Evolution of normal stress and surface roughness in buckled thin films
G. Palasantzas and J. Th. M. De Hosson
a)
Department of Applied Physics, Materials Science Center and Netherlands Institute for Metals Research,
University of Groningen, Nijenborgh 4, 9747 AG Groningen, The Netherlands
͑Received 17 July 2002; accepted 22 October 2002͒
In this work we investigate buckling of compressed elastic thin films, which are bonded onto a
viscous layer of finite thickness. It is found that the normal stress exerted by the viscous layer on the
elastic film evolves with time showing a minimum at early buckling stages, while it increases at later
stages. The normal stress also shows a minimum as a function of applied compressive stress, which
depends strongly on the viscosity of the underlying layer and strain values. Furthermore, with
decreasing viscosity the film roughness amplitude also shows a minimum at early buckling stages.
The effect of the viscosity becomes more pronounced with increasing strain in the film. Finally,
decreasing elastic film thickness and/or increasing viscous layer thickness also enhance buckling
roughness. © 2003 American Institute of Physics. ͓DOI: 10.1063/1.1528299͔
I. INTRODUCTION
Thin films in modern device technology are often in a
state of compression. Actually, the mismatch between ther-
mal expansion coefficients may produce compressive
stresses in thermal barrier coatings and heteroepitaxial
growth may be accompanied by the evolution of compres-
sive stresses. Nevertheless, by adhering a compressed thin
film to a low viscosity glass, the compressive stresses can be
relieved. In particular, this methodology has been explored in
the growth of low dislocation and relaxed heteroepitaxial
semiconductor films.
1
Moreover, atomic force microscopy
and cross-sectional scanning electron microscopy/
transmission electron microscopy have shown buckling of
compressively strained SiGe films ͑deposited on borophos-
phorosilicate͒ during annealing.
2
In general, any freestanding film, which is subject to
compression, will spontaneously display buckling at lateral
length scales that strongly depend on its elastic properties,
the thickness, and the magnitude of the applied stress.
3
The
film expands out of its plane, which leads to buckling, with a
characteristic wavelength that is the result of the competition
of the in-plane strain relaxation and the elastic stress due to
bending.
4,5
So far, a stability analysis of the buckling problem for
thin elastic films with a finite thickness onto a viscous layer
predicted the growth rates of preexisting undulations that
develop in time.
4
However, the former calculations did not
encounter the problem of how the buckling amplitude devel-
ops assuming an initial rough profile of the elastic film, as
well as the how the normal stress that the elastic film feels
from the viscous films changes with progressing film buck-
ling. This will be the topic of the present work where for
simplicity we will consider the case of the initial elastic film
surface roughness to be self-affine type. The latter assump-
tion is based on the fact that the formation of self-affine
roughness has been observed for a wide range of deposited
thin films ͑i.e., metallic, semiconductor, and organic͒.
6–9
II. BUCKLING FILM THEORY
We consider an elastic film of thickness h
f
, Young
modulus E, and Poisson ratio
. The elastic film is assumed
to be bonded onto a viscous substrate with viscosity
and
thickness h
g
, which is also assumed to be bonded onto a
rigid substrate ͑Fig. 1͒. The elastic film is under compressive
stress
, which is related to a misfit strain
⑀
by
ϭE
⑀
/(1
Ϫ
). When the film has buckled under compression and
bending with the vertical displacement h(r)(Ӷh
f
) given by
Refs. 3 and 4 for wavelength much larger than fh
f
Eh
f
3
12
͑
1Ϫ
2
͒
ٌ
4
hϩ
h
f
ٌ
2
hϩ
N
ϭ0. ͑1͒
In Eq. ͑1͒
N
is the normal stress exerted on the film by the
viscous substrate ͑at the elastic/viscous film interface͒, and
rϭ(x,y) the in-plane position vector. Assuming the Fourier
transform h(r)ϭ
͐
h(k,t)e
Ϫik"r
d
2
r, Eq. ͑1͒ yields the normal
stress
N
N
ϭ
͵
ͫ
h
f
k
2
Ϫ
Eh
f
3
12
͑
1Ϫ
2
͒
k
4
ͬ
h
͑
k,t
͒
e
Ϫik"r
d
2
k, ͑2͒
with k
2
ϭk
x
2
ϩk
y
2
. The problem of an infinite viscous layer
(h
g
→ϩϱ) was solved in the past by Mullins.
5
For the linear
boundary value problem the velocity of the elastic film sur-
face
ץ
h/
ץ
t is proportional to a strain rate
N
(k,t)/
for each
Fourier mode h(k,t) of the form
4
ץ
h
͑
k,t
͒
ץ
t
ϭg
N
͑
k,t
͒
2
k
, gϭ
sinh
͑
2h
g
k
͒
Ϫ2h
g
k
1ϩcosh
͑
2h
g
k
͒
ϩ2
͑
h
g
k
͒
2
,
͑3͒
with
N
(k,t)ϭ
͕
h
f
k
2
Ϫ
͓
Eh
f
3
/12(1Ϫ
2
)
͔
k
4
͖
h(k,t). Inte-
gration of Eq. ͑3͒ yields
4
h
͑
k,t
͒
ϭh
͑
k,0
͒
e
a(k)t
a͒
Author to whom correspondence should be addressed; electronic mail:
hossonj@phys.rug.nl
JOURNAL OF APPLIED PHYSICS VOLUME 93, NUMBER 2 15 JANUARY 2003
8930021-8979/2003/93(2)/893/5/$20.00 © 2003 American Institute of Physics