Models for Spatially Resolved Conductivity of Rectangular Interconnects with Integrated Effect of Surface And Grain Boundary Scattering

Abstract: Surface scattering and grain boundary scattering are two prominent mechanisms dictating the conductivity of interconnects and are traditionally modeled using the Fuchs-Sondheimer (FS) and Mayadas-Shatzkes (MS) theories, respectively. In addition to these approaches, modern interconnect structures need to capture the space-dependence of conductivity, for which a spatially resolved FS (SRFS) model was previously proposed to account for surface scattering based on Boltzmann transport equations (BTE). In this paper, we build upon the SRFS model to integrate grain-boundary scattering leading to a physics-based SRFS-MS model for the conductivity of rectangular interconnects. The effect of surface and grain scattering in our model is not merely added (as in several previous works) but is appropriately integrated following the original MS theory. Hence, the SRFS-MS model accounts for the interplay between surface scattering and grain boundary scattering in dictating the spatial dependence of conductivity. We also incorporate temperature (T) dependence into the SRFS-MS model. Further, we propose a circuit compatible conductivity model (SRFS-MS-C3), which captures the space-dependence and integration of surface and grain boundary scattering utilizing an analytical function and a few (three or four) invocations of the physical SRFS-MS model. We validate the SRFS-MS-C3 model across a wide range of physical parameters, demonstrating excellent agreement with the physical SRFS-MS model, with an error margin of less than 0.7%. The proposed SRFS-MS and SRFS-MS-C3 models explicitly relate the spatially resolved conductivity to physical parameters such as electron mean free path ($\lambda_0$), specularity of surface scattering (p), grain boundary reflectance coefficient (R), interconnect cross-section geometry and temperature (T).