3.2.2 Anisotropic Diffusivity: Diffusion Dependence on Stress

Residual mechanical stresses are introduced on interconnect lines as a result of the fabrication process flow [141]. These stresses can be very high, significantly affecting the diffusion coefficient of the interconnect metal atoms [142].

Typically, the diffusion dependence on stress is taken into account through a simple modification of the usual diffusion coefficient expression (2.10) to [54,97]

In this way, (3.13) describes the effect of a scalar hydrostatic stress on the scalar diffusion coefficient. However, Flynn [143] pointed out that, for a homogeneously deformed cubic crystal with strain field , there is an additional contribution proportional to to the diffusion of the form

where is the elastodiffusion tensor, which can be experimentally determined. Here, the simple scalar diffusivity is replaced by a tensorial quantity. Consequently, in the presence of a stress field the diffusivity can now become anisotropic, depending on the properties of .

Based on microscopic lattice theory, Dederichs and Schroeder [142] showed that the diffusivity in a simple face-centered cubic crystal (fcc) follows

where is the jump vector for a hop and is the corresponding jump rate. From classical thermodynamics the jump rate is given by [59,144]

where is the Debye frequency ( s for metals [144]) and is the migration energy barrier. The presence of an external strain field affects the jump rate by changing the migration energy barrier according to [145]

where is the migration energy barrier in the absence of an external field, is the induced strain, is the elasticity tensor, and is the applied external strain. For a fcc crystal the elasticity tensor has the form [146]

Combining (3.16) and (3.17) one obtains

with

Considering a single vacant point defect in a crystal lattice, the local volumetric strain induced by the presence of this vacancy is given by

where is the vacancy volume and , , and are the induced strains in the given directions. Due to the symmetry of a vacant point defect

which together with (3.21) yields

Thus, the induced strain tensor becomes

and the induced strain vector is determined by

where is a jump unit vector for a hop .

Normalizing (3.15) with the jump distance and summing over the nearest neighbors ( for fcc crystals) one obtains

Substituting (3.19) in this equation, the diffusivity tensor in the presence of a external stress field is determined through the expression

where is given by [144]

In order to take into account the anisotropy of diffusion, the scalar diffusivity, , in the flux equation (3.9) is replaced by the stress dependent diffusivity tensor yielding

with calculated via (3.27). In this way, the development of mechanical stress can lead to anisotropic diffusion in the interconnect line, therefore, affecting the material transport along the line under electromigration.

R. L. de Orio: Electromigration Modeling and Simulation