Integrating nanometer sized pores into low-k ILD films is one of the approaches to lower the RC signal delay and thus help sustain the continued scaling of micro-electronic devices. While increasing porosity of porous dielectrics lowers the dielectric constant (k), it also creates many reliability and implementation issues. One of the problems is the little understood metal ion drift in porous media. Here, we extend the rigorous simulation method of Cu diffusion in porous dielectrics based on elementary jump probabilities (presented at 231^st ECS Spring Meeting 2017 [1]) to include the ion drift in electric fields and explore the conditions for metal filament formation in porous media. The present atomistic approach allows a consistent co-implementation of Cu diffusion and Cu ion drift in electric field by lowering and raising of the diffusion activation energy E_a along the electric field E_el direction as exp(-(E_a ±1/2qlE_el)/kT), where l is the elementary jump distance, q is the elementary electronic charge, "-" denotes the enhancement factor in the field direction and "+" the retardation factor against the field. Our analysis of resistive switching behavior in porous dielectrics [2] shows that electric fields responsible for Cu filament formation are in excess of 10⁶ V/cm. At such high fields, the drift transport based mobility (m) concept where drift velocity v=m´E_el is no longer valid since the required inequality E_el<< kT/(ql) = 1.2´10⁵ V/cm for T=300K and l=21 A, does not hold, and, consequentially, all modeling methods based on drift-diffusion continuum equations fail to describe ion drift adequately. As described in [1] as a numerical method the diffusion tensor technique (4^th rank tensor in 2D, 9^th rank in 3D) has been used with elementary diffusion/drift jump probabilities at each lattice site.

Simulation shows that the Cu drift transport across porous materials depends more on the type of morphology than on the level of porosity. 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 47%. We find that, for spherical pores distributed more or less uniformly, the effective drift transport of Cu across dielectric at a room temperature comes to a stop at a porosity of around 40%. At room temperature, uniform spherical pore morphologies at certain porosity levels will block the drift transport of Cu across the dielectric entirely. However, for the same pore morphology at elevated temperatures, the relative jump frequency enhancement factor caused by the electric field is considerably reduced and now the increased lateral jump frequencies, owing to pure diffusion, help circumnavigate the pores causing a significant surge of drift transport of Cu ions. This is important technologically in CMOS back-end Cu lines when those are operated at high currents or at high switching frequencies. Under such circumstances, the local temperature of the dielectric may rise significantly over the room temperature. We have performed atomistic drift-diffusion simulations in the temperature interval between 27 ^oC and 400 ^oC. We find that although the relative electric field enhancement factor has been substantially reduced, the increased lateral jump frequencies give rise to a significant drift transport of Cu ions across the porous dielectric. One consequence of this is that resistive switching in porous media is facilitated by rising temperature which may pose a reliability issue.

In case of columnar pore morphology, when the electric field is aligned to the main pore orientation, the so called “channeling effect” [1] is simply enhanced by the drift component. However, the drift enhancement factor is decreasing with increasing temperature. In most of morphologies, the increasing temperature makes the drift-diffusion transport more efficient. However, in the case of the columnar morphology with its main orientation perpendicular to the electric field, the effective drift-transport may decrease with the increasing temperature. The reason for this unusual transport retardation at elevated temperatures is the deflection of the Cu ions due to increase of lateral movement into the quasi-trapping regions between the columnar pores which weakens the overall perpendicular drift diffusion transport from the Cu electrode to the counter-electrode. In our simulation grain boundary diffusion can be accommodated as a particular embodiment of interconnected pore morphology.

[1] R. Ali et al, “Modeling and Simulation of Cu Diffusion in Porous low-k Dielectrics’ 231^stECS Spring MTG, 2017

[2] Y. Fan et al, “Characterization of Porous BEOL Dielectrics for Resistive Switching”, 229^th ECS Spring MTG, 2016