Resistivity scaling and electron relaxation times in metallic nanowires

Abstract

We study the resistivity scaling in nanometer-sized metallic wires due to surface roughness and grain-boundaries, currently the main cause of electron scattering in nanoscaled interconnects. The resistivity has been obtained with the Boltzmann transport equation, adopting the relaxation time approximation (RTA) of the distribution function and the effective mass approximation for the conducting electrons. The relaxation times are calculated exactly, using Fermi's golden rule, resulting in a correct relaxation time for every sub-band state contributing to the transport. In general, the relaxation time strongly depends on the sub-band state, something that remained unclear with the methods of previous work. The resistivity scaling is obtained for different roughness and grain-boundary properties, showing large differences in scaling behavior and relaxation times. Our model clearly indicates that the resistivity is dominated by grain-boundary scattering, easily surpassing the surface roughness contribution by a factor of 10.Comment: 19 pages, 5 figure

Modeling surface roughness scattering in metallic nanowires

“…10 together with the approximated solution by putting the relaxation time ratio on the right-hand-side equal to 1, assuming equal lifetimes for the initial and final states. 16 The resistivity drop is only apparent in the self-consistent solution of the BTE because it is the only solution that solves the coupled system of equations, determining the scattering lifetimes correctly. Two data points of Fig.…”

“…It was shown that the resistivity due to SR can have very different values and scaling behavior for different values of SR standard deviation and correlation length. 16 A disadvantage of the Prange-Nee approximation is that it takes the infinite barrier limit of an expansion for small roughness sizes, which often falls short to determine the wave function overlap, especially in the case of large roughness sizes, highly oscillating wave functions or low barrier heights. For metallic nanowires one should definitely be careful, as there are many subbands and, hence, high wave vector values in the confinement direction are generally present.…”

Dielectric Barrier, Etch Stop, and Metal Capping Materials for State of the Art and beyond Metal Interconnects

“…[3][4][5][6] This reverse effect of dimensional scaling became particularly apparent in the late 90's as the industry left the realm of the micron (i.e. micro-electronics) and entered into the realm of the nanometer (i.e.…”

Modeling and tackling resistivity scaling in metal nanowires

“…In this work we present an alternative approach, based on an analytic solution of the multi-subband Boltzmann transport equation (BTE) within the effective mass approximation for the conduction electrons. The relaxation times (RTs) are calculated self-consistently and Fermi's golden rule (FGR) is used to obtain the scattering rates [7]. The FGR matrix elements only rely on (statistical) physical parameters of the scattering mechanisms, such as the shape, density and energy barrier strength for GBs and the standard deviation and correlation length for SR.…”