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# Constrained Grain Boundary diffusion in thin copper films

(Parte **1** de 6)

CHAPTER 13

Constrained Grain Boundary Diffusion in Thin Copper Films

Markus J. Buehler, T. John Balk, Eduard Arzt, Huajian Gao Max Planck Institute for Metals Research, Stuttgart, Germany

1. Introduction | 2 |

2. Continuum Modeling | 3 |

2.1. Basics of the Continuum Modeling | 3 |

2.2. Average Stress and Thermal Cycling Experiments | 9 |

2.3. Single Edge Dislocations in Nanoscale Thin Films | 10 |

2.4. Initiation Condition for Diffusion | 10 |

2.5. Nucleation Criterion for Parallel Glide Dislocations | 1 |

Threshold Stress | 13 |

2.7. Discussion and Summary of Continuum Modeling | 14 |

3. Atomistic Modeling | 15 |

Polycrystalline Thin Films | 15 |

3.2. Atomistic Modeling of Diffusional Creep | 17 |

Dislocations from Diffusion Wedges | 18 |

3.4. Discussion of Atomistic Simulation Results | 20 |

4. Experimental Studies | 21 |

4.1. Thermomechanical Behavior of Thin Copper Films | 21 |

Dislocation Behavior | 24 |

4.3. Interpretation of Experimental Observations | 26 |

CONTENTS 2.6. Constrained Grain Boundary Diffusion with 3.1. Large-Scale Atomistic Simulations of Plasticity in 3.3. Atomistic Modeling of Nucleation of Parallel Glide 4.2. Transmission Electron Microscopy Observations of

Continuum Theory | 28 |

5.1. Experimental Estimate of the Threshold Stress | 29 |

to Experimental Results | 29 |

5. Modeling the Experimental Results with 5.2. Fit of the Continuum Theory

ISBN: 1-58883-042-X/$35.0 Copyright © 2005 by American Scientiﬁc Publishers All rights of reproduction in any form reserved.

Handbook of Theoretical and Computational Nanotechnology

Edited by Michael Rieth and Wolfram Schommers Volume X: Pages (1–35)

6. Mapof Plastic Deformation Mechanisms | 31 |

7. Summary and Conclusions | 34 |

References | 34 |

2 Constrained Grain Boundary Diffusion in Thin Copper Films

1. INTRODUCTION

Materials in small dimensions have become an increasingly important topic of research in the last decade. Changes in material behavior resulting from the effects of surfaces, interfaces, and constraints are still not completely understood. The focus of this chapter is on the mechanical properties of ultrathin submicron copper ﬁlms on substrates. In such materials, important effects on the ﬁlm surface and grain boundaries occur, and the constraint of the ﬁlm–substrate interface can govern the mechanical behavior.

Polycrystalline thin metal ﬁlms, shown schematically in Fig. 1, are frequently deposited on substrate materials to build complex microelectronic devices and are a relevant example of materials in small dimensions. In many applications and during the manufacturing process, thin ﬁlms are subjected to stresses arising from thermal mismatch between the ﬁlm material and the substrate. This can have a signiﬁcant effect on the production yield as well as on the performance and reliability of devices in service. In past years, an ever-increasing trend toward miniaturization in technology has been observed, stimulating a growing interest in investigating the deformation behavior of ultrathin ﬁlms with ﬁlm thicknesses well below 1 m.

Different inelastic deformation mechanisms operate to relax the internal and external stresses on a thin ﬁlm. Experiments have shown that for ﬁlms of thicknesses between approximately 2 and 0 5 m, the ﬂow stress increases in inverse proportion to the ﬁlm thickness (see, for example, Refs. [1–3]). This has been attributed to dislocation channeling through the ﬁlm [4–6], where a moving threading dislocation leaves behind an interfacial segment. The relative energetic effort to generate these interfacial dislocations increases with decreasing ﬁlm thickness, which explains the higher strength of thinner ﬁlms. This model, however, could not completely explain the high strength of the thin ﬁlms that was found in experiments [1]. More recent theoretical and experimental work [7–1] indicates that the strength of thin metal ﬁlms often results from a lack of active dislocation sources rather than from the energetic effort associated with dislocation motion.

Figure 1. A polycrystalline thin ﬁlm constrained by a substrate. The thin ﬁlm is subject to biaxial loading as a result of thermal mismatch of ﬁlm and substrate material.

Constrained Grain Boundary Diffusion in Thin Copper Films 3

The regime in which plastic relaxation is limited by dislocation nucleation and carried by the glide of threading dislocations reaches down to ﬁlm thicknesses of about hf ≈ 400 nm. For yet-thinner ﬁlms, experiments have revealed a ﬁlm-thickness-independent ﬂow stress [7].

In-situ transmission electron microscopy observations of the deformation of such ultrathin ﬁlms reveal dislocation motion parallel to the ﬁlm–substrate interface [7, 8]. This glide mechanism is unexpected, because in the global biaxial stress ﬁeld, there is no resolved shear stress on parallel glide planes, indicating that there must be a mechanism involving long-range internal stresses that vary slowly on the length scale of the ﬁlm thickness. For sufﬁciently thin ﬁlms, these internal stresses have a pronounced effect on the mechanical behavior. Constrained diffusional creep[12] has provided what is so far the only feasible mechanism for such internal stresses to cause nucleation of dislocations on glide planes parallel to the surface.

This chapter contains three main parts. The ﬁrst part reviews continuum mechanics modeling of constrained diffusional creep. The second part contains the results of molecular dynamics simulations of diffusional creep and plasticity in polycrystalline thin ﬁlms. In the third part, we discuss relevant experimental results. Finally, we summarize all the results to developa deformation mechanism mapfor ultrathin ﬁlms on substrates.

2. CONTINUUM MODELING

The continuum model of constrained diffusional creepwas developed by Gao and coworkers [12]. Diffusion can have a fundamentally different nature in thin ﬁlms than in bulk materials. The constraint imposed by the strong bonding between ﬁlm and substrate implies that no sliding can occur at the ﬁlm–substrate interface. In contrast to previously proposed models of diffusion in thin ﬁlms [13], the constraint of no sliding at the ﬁlm–substrate interface renders diffusion in thin ﬁlms an inherently transient phenomenon [12]. Therefore, steady-state solutions frequently used to describe grain boundary diffusion may not be applied. An additional constraint is that material transport cannot proceed in the substrate, and diffusion must therefore stopat the ﬁlm–substrate interface. It was shown by Gao and coworkers [12] that constrained grain boundary diffusion leads to a new material defect, referred to as the grain boundary diffusion wedge.

Figure 2 illustrates the basic mechanism of constrained diffusional creep[12] in three stages. In stage 1, material is transported from the surface into the grain boundary. In stage 2, mass transport leads to the formation of a diffusion wedge, as more and more material ﬂows into and accumulates in the grain boundary. The continuum model predicts that the traction along the grain boundary diffusion wedge becomes fully relaxed and cracklike on the scale of a characteristic time . This leads to extraordinarily large resolved shear stresses on glide planes that are parallel and close to the ﬁlm–substrate interface, and that scan cause emission of parallel glide dislocations in the last stage. Although the resolved shear stresses on the parallel glide planes induced by a diffusion wedge are similar to those generated by a crack, differences in the dislocation nucleation process may arise.

This section is brieﬂy outlined as follows. We discuss the continuum mechanics model of constrained diffusional creepfor different cases, including the homogeneous case, bimaterial interface, and coupled surface and grain boundary diffusion. We also present an extension of the continuum model to include a threshold stress for diffusion. Finally, we discuss a model of the critical condition for initiation of diffusion and a model for parallel glide dislocation nucleation from the diffusion wedge in the spirit of a model by Rice and Thomson for dislocation nucleation at crack tips [14].

2.1. Basics of the Continuum Modeling

In the continuum model, diffusion is treated as a dislocation climb in the grain boundary. The basis for this decision is the solution for the normal traction x along the grain boundary resulting from the insertion of a single dislocation (material layer of thickness b) along

0 | (corresponding to a climb edge dislocation). The coordinate system for the problem |

is depicted in Fig. 3. The solution for a single edge dislocation near a surface is used as the

4 Constrained Grain Boundary Diffusion in Thin Copper Films Stage 1

Stage 2

Stage 3 substrate substrate substrate diffusion wedge

Figure 2. Mechanism of constrained diffusional creep. Material is transported from the surface into the grain boundary, leading to the formation of a diffusion wedge as material accumulates in the grain boundary. The diffusion wedge can be modeled as a pile-up of climb edge dislocations near the ﬁlm–substrate interface, causing a cracklike singular stress concentration near the root of the grain boundary.

Green’s function to construct a solution with inﬁnitesimal Volterra edge dislocations [15–17]. The traction at position resulting from a dislocation at is

x | = Eb |

K | (1) |

K | = 1 |

is the Cauchy kernel function for this particular problem, with E denoting Young’s modulus and Poisson’s ratio. For an arbitrary opening function 2u in a ﬁlm with thickness hf, the stress along the grain boundary is given by

S |

ζ=b ζ d/2 film substrate

Figure 3. Coordinate system for the continuum mechanics analysis. The schematic shows climb edge dislocations in the grain boundary of a thin ﬁlm on substrate.

where 0 is the stress in the absence of diffusion and S | is a Green’s function kernel for |

Constrained Grain Boundary Diffusion in Thin Copper Films 5 the continuous dislocation problem (Cauchy kernel). For a dislocation near a free surface,

S = K . The kernel function S can also be established for a dislocation near a bimaterial interface [12, 18] or for periodic wedges [12]. The calculations in this chapter are mostly for an elastic ﬁlm on a rigid substrate, although we do not give the Green’s function kernel explicitly here.

The chemical potential relative to the free surface (with atomic volume = a30/4 in facecentered cubic crystals where a0 is the lattice parameter) is given by where 0 is an arbitrary reference constant. The corresponding atomic ﬂux per unit thickness in the boundary is

where k is the Boltzmann constant and T the absolute temperature. The parameter gbDgb denotes temperature-dependent grain boundary diffusivity. Equations (4) and (5) are cou- pled via mass conservation, as the ﬂux divergence is related to the displacement rate through

which can be combined with Eq. (5) as

The derivative of x t with respect to time is given by

S |

and inserting Eq. (7) into Eq. (8) yields the main governing equation

S |

for the grain boundary traction. Boundary and initial conditions are given as follows: for the continuity of chemical potential near the free surface, x = 0 t = 0 (10) and for no sliding and no diffusion at the interface,

sets the initial condition for the transient problem.

The problem given by Eq. (9) can be expressed by the method of separation of variables in the form of an expansion series

6 Constrained Grain Boundary Diffusion in Thin Copper Films where

is a characteristic time and n, fn, and cn denote the eigenvalues, eigenfunctions, and coefﬁcients. The coefﬁcients are determined from the initial condition at t = 0. It is important to note that ∼ h3f, similar to the classical Coble creepequation [19]. To solve the equations numerically, the problem is transformed into a standard Cauchy-type singular equation for f′′′n [12]. The Gauss–Chebyshev quadrature developed by Erdogan et al. [20, 21] can be used to solve such equations. The opening displacement u z t is given by

The solution procedure can be summarized in the following steps: ﬁrst, ﬁnd eigenvalues and eigenfunctions; second, ﬁnd the coefﬁcients cn, and third, calculate the traction and displacement from Eqs. (13) and (15).

The dislocations “stored” in the grain boundary represent additional material in the boundary. With respect to the lattice distortion around the diffusion wedge, the dislocations in the grain boundary exemplify a type of geometrically necessary dislocation [2] that causes nonuniform plastic deformation in the thin ﬁlm. The eigenvalues measure the rate of decay of each eigenmode. The results showed that the higher eigenmodes decay much faster than the ﬁrst eigenmode, so the diffusion process is dominated by the ﬁrst eigenmode [12].

Figures 4, 5, and 6 show several numerical examples. Figure 4 shows the stress intensity factor normalized by the corresponding value for a crack versus the reduced time t∗ = t/ for identical elastic properties of substrate and ﬁlm material (homogeneous case), rigid substrate (copper ﬁlm and rigid substrate), and soft substrate (aluminum ﬁlm and epoxy substrate).

Table 1 summarizes the material parameters used for the calculation. In the table, ﬁlm/ subs is the ratio of the shear moduli.

Figure 5 shows the opening displacement along the grain boundary for several instants in time for the case of soft ﬁlm on rigid substrate, and Fig. 6 shows the traction along the boundary for various instants in time. These examples illustrate that in the long time limit t → , the solution approaches the displacement proﬁle of a crack.

For constrained diffusional creepto proceed, both grain boundary diffusion and surface diffusion need to be active. In the original paper on constrained diffusional creep [12], copper-rigid homogeneous aluminum-epoxy

K(t)/K ∞

Time t/τ

Figure 4. Development of stress intensity factor over time for the cases of an elastically homogeneous ﬁlm on substrate, a rigid substrate, and a soft substrate. The convergence to the stress intensity factor of a crack is fastest for a rigid substrate and slowest for a soft substrate.

Constrained Grain Boundary Diffusion in Thin Copper Films 7

film thickness ζ /h f displacement ux/b

Figure 5. Development of grain boundary opening u normalized by a Burgers vector over time, in the case of a copper ﬁlm on a rigid substrate. The loading is chosen such that the opening displacement at the ﬁlm surface ( = 0) at t → is one Burgers vector.

surface diffusion was assumed to be inﬁnitely fast relative to grain boundary diffusion. This implies that grain boundary diffusion is the rate-limiting factor. To account for cases when surface diffusion is slower than grain boundary diffusion, or for cases when surface and grain boundary diffusion occur on comparable timescales, the model was extended in a later publication to also include the effect of surface diffusion [18]. We now brieﬂy review this model. The atomic ﬂux on a free surface may be expressed as

Here s/ s refers to the gradient in chemical potential along the free surface, and sDs is the temperature-dependent coefﬁcient of surface diffusion. The chemical potential of an

film thickness ζ /h f

Figure 6. Development of grain boundary traction / over time for the case of a copper ﬁlm on a rigid substrate. The grain boundary traction approaches when t → 0, and the grain boundary traction relaxes to zero when t → .

8 Constrained Grain Boundary Diffusion in Thin Copper Films

Table 1. Material parameters for calculation of stress intensity factor over the reduced time.

where the contribution from elastic energy is neglected, following Rice and Chuang [23].

Here, #s is the surface energy and $ s t is the local surface curvature. Assuming that the surface slope is small, the governing equation for the surface evolution can be expressed by

Mullins’ equation [24]:

where y measures the deviation of the surface relative to the ﬂat surface and y is the time derivative of y. The boundary and initial conditions for the present problem are given as where d is the grain size in the x-direction, with d/2 referring to the midpoint of the grain, as shown in Fig. 3. The parameter

is the angle of grain boundary groove to be calculated from #s and the grain boundary energy #gb. Solving the fully coupled grain boundary and surface diffusion problem is rather difﬁcult.

(Parte **1** de 6)