Decoupling of Ion Diffusivity and Electromobility in Porous Dielectrics
Authors: Fan Ye , Ali Rizwan , King Sean W. , Bielefeld Jeff , Orlowski Marius K 
Published: February 2020

In an effort to lower the interconnect time delay and address the latency problem, porous low-k dielectrics are being currently explored as a low-k BEOL dielectric. Because of low density, porous dielectrics suffer from excessive impurity diffusion and from ion electromigration when the interconnect lines are powered. This paper investigates the relation of ion diffusivity and electro-mobility in porous dielectrics. Metal-Insulator-Metal (MIM) structures have been manufactured with bottom Cu electrode and with top island-shaped W or Pt counter-electrode. The porous dielectric was a-SiC:H or a-SiOC:H with level of porosity varying between 8% to 25%. The thickness of the porous dielectric is 20 nm and the dielectric constant k varied between 3.1 (@ 8% porosity) and 2.5 (@25% porosity). Some devices have a 2 nm SiCN diffusion barrier (DB), either at the Cu or W/Pt electrode or at both electrodes. The SiCN layers serve a dual-purpose: i) to prevent Cu diffusion, and ii) to prevent Cu, W, and Pt protrusion into pores and cavities of porous dielectrics. A bulk majority of devices have been found intrinsically conductive, i.e. conductive from the beginning and less than 10% - only those with at least one DB - displayed resistive switching (RS) behavior. Of those devices that showed resistive switching behavior, devices with higher porosity showed consistently higher forming voltage Vform of Cu filaments than dielectrics with lower porosity. This is strange in view of the fact that most of the devices even of high porosity where intrinsically conductive indicating a very high degree of Cu diffusion. For a-SiC:H film with 8% porosity we find Vform ≈ 0V, compared with Vform= 0.4 - 0.5V for  12% porosity and Vform= 1.4 V- 2.3 V for 25% porosity. Why should electromigration require higher electric field when the Cu diffusivity appears to be equally strong?

In nonporous materials the diffusivity D and mobility m of charged particles such as an ion is related by the Einstein relation:, where k is the Boltzmann constant, T the absolute temperature and q the charge of the ion. In terms of microscopic mechanisms, diffusivity and mobility are very related phenomena. The difference between diffusion and mobility is that the applied electric field lowers the barrier for the forward jump and increases the barrier for the reverse jump in the direction of the electric field. In two and three dimensions the difference between diffusion and electromigration is that diffusion is isotropic in homogeneous materials and electromigration is directional. We postulate that in porous materials the diffusivity and mobility are decoupled and the Einstein relation does not hold globally but only locally within a contiguous patch of material outside of the voids constituting the porosity. In porous materials, the voids lying in the path of the migrating ion present an impenetrable barrier for electromigration. They can be only overcome by a detour around the obstacle that can be brought about only by the driving force of diffusion. Thus an ion encountering a void in its path has to rely on diffusion to circumnavigate the obstacle. We propose that the relation between diffusivity and mobility in porous materials should be described by a revised relation: where a porosity factor fp <1 is introduced. The factor fp will depend on porosity, such that fp = f(porosity) will decrease with increasing porosity. The revised relation between diffusivity and mobility has to be understood as a macroscopic relation. Within the dense matrix of the material between the voids, the classic Einstein relation will still hold microscopically. But, in porous materials the obstacles posed by the voids compromise the directional nature of the mobility. We propose a numerical model where the porosity factor fp of the revised Einstein relation can be modeled as a function of porosity in two dimensions. An extension to three dimensions is also feasible, but requires larger computational resources.