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# Constrained Grain Boundary diffusion in thin copper films

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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 represents 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.

×108 1st cycle experiment

2nd cycle experiment

1st cycle continuum model 2nd cycle continuum model

Film stress σ (Pa)

Temperature T (K)

Figure 26. Fit of continuum model with threshold stress to experimental data. The ﬁlm thickness is h = 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 diffusivities 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 discussion 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

The characteristic time for a ﬁlm with hf = 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, 3].

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 hf ≈ 100 nm. In Fig. 29 we show a comparison of experiment and the continuum model with threshold stress for ﬁlm thickness hf ≈ 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 submicron thin ﬁlms. It was proposed in Ref. [30] that there exist four different deformation regimes: (1) deformation with threading dislocations, (2) constrained diffusional creepwith

Film stress σ (Pa)

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

1st cycle experiment

2nd cycle experiment

1st cycle continuum model

2nd cycle continuum model

Film stress σ (Pa)

Temperature T (K)

Figure 28. Fit of continuum model with threshold stress to experimental data. The ﬁlm thickness is h = 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 dislocation 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/hf [4–6]. The

1/hf 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

×108 1st cycle experiment

2nd cycle experiment

1st cycle continuum model

2nd cycle continuum model

Film stress σ (Pa)

Temperature T (K)

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

Constrained Grain Boundary Diffusion in Thin Copper Films 3 lateral stress σ 0 theoretical strength

film thickness hf D

Figure 30. Deformation mapof 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/hsf (s = 0 5 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 hf, as seen in experiment [7] and as later shown by the discrete dislocation model [31]. However, for ﬁlms thinner than hf ≈ 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 yield stress inverse film thickness 1/hf threading dislocations 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 ≈ 400 nm, as well as the plateau yield stress of 0 64 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, simulation, 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.

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 creepwith molecular dynamics. The simulations reported in this work were carried out at the Max Planck Society Supercomputer Center in Garching.

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