Constrained Grain Boundary diffusion in thin copper films

Constrained Grain Boundary diffusion in thin copper films

(Parte 5 de 6)

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 parallel 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 6 (i.e., the 60 ambiguity inherent to the comparison of [1] 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 creepand 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 valleys, with a peak-to-valley height of approximately 2 nm. Given the observation that parallel glide occurs over large distances, without cross-slipevents, 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. 2. Three families, each oriented roughly perpendicular to one of the three 110 directions within the (1) 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. 2 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 21113 planes, they deposit straight segments along 110 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 diffusional 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 constrained diffusional creepmodel. 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 (1) 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 diffusional 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 2. 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 110 directions within the (1) film plane.

Constrained Grain Boundary Diffusion in Thin Copper Films 27 g = [1]

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 sliptraces 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:

where dgrain is the grain length parallel to b, 4plastic is the plastic strain measured from the stress–temperature plot for a thermal cycle, and Nd is the number of parallel glide dislocations.

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 dgrain = 400 nm and 4plastic = 0.30% for a 200-nm copper film. The required number of dislocations is thus Nd = 4 7. 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 multiple 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:

For larger grains, where many dislocations should undergo glide, the difference between Eqs. (54) and (5) may be insignificant. However, as grain size decreases toward the nanoregime, the exact form of Eq. (5) 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 (5) is that there is a critical length scale for the formation of a parallel glide dislocation; namely, roughly b/4plastic. 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/4plastic = 102 nm. The median grain size in a 50-nm copper film is only 9 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. 2 indicate that plastic strain is accommodated along all possible 110 directions within the (1) 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 deformation 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 slipand 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 constrained diffusional creepmodel 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 discrepancies 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 500 C, the sample was held at 250 C 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 diffusional 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 thicknessindependent 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 threshold 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

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

iment and the continuum model with threshold stress for steady-state thermal cycling of thin films of thickness hf ≈ 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 creepdominates 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 250 C for 16 h.

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

1st cycle experiment

2nd cycle experiment

1st cycle continuum model

2nd cycle continuum model

Film stress σ (Pa)

Temperature T (K)

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

(Parte 5 de 6)

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