^{th}rank tensor in 2D, and 9

^{th}rank in 3D), 2) sequential lattice point evaluation of elementary diffusion jumps. The 2

^{nd}method is very efficient in 3D, where the simulation time increases only linearly with size of the array, while the 1

^{st}method allows an easy implementation of jump frequencies at the boundaries of the film and pore boundaries.

Simulation shows that the Cu transport across porous materials depends as much on the level of porosity as on the pore morphology. Three different basic pore morphologies are considered: 1) columnar pores laterally oriented to the Cu transport, 2) columnar pores vertically oriented to the Cu transport, and 3) uniform distribution of pores of the same size across the dielectric. For all morphologies the porosity is varied between 0% and 46%. We find that, in general, increasing porosity reduces the Cu transport, i.e. reduces the effective diffusivity, except for the vertically oriented columnar pores. The latter effect is explained by a “channeling effect”. The vertically oriented columnar pores restrict the lateral diffusion and hence channel Cu atoms from the Cu electrode to the counter-electrode along the columnar pores. In contrast, laterally oriented columnar pores are most effective in suppressing Cu diffusion across the dielectric. For laterally oriented columnar nano-pores we observe the lowest effective diffusivity for porosity around 20%-25%. This effect is the result of two opposite mechanisms: at low porosity the laterally oriented columnar pores act as diffusion barriers to the diffusion. Thus, with higher porosity more barriers are added decreasing thus the effective diffusivity. However, when the porosity increases further, the trapping volume between the barriers becomes very small reducing the probability of a Cu atom to enter the space between the pores and being trapped there. Thus, the effective diffusivity increases again. We observe also a difference between a finite and infinite Cu supply at the interface. The finite supply is dictated by the rate of dissolution of Cu into the dielectric material. In terms of effective diffusivity there is no difference between the infinite and finite supply for uniform distribution of more or less spherical pores. However, for laterally oriented columnar pores, the effective diffusivity for finite supply is substantially smaller then for the case of infinite supply. In porous materials depending on the deposition method the contact area of Cu electrode with the dielectric can be larger or smaller than the contact area with non-porous material. The contact area effect is fully implemented. In our simulation the Cu segregation into the pores boundary and diffusion along the pore surface is considered as well. Depending on the ratio of the diffusion jump frequencies in the matrix material and on the pore surfaces, on the strength of the segregation of Cu to the pore surfaces, and on the porosity level, and on the morphology of the pores, the Cu transport is strongly suppressed (most of the cases). In some rare cases, however, it can be enhanced as in the case of vertically oriented columnar pores with much faster diffusion along the pore surface than in the matrix material and weak segregation at the pore interface. Thus, in principle, our method allows identification of detailed quantities of the diffusional transport from comparison with the experiment.

^{th}rank tensor in 2D, 9

^{th}rank in 3D), 2) Sequential lattice point evaluation of elementary diffusion jumps. The 2

^{nd}method is very efficient in 3D, where the simulation time increases only linearly with the dimension of the matrix, while the 1

^{st}method allows an easy implementation of jump frequencies at the boundaries of the film and pore boundaries.

Simulation shows that the Cu transport across porous materials depends as much on the level of porosity as on the pore morphology. Three different basic pore morphologies are considered: 1) Columnar pores laterally oriented to the Cu transport, 2) Columnar pores vertically oriented to the Cu transport, and 3) Uniform distribution of pores of the same size across the dielectric. For all morphologies the porosity is varied between 0% and 46%. We find that, in general, increasing porosity reduces the Cu transport, i.e. reduces the effective diffusivity, except for the vertically oriented columnar pores. The latter effect is explained by a “channeling effect”. The vertically oriented columnar pores restrict the lateral diffusion and hence channel Cu atoms from the Cu electrode to the counter-electrode along the columnar pores. In contrast, laterally oriented columnar pores are most effective in suppressing Cu diffusion across the dielectric. For laterally oriented columnar nano-pores we observe the lowest effective diffusivity for porosity around 20%-25%. This effect is the result of two opposite mechanisms: at low porosity the laterally oriented columnar pores act as diffusion barriers to the diffusion. Thus, with higher porosity more barriers are added decreasing thus the effective diffusivity. However, when the porosity increases further, the trapping volume between the barriers becomes very small reducing the probability of a Cu atom to enter the space between the pores and being trapped there. Thus, the effective diffusivity increases again. We observe also a difference between a finite and infinite Cu supply at the interface. The finite supply is dictated by the rate of dissolution of Cu into the dielectric material. In terms of effective diffusivity there is no difference between the infinite and finite supply for uniform distribution of more or less spherical pores. However, for laterally oriented columnar pores, the effective diffusivity for finite supply is substantially smaller then for the case of infinite supply. In porous materials depending on the deposition method the contact area of Cu electrode with the dielectric can be larger or smaller than the contact area with non-porous material. The contact area effect is fully implemented. In our simulation the Cu segregation into the pores boundary and diffusion along the pore surface is considered as well. Depending on the ratio of the diffusion jump frequencies in the matrix material and on the pore surfaces, on the strength of the segregation of Cu to the pore surfaces, and on the porosity level, and on the morphology of the pores, the Cu transport is strongly suppressed (most of the cases). In some rare cases, however, it can be enhanced as in the case of vertically oriented columnar pores with much faster diffusion along the pore surface than in the matrix material and weak segregation at the pore interface. Thus, in principle, our method allows identification of detailed quantities of the diffusional transport from comparison with the experiment.