Gupta D. (Ed.). Diffusion Processes in Advanced Technological Materials
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362 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS
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8 Measurement of Stresses in Thin
Films and Their Relaxation
Oliver Kraft
Institut für Materialforschung II, Forschungszentrum Karlsruhe
und Institut für Zuverlässigkeit von Bauteilen und Systemen
Universität Karlsruhe, Germany
Huajian Gao
Max-Planck-Institut für Metallforschung
Stuttgart, Germany
8.1 Introduction
Thin films are used in many technological applications because they
provide special chemical, mechanical, and electrical properties, as and
where needed, using only small amounts of materials. For example, the
high density and fast performance of modern computers have been possi-
ble due to the incorporation of multilayer thin-film structures coupled
with submicron photolithography on the back of the active devices formed
in Si chips. Because thin-film structures are always deposited onto rela-
tively rigid substrates, they are heavily stressed and undergo relaxation
when subjected to fabrication processes involving higher temperatures
than those used for the deposition. This chapter discusses techniques for
measuring stresses in thin metal films attached to substrates and their
relaxation by plastic deformation and diffusional creep.
Figure 8.1 shows the origin of stresses in thin films as described by
Nix.
[1]
As a start, we consider a film/substrate composite that is free of
stress [Fig. 8.1(a)]. If we remove the film from the substrate, its lateral
dimensions will match those of the substrate [Fig. 8.1(b)] and the film
can be reattached without causing any stress. However, if we assume that
the film is subject to a dilatational strain relative to the substrate, then the
lateral dimensions of the film no longer match those of the substrate
[Fig. 8.1(c)]. Astress must be applied to the film in order to make it fit to
the substrate again [Fig. 8.1(d)]. Now the film can be rigidly reattached
to the substrate and the applied stress removed. As a result, shear stresses
are produced near the edge of the film/substrate composite, which main-
tains a biaxial stress in the film [Fig. 8.1(e)]. In addition, the shear
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366 DIFFUSION PROCESSES IN ADVANCED TECHNOLOGICAL MATERIALS
Figure 8.1 Illustration origin of biaxial stress in a thin film: (a) stress-free film on
the substrate; (b) remove film from the substrate; (c) film dimension change rela-
tive to the substrate; (d) external stress is applied to the film to return to the sub-
strate dimension; (e) film is reattached to the substrate and external stresses are
removed. As a result, (1) an internal biaxial stress remains in the film; (2) the inho-
mogeneous stress state near the edges leads to a bending moment, which causes
the substrate to bend; and (3) a homogeneous stress in the substrate is devel-
oped, which can be neglected for the thin-film limit. After Nix.
[1]
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STRESSES IN THIN FILMS AND THEIR RELAXATION, KRAFT, GAO 367
stresses cause a bending moment that will lead to a curvature of the sub-
strate. As discussed below, the measurement of this curvature is a com-
mon technique to determine the average stress in the film. It should be
noted that, under the assumed condition that the film thickness t
f
is much
smaller than the thickness t
s
of the substrate (100 t
f
t
s
), stresses in the
substrate are of minor importance.
For pure dilatational strain, that is, a volume change of the film with respect
to the substrate, with the strain components being ∆e
xx
∆e
yy
∆e
zz
∆e and
considering only elastic deformation for an isotropic material, the stress state
in the film is given by:
s s
xx
s
yy
1
E
f
n
f
e and s
zz
0, (1)
where E
f
is Young’s modulus and n
f
is Poisson’s ratio of the film material.
The stress component s
zz
perpendicular to the film surface equals zero
because the film is free to expand or contract in this direction. There are many
possible reasons for a relative volume change of the film material with respect
to the substrate. The most important is change of temperature if the thermal
expansion coefficients of film and substrate are different:
e (a
f
a
s
)T aT, (2)
where ∆T is the imposed temperature change, and a
f
and a
s
are the thermal
expansion coefficients of the film and the substrate, respectively. Other
mechanisms that cause dilatational strains include the annihilation of lattice
defects such as excess vacancies, dislocations, and grain boundaries; phase
transformations; and composition changes. These mechanisms typically
occur during film growth or during an initial heat treatment until a stable
microstructure has evolved. For the special case of heteroepitaxial films, the
film is strained to accommodate the mismatch of the lattice parameters.
For crystalline materials with an elastic anisotropy, the film stress
depends on the orientation of a single crystalline film or the texture of a
polycrystalline film. For cubic materials, grains oriented with both (111)
and (100) directions parallel to the film normal are biaxially isotropic in
the film plane. For single crystal films with one of these orientations, a
uniform elastic strain leads to a uniform stress state, and Eq. (1) becomes:
s
111
Y
111
e and s
100
Y
100
e, (3)
with
Y
111
C
6
1
C
1
4
4
(C
2
1
C
1
1
2
2C
4
1
C
2
)
44
(4)
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and
Y
100
C
11
C
12
, (5)
where Y
111
and Y
100
are the biaxial moduli for (111) and (100) out-of-plane
oriented crystals, respectively, and C
11
, C
12
, and C
44
are the anisotropic
stiffness coefficients of the cubic material. For instance, for Al and Cu at
room temperature, the biaxial moduli calculate to Y
111
116 and 261
GPa, and Y
100
100 and 115 GPa, so that Y
111
Y
100
1.2 and 2.3, respec-
tively. From Eq. (3), it is evident that for a given ∆e, the ratio of stresses
for the two orientations is s
111
s
100
Y
111
Y
100
. For a polycrystalline
film, however, stress continuity requires s
111
s
100
at the grain bound-
aries.
[2]
Let us consider stress and strain states in a film with a mixed tex-
ture of (100) and (111) grains as shown in Fig. 8.2. In large grains, the
stress in the interior of the grains reaches the expected value for the
respective orientation [Fig. 8.2(a)]. Near the grain boundaries, the
required stress variation is confined to roughly the film thickness.
[3]
Thus, in the limit of d t
f
(d is the grain size), the volume fraction of
the film with inhomogeneous strain is small (isostrain condition). In fine-
grained films [Fig. 8.2(b)], nearly complete stress accommodation is
required, the stress will approach a uniform value everywhere (isostress
condition), and the strains will follow e
100
e
111
Y
111
Y
100
. Note that near
the interface, if perfect adhesion is assumed, a stress singularity would
arise and shear stresses must develop on the grain boundaries. In a real
polycrystalline metal film, the grain size is typically on the order of the
film thickness and, as a result, the average stresses in each orientation are
expected to lie between the isostrain and isostress limits. Finally, note
that the considerations above are valid for purely elastic behavior and,
once plastic deformation of the film material is involved, an even broader
stress distribution can be expected.
8.2 Measurement Techniques
Stresses can never be measured directly. Any experimental determination
of a stress involves the measurement of an elastic strain and the use of appro-
priate elastic constants to calculate the stress. Two main methods are used to
determine the stresses or strains in thin films. The first method is based on the
measurement of the curvature of the substrate, which is bent as a result of the
film stress. The second method is based on the measurement of the elastic
strain directly in the film by x-ray diffraction. For the first technique, the elas-
tic properties of the substrate material need to be known to determine the film
stress; for the second, those of the film material itself need to be known.
2C
2
12
C
11
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STRESSES IN THIN FILMS AND THEIR RELAXATION, KRAFT, GAO 369
Figure 8.2 Schematics showing the stress and strain distributions during elastic
deformation for a film microstructure consisting of (a) large (111) and (100) grains
(d W t
f
), and (b) alternating narrow (111) and (100) grains (d ≈ t
f
). In the limit of
large grains, the average strains in both components are the same (isostrain
case).In the limit of small grains, we assume that the grains are narrower than the
film thickness such that the stress is the same in both orientations (isostress
case). After Baker et al.
[2]
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8.2.1 Substrate Curvature
The substrate curvature technique makes use of the fact that the radius
of curvature, R, of the substrate is proportional to the stress s in the film,
as quantitatively expressed by Stoney’s equation:
[4]
s
, (6)
where E
s
and n
s
are the Young’s modulus and Poisson’s ratio of the sub-
strate material; t
f
and t
s
are the thickness of the film and the substrate,
respectively; and t
f
must be much less than t
s
for Eq. (6) to be valid. It
should also be noted that the substrate is usually not perfectly flat and R
o
is the radius of curvature of the substrate without the film. As Eq. (6)
shows, an advantage of this method is that the only film property that must
be known to evaluate the stress is the film thickness; knowledge of the
elastic properties of the film is not required. However, the value measured
for s represents the average stress in the film, and a statement on the uni-
formity of the stress is not possible.
Another sample geometry, which is also frequently used, consists of a
cantilever beam with one fixed end. Then the film stress is given by:
s E
s
u, (7)
where L is the length and u is the deflection of the free end of the can-
tilever beam.
To determine stresses accurately, the radius of curvature needs to be
measured over a range from a few meters to several kilometers, or the dis-
placement of the cantilever beam must be determined in the micrometer
regime. This has been achieved experimentally in many different ways;
for example, by optical methods;
[5]
profilometric, interferometric, or capac-
itive methods;
[6]
and, for different conditions, during film deposition,
[7, 8]
in vacuum, or at elevated temperatures.
[9–12]
Figure 8.3 shows an experimental setup that has been used by a number
of authors
[5, 12, 13]
to measure film stresses as a function of temperature for dif-
ferent thin-film materials. Alaser beam is used to scan the surface of the sam-
ple while it is heated. The curvature is determined from the orientation of the
sample surface as measured by the angle of reflection of the laser beam using
a position sensitive detector. Figure 8.4 shows some typical results for the
stress as a function of temperature for a thin metal film (a) and a thin glass
t
s
2
3L
2
t
f
1
R
o
1
R
t
s
2
6t
f
E
s
1 n
s
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STRESSES IN THIN FILMS AND THEIR RELAXATION, KRAFT, GAO 371
Figure 8.3 Schematic showing an experimental setup for wafer curvature
measurements.
Figure 8.4 Film stress as a function of temperature as determined from wafer cur-
vature measurements for (a) a 1.0-µm-thick Cu film and (b) SiO
2
films made by
chemical vapor deposition on Si substrates.
Notes: 1. Data for (a) from Keller et al.
[13]
2. Data for (b) from C. A. Volkert, unpublished work (1997)
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