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# Constrained Grain Boundary diffusion in thin copper films

(Parte **2** de 6)

surface retains a small slope such that the kernel function Sz | remains invariant during |

In [18] it was proposed that the two problems be solved separately by assuming that the 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

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

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 ∼ h3f. The main result reported in Ref. [18] is that an effective diffusivity could be deﬁned according to an empirical mixing rule

where

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

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 0 2 ≤ hf/d ≤ 10. The loading rate of applied stress with respect to temperature is given by

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

T d

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

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

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

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 hf 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 ≈ 0 4hf. 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 conditions 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 dislocation 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

–2 ζ/hf

30 nm 100 nm

Image stress 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 1 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 KPG 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

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

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 corresponding Dundur’s parameters for this case are ( =− 1 and . =− 0 2647. The singularity exponent is found to be s ≈ 0 31 for the material combination considered in our simulations (compared to s = 0 5 in the case of a homogeneous material). Close to the bimaterial interface, we calculate the stress intensity factor

where

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 dFd = · b × dl, where dl 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 diffusion wedge is shown in Fig. 8. The crack case is treated similarly as in Ref. [14], and the forces involved are Fc because of the crack tipstress ﬁeld, Fimage from the crack surface (image dislocation), and Fstep because of the creation of a surface step, which is assumed to be negligible in comparison of Fimage.

Close to a diffusion wedge, Fstep = 0 because no surface stepis 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 Fdipole. 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 tipat a rigid bimaterial interface leads to the critical stress intensity factor for dislocation nucleation from a crack

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

For copper with b = 2 56 Å, E ≈ 150 GPa, and = 0 3, typical values are KPGcr ≈ 12 5MPa× ms and KPGdw ≈ 25 MPa × ms. We note a factor of 2 difference in the critical K-values, KPGdw /KPGcr = 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 creepin 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 and

The initial condition remains the same

We consider the possibility that t may be different under tension and compression. That is, no grain boundary diffusion can occur when where x 0 t is the stress at the site of initiating climb edge dislocations near the entrance to the grain boundary.

Using superposition, we obtain the governing equation for t as

S |

14 Constrained Grain Boundary Diffusion in Thin Copper Films

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

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]

Then,

T d

The average stress in the ﬁlm as measured in experiment is calculated from gb t 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 ∼ h3f, similar to the classical Coble creep[19]. A new defect referred to as a “diffusion wedge” is built upby the pileupof 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

Model without threshold stress Model with threshold stress 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) Experimental results of stress decay at 250 C for h = 200 nm and h = 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].

(Parte **2** de 6)