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CHAPTER 13

Constrained Grain Boundary Diffusion in Thin Copper Films

Markus J. Buehler, T. John Balk, Eduard Arzt, Huajian Gao Max Planck Institute for Metals Research, Stuttgart, Germany

CONTENTS

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2. Continuum Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1. Basics of the Continuum Modeling . . . . . . . . . . . . . . . . 3 2.2. Average Stress and Thermal Cycling Experiments . . . . . 9 2.3. Single Edge Dislocations in Nanoscale Thin Films . . . . 10 2.4. Initiation Condition for Diffusion . . . . . . . . . . . . . . . . 10 2.5. Nucleation Criterion for Parallel Glide Dislocations . . . 11 2.6. Constrained Grain Boundary Diffusion with

Threshold Stress . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.7. Discussion and Summary of Continuum Modeling . . . . 14

3. Atomistic Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.1. Large-Scale Atomistic Simulations of Plasticity in

Polycrystalline Thin Films . . . . . . . . . . . . . . . . . . . . . . 15 3.2. Atomistic Modeling of Diffusional Creep . . . . . . . . . . . 17 3.3. Atomistic Modeling of Nucleation of Parallel Glide

Dislocations from Diffusion Wedges . . . . . . . . . . . . . . . 18 3.4. Discussion of Atomistic Simulation Results . . . . . . . . . 20

4. Experimental Studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 4.1. Thermomechanical Behavior of Thin Copper Films . . . 21 4.2. Transmission Electron Microscopy Observations of

Dislocation Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3. Interpretation of Experimental Observations . . . . . . . . 26

5. Modeling the Experimental Results with Continuum Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.1. Experimental Estimate of the Threshold Stress . . . . . . 29 5.2. Fit of the Continuum Theory

to Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . 29

ISBN: 1-58883-042-X/$35.00 Copyright © 2005 by American Scientific Publishers All rights of reproduction in any form reserved.

**1
**

*Handbook of Theoretical and Computational Nanotechnology
*Edited by Michael Rieth and Wolfram Schommers

Volume X: Pages (1–35)

2 Constrained Grain Boundary Diffusion in Thin Copper Films

6. Map of Plastic Deformation Mechanisms . . . . . . . . . . . . . . . 31 7. Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . 34

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

1. INTRODUCTION Materials in small dimensions have become an increasingly important topic of research in the last decade. Changes in material behavior resulting from the effects of surfaces, interfaces, and constraints are still not completely understood. The focus of this chapter is on the mechanical properties of ultrathin submicron copper films on substrates. In such materials, important effects on the film surface and grain boundaries occur, and the constraint of the film–substrate interface can govern the mechanical behavior. Polycrystalline thin metal films, shown schematically in Fig. 1, are frequently deposited on

substrate materials to build complex microelectronic devices and are a relevant example of materials in small dimensions. In many applications and during the manufacturing process, thin films are subjected to stresses arising from thermal mismatch between the film material and the substrate. This can have a significant effect on the production yield as well as on the performance and reliability of devices in service. In past years, an ever-increasing trend toward miniaturization in technology has been observed, stimulating a growing interest in investigating the deformation behavior of ultrathin films with film thicknesses well below 1 m. Different inelastic deformation mechanisms operate to relax the internal and external

stresses on a thin film. Experiments have shown that for films of thicknesses between approx- imately 2 and 05 m, the flow stress increases in inverse proportion to the film thickness (see, for example, Refs. [1–3]). This has been attributed to dislocation channeling through the film [4–6], where a moving threading dislocation leaves behind an interfacial segment. The relative energetic effort to generate these interfacial dislocations increases with decreas- ing film thickness, which explains the higher strength of thinner films. This model, however, could not completely explain the high strength of the thin films that was found in experi- ments [1]. More recent theoretical and experimental work [7–11] indicates that the strength of thin metal films often results from a lack of active dislocation sources rather than from the energetic effort associated with dislocation motion.

**d
**

hf

Figure 1. A polycrystalline thin film constrained by a substrate. The thin film is subject to biaxial loading as a result of thermal mismatch of film and substrate material.

Constrained Grain Boundary Diffusion in Thin Copper Films 3

The regime in which plastic relaxation is limited by dislocation nucleation and carried by
the glide of threading dislocations reaches down to film thicknesses of about *h*f ≈ 400 nm.
For yet-thinner films, experiments have revealed a film-thickness-independent flow stress [7].
*In-situ *transmission electron microscopy observations of the deformation of such ultrathin
films reveal dislocation motion parallel to the film–substrate interface [7, 8]. This glide mech-
anism is unexpected, because in the global biaxial stress field, there is no resolved shear stress
on parallel glide planes, indicating that there must be a mechanism involving long-range
internal stresses that vary slowly on the length scale of the film thickness. For sufficiently
thin films, these internal stresses have a pronounced effect on the mechanical behavior.
Constrained diffusional creep [12] has provided what is so far the only feasible mechanism
for such internal stresses to cause nucleation of dislocations on glide planes parallel to the
surface.
This chapter contains three main parts. The first part reviews continuum mechanics mod-

eling of constrained diffusional creep. The second part contains the results of molecular dynamics simulations of diffusional creep and plasticity in polycrystalline thin films. In the third part, we discuss relevant experimental results. Finally, we summarize all the results to develop a deformation mechanism map for ultrathin films on substrates.

2. CONTINUUM MODELING The continuum model of constrained diffusional creep was developed by Gao and coworkers [12]. Diffusion can have a fundamentally different nature in thin films than in bulk materials. The constraint imposed by the strong bonding between film and substrate implies that no sliding can occur at the film–substrate interface. In contrast to previously proposed models of diffusion in thin films [13], the constraint of no sliding at the film–substrate interface ren- ders diffusion in thin films an inherently transient phenomenon [12]. Therefore, steady-state solutions frequently used to describe grain boundary diffusion may not be applied. An addi- tional constraint is that material transport cannot proceed in the substrate, and diffusion must therefore stop at the film–substrate interface. It was shown by Gao and coworkers [12] that constrained grain boundary diffusion leads to a new material defect, referred to as the grain boundary diffusion wedge. Figure 2 illustrates the basic mechanism of constrained diffusional creep [12] in three

stages. In stage 1, material is transported from the surface into the grain boundary. In stage 2,
mass transport leads to the formation of a diffusion wedge, as more and more material
flows into and accumulates in the grain boundary. The continuum model predicts that the
traction along the grain boundary diffusion wedge becomes fully relaxed and cracklike on
the scale of a characteristic time * *. This leads to extraordinarily large resolved shear stresses
on glide planes that are parallel and close to the film–substrate interface, and that scan cause
emission of parallel glide dislocations in the last stage. Although the resolved shear stresses
on the parallel glide planes induced by a diffusion wedge are similar to those generated by
a crack, differences in the dislocation nucleation process may arise.
This section is briefly outlined as follows. We discuss the continuum mechanics model of

constrained diffusional creep for different cases, including the homogeneous case, bimaterial interface, and coupled surface and grain boundary diffusion. We also present an extension of the continuum model to include a threshold stress for diffusion. Finally, we discuss a model of the critical condition for initiation of diffusion and a model for parallel glide dislocation nucleation from the diffusion wedge in the spirit of a model by Rice and Thomson for dislocation nucleation at crack tips [14].

2.1. Basics of the Continuum Modeling

In the continuum model, diffusion is treated as a dislocation climb in the grain boundary. The
basis for this decision is the solution for the normal traction *xx *along the grain boundary
resulting from the insertion of a single dislocation (material layer of thickness *b*) along
0(corresponding to a climb edge dislocation). The coordinate system for the problem
is depicted in Fig. 3. The solution for a single edge dislocation near a surface is used as the

4 Constrained Grain Boundary Diffusion in Thin Copper Films

Stage 1

Stage 2

Stage 3

substrate

substrate

substrate

diffusion wedge

Figure 2. Mechanism of constrained diffusional creep. Material is transported from the surface into the grain boundary, leading to the formation of a diffusion wedge as material accumulates in the grain boundary. The diffusion wedge can be modeled as a pile-up of climb edge dislocations near the film–substrate interface, causing a cracklike singular stress concentration near the root of the grain boundary.

Green’s function to construct a solution with infinitesimal Volterra edge dislocations [15–17]. The traction at position resulting from a dislocation at is

*xx
*=
*Eb
*

41− 2*K
*(1)

where

*K
*= 1
− −

1 + −

2*
*− *
*+ 3 (2)

is the Cauchy kernel function for this particular problem, with *E *denoting Young’s modulus
and * *Poisson’s ratio. For an arbitrary opening function 2*u
*in a film with thickness *h*f ,
the stress along the grain boundary is given by

*xx
t *= 0 −
*E
*

21− 2∫ *h*f
0

*S
u t
*

*
d *(3)

ζ=0

σ0σ0

ζ=b

ζ

d/2

film

x

substrate

Figure 3. Coordinate system for the continuum mechanics analysis. The schematic shows climb edge dislocations in the grain boundary of a thin film on substrate.

Constrained Grain Boundary Diffusion in Thin Copper Films 5

where 0 is the stress in the absence of diffusion and *S
*is a Green’s function kernel for
the continuous dislocation problem (Cauchy kernel). For a dislocation near a free surface,
*S
*= *K
*. The kernel function *S
*can also be established for a dislocation near
a bimaterial interface [12, 18] or for periodic wedges [12]. The calculations in this chapter
are mostly for an elastic film on a rigid substrate, although we do not give the Green’s
function kernel explicitly here.
The chemical potential relative to the free surface (with atomic volume * *= *a*30*/*4 in face-

centered cubic crystals where *a*0 is the lattice parameter) is given by

*
t *= 0 − *xx
t *(4)
where 0 is an arbitrary reference constant. The corresponding atomic flux per unit thickness
in the boundary is

*j
t *= −gb*D*gb
k*T
*

*
t
*

*
*= gb*D*gb

k*T
xx
t
*

*
*(5)

where k is the Boltzmann constant and *T *the absolute temperature. The parameter gb*D*gb
denotes temperature-dependent grain boundary diffusivity. Equations (4) and (5) are cou-
pled via mass conservation, as the flux divergence is related to the displacement rate through

2
*u
*

*t
*= −*j
t
*

*
*(6)

which can be combined with Eq. (5) as

*u
*

*t
*= −gb*D*gb*
*

2k*T
*2*xx
t
*

*
*2
(7)

The derivative of *xx
t *with respect to time is given by

*xx
t
*

*t
*= − *E
*

21− 2∫ *h*f
0

*S
*2*u
*

*t
d *(8)

and inserting Eq. (7) into Eq. (8) yields the main governing equation

*xx
t
*

*t
*= *E*gb*D*gb*
*

41− 2k*T
*∫ *h*f
0

*S
*3*xx t
*

3
*d *(9)

for the grain boundary traction. Boundary and initial conditions are given as follows: for the continuity of chemical potential near the free surface,

*xx
*= 0* t *= 0 (10)
and for no sliding and no diffusion at the interface,

*xx
*

*
*= *h*f* t *=
2*xx
*2

*
*= *h*f* t *= 0 (11)

Finally,

*xx
t *= 0= 0 (12)
sets the initial condition for the transient problem.
The problem given by Eq. (9) can be expressed by the method of separation of variables

in the form of an expansion series

*xx
t *= 0
∑
*n*=1

*cn *exp −*nt/fn
/h*f(13)

6 Constrained Grain Boundary Diffusion in Thin Copper Films

where

* *= 41− *
*2k*Th*3f

*ED*gbgb*
*(14)

is a characteristic time and *n*, *fn*, and *cn *denote the eigenvalues, eigenfunctions, and coeffi-
cients. The coefficients are determined from the initial condition at *t *= 0. It is important to
note that * *∼ *h*3f , similar to the classical Coble creep equation [19]. To solve the equations
numerically, the problem is transformed into a standard Cauchy-type singular equation for
*f *′′′*n *[12]. The Gauss–Chebyshev quadrature developed by Erdogan et al. [20, 21] can be used
to solve such equations. The opening displacement *uz t *is given by

*u
t *= −21− *
*2*h*f0

*E
*

∑
*n*=1

*cn
*−1
*n *1− exp −*nt/ f *′′*n
/h*f(15)

The solution procedure can be summarized in the following steps: first, find eigenvalues
and eigenfunctions; second, find the coefficients *cn*, and third, calculate the traction and
displacement from Eqs. (13) and (15).
The dislocations “stored” in the grain boundary represent additional material in the

boundary. With respect to the lattice distortion around the diffusion wedge, the dislocations in the grain boundary exemplify a type of geometrically necessary dislocation [22] that causes nonuniform plastic deformation in the thin film. The eigenvalues measure the rate of decay of each eigenmode. The results showed that the higher eigenmodes decay much faster than the first eigenmode, so the diffusion process is dominated by the first eigenmode [12]. Figures 4, 5, and 6 show several numerical examples. Figure 4 shows the stress intensity

factor normalized by the corresponding value for a crack versus the reduced time *t*∗ = *t/ *for
identical elastic properties of substrate and film material (homogeneous case), rigid substrate
(copper film and rigid substrate), and soft substrate (aluminum film and epoxy substrate).
Table 1 summarizes the material parameters used for the calculation. In the table, *film/subs
*is the ratio of the shear moduli.
Figure 5 shows the opening displacement along the grain boundary for several instants

in time for the case of soft film on rigid substrate, and Fig. 6 shows the traction along the
boundary for various instants in time. These examples illustrate that in the long time limit
*t *→ , the solution approaches the displacement profile of a crack.
For constrained diffusional creep to proceed, both grain boundary diffusion and surface

diffusion need to be active. In the original paper on constrained diffusional creep [12],

copper-rigid homogeneous aluminum-epoxy

0 0.5 1 1.5 2

1

0.8

0.6

0.4

0.2

0

K (t )/ K ∞

Time t/τ

Figure 4. Development of stress intensity factor over time for the cases of an elastically homogeneous film on substrate, a rigid substrate, and a soft substrate. The convergence to the stress intensity factor of a crack is fastest for a rigid substrate and slowest for a soft substrate.

Constrained Grain Boundary Diffusion in Thin Copper Films 7

fil m

th ic

kn es

s ζ/

h f

displacement ux/b

–0.2 0 0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1 1

1.2

0

0.0001 τ 0.01 τ

0.05 τ ∞

Figure 5. Development of grain boundary opening *ux *normalized by a Burgers vector over time, in the case of a
copper film on a rigid substrate. The loading 0 is chosen such that the opening displacement at the film surface
(= 0) at *t *→ is one Burgers vector.

surface diffusion was assumed to be infinitely fast relative to grain boundary diffusion. This implies that grain boundary diffusion is the rate-limiting factor. To account for cases when surface diffusion is slower than grain boundary diffusion, or for cases when surface and grain boundary diffusion occur on comparable timescales, the model was extended in a later publication to also include the effect of surface diffusion [18]. We now briefly review this model. The atomic flux on a free surface may be expressed as

*jss t *=
*sDs
*k*T
*

*s
s
*

(16)

Here *s/s *refers to the gradient in chemical potential along the free surface, and *sDs
*is the temperature-dependent coefficient of surface diffusion. The chemical potential of an

fil m

th ic

kn es

s ζ/

h f

traction σxx(ζ)/σ0

0 0.2

0.2

0.4

0.4

0.6

0.6

0.8

0.8

1 1

1.2 1.4

0

0.01 τ

0.0001 τ

0.05 τ

Figure 6. Development of grain boundary traction *xx/*0 over time for the case of a copper film on a rigid
substrate. The grain boundary traction approaches 0 when *t *→ 0, and the grain boundary traction relaxes to zero
when *t *→ .

8 Constrained Grain Boundary Diffusion in Thin Copper Films

Table 1. Material parameters for calculation of stress intensity factor over the reduced time.

*film *subs *film/subs
*

Cu/rigid 032 — 0 Al/epoxy 03 035 2308 isotropic — — 1

atom at surface is

*s *= 0 −*#s$s t *(17)

where the contribution from elastic energy is neglected, following Rice and Chuang [23].
Here, *#s *is the surface energy and *$s t *is the local surface curvature. Assuming that the
surface slope is small, the governing equation for the surface evolution can be expressed by
Mullins’ equation [24]:

*ẏx t *= −*sDs#s
*k*T
*

*y*′′′′*x t *(18)

where *y *measures the deviation of the surface relative to the flat surface and *ẏ *is the time
derivative of *y*. The boundary and initial conditions for the present problem are given as

*y*′0* t *= cos*& y*′*d/*2* t *= *y*′′′*d/*2* t *= 0 and *yx *0= 0 (19)

where *d *is the grain size in the *x*-direction, with *d/*2 referring to the midpoint of the grain,
as shown in Fig. 3. The parameter

*& *= cos−1
(
*#*gb

2*#s
*

) (20)

is the angle of grain boundary groove to be calculated from *#*s and the grain boundary
energy *#*gb.
Solving the fully coupled grain boundary and surface diffusion problem is rather difficult.

In [18] it was proposed that the two problems be solved separately by assuming that the
surface retains a small slope such that the kernel function *Sz
*remains invariant during
surface diffusion. The link between the two governing Eqs. (9) and (18) is the continuity of
chemical potential at the intersection of the grain boundary and the free surface

gb0* t *= *#s$*0* t *= *#sy*′′0* t *(21)

and that of atomic flux

gb*D*gb*
*′
gb0* t *= −2*sDs#sy*′′′0* t *(22)

For the case of infinitely fast grain boundary diffusion, a characteristic time for stress decay can be defined as

*s *=
k*Th*3f 1− 2

*E*s*D*s*
*(23)

Note that *s *is defined 4* *larger than that adopted in Ref. [18].
In Ref. [18], solutions are reported for different ratios of the rate of surface diffusion

relative to grain boundary diffusion and vice versa. The ratio *' *= gb*/s *determines the
relative importance of grain boundary to surface diffusion, where gb is equal to * *defined
in Eq. (14). For all values of *'*, constrained grain boundary diffusion leads to exponential
relaxation of grain boundary traction, and the relaxation results in a singular stress field at
the root of the grain boundary. Another important finding was that in all cases (even if grain
boundary diffusion is comparable in rate to surface diffusion), the characteristic timescales

Constrained Grain Boundary Diffusion in Thin Copper Films 9

with the cube of the film thickness * *∼ *h*3f . The main result reported in Ref. [18] is that an
effective diffusivity could be defined according to an empirical mixing rule

eff =
4k*Th*3f 1− 2

eff*D*eff*E
*(24)

where

eff*D*eff = gb*D*gb
*'
*

*'*+ *( *(25)

and *( *is a constant to be determined by matching the relaxation behavior for different values
of *'*. In Ref. [18], *( *was determined to be around 1*/*3.

2.2. Average Stress and Thermal Cycling Experiments

Weiss and coworkers [25] extended the continuum model based on a convolution procedure to model their thermal cycling experiments. The average stress in the film is calculated and compared to direct measurement in the laboratory. Because the averaged grain boundary traction follows an exponential time decay [12], the

average stress in the grain boundary can be approximated as

*̂gbt *= 0 exp −0*t/ *(26)
with a geometry-dependent constant

0 = 810+ 3065*h*f*/d *(27)
Here, *d *is the grain size, and 0 is the reference stress in the absence of diffusion, as
discussed in Section 2.1. Equation (27) is an empirical formula and is valid for 02 ≤ *h*f*/d *≤
10. The loading rate of applied stress with respect to temperature is given by

*d*0
*dT
*

= −*'(M*f (28)

Then, with *Ṫ *as the time derivative of the temperature (heating or cooling rate),

*̂gbt *= 0 exp
[
−*
*

∫ *T
T*start

*d
*

*Ṫ
*

] −Mf'Ṫ

∫ *T
T*start

exp
[
− *
Ṫ
*

(∫ *T
*

*d
*

*
*

)] 1

*Ṫ
d
*(29)

where *T *is the temperature at time *t*, and *T*start is the starting temperature. The average
stress * *in the film is related to 0 and *̂gb *as

* *= 0 − 0 − *̂gb* *(30)
where

** *= 4*h*f
*d
*

tanh (

*d
*

4*h*f

) (31)

Equation (30) was obtained from an analysis of a periodic array of cracks in a thin elastic
film analyzed by Xia and Hutchinson [26]. For a given experimentally measured stress * *,
the average stress in the grain boundary *̂gb *is given by

*̂gb *=
1
**
*

* *− 0+ 0 (32)

The average film stress depends on the ratio of grain size to film thickness. With full grain boundary relaxation and no further diffusion, the average stress follows a

thermoelastic line with a reduced slope of

*d
*

*dT
*= −*'(M*f 1−** *(33)

10 Constrained Grain Boundary Diffusion in Thin Copper Films

2.3. Single Edge Dislocations in Nanoscale Thin Films

Mass transport from the film surface into the grain boundary was modeled as a climb of infinitesimal edge dislocations [12, 18]. However, diffusion into the grain boundary must proceed with at least one atomic column. At the nanoscale, dislocation climb in the grain boundary becomes more of a discrete process. Grain boundary diffusion requires insertion of discrete climb dislocations into the grain boundary, a fact that has not been accounted for by the continuum model. To investigate this effect, we consider a single edge dislocation climbing along a grain

boundary in an elastic film of thickness *h*f on a rigid substrate. The elastic solution of edge
dislocations in such a film can be obtained using the methods described in Refs. [15, 27].
The geometry is shown in Fig. 3. A dislocation placed inside the film is subjected to

image forces arising from the surface and the film–substrate interface. The image stress on a
dislocation for different film thicknesses is shown in Fig. 7. Between the film surface and the
film–substrate interface, the image force is found to attain a minimum value at EQ ≈ 04*h*f .

From the energetic point of view, a minimum critical stress is required to allow a single climb edge dislocation to exist in the grain boundary. The thicker the film, the smaller this critical stress. This analysis suggests that consideration of single, discrete dislocations could become very important for nanoscale thin films. For films around 5 nm in thickness, the minimum critical stress for one stable dislocation in the grain boundary approaches 1 GPa. The requirement that an edge dislocation in the film be in a stable configuration could be regarded as a necessary condition for the initiation of constrained grain boundary diffusion.

2.4. Initiation Condition for Diffusion

Considerations similar to those in the previous section were employed to determine condi- tions under which diffusion could initiate. In Ref. [28], a criterion for the initiation of grain boundary diffusion is proposed, following the spirit of the Rice–Thomson model [14]. It is postulated that the condition for initiation of diffusion is a local criterion, independent of the film thickness. The main assumption is that grain boundary diffusion is initiated when a test climb dislo-

cation near the surface is spontaneously inserted into the grain boundary. Considering the force balance on the critical configuration of an edge dislocation located one Burgers vector away from the free surface, as shown in Fig. 3, a critical lateral stress

* *crit0 *>
E
*

81− 2(34)

1 0.8 0.6 0.4 0.2 0

0

–0.5

–1

–1.5

–2

ζ/hf

h*f*=5 nm

10 nm

× 109 Pa

30 nm

100 nm

Im ag

e st

re ss

σ im

Figure 7. Image stress on a single edge dislocation in a thin film for different film thicknesses ranging from 5 to 100 nm.

Constrained Grain Boundary Diffusion in Thin Copper Films 11

can be defined for spontaneous insertion of the test dislocation into the grain boundary.
Note that *E *should be interpreted here as the local modulus of the grain boundary near
the surface, which could be much smaller than the bulk modulus. This result implies that a
critical stress independent of the film thickness is required for the onset of grain boundary
diffusion.

2.5. Nucleation Criterion for Parallel Glide Dislocations

A criterion based on a critical stress intensity factor *K*PG was proposed for the nucleation
of parallel glide dislocations at grain boundary diffusion wedges [28].
The stress intensity factor at the root of a crack or diffusion wedge is defined as

*K *= lim
→*h*f

{
*,*2*
*− *h*f*-s xx*0

} (35)

where *s *refers to the stress singularity exponent that can be determined from Ref. [29].

cos*s*− 2*(*− *.
*1− *. *1− *s
*

2 + *(*− *.
*2

1− *.*2 = 0 (36)

The parameters *( *and *. *are Dundur’s parameters, which measure the elastic mismatch of
film and substrate material.
It is assumed that the diffusion wedge is located close to a rigid substrate, and the corre-

sponding Dundur’s parameters for this case are *( *= −1 and *. *= −02647. The singularity
exponent is found to be *s *≈ 031 for the material combination considered in our simula-
tions (compared to *s *= 05 in the case of a homogeneous material). Close to the bimaterial
interface, we calculate the stress intensity factor

*K *= *A*× lim
→*h*f

(
*ux
*

*
*1− *
/h*f2*s
*

)
*h*f

*s *(37)

where

*A *= *E
*1− 2

1− *(
*4 sin*s
*

(
3− 2*s
*1+ *. *−

1− 2*s
*1− *.
*

) (38)

The stress intensity factor provides an important link between the atomistic results and continuum mechanics. To calculate the stress intensity factor from atomistic data, the atomic displacements of the lattice close to the diffusion wedge are calculated and the stress intensity factor is then determined using Eq. (37). A common approach to study the nucleation of dislocations from defects is to balance

the forces on a test emergent dislocation [14]. The force on dislocations element is referred
to as Peach–Koehler force, which can be written as *d***F***d *= * *· **b**× *d***l**, where *d***l **is the
element length vector and * *is the local stress [16]. A dislocation is assumed to be in an
equilibrium position when *Fd *= 0. Following the Rice-Thomson model [14], we consider the
force balance on a probing dislocation in the vicinity of a dislocation source to define the
nucleation criterion. The probing dislocation is usually subject to an image force attracting
it toward the source, as well as a force resulting from applied stress driving it away from
the source. The image force dominates at small distances, and the driving force caused by
applied stress dominates at large distances. There is thus a critical distance between the
dislocation and the source at which the dislocation attains unstable equilibrium. Spontaneous
nucleation of a dislocation can be assumed to occur when the unstable equilibrium position
is within one Burgers vector of the source.
Nucleation of parallel glide dislocations from a crack in comparison to that from a dif-

fusion wedge is shown in Fig. 8. The crack case is treated similarly as in Ref. [14], and the
forces involved are *F*c because of the crack tip stress field, *F*image from the crack surface
(image dislocation), and *F*step because of the creation of a surface step, which is assumed to
be negligible in comparison of *F*image.
Close to a diffusion wedge, *F*step = 0 because no surface step is involved, and a dipole

must be created to nucleate a parallel glide dislocation from the wedge. This leads to a

12 Constrained Grain Boundary Diffusion in Thin Copper Films

Fstep

Fimage

Fc

FcFcFdipole

(a)

(b)

Figure 8. Force balance on a dislocation near (a) a crack (top) and (b) a diffusion wedge (bottom).

dipole interaction force *F*dipole. The dipole consists of a pair of dislocations of opposite signs,
one pinned at the source and the other trying to emerge and escape from the source. The
pinned end of the dipole has the opposite sign of the climb dislocations in the diffusion
wedge and can be annihilated via climb within the grain boundary. Such annihilation breaks
the dipole free and eliminates the dipole interaction force so that the emergent end of
the dipole moves away to complete the nucleation process. Therefore, it seems that there
could be two possible scenarios for dislocation nucleation at a diffusion wedge. In the first
scenario, the nucleation condition is controlled by a critical stress required to overcome
the dipole interaction force. In the second scenario, the nucleation criterion is controlled
by the kinetics of climb annihilation within the grain boundary, which breaks the dipole
interaction by removing its pinned end and setting the other end free. The force balance on
the dislocation is illustrated in Fig. 8 for two different, subsequent instants in time.
We assume that dislocation nucleation at a diffusion wedge is stress controlled (rather

than kinetics controlled), and the first scenario of dislocation nucleation described above is adopted. This assumption has been verified by recent molecular dynamics simulations [28]. With this assumption, it is possible to define a nucleation criterion in terms of a critical stress intensity factor. We illustrate the critical condition for dislocation nucleation in Fig. 9. A force balance on a dislocation near a crack tip at a rigid bimaterial interface leads to the critical stress intensity factor for dislocation nucleation from a crack

*K*PGcr =
*E*2*bs
*

81− 2(39)

2b b

(a) (b)

Figure 9. Nucleation of parallel glide dislocations from (a) a crack and (b) a diffusion wedge.

Constrained Grain Boundary Diffusion in Thin Copper Films 13

In comparison, a balance of critical stress required to break the dipole interaction in front of a diffusion wedge yields a similar nucleation criterion

*K*PGdw =
*E*2*bs
*

41− 2(40)

For copper with *b *= 256 Å, *E *≈ 150 GPa, and * *= 033, typical values are *K*PGcr ≈ 125MPa×
m*s *and *K*PGdw ≈ 25 MPa × m*s *. We note a factor of 2 difference in the critical *K*-values,
*K*PGdw */K
*

PG cr = 2, for dislocation nucleation at a diffusion wedge and at a crack tip.

2.6. Constrained Grain Boundary Diffusion with Threshold Stress

In recent molecular dynamics and discrete dislocation simulations [28, 30, 31] of constrained
diffusional creep in thin films, it has been shown that the stress does not relax to zero in the
film. Also, a critical stress * *crit0 is found to be necessary to initiate diffusion. In addition to
the theoretical considerations (see Section 2.4), experimental results of stress–temperature
plots during thermal cycling also indicate a threshold stress for diffusion [8]. In the following,
we extend the continuum model to include a threshold stress.
Equation (9) is the governing equation for the problem of constrained diffusional creep.

Assuming a threshold stress *t *for grain boundary diffusion, the boundary conditions are
now modified as

*xx
*= 0* t *= *t *(41)

and
*xx
*

*
*= *h*f* t *=
2*xx
*2

*
*= *h*f* t *= 0 (42)

The initial condition remains the same

*xx
t *= 0= 0 (43)

We consider the possibility that *t *may be different under tension and compression. That is,
no grain boundary diffusion can occur when

−*t < xx*0* t <
*+
*t *(44)

where *xx*0* t *is the stress at the site of initiating climb edge dislocations near the entrance
to the grain boundary.
Using superposition,

*xx
t *= *t *+ *̃
t *(45)

we obtain the governing equation for *̃
t *as

*̃xx
t
*

*t
*= *ED*gbgb*
*

41− 2k*T
*∫ *h*f
0

*S
3̃xx t
*

3
*d *(46)

The boundary conditions are

*̃xx
*= 0* t *= 0 (47)
*̃xx
*

*
*= *h*f* t *=
*2̃xx
*2

*
*= *h*f* t *= 0 (48)

and the initial condition is

*̃xx
t *= 0= 0 − *t *(49)

14 Constrained Grain Boundary Diffusion in Thin Copper Films

These equations are identical to those of Ref. [12], except that the initial condition is effec- tively reduced. Therefore, we can simply use the previous solution [see Eq. (26)] to obtain the average stress along the grain boundary as

*̂gbt *= 0 − *t *exp −0*t/*+ *t *(50)
Figure 10(a) shows a numerical example of the decay of the average stress in the film as
given by Eq. (30) both with and without threshold stress. The parameter +*t *is assumed to
be 65 MPa. For comparison, experimental stress relaxation curves for 200- and 800-nm Cu
films are presented in Fig. 10(b). The experimental results show that the film stress does not
decay to zero but, instead, approaches a plateau value.
We can now generalize the convolution procedure described by Weiss et al. [25]

*d*0
*dT
*

= −*'(M*f (51)

Then,

*̂gbt *= *t *+
(
* *start0 − *t
*

) exp

[
−*
*

∫ *T
T*start

*d
*

*Ṫ
*

]

−Mf'Ṫ
∫ *T
T*start

exp
[
− *
Ṫ
*

(∫ *T
*

*d
*

*
*

)] 1

*Ṫ
d
*(52)

The average stress in the film as measured in experiment is calculated from *̂gbt *by using
Eq. (30).

2.7. Discussion and Summary of Continuum Modeling

We summarize the main results of the continuum modeling: first, the continuum model
predicts an exponential decay of stress in the thin film with a characteristic time * *∼ *h*3f ,
similar to the classical Coble creep [19]. A new defect referred to as a “diffusion wedge”
is built up by the pileup of edge dislocations in the grain boundary [12] on the order of
the characteristic time * *. Second, the continuum model predicts that the deformation field
near a diffusion wedge becomes cracklike on the order of the characteristic time * *[12, 18].
Third, for nanoscale thin films, the effect of discrete dislocations becomes important as the
image stress on a single dislocation reaches several hundred MPa [32]. Fourth, treatment of
atomic diffusion from a free surface into the grain boundary by the climb of discrete edge
dislocations suggests a threshold stress for diffusion initiation independent of film thickness
[28, 30, 32]. Finally, a condition for the nucleation of parallel glide dislocations from a

×108 1.8

1.6

1.4

1.2

1

0.8

0.6

0.4

0.2 0 0.1 0.2 0.3 0.4 0.5

t/t0

σ

Model without threshold stress Model with threshold stress

×108

×104

2.5

time t (sec) 0 1 32 54 6

σ

2

1.5

1

0.5

0

hf=800 nm

hf=200 nm

(a) (b)

Figure 10. (a) Prediction of stress decay by the continuum model with and without threshold stress. (b) Experi-
mental results of stress decay at 250C for *h*f = 200 nm and *h*f = 800 nm.

Constrained Grain Boundary Diffusion in Thin Copper Films 15

grain boundary diffusion wedge has been derived based on the force balance on a probing dislocation, in the spirit of the Rice–Thomson model [32]. The continuum modeling provided key input toward the understanding of the plasticity of

ultrathin films. However, some of the atomic aspects of diffusion could not be understood from the standpoint of continuum modeling alone. This lack of understanding motivated the development of atomistic models.

3. ATOMISTIC MODELING Under the guidance of the continuum model [12, 18, 27], atomistic simulations have proven to be capable of providing a detailed description of how parallel glide dislocations are nucle- ated near a diffusion wedge. In the following sections we summarize the main results of recent molecular dynamics simulations of constrained grain boundary diffusion [28, 30, 32]. Large-scale atomistic modeling provides a helpful tool for investigating material phenom-

ena from a fundamental perspective. The methodology of atomistic simulations together with their strengths and shortcomings are discussed in another chapter of this book [33].

3.1. Large-Scale Atomistic Simulations of Plasticity in Polycrystalline Thin Films

It was proposed that traction relaxation along grain boundaries in thin films significantly alters the mechanism of dislocation nucleation and motion [7, 12]. In a recent paper [30], the effect of grain boundary traction relaxation on the plasticity of polycrystalline thin films was investigated, with the focus being the dynamics of defect evolution in films both with and without traction relaxation. The molecular dynamics simulations showed that the relaxation of grain boundary

tractions changes the dislocation microstructure and triggers different stress relaxation mechanisms in thin films. Threading dislocations dominate when tractions along the grain boundaries are not relaxed, as shown in Fig. 11(a). If the grain boundary tractions are relaxed, parallel glide dislocations dominate, as in Fig. 11(b). Threading dislocations were found to be mostly complete dislocations, whereas a strong tendency to nucleate partial dislocations is seen in the case of parallel glide dislocations in the nanometer-sized grains. This is qualitatively consistent with the results of atomistic modeling the deformation of nanocrystalline materials [34, 35]. Twinning along parallel planes becomes an active defor- mation mechanism at high strain rates. Figure 12 shows that threading dislocations dominate

partial dislocation

stacking fault

(a) (b)

Figure 11. Plasticity in a polycrystalline thin film (a) without and (b) with grain boundary traction relaxation. When grain boundary tractions are not relaxed, threading dislocations dominate plasticity. In contrast, when the tractions are relaxed, parallel glide dislocations dominate. The plot shows dislocations as curved lines, and stacking faults as dark areas.

16 Constrained Grain Boundary Diffusion in Thin Copper Films

Figure 12. Plasticity in a polycrystalline thin film without grain boundary traction relaxation. The threading dislo- cations dominate. The plot shows a view on the surface, showing creation of surface steps as dark lines.

plasticity when grain boundary relaxation is shut down. Threading dislocations leave surface steps as they glide through the crystal. Further studies in Ref. [30] focused on the influence of grain boundary structure on the

nucleation of parallel glide dislocations. A tricrystal model was considered with different types of grain boundaries, as shown in

Fig. 13(a), to investigate the difference between low-energy (symbol A in the plot) and high- energy grain boundaries (symbol B in the plot). Low-energy grain boundaries are composed of an array of misfit grain boundary dislocations that serve as multiple nucleation sites for dislocations. It was observed that low-energy grain boundaries provide more fertile sources for threading dislocation nucleation. At such a boundary, dislocations are often observed to nucleate close to the misfit dislocations. This could be referred to as an intrinsic condition, as the structure of the grain boundary leads to local stress magnification at the misfit dislo- cations that serve as the source of parallel glide dislocations. Because it is possible that the incipient dislocations are nucleated on different glide planes, dislocation reactions may take place when several of them combine to form a single dislocation. The formation of jogs was observed in the dislocation line, generating trails of point defects that significantly hinder the motion of dislocations. In the more homogeneous high-energy grain boundaries, there is inherently no preferred

nucleation site, in which case triple junctions of grain boundaries serve as the nucleation

partial dislocation

stacking fault

grain 2

grain 3

A B

B(a)

grain 1

(b) (c)

Figure 13. A temporal sequence of nucleation of parallel glide dislocations from grain boundaries in a triple junc- tion model. Grain boundary type A refers to low-energy grain boundaries, and B corresponds to high-energy grain boundaries.

Constrained Grain Boundary Diffusion in Thin Copper Films 17

site. A temporal sequence of dislocation nucleation from a grain boundary triple junctions is shown in Fig. 13(a)–(c).

3.2. Atomistic Modeling of Diffusional Creep

Atomistic modeling of thin-film mechanics may become routine with the advent of massively
parallel computers on time- and length-scales comparable with those usually attained in
experimental investigations. Because of the time limitation of the classical molecular dynam-
ics method (time intervals typically *<*10−8 s), simulations of diffusional creep were performed
at elevated temperatures to accelerate the dynamics of grain boundary diffusion [33].
The phenomenon of grain boundary diffusion wedge and the associated dislocation mech-

anisms persist at very high temperatures, making it possible to simulate this phenomenon
using classical molecular dynamics [36, 37]. At elevated temperatures, grain boundary dif-
fusion in a bulk material was recently successfully modeled [36], where grain sizes up to
15 nm were considered in a model system of palladium. Recent work [36, 38, 39] suggests
that at elevated temperatures, the grain boundary structure of metals may transform into a
liquid-like structure with a width up to several nanometers, which was referred to as a “glassy
phase.” Glassy phases in grain boundaries were found in copper at homologous tempera-
tures as low as *Th *≈ 04 [39, 40]. Experimental evidence for glassy intergranular phases was
discussed in Ref. [41]. Such phase transformation at the grain boundary plays a significant
role in the plastic properties at elevated temperatures because each different grain boundary
structure has significant influence on the diffusion [38, 39].
To investigate creep behavior, the sample was loaded according to a prescribed strain

field. An important result of the simulations was that diffusion cannot relax stresses in such thin films completely. These observations are qualitatively consistent with the predictions of Eq. (34) and are in agreement with experimental results. The snapshots in Fig. 14 show how the displacement changes as material diffuses into

the grain boundary. The horizontal coordinates have been stretched by a factor of 10 in
*x*-direction to make the crystal lines clearly visible; that is,

*,x y z-*new = *,*10 *x y z-*orig (53)
This technique helps to highlight the additional half-planes of atoms close to the grain
boundary.
To illustrate diffusional motion of atoms in the grain boundary, we color each atom with

diffusive displacement *z *larger than a few Burgers vectors. Figure 15 plots these atoms for
several instants in time. Diffusion leads to significant surface grooving, with groove depths
of up to several nanometers. One can clearly identify the wedge shape of the diffused
atoms. The atomistic simulations show that atoms inserted into the grain boundary instanta-
neously crystallize, rendering the structure of the grain boundary invariant (this was observed
for temperatures below 1150 K; at higher temperatures, the width of the grain boundary
increases slightly). The atoms transported along the grain boundary may add to either one
of the two grains, illustrating that the continuum mechanics assumptions [12, 18, 27] are also
valid on the atomistic level. A frequently observed phenomenon is the emission of disloca-
tions from the grain boundary on inclined 111 glide planes [6, 34], corresponding to the
“classical” threading dislocations that become operative when stresses in the film are high
enough to allow nucleation of dislocations [34, 36].

1 2 3

Figure 14. Lattice distortion near a diffusion wedge. The field becomes cracklike as the diffusion wedge builds up.
The black lines correspond to the continuum solution of constrained grain boundary diffusion [12, 18] for *t *→ .

18 Constrained Grain Boundary Diffusion in Thin Copper Films

**1 2 3
**

**6
**

**98
**

**54
**

**7
**

Figure 15. Formation of a grain boundary diffusion wedge. Atoms with significant diffusive displacement *z *over
time are highlighted. The plot shows that a diffusion wedge is formed by insertion of material into the grain
boundary. The diffusion wedge develops at a relatively short timescale.

3.3. Atomistic Modeling of Nucleation of Parallel Glide Dislocations from Diffusion Wedges

Here we summarize the main results of the atomistic simulations that focused on the nucle- ation of parallel glide dislocations from diffusion wedges. It was confirmed that thinner films require a higher critical stress for threading disloca-

tion nucleation from the grain boundary, in qualitative agreement with the notion that the strength of thin films increases inversely to the film thickness [6]. In films thinner than 10 nm, extremely high stresses are required to nucleate inclined dislocations, which renders this mechanism almost impossible. This was also verified in a recent paper by Shen [42], in which the scaling of the strength of thin films with respect to the film thickness was investigated. Continuum theory assumes that parallel glide dislocations are nucleated when the stress

field around the diffusion wedge becomes cracklike. Critical stress intensity factors (SIFs)
for dislocation nucleation measured from the atomistic simulations are shown in Table 2
for different simulations. Equation (37) was used to determine the SIF from the atomistic
simulation results of the displacement field near the diffusion wedge. The critical SIF for
the nucleation of parallel glide dislocations appears to be independent of film thickness and
had similar values at *Th *= 08 and *Th *= 09.

For films thinner than 20 nm, parallel glide dislocations are not observed in our bicrys- tal model, as the cohesive strength of the film material is reached before the nucleation condition is met. As reported in Refs. [28, 32], dislocation nucleation at a diffusion wedge can be divided

into different stages, as shown in Fig. 16(a). After the critical SIF is achieved, a dislocation

Table 2. Critical stress intensity factors *K*PG for nucleation of parallel glide dislocations
near a diffusion wedge and a crack.

Temperature *T *(K) Film thickness *h*f (nm) *K*PG (MPa ×m*s*)
Crack

300 27.2 495

Diffusion wedge

1150 27.2 1191 1250 27.2 1135 1250 34.2 1123

*Source: *The data are taken from Ref. [28].

Constrained Grain Boundary Diffusion in Thin Copper Films 19

1

2

3

4

5

1

2

3

4

5

(a) (b)

Figure 16. Atomistic details of the nucleation process of parallel glide dislocations from (a) a diffusion wedge
and (b) a crack. The black line in the first snapshot of column (a) gives the solution of the continuum theory of
constrained grain boundary diffusion [12, 18] for *t *→ .

dipole is formed. One end of the dipole is pinned in the grain boundary, whereas the dislo- cation at the other end of the dipole slides away from the grain boundary. Subsequently, the pinned dislocation is annihilated or “dissolves into” the grain boundary, and the dislocation at the right end of the dipole begins to move away from the nucleation site. The parallel glide dislocation glides on a slip plane parallel to the plane of the film at a distance of a few Burgers vectors above the film–substrate interface (and is therefore completely inside the film material). The dislocation moves a small distance away from the grain boundary to its equilibrium position. When stresses in the film become larger, the film responds by moving farther away from the grain boundary. The nucleation process is highly repeatable. It was found that every time a parallel glide dislocation is nucleated, one climb edge dislocation is annihilated, leading to a decay in stress intensity. After a nucleation event, the time required to nucleate the next parallel glide dislocation is determined by the time required for diffu- sion to recover the critical stress intensity. This time increment is much less than the initial time required to form the diffusion wedge. After the first dislocation is nucleated, more and more parallel glide dislocations are observed. In our confined, finite simulation geometry, the emitted parallel glide dislocations form a “secondary pileup” close to the boundary of the simulation cell. The nucleation of parallel glide dislocations from a crack in a similar geometry is shown in

Fig. 16(b). The chosen loading rate was higher than in the previous case, and the temperature in the simulations was about 300 K. After an incipient dislocation is formed, a dislocation nucleates and moves away from the crack tip. The crack tip is blunted, and each time a parallel glide dislocation is nucleated, one surface step is formed. This process is also highly repeatable. The nucleation of parallel glide dislocations from a crack tip was observed at a loading rate a few orders of magnitude higher than in the case of a diffusion wedge, and there seemed to be no rate limitation in the case of a crack. As in the case of a diffusion

20 Constrained Grain Boundary Diffusion in Thin Copper Films

wedge, the dislocation glides on a parallel glide plane a few Burgers vectors above the film– substrate interface. The critical stress intensity factor for parallel glide dislocation nucleation from a diffusion wedge is about 2.3 times larger than that for a crack. This value was in good agreement with the estimated factor of two based on the Rice–Thomson model.

3.4. Discussion of Atomistic Simulation Results

When classical mechanisms of plastic deformation based on creation and motion of disloca- tions are severely hindered in thin films on substrates, constrained diffusional creep provides an important mechanism for stress relaxation, leading to the formation of a new type of defects called the grain boundary diffusion wedges. We have performed a set of large-scale atomistic simulations to investigate the properties of such diffusion wedges. Atomistic simu- lations show that material is indeed transported from the surface into grain boundaries and that such transport leads to a crack–like stress field causing the nucleation of parallel glide dislocations near the film–substrate interface. The atomistic simulations of parallel glide dis- locations emitted near the root of the grain boundary have further clarified the mechanism of constrained grain boundary diffusion in thin films. Computer simulations provide evidence that diffusion initiation occurs at a critical applied

stress * *crit0 ≈ 16 GPa in copper, independent of the film thickness. Continuum analysis in
Eq. (34) at *T *= 0 K for initiation of diffusion supports this finding and predicts a criti-
cal stress 0 ≈ 6 GPa, also independent of the film thickness. The fact that the continuum
analysis indicates a higher value could be explained by the higher temperature used in the
simulations, in contrast to the continuum mechanics analysis at 0 K. It was found in Ref. [43]
that the critical stresses for dislocation nucleation are about five times smaller at room tem-
perature than at 0 K, thus matching the value measured in the atomistic simulations. It has
recently been shown that stress distribution in thin films over different grains is highly inho-
mogeneous [44]. In some grains, extremely high stresses of several GPa are observed. This
provides sufficiently large stresses to initiate diffusion and a possible explanation of why par-
allel glide dislocation nucleation only occurs at specific grain boundaries in the experiments.
In films thinner than 10 nm, image stresses on climb dislocations can be as large as 1 GPa

(see Section 2.3). This can severely hinder climb-mediated diffusional creep and indicates that the behavior of discrete dislocations needs to be considered in nanoscale thin films. This is supported by the atomistic results showing that stress cannot be completely relaxed in extremely thin films. These results are in qualitative agreement with experimental results that often show large residual stresses in extremely thin films [2]. The diffusion wedge has properties similar to a crack, but it requires a larger SIF to

nucleate a dislocation. The reason for this is that in the case of a diffusion wedge, a dis-
location dipole needs to be formed, and the dipole interaction force is twice as strong as
the image force on an emergent dislocation near a crack tip. We have studied common and
distinct properties observed in atomistic simulations of the two kinds of defects (crack ver-
sus diffusion wedge) along the grain boundary at elevated temperatures. For a crack, the
following was observed: as the applied stress 0 is increased, the normal stress *xx *along
the grain boundary remains zero throughout the film thickness, consistent with the traction-
free condition along the crack faces; the loading rate can be much higher than in the case
of diffusional creep (there is no rate-limiting factor on dislocation nucleation); dislocation
nucleation occurs at a relatively small stress intensity factor; and dislocation nucleation starts
with an incipient dislocation close to the crack tip.
In contrast, for a diffusion wedge the following can be summarized: the loading rate must

be slow enough for diffusion to remain as the dominant relaxation mechanism—at high load-
ing rates, dislocation activities on inclined planes are observed instead of grain boundary dif-
fusion; to nucleate a parallel glide dislocation, dislocation climb in the grain boundary must
occur to annihilate part of an incipient dipole (on atomistic timescales, this is an extremely
slow process—(for a crack, dislocation nucleation is fast); dislocation nucleation starts when
the stress intensity factor reaches a critical value required to create a dislocation dipole near
the diffusion wedge; and there exists a minimal thickness for parallel glide dislocation nucle-
ation: If the film is very thin, the applied stress reaches the cohesive strength of the material
before the critical stress intensity factor *K*PG for dislocation nucleation is reached.

Constrained Grain Boundary Diffusion in Thin Copper Films 21

The two types of defects have significantly different timescales associated with dislocation
nucleation. A crack is a ready source for dislocations, whereas a diffusion wedge has an
intrinsic characteristic time associated with the dislocation climb. No essential difference
in the mechanism of parallel glide dislocation nucleation from a diffusion wedge has been
observed in the temperature range 08 *< Th < *09.

4. EXPERIMENTAL STUDIES Since its proposal in 1999, the model for constrained diffusional creep [12] has been further developed in light of experimental results [7, 35]. Weiss et al. [25] invoked the constrained diffusional creep model to explain the occurrence of a stress drop observed during the first heating cycle of thin copper films. They also used constrained diffusional creep to model the stress–temperature curves measured during thermal cycling of several films and found good agreement for a 500 nm copper film, albeit by assuming a very large grain size. Later, the dis- covery of parallel glide dislocations [45] provided experimental support for the constrained diffusional creep model. In turn, constrained diffusional creep furnished the basis for the interpretation of certain experimental results, especially in regard to the mechanisms for the creation and emission of parallel glide dislocations. In the following sections, we present key experimental results and their connection to the model of constrained diffusional creep.

4.1. Thermomechanical Behavior of Thin Copper Films

A series of thin copper films, ranging in thickness from 2 m down to 35 nm, was pro- duced by magnetron sputtering under ultrahigh vacuum. All films were annealed immediately thereafter in the deposition chamber, without breaking vacuum, yielding oxide-free unpas- sivated copper films. The substrates used for the films were (001)-oriented single crystalline silicon that had been coated with 50 nm amorphous silicon oxide and 50 nm amorphous silicon nitride before copper deposition. Thus, the interface between film and substrate is crystalline/amorphous. The average grain size, as determined from focused ion beam (FIB) and transmission electron microscopy (TEM) measurements, ranged from 1.5 times the film thickness (for the 2-m film) to 2.5 times the film thickness (for the 100-nm film). X-ray diffraction measurements revealed that all films had a strong [111] fiber texture, albeit with a small population of [100]-oriented grains for films thicker than 200 nm. Thus, the vast majority of grains within each film, including the grains imaged using TEM and presented below, are oriented with a (111) plane parallel to the substrate. The thermomechanical behavior of each film was determined by subjecting it to a thermal

cycle between room temperature and 500C. Because copper possesses a higher coefficient of thermal expansion than silicon, an equibiaxial compressive stress evolves in the film during heating, and a tensile stress evolves during cooling. This coupling of stress and tempera- ture in the film/substrate system enables the thermomechanical loading of thin films, and a variety of thermal paths can be employed to generate compressive and tensile stresses of varying magnitude. In the examples presented below, simple thermal cycles starting at room temperature, progressing up to 500C and then returning to 50C were performed. Heating and cooling rates were 6 K/min, except for final cooling below 100C, which proceeded at 4 K/min because of the limited cooling capacity of the oven. An example of the thermomechanical behavior of an ultrathin copper film is presented

in Fig. 17. The stress within this 100-nm film was initially 360 MPa. During heating, the tensile stress decreased and a compressive stress was achieved by 210C. The compressive stress rose in magnitude with further heating but then dropped after reaching a maximum at 260C. The stress level decreased slightly during heating to 320C and then became more compressive during heating to 500C. From this point on, the stress evolution was monotonic with temperature: cooling produced a tensile stress and subsequent heating in the second cycle produced a compressive stress again. No stress drops were observed in the second cycle or later. The stress drop that occurs above 250C in the 100-nm film, as well as in all thicker films

measured in this study, appears to be the result of constrained diffusional creep, as previously

22 Constrained Grain Boundary Diffusion in Thin Copper Films

800

600

400

200

0

–200

–400 0 100 200 300 400 500

Temperature (°C)

S tr

es s

(M P

a)

Cycle 1 Cycle 2

Figure 17. The first two thermal cycles of a 100-nm copper film. During the first heating segment, a stress drop is observed at 260C, which is attributed to constrained diffusional creep. Subsequent thermal cycles are highly repeatable, with no stress drop.

explained by Weiss et al. [25]. The temperature of 250C, which represents a homologous temperature of 0.4, is apparently sufficient to activate surface and grain boundary diffusion and lead to the formation of diffusion wedges during heating. Only when the relaxation of stress at the grain boundaries is saturated, in this case at a temperature well above 300C, does stress increase again. The lack of the stress drop during the second and later thermal cycles is explained by the stress level: at 250C in the second cycle, film stress is only −26 MPa, which may be insufficient to cause outward diffusion of copper atoms from the grain boundaries, and therefore no stress relaxation occurs. The second cycle in Fig. 17 deviates from the first cycle during the final stages of cooling

(below ∼200C), perhaps because of the thermal drift of the sample. This, however, is an inherent error in the experimental technique and gives a good measure of the error one should assign to the absolute stress values that are reported. The film strength is typically taken to be the flow stress measured at the end of a thermal cycle. Thus, in this case, the strength of the 100-nm film would simply be given by the average of the two stresses measured at 50C at the end of cooling: 676± 39 MPa. Although film strength generally increases with decreasing film thickness [2, 25, 46], unpas-

sivated copper films deviate from this trend below 400 nm [7]. Instead of rising continuously, the film strength exhibits a plateau between 650 and 700 MPa for films 400 nm and thinner. This is apparent in Fig. 18, which shows the second thermal cycle for a 200-nm copper film, along with the second cycle for the 100-nm film from Fig. 17. The two films exhibit virtu- ally identical thermomechanical behavior, with the film strengths at 50C (695± 9 MPa for the 200-nm film) and the flow stresses at 500C matching very closely. Although the 500C flow stress does become slightly more compressive in thinner films, the plateau in 50C film strength persists down to a thickness of 35 nm. To explore whether the stress–temperature cycles, such as those shown in Figs. 17 and 18,

represent steady-state behavior, a 50-nm copper film was thermally cycled at two different rates. In addition to the standard 6 K/min rate, a complete cycle was performed at 1 K/min; both are portrayed in Fig. 19. Because of the extremely small film thickness, this comparison allows us to investigate whether grain boundary diffusion and constrained diffusional creep are able to proceed to completion within the time frame of these experiments. The heat- ing segments of each cycle are somewhat different, but they yield roughly the same 500C

Constrained Grain Boundary Diffusion in Thin Copper Films 23

800

600

400

200

0

–200

–400 0 100 200 300 400 500

Temperature (°C)

S tr

es s

(M P

a)

100 nm Cu

200 nm Cu

Figure 18. The second thermal cycle of a 200-nm copper film, along with the second cycle of the 100-nm film from Fig. 18. Both films exhibit the same thermomechanical behavior, with the same stress levels at 500 and 50C.

flow stress; namely, 322 ± 14 MPa. However, there is a large discrepancy in stress evolu- tion during cooling, especially between 350 and 200C. Within this temperature range, the lower cooling rate permits significantly more stress relaxation. This indicates that the stan- dard 6 K/min used throughout this study is too fast to allow constrained diffusional creep to reach steady state and that grain boundary diffusion may be the rate-limiting step in this

800

600

400

200

0

–200

–400 0 100 200 300 400 500

Temperature (°C)

S tr

es s

(M P

a)

1 K/min cooling

6 K/min heating 6 K/min cooling 1 K/min heating

Figure 19. Two thermal cycles of a 50 nm copper film, conducted at different heating and cooling rates. Although the same 500C flow stress is reached in both cases, the stress evolution during cooling is seen to depend strongly on cooling rate, indicating that the standard rate of 6 K/min is too fast to allow constrained diffusional creep to proceed to completion.

24 Constrained Grain Boundary Diffusion in Thin Copper Films

temperature regime. The final cooling curves below 200C appear to be offset by approx- imately 200 MPa, but they exhibit the same slope, indicating that constrained diffusional creep is equally slow in both cases, or that constrained diffusional creep is no longer able to provide stress relaxation below 200C.

4.2. Transmission Electron Microscopy Observations of Dislocation Behavior

Thermal cycling experiments were also conducted *in-situ *in the TEM to elucidate the defor-
mation mechanisms underlying the thermomechanical behavior illustrated in Figs. 17–19.
These figures clearly showed that films in this thickness range (≤200 nm) exhibit a plateau
in strength, which indicates that the Mathews–Freund–Nix mechanism involving threading
dislocations is saturated or has been replaced. As discussed below, *in-situ *observations of
dislocation motion revealed that a new, alternative deformation mechanism does indeed
dominate in ultrathin, unpassivated copper films.
Figure 20 shows an example of parallel glide dislocations in a small grain within a 200-nm

copper film. There are four dislocations oriented roughly perpendicular to the diffraction
vector **g **in the TEM micrograph. During subsequent heating, these dislocations moved in
the direction of **g **and underwent glide across the entire width of the grain. In this image of
the [111]-oriented grain, the projected width of all inclined *2*111*3 *planes is 70 nm. On the
basis of their length and the extent of their motion (up to 300 nm), the dislocations must
therefore have moved on the (111) plane parallel to the film/substrate interface. Because of
this, the observed motion was termed “parallel glide” [7].
An important consequence of glide occurring parallel to the film/substrate interface is that

the Burgers vector of the dislocations, which must lie within the (111) glide plane, is parallel to the plane of the film. However, the biaxial film stress that evolves in the film during thermal cycling does not produce a resolved shear stress on this plane, and it therefore cannot be directly responsible for the motion of parallel glide dislocations. Both of these points, however, are addressed by the constrained diffusional creep model [12], which aids the interpretation of these experimental results. As predicted by the theoretical simulations and discussed further below, parallel glide is a consequence of constrained diffusional creep. Additional experimental observations have clarified several details of this deformation

mechanism. Figure 21 shows another example of parallel glide dislocations, extending from

**g** = [220]

50 nm

Figure 20. Transmission electron micrograph of parallel glide dislocations in a 200-nm copper film. These edge-
oriented dislocations underwent glide in the direction of the diffraction vector **g**. On the basis of the extent of their
motion, the only possible glide plane is the (111) plane parallel to the film–substrate interface.

Constrained Grain Boundary Diffusion in Thin Copper Films 25

**g** = [111]

200 nm

**g** = [220]

200 nm

(a) (b)

Figure 21. (a) Three parallel glide dislocations are visible in this transmission electron (TEM) micrograph of a
200-nm copper film. Their (111) glide plane, which is also the plane of this image, contains the Burgers vector
**b**. (b) TEM of the same location, but with a different diffraction vector. The three parallel glide dislocations are
invisible here, indicating that **b **is perpendicular to *,111̄-. *On the basis of these images, **b **must be parallel to *,2̄20-,
*and the dislocations thus have predominantly edge character.

the twin in the middle of the micrographs to the grain boundary at the right. In Fig. 21(a),
three dislocations are visible, but in Fig. 21(b), they are invisible, indicating that the Burg-
ers vector is perpendicular to the *,111̄- *direction. Moreover, the Burgers vector lies within
the (111) plane and must therefore be parallel to the *,2̄20- *diffraction vector in Fig. 21(a).
The dislocations thus have predominantly edge character, as they are oriented roughly per-
pendicular to **b **and **g **in Fig. 21(a). This is a general feature of parallel glide dislocations,
which are always emitted with edge character. During subsequent glide, which can involve
interaction with other dislocations and with obstacles such as twins and grain boundaries,
the parallel glide dislocations may bow into a curved shape and attain mixed character.
*In-situ *TEM observations [7] have revealed that parallel glide dislocations are emitted

from sources at triple junctions and grain boundaries. Typically, these grain boundaries (or one of the grain boundaries leading to the triple junction) are oriented roughly perpendicular to the Burgers vector. This indicates that the diffusion wedge added to the grain boundary is indeed essentially an array of edge dislocations, which are eventually emitted and that undergo parallel glide. In all cases in which it was measured [7, 45], the tilt angle between grains bounding a par-

allel glide source was found to lie between 6 and 6.5. Tilting and diffraction experiments near a 112 zone axis revealed that the tilt angle is 6, not 66 (i.e., the 60 ambiguity inher- ent to the comparison of [111] electron diffraction patterns was eliminated). This provided a key piece of information for input into the atomistic simulations [28, 30, 32] and allowed the introduction of grain boundaries that were suitable for constrained diffusional creep and the emission of parallel glide dislocations in the simulated cooling experiments. By tilting the specimen, one can image the inclined grain boundaries and observe the

position of parallel glide dislocations by determining exactly where they terminate. TEM micrographs [7] reveal that the dislocations meet the grain boundaries at the film–substrate interface. Thus, the parallel glide plane is located at the bottom of the grain, albeit some small distance above the silicon nitride. From high-resolution TEM micrographs [47], it was observed that the silicon nitride layer is not atomically flat but, instead, contains hills and val- leys, with a peak-to-valley height of approximately 2 nm. Given the observation that parallel glide occurs over large distances, without cross-slip events, the glide plane cannot lie directly at the film–substrate interface, but instead must be situated several nanometers above it.

26 Constrained Grain Boundary Diffusion in Thin Copper Films

Multiple families of parallel glide dislocations are sometimes also observed, as shown in Fig. 22. Three families, each oriented roughly perpendicular to one of the three 1̄10 directions within the (111) film plane, are located at the bottom, right, and top-left corners of the grain. In contrast to other examples of parallel glide (e.g., Fig. 19), the dislocations in Fig. 22 accommodate strain symmetrically within the film plane. This appears to be a result of the grain size, as smaller grains rarely exhibit multiple families of dislocations. Finally, copper films passivated with an aluminum oxide layer were also investigated for

comparison with the observations of the unpassivated films presented above. Although their
thickness and grain size are identical to the unpassivated films, passivated films do not exhibit
parallel glide. Instead, threading dislocations are generated. As they glide through the film
on inclined *2*111*3 *planes, they deposit straight segments along 1̄10 directions at the film–
substrate interface, as shown in Fig. 23. Passivation of the film surface completely shuts down
parallel glide, providing strong evidence that surface diffusion is an essential component of
this deformation mechanism.

4.3. Interpretation of Experimental Observations

The observations of parallel glide can be explained with the help of the constrained diffu- sional creep model. The finding that an unpassivated film surface is required for parallel glide supports the claim that constrained diffusional creep is the mechanism responsible for stress relaxation and plastic deformation in ultrathin copper films. The stress drop observed during the first heating cycle of a copper film (e.g., in Fig. 17) is explained nicely by the con- strained diffusional creep model. Finally, other models of thin film deformation are unable to explain parallel glide, as the biaxial film stress that evolves during thermal cycling does not generate a resolved shear stress on the (111) plane parallel to the substrate and therefore cannot directly drive the dislocation motion presented here. Because parallel glide is the dominant mode of dislocation motion in copper films 200 nm

thick and thinner, and as parallel glide appears to be a consequence of constrained diffu- sional creep, we can check whether constrained diffusional creep mediates the entire amount of plasticity during a thermal cycle. According to the constrained diffusional creep model, atoms diffuse from the surface into the grain boundaries of a film under tension, assuming that temperature and stress are sufficiently high. For an appropriately oriented grain, this

**g** = [220]

200 nm

Figure 22. Transmission electron micrograph of a grain exhibiting multiple families of parallel glide dislocations, one each at the bottom, right, and top left corners of the grain. The families are each oriented roughly perpendicular to one of the three 1̄10 directions within the (111) film plane.

Constrained Grain Boundary Diffusion in Thin Copper Films 27

**g** = [111]

500 nm

Figure 23. When passivated with an aluminum oxide layer, 200-nm copper films no longer exhibit parallel glide. Instead, threading dislocations are activated, which glide through the film and generate straight slip traces at the film–substrate interface.

is equivalent to adding edge dislocations at the grain boundary. For each edge dislocation,
the addition of the extra half-planes increases the grain length by *b*, the magnitude of the
Burgers vector. We can therefore calculate how many parallel glide dislocations should exist
within a given grain:

*d*grain*4*plastic = *Ndb *(54)
where *d*grain is the grain length parallel to **b**, *4*plastic is the plastic strain measured from the
stress–temperature plot for a thermal cycle, and *Nd *is the number of parallel glide disloca-
tions.
Application of Eq. (54) to TEM observations yields excellent agreement. In Fig. 24, a

schematic of the grain and parallel glide dislocations from Fig. 20, tilted back to simulate a
three-dimensional view, has been drawn. The parallel glide plane, shaded gray in the figure,
is located at the bottom of the grain, near the film–substrate interface. The dislocations are
roughly perpendicular to **b **and are distributed over the length of the grain. The relevant
values for Eq. (54) are *d*grain = 400 nm and *4*plastic = 0.30% for a 200-nm copper film. The
required number of dislocations is thus *Nd *= 47. Four dislocations were observed in the
grain. The remainder, 0.7, is insufficient to necessitate the emission of a fifth dislocation.

**b **hf

Figure 24. Schematic of the grain and parallel glide dislocations from Fig. 20. The glide plane is at the bottom of the grain, near the film–substrate interface, and the dislocations are distributed across the width of the grain. The number of parallel glide dislocations corresponds to the average amount of plastic strain exhibited by the film during thermal cycling.

28 Constrained Grain Boundary Diffusion in Thin Copper Films

More specifically, however, this “missing” partial dislocation should simply remain in the grain boundary, as there is no driving force for its emission. In other words, a criterion for the occurrence of parallel glide is the presence of multi-

ple dislocations in a grain boundary. Once an additional dislocation, or perhaps a partial dislocation, is formed in the grain boundary, the repulsive force between them causes the emission of the first dislocation into the grain. Based on comparisons between Eq. (54) and TEM observations, the presence of a partial dislocation appears to be the more likely case. This leads to a slightly modified equation that accounts for the emission criterion:

*d*grain*4*plastic =
(
*Nd *+

1 2

)
*b *(55)

For larger grains, where many dislocations should undergo glide, the difference between
Eqs. (54) and (55) may be insignificant. However, as grain size decreases toward the
nanoregime, the exact form of Eq. (55) becomes important. The additional 1*/*2 may need
to be replaced by a different quantity, but this possibility requires further investigation.
A consequence of Eqs. (54) and (55) is that there is a critical length scale for the formation

of a parallel glide dislocation; namely, roughly *b/4*plastic. In grains smaller than this quantity,
less than a single dislocation is required to mediate the entire amount of plasticity. In this
case, the partial dislocation remains in the grain boundary. Indeed, almost no dislocation
motion is observed in 50-nm copper films, which undergo a plastic strain of 0.25% during
a thermal cycle and therefore have *b/4*plastic = 102 nm. The median grain size in a 50-nm
copper film is only 99 nm (i.e., more than half of the grains require less than a single parallel
glide dislocation, and only a small minority of the grains are sufficiently large [*>*150 nm] for
parallel glide to occur at all). In 50-nm films, parallel glide was observed in relatively large
grains (i.e., *>*200 nm), supporting the concept of a critical length scale.
The multiple families of parallel glide dislocations in Fig. 22 indicate that plastic strain

is accommodated along all possible 1̄10 directions within the (111) film plane, as should be the case for an equibiaxial stress state. In many cases, especially in smaller grains, parallel glide occurs in only one or two directions. This is likely a result of the character of the remaining grain boundaries, which may allow some degree of diffusion but still be unsuitable for the emission of parallel glide dislocations. In addition, in small grains, the emission of parallel glide dislocations from a grain boundary may alter the stress state sufficiently to hinder the insertion of diffusion wedges into the remaining grain boundaries. Copper films with 200 nm thickness have been presented here to illustrate parallel glide.

Although these films, which yielded the TEM observations presented above, are thicker than those that were simulated, it has been experimentally confirmed [7] that parallel glide is the dominant mode of dislocation motion in films between 50 and 200 nm thick. Note that 50-nm films exhibit little dislocation activity, but that which does occur always takes place via parallel glide. Parallel glide is a consequence of constrained diffusional creep (i.e., the actual deforma-

tion mechanism is constrained diffusional creep). The motion of parallel glide dislocations does not mediate plasticity. Rather, it redistributes, across the width of the grain, the strain fields of the edge dislocations contained within the diffusion wedges. By the time parallel glide occurs, constrained diffusional creep has already provided plastic strain by increasing the grain dimension. Thus, in terms of strain accommodation, it does not matter where the parallel glide dislocations are situated within the grain. This is fundamentally different from conventional dislocation slip and threading dislocation motion, which progressively mediate plastic strain as glide occurs.

5. MODELING THE EXPERIMENTAL RESULTS WITH CONTINUUM THEORY

In addition to aiding the interpretation of the TEM observations of parallel glide, the con- strained diffusional creep model can be used to generate the stress–temperature curve for a thermal cycle. This was done by Weiss et al. [25] for copper films, although certain discrep- ancies between simulated and experimental curves could not be overcome. The model has

Constrained Grain Boundary Diffusion in Thin Copper Films 29

since been modified to include a threshold stress for constrained diffusional creep, which has led to better agreement in the high-temperature regime, as presented below.

5.1. Experimental Estimate of the Threshold Stress

Experimental results provided an estimate of the threshold stress in thin copper films. Constant-temperature stress relaxation experiments were performed with films ranging in thickness from 100 nm to 2 m. During cooling from 500C, the sample was held at 250C for 16 h, and the amount of stress reduction was monitored. The final stress measured in each film is listed in Table 3. For film thicknesses between 100 nm and 1 m, the relaxed stress level is very consistent, with an average value of 81 MPa [see also Fig. 10(b)]. This is surprising, given the large variation in stress as a function of film thickness, as determined from standard thermal cycles. Only the 2-m film exhibits a lower stress. The data in Table 3 indicate that the concept of a threshold stress for constrained dif-

fusional creep is valid for all copper films below micrometer size. It is also in agreement with the notion of an athermal stress component in thin films, as discussed by Kobrinsky et al. [2, 48, 49]. The athermal stress cannot be relaxed by thermally activated mechanisms such as constrained diffusional creep. Kobrinsky’s results [2, 48] also indicate a thickness- independent threshold stress for silver films between 250 and 860 nm.

5.2. Fit of the Continuum Theory to Experimental Results

In this section we discuss some fits of the extended continuum mechanics model with thresh- old stress to the experimental results.

5.2.1. Modeling of Thermal Cycling Experiments As reported in Refs. [25, 50], the grain boundary diffusivity for copper is given by

gb*D*gb*T *= 5× 10−15 exp
(
− *Qb
*R*T
*

) m3 s−1 (56)

with activation energy *Qb *= 104 kJ/mole. We consider two film thicknesses of 100 and
600 nm. The threshold stresses for the grain boundary average stress is estimated from the
experimentally measured average stress in the film by Eq. (32) and are approximated to
be +*t *= 65 MPa and −*t *= −65 MPa. In the following text we assume that *E *= 124 GPa,
* *= 034 and *'( *= 13× 10−5 K−1. In Figs. 25 and 26 we show a comparison between exper-
iment and the continuum model with threshold stress for steady-state thermal cycling of thin
films of thickness *h*f ≈ 100 nm (Fig. 25) and 600 nm (Fig. 26).
In both cases, some qualitative agreement is found, especially at elevated temperatures.

For the 600-nm film, the agreement at low temperatures during the cooling cycle (upper
curve) is not as good as for high temperatures. This could partly be explained by the fact that,
because of the rather large film thickness, not only parallel glide dislocations present, but
threading dislocations are also nucleated and relieve stresses, as observed in the experiment
in Ref. [7]. During cooling of the film, diffusional creep dominates at high temperatures, but
threading dislocations may dominate at low temperatures. Because the continuum model
does not account for threading dislocations, the stresses at low temperatures (upper left
corner in the plot) are overestimated. At high temperatures during the heating cycle, the
stress at the grain boundary relaxes to the value associated with the threshold stress −*t *, and

Table 3. Average film stresses measured after isothermal relaxation at 250C for 16 h.

*h*f (nm) 100 200 800 1000 2000

250Crelaxed 80 73 85 85 43

*Note: *All films between 100 nm and 1 m exhibit nearly the same stress, with an average value of
81 MPa. On the basis of the experimental data, 65 MPa was taken to be the threshold stress at the
grain boundary; that is, the high-temperature flow stress of thin copper films following maximum
relaxation by constrained diffusional creep.

30 Constrained Grain Boundary Diffusion in Thin Copper Films

300 400 500 600 700 800 –4

–2

0

2

4

6

8 ×108

1st cycle experiment

2nd cycle experiment

1st cycle continuum model

2nd cycle continuum model

F ilm

s tr

es s σ

(P a)

Temperature T (K)

Figure 25. Fit of continuum model with threshold stress to the experimental data from Fig. 17. The film thickness
is *h*f = 100 nm and grain boundary diffusivities are as in Ref. [50].

the experimental measurements and continuum modeling of the average stress in the film agree quite reasonably. The fits show that the introduction of a threshold stress based on experimental data rep-

resents an improvement of the previous model. Unlike in Ref. [25], where the grain size had to be adjusted to improve agreement, the parameters used here are identical to those used in the experiments (same grain size, same ratio of grain size to grain diameter, same cooling and heating rates, etc.). The most important difference is that without the threshold stress, the numerically estimated curves are very “thin” in contrast to the experimental results. The thermal slope, which is apparent at high temperatures during the cooling cycle, can only be reproduced with the generalized model, including a threshold stress.

300200 400 500 600 700 800 –5

0

5

10 ×108

1st cycle experiment

2nd cycle experiment

1st cycle continuum model

2nd cycle continuum model

F ilm

s tr

es s σ

(P a)

Temperature T (K)

Figure 26. Fit of continuum model with threshold stress to experimental data. The film thickness is *h*f = 600 nm
and grain boundary diffusivities are as in Ref. [50].

Constrained Grain Boundary Diffusion in Thin Copper Films 31

5.2.2. Estimation of Diffusivities from Experimental Data
Another interesting aspect is the estimation of diffusivities from the experimental data. It
can be observed in Fig. 26 that the stress decrease in the first heating cycle is larger in the
experiment than in the simulation. When the diffusivities reported in the literature are used,
the maximum compressive stress is overestimated by a factor of two (a similar observation
was also made in the paper by Weiss et al. [25], Fig. 10). This could be because the diffusivi-
ties used in the continuum mechanics model are smaller than the actual experimental value.
Figure 27 shows a fit of the continuum mechanics model to the experimental data, with a
threshold stress −*t *= −65 MPa for a 600-nm film, as obtained from experiment (see discus-
sion above). Fitting the maximum compressive stress to the experimental results by adjusting
the diffusivity yields a diffusivity 80 times higher than in the literature. Possible reasons for
this could be the fact that the grain boundary structure strongly influences the diffusivities.
In the literature, a dependence of the diffusivities on grain boundary structure has been
proposed (for further discussion, see, for instance, Refs. [28, 35]). Such considerations, have
not been taken into account in modeling diffusion in thin films so far, but they may explain
the observations discussed above.
The fitted grain boundary diffusivity is given by

gb*D*gb*T *= 4× 10−13 exp
(
− *Qb
*R*T
*

) m3 s−1 (57)

The characteristic time for a film with *h*f = 20 nm at a temperature of about 90% of the
melting point is then on the order of 10−8 s. This supports the idea that constrained grain
boundary diffusion can be modeled with classical molecular dynamics, as such simulations
are typically limited to a timescale of around 1× 10−8 s [28, 33].
Finally, we compare calculations with the new estimate for diffusivities to the experimental

results. In Fig. 28 we show a comparison of experiment and the continuum model with
threshold stress for steady-state thermal cycling of thin films with film thickness *h*f ≈ 100 nm.
In Fig. 29 we show a comparison of experiment and the continuum model with threshold
stress for film thickness *h*f ≈ 600 nm.

6. MAP OF PLASTIC DEFORMATION MECHANISMS The results from the numerical modeling together with the experimental findings [7] were used to qualitatively describe different deformation mechanisms that occur in sub- micron thin films. It was proposed in Ref. [30] that there exist four different deformation regimes: (1) deformation with threading dislocations, (2) constrained diffusional creep with

400300200 500 600 700 800

–4

–2

0

2

4

×108

F ilm

s tr

es s σ

(P a)

Temperature T (K)

experimental data high diffusivity low diffusivity

Figure 27. Fit of diffusivities to the experimental data based on the first heating curve of the 600-nm film.

32 Constrained Grain Boundary Diffusion in Thin Copper Films

300 400 500 600 700 800 –4

–2

0

2

4

6

8 ×108

1st cycle experiment

2nd cycle experiment

1st cycle continuum model

2nd cycle continuum model

F ilm

s tr

es s σ

(P a)

Temperature T (K)

Figure 28. Fit of continuum model with threshold stress to experimental data. The film thickness is *h*f = 100 nm,
and grain boundary diffusivities are fitted to experimental data.

subsequent parallel glide, (3) constrained diffusional creep without parallel glide, and finally,
for the thinnest films, and (4) no stress relaxation mechanism (no diffusion and no dislo-
cation motion possible). A schematic “deformation map” is plotted in Fig. 30. This plot
shows the critical applied stress to initiate different mechanisms of deformation as a function
of the film thickness. Note that it is assumed that loading is applied very slowly and that
temperature is sufficiently high for diffusive processes to occur.
The critical applied stress to nucleate threading dislocations scales with 1*/h*f [4–6]. The

1*/h*f scaling has also been found in two-dimensional molecular dynamics simulations [34].
For films thicker than a material-dependent value, regime (1) is the dominant deformation
mechanism. For thinner films, the stress necessary to nucleate threading dislocations is larger
than that required to initiate grain boundary diffusion. In this regime (2), diffusion dominates
stress relaxation and causes a plateau in the flow stress, as shown by experimental results

300200 400 500 600 700 800 –3

–2

–1

0

2

4

×108 1st cycle experiment

2nd cycle experiment

1st cycle continuum model

2nd cycle continuum model

F ilm

s tr

es s σ

(P a)

Temperature T (K)

1

3

5

6

Figure 29. Fit of continuum model with threshold stress to experimental data. The film thickness is *h*f = 600 nm,
and grain boundary diffusivities are fitted to experimental data.

Constrained Grain Boundary Diffusion in Thin Copper Films 33

la te ra l

st re ss

σ 0

theoretical strength

film thickness hf

D

A

BC

≈25 nm ≈400 nm

≈1.6 GPa

Figure 30. Deformation map of thin films as a function of film thickness, summarizing different stress relaxation mechanisms in ultrathin films on substrates [30].

[7, 8] and later by discrete dislocation modeling [31]. Parallel glide helps to sustain grain
boundary diffusion until the overall stress level is below the diffusion threshold, which is
independent of the film thickness. For yet thinner films, our MD simulations revealed that
grain boundary diffusion stops before a sufficient stress concentration for triggering parallel
glide is obtained. The onset of regime (3) can be described by the scaling of the critical
nucleation stress for parallel glide with 1*/hs*f (*s *= 05 for the homogeneous film/substrate
case). In this regime, the flow stress again increases for smaller films because of due to the
back stress of the climb dislocations in the grain boundary, effectively stopping further grain
boundary diffusion. If the applied stress is lower than the critical stress for diffusion, no
stress relaxation is possible. This is referred to as regime (4).
The yield stress of thin films resulting from these considerations is summarized in Fig. 31

for different film thicknesses. For thicker films, the strength increase is inversely proportional
to the film thickness, as has been shown in several theoretical and experimental studies
[1, 5, 7, 9]. If the film thickness is small enough such that grain boundary diffusion and
parallel glide are the prevailing deformation mechanisms, the film strength is essentially
independent of *h*f , as seen in experiment [7] and as later shown by the discrete dislocation
model [31]. However, for films thinner than *h*f ≈ 30 nm, discrete dislocation modeling [31]
predicts again an increase in strength with decreasing film thickness.

≈0.04 GPa

≈1/400 nm

≈0.64 GPa

yi el

d st

re ss

inverse film thickness 1/hf

th re

ad ing

dis loc

at ion

s

A B C

constrained grain boundary diffusion and parallel glide dislocations

1/hf

Figure 31. Strength of thin films as a function of film thickness. The film thickness of *h*f ≈ 400 nm, as well as the
plateau yield stress of 064 GPa, are taken from experimental results [7, 8]. Further details are given in Ref. [30].

34 Constrained Grain Boundary Diffusion in Thin Copper Films

The existing understanding is mostly qualitative. A few data points from experiment, sim- ulation, or theory were taken to draw the deformation map. Future studies could focus on a more quantitative study of the different deformation mechanisms and provide a systematic study of transition from one regime to the other. Mesoscopic simulations [31], as applied to constrained diffusional creep and nucleation of parallel glide dislocations, could play an important role in these studies.

7. SUMMARY AND CONCLUSIONS In this chapter we have reviewed recent theoretical, numerical, and experimental activities investigating the mechanical properties of submicron thin films on substrates. Theoretical modeling predicted the existence of so-called grain boundary diffusion wedges.

This model was later successfully used to explain experimental observations of parallel glide dislocations, which verified the prediction that dislocations with Burgers vector parallel to the film plane could be emitted from the tip of a diffusion wedge. The experimental results, together with the theoretical predictions, guided molecular dynamics studies that confirmed this concept at the atomic scale. A new and significant result reported in this chapter is the modeling of thermal cycling

of ultrathin films using continuum theories. With the new concept of a threshold stress, we qualitatively reproduced experimental results. Additional discussion was devoted to estimates of the grain boundary diffusivity from experimental data. It was found that diffusivities are higher than reported in the classical literature [50], which may be attributed to the dependence of diffusivities on the grain boundary structure. A deformation map describing numerous novel deformation mechanisms in thin metal films was developed to summarize the main findings. The examples reported in this chapter exemplify the richness of phenomena that can occur

as the dimensions of materials shrink to nanometer scale.

ACKNOWLEDGMENTS We thank Gerhard Dehm and Alexander Hartmaier for many helpful discussions on the experimental and modeling work described here. M.B. acknowledges fruitful discussions with Dieter Wolf on modeling diffusional creep with molecular dynamics. The simulations reported in this work were carried out at the Max Planck Society Supercomputer Center in Garching.

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