Constrained Grain Boundary diffusion in thin copper films

Constrained Grain Boundary diffusion in thin copper films

(Parte 3 de 6)

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.


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 nucleated 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 phenomena from a fundamental perspective. The methodology of atomistic simulations together with their strengths and shortcomings are discussed in another chapter of this book [3].

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. 1(a). If the grain boundary tractions are relaxed, parallel glide dislocations dominate, as in Fig. 1(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 deformation mechanism at high strain rates. Figure 12 shows that threading dislocations dominate

partial dislocation stacking fault

(a) (b)

Figure 1. 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 dislocations 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 highenergy 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 dislocations 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 grain 1

(b) (c)

Figure 13. A temporal sequence of nucleation of parallel glide dislocations from grain boundaries in a triple junction 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 dynamics method (time intervals typically <10−8 s), simulations of diffusional creepwere performed at elevated temperatures to accelerate the dynamics of grain boundary diffusion [3].

The phenomenon of grain boundary diffusion wedge and the associated dislocation mechanisms persist at very high temperatures, making it possible to simulate this phenomenon using classical molecular dynamics [36, 37]. At elevated temperatures, grain boundary diffusion in a bulk material was recently successfully modeled [36], where grain sizes upto 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 upto 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 ≈ 0 4 [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,

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 upto several nanometers. One can clearly identify the wedge shape of the diffused atoms. The atomistic simulations show that atoms inserted into the grain boundary instantaneously 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 dislocations from the grain boundary on inclined 1 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].

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 12 3

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 nucleation of parallel glide dislocations from diffusion wedges.

It was confirmed that thinner films require a higher critical stress for threading dislocation 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 = 0 8 and Th = 0 9. 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 for nucleation of parallel glide dislocations near a diffusion wedge and a crack.

Temperature T (K) Film thickness h (nm) K (MPa × m )

Diffusion wedge

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

Constrained Grain Boundary Diffusion in Thin Copper Films 19 1

(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 dislocation 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 diffusion 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 dislocations are severely hindered in thin films on substrates, constrained diffusional creepp rovides 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 simulations 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 dislocations 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 ≈ 1 6 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 temperature 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 inhomogeneous [4]. 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 parallel glide dislocation nucleation only occurs at specific grain boundaries in the experiments.

(Parte 3 de 6)