Molecular-dynamics simulations of stacking-fault-induced dislocation annihilation in pre-strained ultrathin single-crystalline copper films

We report results of large-scale molecular-dynamics (MD) simulations of dynamic deformation under biaxial tensile strain of pre-strained single-crystalline nanometer-scale-thick face-centered cubic (fcc) copper films. Our results show that stacking faults, which are abundantly present in fcc metals, may play a significant role in the dissociation, cross-slip, and eventual annihilation of dislocations in small-volume structures of fcc metals. The underlying mechanisms are mediated by interactions within and between extended dislocations that lead to annihilation of Shockley partial dislocations or formation of perfect dislocations. Our findings demonstrate dislocation starvation in small-volume structures with ultra-thin film geometry, governed by a mechanism other than dislocation escape to free surfaces, and underline the significant role of geometry in determining the mechanical response of metallic small-volume structures.


I. Introduction
A broad range of modern technologies relies increasingly on the use of nanometer-scale-thick metallic films. Typically, over 500 processes are involved in the development of micro-and nano-scale devices. During these processes, the constituent materials are subjected to various thermo-mechanical environments, which introduce strain in the nano-scale-thick metallic films, causing them to undergo plastic deformation.
Such materials response to strain has significant impact on the ductility, hardness, and strength of the metallic films and may lead eventually to their failure; the mechanical response of device interconnects in microelectronics offers typical examples. [1][2][3][4] The mechanical behavior of single-crystalline metallic materials undergoing plastic deformation is a direct consequence of the nucleation and motion of dislocations, as well as a variety of dislocation-dislocation interactions. For bulk materials, such mechanisms are well understood and phenomenological models have been developed for their deformation. These phenomenological models, however, cannot be extended to the nanoscale and, more specifically, to describing mechanical response of ultra-thin metallic films.
Unlike bulk metals, metallic nanostructures and other small-volume structures exhibit, for example, ultra-high strength. Examples range from whiskers with small diameters that were reported (as early as almost 6 decades ago) to have much higher strength than bulk metals 5,6 to the uniaxial compression of nanometer-scale pillars of face-centered cubic (fcc) metals that yielded flow strengths comparable to the materials' ideal shear strength. [7][8][9][10][11] In bulk metals subjected to external stress, work hardening is caused by an increase in the dislocation density and by a sharp decrease in plastic flow; the former is due to rapid dislocation multiplication through double cross-slip and Frank-Read-type mechanisms, while the latter is due to interactions between dislocations in multiple slip planes that lead to the formation of sessile dislocations that act as barriers to dislocation motion, as well as due to pile-up of dislocations at obstacles such as grain boundaries and sessile dislocations. In metallic nano-meter-thick films and (in general) small-volume structures, however, it has been argued that depletion of dislocations causes the observed ultra-high strength. [7][8][9][10][11][12][13] For example, in compression experiments of Ni single-crystalline nanopillars with a high initial dislocation density, the dislocation density always decreased during the application of strain; 11 subsequently, the nearly defect-free crystal deformed elastically until new dislocations nucleated. In ultra-thin metallic films, a fundamental understanding of the atomistic mechanisms of deformation dynamics is essential for accurate property predictions, better reliability analysis, and design of higher-quality devices.
The deformation mechanisms, however, are particularly difficult to study experimentally; for example, in high-strain-rate deformation of single-crystalline fcc metals such as Cu, certain strain relaxation time scales are in the sub-nanosecond range. 14 Atomic-scale and other fine-scale dynamical simulation methods provide ideal means for analyzing the deformation mechanisms in ductile thin films by direct monitoring of the dynamical response of the materials at the atomic/microscopic scale. 15,16 First-principles density-functional-theory calculations and large-scale moleculardynamics (MD) simulations have been used extensively toward a fundamental understanding of plastic flow initiation and nucleation of dislocations, as well as investigations of plastic deformation during nanoindentation of metal surfaces. 17,18 These techniques also have been used to explore and analyze dislocation intersections, 19,20 deformation processes at shock loading conditions in metallic bulk 21 and thin-film [22][23][24] 4 materials, as well as amorphization of fcc crystalline metallic nanowires under strainrate-limited shock loading at controlled temperatures. 25 In addition, discrete dislocationdynamics simulations have been used to study size effects in nanopillars of fcc metals; these simulations of fcc nanopillars have shown that dislocation escape through the free surfaces and depletion or inactivation of dislocation sources play a significant role in the plastic response of small-volume structures. 13,[26][27][28] The fundamental mechanisms of dislocation interactions and their possible annihilation in ultra-thin metallic films, however, remain largely unexplored and, therefore, elusive.
In this article, we report results of an atomic-scale analysis of dislocationdislocation interaction mechanisms based on MD simulations of dynamic deformation of free-standing single-crystalline ultra-thin fcc copper films. The thin-film samples used in our analysis were pre-strained to introduce dislocations, consistently with samples used in experimental studies of the mechanical behavior of small-volume structures that had a high initial dislocation density. 11 Our results indicate that the interactions within and between simple extended dislocations, each of which consists of two Shockley partial dislocations with complementary Burgers vectors separated by a stacking fault, may contribute significantly to dislocation depletion in nanometer-scale fcc metals. We identify different classes of stacking fault-dislocation interactions and present a detailed analysis of dislocation depletion in these films mediated by these mechanisms. Our findings suggest that dislocation escape to free surfaces may not be the primary mechanism of dislocation depletion in small-volume structures of fcc metals with ultrathin film geometry.
The article is structured as follows. The simulation methods and analysis

II. Simulation Methods and Analysis of Simulation Results
In our MD simulations, we employed slab supercells consisting of singlecrystalline copper films with periodic boundary conditions applied in the x-and ydirections in a Cartesian representation; the film planes were oriented normal to the z axis. Vacuum space above the top and below the bottom surfaces of the films was used in order to generate free surfaces and mimic free-standing thin films. The Cartesian x, y, and z axes were taken along the [110] , [112] , and The interatomic interactions were modeled using an embedded-atom-method (EAM) potential, parameterized for copper. 29 This potential was validated by comparison 6 of its predictions with those of ab initio calculations. 30 For our MD simulations of thinfilm dynamic deformation under biaxial tensile strain, we used the public-domain computer software LAMMPS. 31 In our simulations, the temperature was kept constant at T = 100 K by explicitly rescaling the atomic velocities at each time step. In the MD simulations of thin-film dynamic deformation under biaxial tensile strain, all the films used were pre-treated before conducting the MD simulations of dynamical straining in order to generate the initial dislocation distributions. Specifically, the thin films were pre-strained in biaxial tension to a specified strain level at a constant true strain rate of 2.01 ! 10 11 s -1 (equally in each lateral direction). Under this mode of loading, three {111} crystallographic planes are activated; the Schmidt factor for all possible slip directions in the (111) plane, which is also the orientation of the thin film surface, is negligibly small. After the desired strain level was reached, isothermal-isostrain MD simulations were performed in order to equilibrate the thin-film structure. We analyzed the mechanical response of three thin-film samples; two of these, samples 1 and 2, had initial (unstrained-state) thicknesses of 4.38 nm and 11.27 nm, respectively. The third one, sample 3, was prepared by compressing sample 1 after its tensile straining back to its original lateral dimensions at the same constant high strain rate and then equilibrating the film's structure by isothermal-isostrain MD simulation. In their pre-treatment, the thinfilm samples discussed in this article were subjected to biaxial tensile strain at a level of 8%. For pre-treatment under biaxial tensile strain at strain levels over the range from 6% to 10%, the resulting microstructure of the equilibrated thin films remained qualitatively identical. The final thicknesses of samples 1, 2, and 3 were 3.84 nm, 9.76 nm, and 4.43 nm, respectively.
These samples contained predominantly simple extended dislocations that either extended across the film thickness or were threading dislocation half loops emanating from the film surfaces. The initial dislocation density, ! , of these films ranged from 1.0 ! 10 17 m -2 to 3.2 ! 10 17 m -2 ; ! = l/V where l is the total length of all the dislocations in the film and V is the film's volume. In samples 1 and 3, many extended dislocations spanned across the film thickness, while only a few threading dislocation half loops were present.
In the thicker film, however, the population of threading dislocation half loops was comparable with that of the extended dislocations that spanned across the film thickness.
We note that this initial film microstructure is similar to that of the Ni nanopillars used in mechanical annealing experiments; 11 in those experiments, similar dislocation types were observed, i.e., most of them extended across the thickness of the nanopillar and a few of them were threading dislocation half loops at the surface of the nanopillar, and the dislocation density was ~ 10 15 m -2 . After this film pre-treatment stage, the final steadystate values of the von Mises shear stress, " vM , for samples 1, 2, and 3 were 530, 460, and 105 MPa, respectively; " vM is defined as the second invariant of the stress tensor and is equally in each lateral direction) and the evolution of the film structure was monitored in detail.
For monitoring the structural response of the thin film to the applied biaxial strain, we employed common neighbor analysis (CNA) in order to identify atoms in locally perfect hexagonal close-packed (hcp) and fcc lattice arrangements. 33 We used the publicdomain computer software ATOMEYE 32  The stress in the material was calculated using the Virial formula. 34 We implemented two methods for the computation of the film's volume (involved in both the stress and the dislocation density computation). In the first method, we constructed on the thin-film surface a grid of multiple elements of known lateral dimensions; we computed the thickness of each such element as the distance, in z, between the two atoms on the opposite surfaces of the film that were the farthest located from its center plane on each surface within the element. The total volume of the film was then computed as the sum of the element volumes. In the second method, we estimated the thickness of the film as the distance, in z, between the two atoms on each of the opposite film surfaces that were the farthest located from the center plane of the film. The first method underestimates the total film volume, while the second method overestimates it. The stress values reported 9 in the stress-strain curve plots of this article were computed using the second method.
Irrespective of the method used for film volume computation, the corresponding qualitative features of the stress evolution in dynamic deformation remained identical except at a stage close to failure; at this stage, the volume computed according to the first method excludes (incorrectly) the pits formed on the surface. The difference in the stress values computed by the two methods were ~100-200 MPa; the first method yielded consistently higher stress values. The mechanical response of the thin film to the applied biaxial strain exhibits three different stages; these stages are indicated in Fig. 1(a) and are termed as Stage I, Stage II, and Stage III. Figure 1(b) shows three-dimensional views of the MD simulation cell at various states, which are also marked in Fig. 1 to a sharp decrease in the thin-film stress. In sample 1 (Fig. 1), the propagation of the newly nucleated dislocations leads quickly to film failure; we note that, due to film failure, we do not observe successive strain burst events similar to those observed in experiments on metallic nanopillars. [9][10][11][12][13] Sample 3 also responds to applied biaxial strain in a similar fashion. The evolution of " vM , !, and A sf in samples 2 and 3 is shown in Fig. 2(a) and Fig. 2(b), respectively. In samples 1 and 3, the stress increases rapidly and then remains constant until significant dislocation depletion and annihilation of stacking faults occur.

III. Mechanical Behavior and Underlying Physical Mechanisms
Reduction in plastic flow due to dislocation depletion leads to build-up of stress in the thin films and, eventually, to nucleation of new dislocations. As dislocation density in sample 3 reaches a minimum faster (due to much higher initial dislocation density than in sample 1), the regime of constant stress is smaller in sample 3 than in sample 1. Samples 1 and 3 are considerably thin; in these samples, the nucleation and propagation of new dislocations lead quickly to thin-film failure. In sample 2, however, the newly nucleated dislocations propagate without causing failure; instead, they lead to successive cycles of dislocation dissociation and annihilation followed by dislocation nucleation events.
Consequently, the strain bursts that correspond to nucleation of new dislocations and the associated sharp decrease in the thin-film stress are identified clearly in Fig. 2(a); here, it is worth noting that these strain bursts are similar to those observed in the nanopillar experiments. [9][10][11][12][13] In Fig. 2(a), green shaded zones correspond to deformation stages associated with plateaus and local minima in the stress curve; these stages are  Effectively, this reaction leads to annihilation of the gliding extended dislocation and break-up of the obstacle dislocation into two extended dislocations. This reaction mechanism is similar to the Fleischer cross-slip mechanism. 36 As shown in Fig. 3(c), one of these two dislocations is annihilated.
In the thicker film (sample 2), we also observed dislocation nucleation any dissociation or transmission. The former of these interactions belongs to category II, while the latter one belongs to category I(b). These interaction mechanisms, however, were observed only rarely.

IV. Discussion
A stage of constant or decreasing stress during dynamic deformation, known as the "easy glide" stage, also has been observed in the compression experiments of Ni and Au nanopillars. [7][8][9][10][11][12] In the nanopillar experiments, this regime was followed by a regime of increasing stress. The stress increase in these nanopillars has been attributed to reduced plastic flow due to a reduction in the dislocation density. In the case of Ni nanopillars which had a very high initial dislocation density (~ 10 15 m -2 ), the experiments showed that the dislocation density at the end of the "easy glide" stage was always far lower than that at the start; in some cases, the nanopillars were dislocation free before a steep increase in the stress. 11 Even though the dislocation annihilation mechanisms that In our MD simulations, the highest stresses computed were far below the ideal shear strength of the material; the maximum shear stresses observed were up to 45% of the ideal shear stress. Moreover, the maximum dislocation velocities computed in our simulations ranged between 400 m/s and 500 m/s, which is well below the sound velocity in the material (3703 m/s for this potential under unstrained conditions). These comparisons suggest that the high strain rates applied in our MD simulations of dynamic deformation do not affect the underlying dislocation dynamics. The initial dislocation density in our study is on the order of 10 17 m -2 , which is considerably higher than the initial dislocation density of 10 15 m -2 in the submicron-diameter Ni pillar compression experiments. 11 This high dislocation density is mainly due to the nanometer-scale film thickness and the thin-film geometry, where dislocation escape to the free surfaces is not likely since the dislocation lines terminate at and are confined between the parallel free surfaces of the thin film. Furthermore, high dislocation densities in MD simulations are inevitable; for example, the dislocation density in the thin films used in our simulations with only a single dislocation extended across the film thickness is ~7.9 ! 10 14 m -2 . The reported mechanisms of dislocation depletion, however, are not artificially induced due to the high initial dislocation density since these mechanisms are similar to those identified in recent studies of dislocation interactions with twin boundaries in samples with lower dislocation densities. [40][41][42][43] Finally, the initially deformed (pre-strained) structures preceding the dynamical loading are not stress free after the thermal equilibration procedure. The resulting residual elastic strain in our thin films at the beginning of the dynamic deformation simulations represents more realistically the residual strains in the metallic thin films in complex and heterogeneous device structures.

V. Summary
We have analyzed the mechanical behavior of nanometer-scale-thick fcc Cu films based on large-scale MD simulations of dynamic deformation under biaxial tensile strain.
Our analysis identified the physical mechanisms of defect dynamics that govern this mechanical behavior. Specifically, the analysis revealed that stacking faults are likely to provide an important source of dissociation, cross-slip, and annihilation of dislocations in small-volume structures of fcc metals. Our findings suggest that in small-volume structures with ultra-thin film geometry, dislocation escape to free surfaces may not be the primary mechanism of dislocation starvation.

20
The mechanisms analyzed in this study lead to dislocation annihilation or