Coupled grain boundary and surface diffusion in a polycrystalline thin film constrained by substrate

Appendix A

Tables of the characteristic relaxation times tgrg and tgrs

The characteristic relaxation time tgrs of the grain boundary traction for the infinitely fast grain boundary diffusion problem is defined by Eq. (33). In another limiting case, the characteristic relaxation time tgrg of the grain boundary traction for the infinitely fast surface diffusion problem is defined by Eq. (34). The ratio of these two characteristic relaxation times is related to the ratio Δ of the diffusional diffusivities by Eq. (35), and the diffusivites depend upon the temperature through

where δSDS0 and δgbDgb0 are the pre-exponential factors, and Q_s and Q_gb are the activation energies for the surface diffusion and grain boundary diffusion, respectively.

Some values of tgrg and tgrs are listed in Tables A1 and A2 for silver and copper thin films with typical thickness h = 1 μm and h = 0:2 μm, respectively.

Appendix B

Diffusion wedges in bimaterial thin film/substrate [19]

Here we consider the effects of possible differences between elastic constants of the film and the substrate. Electronic thin film systems are often made by depositing a metallic material (aluminum, copper, etc.) on a relatively rigid substrate (silicon, silicon carbite, etc.). In other applications, one may wish to protect a soft material by depositing a relatively hard material on it. It is important to consider the bimaterial effect since the Young’s moduli of the films and substrates can sometimes be quite different. The analysis given in the text shows that the fully coupled surface and grain boundary diffusion problem does not provide a significantly different solution as the simpler problem of infinitely fast surface diffusion with grain boundary grooving ignored. Therefore, in investigating the bimaterial effects, we only consider the single wedge problem under infinitely fast surface diffusion with no surface grooving. This problem has the same geometry as the analysis of [1] except that the film may now have different elastic constants as the substrate. The governing equation retains the same form as eq. (13) except that the kernel function P(z,ζ) is replaced by a new kernel function B(z,ζ) for the bimaterial case

where E1*=E1/(1−v12), and E₁ and _m1 are the Young’s modulus and Poisson’s ratio of the thin film, respectively. The kernel function B(z,ζ) corresponds to the elasticity solution to a single array of edge dislocations near a free surface and can be calculated by

where F(z,ζ) and G(z,ζ) are functions of z and ζ, and can be found in [19]. The solution method of Eq. (15) can be generalized to solve the integral Eq. (B1).

For the bimaterial wedge problem, with its wedge tip at the interface, the traction just ahead of the wedge tip is of the form σvv(0,z)=C1σ0hs/(z−h)s where C₁ is nondimensional and a function of the Dundurs’ parameters α and β only. The stress singularity exponent, s, is a function of α and β and satisfies the following equation derived by Zak and Williams [20]

where a value of s = 1=2 would be obtained for the case of identical film and substrate materials.

Applying the finite difference discretization Eq. (17) and the transformation Eq. (18) to Eq. (B1) leads to an integral equation which has the same form as Eq. (20) except that P(¯z,ζ) is replaced by B(z,ζ). The “extending method” described in [1] is employed to solve the unknown function ∂3σgb/∂z3 which is assumed to have the form

where R(z,t+Δt) is a regular function in 0 < z < h. Since s is different from 1=2, except for some specific material combinations, including, of course, two identical materials, the Gauss-Chebyshev quadrature formulae used in [1] are not valid here. The more general Gauss-Jacobi integration quadrature [21, 22] is employed to discretize the integral Eq. (B1) as

where z¯i and ζj(−1≤z,ζ≤1) are the positive roots of Jacobi polynomials

and W_j are the corresponding weights of Jacobi polynomials P 2N+1(−s,−s)(ζ¯). These weights and roots can be easily calculated by using the Fortran programs listed in [23].

The numerical results are presented for the thin film/substrate system of aluminum (v₁ = 0:3) deposited on epoxy (v₂ = 0:35) substrate with the modulus ratio E₁/E₂ = 23:98, where E₂ and v₂ are the Young’s modulus and Poisson’s ratio of the substrate, respectively [24]. From Eq. (B3), the singularity exponent is found to be s = 0:8248. Figure 8 shows the transient solutions to the grain boundary traction and the opening displacement for the diffusion wedge. Even at such large elastic mismatch (the case of Al or Cu deposited on Si would have much smaller elastic mismatch.), the results are qualitatively similar to those in [1].

Fig. 8

For the bimaterial case of constrained grain boundary diffusion in an aluminium film bonded to an epoxy substrate, the evolution of (a) the grain boundary traction and (b) the opening displacement of the grain boundary wedge as snapshots in time. Here τ=t/tgrg and t/tgrg=kTh3/(δgbDgbΩE1*).