Dislocation interactions in thin FCC metal films

Abstract

High strength, high hardening rates, and strong Bauschinger-like effects in thin films have been attributed to constraints on dislocation motion and dislocation interactions. To understand these phenomena, dislocation interactions in (1 1 1) and (0 0 1) oriented single crystal FCC films were studied using dislocation dynamics simulations. Interactions on intersecting glide planes resulted in junction formation, annihilation, or attractive non-junction-forming configurations, while dislocations on parallel glide planes formed dipoles. The configurations adopted by interacting dislocations, and thus the strengths of the interactions, were found to be sensitive to the applied strain, film thickness, crystallographic orientation, and boundary conditions. Different interactions thus dominate film behavior in different ranges of film thickness and applied strain. Interactions are stronger on unloading than on loading. Interactions involving three or more dislocations are found to be different from pairwise interactions. The results suggest that simple analytical calculations are unlikely to describe film phenomena but that full 3-D simulations can be used to understand many features of thin film mechanical behavior.

Introduction

The metallizations in nanofabricated systems, such as integrated circuits, often fail due to stress-driven processes such as decohesion, cracking, void formation, and stress-migration. Thus, in light of the ongoing efforts to reduce characteristic feature sizes in such systems, it is imperative to understand mechanical properties of metals at small scales. Metal thin films have different dimensional and microstructural constraints on deformation than bulk metals and show quite different mechanical properties. It is well known, for example, that thin metal films support much higher stresses than bulk metals, at both low and high temperatures [1], and show higher strain hardening rates [2], [3].

Plastic deformation in metals at low temperatures (less than 0.3T_m, T_m being the melting temperature) is mostly due to motion of dislocations. Freund [4] and Nix [1] developed a model based on the concept of critical strain for the stability of a single dislocation in a strained layer [5], [6], [7] to explain the ability of metal films to sustain high stresses at low temperatures. As shown in Fig. 1, a dislocation moving into a film consists of a threading segment that moves in response to applied stress and stationary misfit segment(s) deposited at the interfaces. In a passivated film, misfit segments are deposited at both film-substrate and film-passivation interfaces (Fig. 1(a)). In an unpassivated film (Fig. 1(b)), only one misfit dislocation is deposited at the film-substrate interface and the threading segment exits at the free surface of the film. The basic concept of the model proposed by Freund and Nix is that such a dislocation will move into a film in response to an applied strain only when the elastic strain energy, W_layer, relieved when the dislocation advances a unit distance provides enough energy, W_dislocation, to deposit a unit length of misfit dislocation. We denote the critical strain required to move a single dislocation into a film of given thickness as the “channeling strain”(ε_ch) [3]. The channeling strain is determined by the condition that

$$\text{W}{}_{\mathrm{<mtext>layer</mtext>}}\text{=W}{}_{\mathrm{<mtext>dislocation</mtext>}}\text{,}$$

where

$$\text{W}{}_{\mathrm{<mtext>layer</mtext>}}\text{=Yεb}\text{cos}\text{φ}\text{cos}\text{λ}\text{h}\text{sin}\text{φ}\text{,}$$

and

$$\text{W}{}_{\mathrm{<mtext>dislocation</mtext>}}\text{=}\text{μb}{}^2\text{4}\text{π(}\text{1−ν}\text{)}\text{4−ν}\text{4}\text{ln}\text{2h}\text{r}{}_0\text{−}\text{3}\text{4}\text{1−2ν}\text{4}\text{(}\text{1−ν}\text{)}\text{−}\text{1}\text{2}\text{cos}\text{2φ}\text{.}$$

Here, Y is the biaxial modulus, b the Burgers vector, h the film thickness, φ the angle between the glide plane normal and the film normal, λ the angle between the Burgers vector and film normal, ε the in-plane biaxial strain, μ the shear modulus, ν Poisson’s ratio, and r₀ the dislocation core radius. Eq. (2) is given by Nix [1], and Eq. (3) is obtained for a 60° dislocation following Freund’s work [4].

This model predicts the often-observed dependence of room temperature strength on the inverse of film thickness (e.g. see [8]), but significantly under-predicts the stress levels [2] and does not account for strain hardening. Various analytical models for strain hardening have been proposed based on particular dislocation-dislocation interactions. Freund [9] considered the blocking of a threading dislocation by a misfit dislocation on an intersecting glide plane. Nix [3] estimated the effect of an array of parallel misfit dislocations on a threading dislocation on an intersecting glide plane. Willis and Jain [10], [11] and Weinacht and Bruckner [12] have calculated the strain required to move a dislocation into a film in the presence of an array of dislocations on parallel glide planes.

Such analytical models use mathematical descriptions that apply only to straight, infinitely long dislocations and depend on assumptions as to what configurations dislocations will adopt. For example, the configuration that is assumed to determine the blocking strength in Freund’s model is depicted schematically in Fig. 2. The threading dislocation is assumed to be repelled by the misfit segment and the blocking effect is accordingly thought to arise from the threading segment being confined to a channel of thickness $\text{h}{}^{\mathrm{\ast }}$ which is less than the film thickness. The critical strain ε_crit needed to just push the threading segment past the misfit dislocation is thus assumed to be the channeling strain for a film of thickness $\text{h}{}^{\mathrm{\ast }}$. In reality, dislocation interactions are very complicated since dislocations can continuously change their shape. If the net force between two dislocations is repulsive in one configuration, the same dislocations may adopt another configuration in which the net force will be attractive. Hence, we cannot expect such analytical models to be accurate, or, to the extent that configuration changes are important, even to predict trends in blocking strength with film thickness correctly.

Dislocation dynamics (DD) simulations provide a better means to study dislocation interactions. The basic idea is to compute the forces on a dislocation arising from the applied stresses, from interactions with other dislocations, and from line tension effects. The dislocations are then moved in response to these forces. To the extent that the simulation is accurate, dislocation interactions and the evolution of dislocation structure can be simulated in a realistic way. One can then observe the configuration changes due to various dislocation interactions and compare the strengths of those interactions. These interactions often generate characteristic structures which can be identified experimentally to confirm the mechanisms [13].

In this paper, we present results from simulations of pairwise dislocation interactions in (111) and (001) oriented single crystal FCC films, excluding dislocation crossing interactions that result in jog formation. We have studied the effect of orientation and film thickness on interaction strength using a dislocation dynamics program (PARANOID) that has been specifically developed to accurately model the behavior of curved dislocations [14]. Our goal is to identify important interaction/reaction mechanisms. In particular, we rank the interactions by strength and look for configurational changes that may help us understand the high strength and strain hardening rates in thin films and nanostructures. Dislocation structure formation is of interest because experimental measurements of stress during thermal cycling show large Bauschinger and memory effects [15] which may be attributed to dislocation behavior.