstrated for homoepitaxial islands on elec-
trically conducting substrates,
22
and current-induced step
meandering on vicinal surfaces has been studied.
26
The role of mechanical stress is crucial in determining
the current-driven dynamics of conducting surfaces, includ-
ing surfaces of voids in metallic thin films.
11,12,18,21,33
It has
been shown that electromechanically driven morphological
evolution of void surfaces depends on the void size, the
strength of the externally applied electric field, the mechani-
cal loading conditions, as well as the strength of the diffu-
sional anisotropy for diffusion on the void surface;
18,33
the
diffusional anisotropy strength depends strongly on tempera-
ture.
18
In particular, we have shown that, for thin films of
face-centered cubic (fcc) metals with h110i-oriented film
planes, upon increasing the level of an applied isotropic
biaxial tensile stress, morphologically stable steady states
transition to time-periodic states through a subcritical Hopf
bifurcation. Further increase of the stress level triggers a
sequence of period-doubling bifurcations that sets the driven
nonlinear system on a route to chaos.
33
For h100i-oriented
film planes, a transition from steady to time-periodic states
also has been found to occur at a critical stress level; in this
case, the corresponding Hopf bifurcation is supercritical and
the nonlinear system is not set on a route to chaos.
33
In this paper, we study the surface morphological
response of voids in thin films to various electromechanical
loadings, aiming at identification of the conditions under
which oscillatory surface dynamics can be induced and stabi-
lized. We place emphasis on a fundamental understanding of
effects associated with the anisotropy of the mechanical load-
ing applied to the films and report results on the void dynami-
cal response as a function of the mechanical loading tensor
anisotropy. We have conducted a systematic computational
analysis of the complex electromechanically driven void dy-
namics in thin films of fcc metals for h100i-oriented film
planes, characterized by four-fold symmetry of surface diffu-
sional anisotropy. The voids are located at the bottom edge of
the metallic films, which are subjected simultaneously to an
a)
Author to whom correspondence should be addressed. Electronic mail:
maroudas@ecs.umass.edu.
0021-8979/2011/110(6)/063705/9/$30.00
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2011 American Institute of Physics110, 063705-1
JOURNAL OF APPLIED PHYSICS 110, 063705 (2011)