Gupta D. (Ed.). Diffusion Processes in Advanced Technological Materials

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array of infinitesimal dislocations, as described by Eshelby.

[56]

The extent

of spreading can be characterized by the half-width, c, of the slipped

region.

The dislocation core spreading illustrated in Fig. 8.20 requires that the

material on one side of the interface be able to slide by small amounts (b)

relative to the material on the other side of the interface. This cannot occur

over large distances in real coherent interfaces. However, for incoherent

interfaces between crystalline and amorphous solids, small relative displace-

ments are possible, and dislocation core spreading may occur over large dis-

tances. When a shear strain is applied across the interface, we can imagine

that some film-substrate bonds are broken and new, energetically more

favorable bonds are formed in their place. Since only a few bonds move, the

mean position of the plane can move by an amount that is much less than b.

Awide range of film-substrate systems have incoherent interfaces and

may be susceptible to dislocation core spreading following a mechanism

like that illustrated in Fig. 8.20. Transmission electron microscopy (TEM)

investigations have shown dislocations disappearing at the interface

between an Al film and its native Al

passivation

[57]

and at the interface

between a Cu film and a SiN

barrier layer.

[43]

Gao et al.

[58]

have developed a mathematical model to describe the

time-dependent process of dislocation core spreading along a crystalline-

amorphous interface. The model is briefly described here. We consider

edge dislocations spreading along an interface characterized by a shear

adhesive strength, t

, below which no core spreading occurs and above

which spreading takes place in a viscous manner. The equilibrium core

width and the rate of core spreading for single dislocations and a periodic

array of dislocations can be calculated as a function of interface adhesion.

We consider an edge dislocation spreading along an interface between

a crystalline and an amorphous solid, as depicted in Fig. 8.20. At time

t  0, the dislocation core structure is compact. Subsequently, deformation

near the film-substrate interface is assumed to be described by a viscous

model. This problem has been treated in detail by Gao et al.

[58]

for a single

or planar array of dislocations in a homogeneous bulk material or at the

interface between a thin film and a semi-infinite substrate, where the film

and substrate may have the same, or different, elastic constants. Some of

the solutions are analytic and others are based on an implicit finite differ-

ence method. For an edge dislocation spreading along an interface between

a crystal and an amorphous material with the same elastic constants, the

solution shows that the equilibrium core width for a single dislocation is:

 , (18)



8(1  n

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STRESSES IN THIN FILMS AND THEIR RELAXATION, KRAFT, GAO 393

and for a period array of edge dislocations with spacing L is:

 sin

1



tanh





. (19)

We observe that the equilibrium core width scales inversely with the inter-

face adhesion t

. The stronger the interface, the more compact the core. The

analyses by Gao et al.

[58]

and Baker et al.

[59]

show that the viscous spreading

of a single dislocation core can be described by the change in the half width,

c(t) 



1  0.3634 exp



 t





, (20)

of the slipped zone, where h and t

are the viscosity and the thickness of

the interface, respectively. At t  0, the half-width

c(0) 



4p(1





(21)

defines a region over which viscous slip occurs immediately because the

shear stress associated with a compact dislocation core exceeds the interface

adhesion t

. Although the slipping width c(t) is somewhat ad hoc in repre-

senting the core width, the evolution Eq. (20) provides the insight that the

time scale over which the core spreading occurs is largely determined by the

ratio ht

. Since viscosity usually decreases with temperature, we conclude

that core spreading occurs faster at higher temperatures and stronger inter-

faces, and slower at lower temperatures and weaker interfaces. It can be fur-

ther argued that dislocation core spreading reduces the effectiveness of

hardening mechanisms, as discussed in Figs. 8.12 and 8.19.

[58]

As a result,

dislocation-mediated plasticity in thin films is immediately related to diffu-

sional processes as well as to the properties of surfaces and interfaces.

However, a complete understanding of dislocation plasticity is still not

achieved and an incorporation of thermally activated core spreading in the

dislocation models described above has not yet been attempted.

8.3.3 Diffusional Creep

As discussed in Sec. 3.1 and shown in Fig. 8.10, wafer curvature

experiments have shown that the thermal mechanical behavior of Cu





8(1  n

pEb



8(1  n



Ch_08.qxd 11/29/04 6:45 PM Page 393

thin films on substrates depends strongly on the passivation of the film

surface. This phenomenon has recently been explained by a special

kind of Coble-creep in thin films on substrates, often referred to as the

constrained diffusional creep.

[31, 60]

The continuum model for con-

strained diffusional creep developed has adopted the hypothesis that

uncapped, pure Cu films are subject to active surface and grain boundary

diffusion, which causes transport of material between the film surface

and the grain boundaries.

[60]

Diffusion in thin films on substrates is fun-

damentally different in nature from that in bulk materials. A constraint

is imposed by the usually perfect bonding between film and substrate

(for example, material sequence Cu-aSiN

-aSiO

-Si,

[61–63]

which

implies that no sliding occurs at the film-substrate interface. In contrast

to previously proposed models of diffusion in thin films,

[64]

the con-

straint that no sliding occurs at the film-substrate interface renders the

diffusion in thin films an inherently transient problem. Steady-state

solutions frequently used to describe grain boundary diffusion may not

be used. An additional constraint is that material transport cannot pro-

ceed in the substrate and diffusion has to stop at the film-substrate

interface. Constrained diffusional creep leads to formation of a new

material defect called the grain boundary diffusion wedge, which also

plays an important role in dislocation nucleation processes in thin

films.

[60]

This process corresponds to the classical Coble creep in a thin-

film structure, except that the roots of the grain boundaries are con-

strained not to slide with respect to the substrate. In the remainder of

this section, we briefly describe the theory of constrained diffusional

creep as well as the grain boundary diffusion wedge and its associated

dislocation mechanisms.

In the model of Gao et al.,

[60]

constrained diffusional creep under a

constant nominal stress in the film is considered and diffusion is modeled

as dislocation climb in the grain boundary. The solution for a single edge

dislocation is used as the Green’s function to construct a solution with

infinitesimal Voltera edge dislocations. For an inserted grain boundary

wedge of width 2u(z,t) in a film with thickness h(0  z  h), the stress

along the grain boundary is given by:

(z,t)  s

 

S(z, z)



∂u

∂

,t)



dz, (22)

where S(z,z) is a Green’s function kernel for the continuous dislocation

problem (Cauchy kernel), G is the shear modulus, n is Poisson’s ratio, and

is the stress in the absence of diffusion. Balancing the diffusive atomic

flux with the displacement rate (mass conservation), the diffusion equation



2p(1  n)

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STRESSES IN THIN FILMS AND THEIR RELAXATION, KRAFT, GAO 395

based on Coble creep,

[65]



∂u

∂

,t)



 , (23)

is woven into the problem. Here, d

denotes the product of grain

boundary thickness and atomic diffusivity, t is time, Ω is the atomic vol-

ume, and kT has the usual meaning. Combining Eqs. (22) and (23) yields

an integro-differential governing equation for constrained diffusional

creep:





S(z,z)



∂

(

z,t)



dz, (24)

which describes the time evolution of grain boundary stress under the fol-

lowing boundary conditions:

(0,t)  0, s

(h,t)  s

(h,t)  0 (25)

and an initial condition

(z,0)  s

. (26)

The boundary conditions in Eq. (25) correspond to continuity of

chemical potential at the junction between the film surface with the grain

boundary, as well as no mass flux and no sliding at the film-substrate

interface. The initial condition in Eq. (26) defines a Green’s function

solution to a step loading on the thin film. The solution can be used later

to construct more general solutions to an arbitrary time-dependent load-

ing s

(t) via a convolution scheme.

[31]

The solution to Eq. (26) shows that

the insertion of a material wedge into the grain boundary causes expo-

nential stress decay with a characteristic time scaling with the cube of the

film thickness,

 . (27)

The diffusion wedge represents columns of atoms corresponding to an

array of climb-edge dislocations that have been transported into the grain

2p(1  n)kTh



Ω



2p(1  n)kT

∂s

(z,t)



∂t

∂

(z,t)



∂z

Ω



2kT

Ch_08.qxd 11/29/04 6:45 PM Page 395

boundary. With respect to the lattice distortion around the diffusion

wedge, the dislocations in the grain boundary exemplify geometrically

necessary dislocations

[66, 67]

that are usually associated with nonuniform

plastic deformation. Gao et al.

[60]

used an eigenmode expansion to obtain

an integro-differential eigenvalue problem that is subsequently trans-

formed into a standard Cauchy-type singular equation to be solved by a

Gauss-Chebyshev quadrature scheme. Gao et al. provide details of the

numerical procedure.

[60]

A simplified solution,

(z,t)  s

ltt

f(zh), (28)

can be used for all practical purposes, where the function f(x) is

defined as:

f(x) 3



1  x



, (29)

the parameter l is related to the film thickness h and grain size d as:

l 



11.72  1.69(hd)  73.09(hd)

0  hd  0.2

8.10  30.65(hd) 0.2  hd  10. (30)

Averaging Eq. (28) over the film thickness shows that the average stress

on the grain boundary decays exponentially as:



(t)  s

ltt

. (31)

Therefore, on the scale of a characteristic time t

l, the traction along the

grain boundary is fully relaxed and the diffusion wedge exhibits the same

traction-free condition as a crack, indicating a singular stress field near

the root of the grain boundary. This leads to extraordinarily large

resolved shear stresses on planes parallel and close to the film-substrate

interface and, in the (111)-textured films, can cause emission of parallel

glide dislocations. Although the resolved shear stresses on the parallel

glide planes induced by a diffusion wedge are similar to those caused by

a crack, differences in the dislocation nucleation process may arise. The

mechanisms involved are still not well understood and are currently

being investigated.

Experimentally, diffusional creep as a major stress relaxation mecha-

nism in uncapped thin copper films has been verified in a number of

studies.

[11, 15, 31, 61, 63]

Experiments have indicated that, while plastic deformation



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STRESSES IN THIN FILMS AND THEIR RELAXATION, KRAFT, GAO 397

in passivated or relatively thick copper films is characterized by the

“classical” threading dislocations, parallel glide dislocations become

increasingly dominant in unpassivated thin films with thicknesses of less

than 400 nm.

[61–63]

The experimental observations of parallel glide dislo-

cation emission strongly indicate that constrained grain boundary diffu-

sion is a dominant stress relaxation mechanism for unpassivated thin films

on substrate. Figure 8.21 shows an in situ TEM observation of parallel

glide dislocations being emitted from a grain boundary source during the

cooling portion of the second thermal cycle of the Cu-aSiN

-aSiO

Si(100) system. The experiments have also indicated that nucleation of

parallel glide dislocations is very selective to the type of grain boundary.

Grain boundaries with 6- and 66-degree tilt angles seem to be preferred

nucleation sites for parallel glide dislocations. The smallest film thickness

investigated is about 35 nm.

In modeling a thermal cycling experiment, both the nominal stress s

and temperature (hence the characteristic time t

also) vary with time, and

Eq. (31) cannot be directly used. Weiss et al.

[31]

have developed an inte-

gral convolution scheme to calculate the grain boundary stress and have

used it to determine the average stress over the entire film using an equation

Figure 8.21 During the cooling portion of the second thermal cycle of the Cu-aSiN

aSiO

-Si(100) film-on-substrate system, parallel glide dislocations were emitted from

the grain boundary source. (a) At 170°C, the first dislocation has been emitted and

is pinned in the middle.(b) The second emitted dislocation has quickly overcome the

pinning point and has pushed the first dislocation forward by the time 150°C is

reached. (c) At 135°C, the third dislocation is pinned, while the first two have

advanced far into the grain (180 and 300 nm).From Dehm et al.,

[63]

with permission.

Ch_08.qxd 11/29/04 6:45 PM Page 397

by Xia and Hutchinson:

[68]

s  s

 (s

 s

)



tanh



. (32)

Weiss et al.

[31]

took the experimentally measured stress in passivated

Cu 1 at.% Al film as the nominal stress s

and found that the stress s cal-

culated from Eq. (32) is in reasonable agreement with the measured stress

in unpassivated pure Cu film. Therefore, they concluded that the observed

difference in thermomechanical behavior between passivated and unpas-

sivated films can be attributed to constrained diffusional creep.

By implicitly assuming that surface diffusivity is much faster than

grain boundary diffusivity, the model of Gao et al.

[60]

does not explicitly

consider mass transport along the surface of the film. Depending on mate-

rials, this assumption may not always be valid. Zhang and Gao

[69]

recently

studied coupled surface and grain boundary diffusion and concluded that

the analyses by Weiss et al.

[31]

and Gao et al.

[60]

remain valid even when sur-

face diffusivity is comparable to grain boundary diffusivity. Specifically,

Zhang and Gao

[69]

have shown that the results in

[31]

and

[60]

become gener-

ally valid when an effective diffusivity

eff

 (33)

is used to replace the grain boundary diffusivity d

, where d

denotes the surface diffusivity. Note that, due to the large constant coeffi-

cient of 12p, the effect of surface diffusion becomes significant only when

the grain boundary diffusivity is much larger than the surface diffusivity.

8.4 Conclusion

In this chapter, techniques for the measurement of stresses in metal-

lic thin films held on substrates and their relaxation by plastic deforma-

tion and diffusional creep are described. All studies mentioned indicate

clearly that the motion of dislocations is constrained in thin metal films

by the presence of interfaces and grain boundaries. As a result, the

mechanical behavior of thin metal films can be qualitatively summarized

by saying that smaller is stronger. On the other hand, diffusional processes

play a more important role with decreasing dimensions as the density

of fast diffusion pathways increases. Therefore, it would be helpful to

12pd



12pd

 d



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STRESSES IN THIN FILMS AND THEIR RELAXATION, KRAFT, GAO 399

describe the deformation behavior of thin films using deformation mech-

anism maps

[70]

similar to the ones established for bulk materials by Frost

and Ashby,

[49]

which include both dislocation-mediated and diffusion-

mediated mechanisms. Unfortunately, clear, quantitative descriptions

have not yet been established for thin films, neither empirically nor the-

oretically, and the use of quantitative descriptions obtained from bulk

materials does not appear to be adequate. This is further accentuated by

the fact that subtle changes in the chemistry of the interfaces, which

might be beyond experimental control, can dramatically influence the

thin-film mechanical behavior. The models discussed for diffusional

creep and dislocation core spreading will contribute to a framework that

will lead eventually to a more complete description of thin-film mechanical

behavior that takes into account the interaction of dislocation plasticity

and diffusional processes. For this, a unified view of the experimental

observations, including mechanical testing as well as in situ electron

microscopy, and of the theory is indispensable.

In most technological applications, thin films are rigidly attached to

a substrate that is typically much thicker than the film itself. As a result

of this configuration, the thin films are commonly subject to high inter-

nal stresses, which may endanger the operation of small-scale devices.

Failure mechanisms discussed in the literature include the following.

(1) Thermal-stress-induced voiding has been a problem for many years

for Al or Cu conductor lines.

[71, 72]

This phenomenon is driven by high

hydrostatic tensile stresses within interconnect lines under a capping

layer during cooling. Even in blanket films, where two-dimensional in-

plane stresses are observed, a few cases of stress-induced voiding have

been reported.

[73, 74]

(2) Stresses may also cause the delamination or

cracking of thin films.

[75]

It has been shown that the plasticity in thin

metal layers affects the interface fracture resistance in thin-film inter-

connect structures.

[76]

Specifically, the TaN/SiO

interface fracture energy

was measured in thin-film Cu/TaN/SiO

structures in which the Cu layer

was varied over a wide range of thickness. It was found that in a regime

of 0.25 to 2.5 µm, the delaminating resistance is dominated by the con-

tribution of plastic deformation in the metallic layer. (3) More recently,

the behavior of thin metal films under cyclic loading conditions has been

given experimental as well as theoretical consideration. In particular,

cracking of a brittle layer caused by ratcheting in an adjacent ductile

layer has been observed in a thin-film system under cyclic thermal load-

ing.

[77]

Moreover, cyclic mechanical or thermomechanical loading can

also lead to the formation of voids and extrusions in thin metal

films.

[78–80]

From these observations, it has been concluded that the role

of point defects and diffusion is much more important for fatigue in thin

films than in bulk materials.

Ch_08.qxd 11/29/04 6:45 PM Page 399

In summary, continuum theories of plasticity or lifetime models that

incorporate dislocation plasticity and diffusional processes with respect to

a physical or microstructural length scale may turn out to be very helpful

if they can be validated by experiments. The predictive capability of such

theories will be very useful in device simulation codes for microelectron-

ics, MEMS, and other applications.

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