Quantitative measurement of the stress transfer function
in nickel/polyimide thin film/copper thin film structures
L. Schadler
Department of Materials Engineering, Drexel University, Philadelphia, Pennsylvania 19104
I. C. Noyan
IBM TJ Watson Research Center, P. O. Box 218, Yorktown Heights, New York 10598
͑Received 4 February 1994; accepted for publication 24 October 1994͒
The stress transfer behavior in multilayer thin film structures ͑nickel/polyimide/copper͒ was
measured using x-ray stress analysis. Copper was deposited in various line lengths, and the stress/
strain transferred from a loaded Ni substrate to the Cu thin film was measured as a function of line
length. It was found that there is incomplete strain transfer from one layer to another, and that the
shape of the stress transfer function is similar to that predicted by the shear lag model. © 1995
American Institute of Physics.
In multilayer interconnect structures, thermal or me-
chanical stress/strain applied to the structure will cause me-
chanical interaction between the layers due to differences in
coefficient of thermal expansion ͑CTE͒, differences in
moduli, and discontinuities at the bimaterial interfaces.
Proper modeling of the stress and strain in these structures is
important in order to design layered structures that are ther-
maly and mechanically stable.
Current models use a shear lag approach
1,2
or finite ele-
ment modeling
3,4
to predict the stress transfer from one layer
to another. In these models, the interface is assumed to be
perfect, and the material is assumed to behave in a linear
elastic manner. In addition, traditional bending methods for
measuring strains in thin films must assume a perfect inter-
face in order to calculate the stress or strain in the film.
5
These assumptions lead to predictions of complete strain
transfer from one layer to the next and a specific shape of the
stress transfer function ͑i.e., the stress profile as a function of
distance from an edge͒. Previous studies using x-ray tensile
testing ͑which make no assumptions about the state of the
interface͒ on nickel/polyimide/copper thin film structures
showed that complete strain transfer is not always
observed.
6,7
This observation was independent of the thick-
ness of the polyimide and the width of the copper layer.
6,7
This implies that the assumption of a perfect interface is not
always valid, and thus predictions of the shape of the stress
transfer function ͑i.e., the stress profile as a function of dis-
tance from an edge͒ may be incorrect. The goal of this study
was to measure the stress transfer function using x-ray ten-
sile testing.
The shear lag solution, for the geometry tested ͑Fig. 1͒,
closely follows the solution of Chen and Nelson.
1
The details
were described in a previous report,
6
and only the results are
shown below. Uniaxial stress is applied to the nickel dog-
bone, and the stress is transferred from the Ni to the Cu
through a shear stress,
, in the polyimide ͑PI͒. The shear
stress, the stress in the Cu,
Cu
, and the stress in the Ni,
Ni
,
due to an applied stress,
A
, are shown in Eqs. ͑1͒–͑3͒. G is
the PI shear modulus, t
0
is the PI thickness, x is the distance
from the center of the sample, and E
Cu
and E
Ni
are the
moduli of the copper and nickel
͑
12
͒
ϭ
G
A
sinh
͑

x
͒
t
0

E
Ni
cosh
͑

l
͒
, ͑1͒
Cu
͑
11
͒
ϭ
A
E
Cu
E
Ni
ͩ
1Ϫ
cosh
͑

x
͒
cosh
͑

l
͒
ͪ
, ͑2͒
Ni
͑
11
͒
ϭ
A
, ͑3͒
where

2
ϭ
G
t
0
ͩ
1
E
Ni
t
Ni
ϩ
1
E
Cu
t
Cu
ͪ
.
From the above equations, the following observations can be
made.
͑1͒ The shear stress is a decreasing hyperbolic sine function,
the shape of which depends on the material properties and
the PI thickness.
͑2͒ The stress in the copper is an increasing hyperbolic co-
sine function, the shape of which also depends on material
properties.
͑3͒ There is a critical length above which there is complete
strain transfer. The maximum stress is transferred to the Cu if
the distance from the edge of the film is greater than this
critical length ͓Eq. ͑4͔͒
FIG. 1. Schematic showing the specimen geometry and the coordinates used
for x-ray stress analysis.
22 Appl. Phys. Lett. 66 (1), 2 January 1995 0003-6951/95/66(1)/22/3/$6.00 © 1995 American Institute of Physics