Validity criteria for Fermi's golden rule scattering rates applied to metallic nanowires

Fermi's golden rule underpins the investigation of mobile carriers propagating through various solids, being a standard tool to calculate their scattering rates. As such, it provides a perturbative estimate under the implicit assumption that the effect of the interaction Hamiltonian which causes the scattering events is sufficiently small. To check the validity of this assumption, we present a general framework to derive simple validity criteria in order to assess whether the scattering rates can be trusted for the system under consideration, given its statistical properties such as average size, electron density, impurity density et cetera. We derive concrete validity criteria for metallic nanowires with conduction electrons populating a single parabolic band subjected to different elastic scattering mechanisms: impurities, grain boundaries and surface roughness.Comment: 23 pages, 8 figures, revised article version accepted for publication in Journal of Physics: Condensed Matte

Analytic solution of Ando's surface roughness model with finite domain distribution functions

Ando's surface roughness model is applied to metallic nanowires and extended beyond small roughness size and infinite barrier limit approximations for the wavefunction overlaps, such as the Prange-Nee approximation. Accurate and fast simulations can still be performed without invoking these overlap approximations by averaging over roughness profiles using finite domain distribution functions to obtain an analytic solution for the scattering rates. The simulations indicate that overlap approximations, while predicting a resistivity that agrees more or less with our novel approach, poorly estimate the underlying scattering rates. All methods show that a momentum gap between left- and right-moving electrons at the Fermi level, surpassing a critical momentum gap, gives rise to a substantial decrease in resistivity.Comment: 5 pages, 5 figure

Modeling and tackling resistivity scaling in metal nanowires

A self-consistent analytical solution of the multi-subband Boltzmann transport equation with collision term describing grain boundary and surface roughness scattering is presented to study the resistivity scaling in metal nanowires. The different scattering mechanisms and the influence of their statistical parameters are analyzed. Instead of a simple power law relating the height or width of a nanowire to its resistivity, the picture appears to be more complicated due to quantum-mechanical scattering and quantization effects, especially for surface roughness scattering.Comment: 6 pages, 5 figure

Modeling surface roughness scattering in metallic nanowires

Ando's model provides a rigorous quantum-mechanical framework for electron-surface roughness scattering, based on the detailed roughness structure. We apply this method to metallic nanowires and improve the model introducing surface roughness distribution functions on a finite domain with analytical expressions for the average surface roughness matrix elements. This approach is valid for any roughness size and extends beyond the commonly used Prange-Nee approximation. The resistivity scaling is obtained from the self-consistent relaxation time solution of the Boltzmann transport equation and is compared to Prange-Nee's approach and other known methods. The results show that a substantial drop in resistivity can be obtained for certain diameters by achieving a large momentum gap between Fermi level states with positive and negative momentum in the transport direction.Comment: 25 pages, 11 figure

Electron relaxation times and resistivity in metallic nanowires due to tilted grain boundary planes

We calculate the resistivity contribution of tilted grain boundaries with varying parameters in sub-10nm diameter metallic nanowires. The results have been obtained with the Boltzmann transport equation and Fermi's golden rule, retrieving correct state-dependent relaxation times. The standard approximation schemes for the relaxation times are shown to fail when grain boundary tilt is considered. Grain boundaries tilted under the same angle or randomly tilted induce a resistivity decrease.Comment: 5 pages, 3 figures in 2015 Joint International EUROSOI Workshop and International Conference on Ultimate Integration on Silicon (EUROSOI-ULIS

Finite Size Effects in Highly Scaled Ruthenium Interconnects

Ru has been considered a candidate to replace Cu-based interconnects in VLSI circuits. Here, a methodology is proposed to predict the resistivity of (Ru) interconnects. First, the dependence of the Ru thin film resistivity on the film thickness is modeled by the semiclassical Mayadas-Shatzkes (MS) approach. The fitting parameters thus obtained are then used as input in a modified MS model for nanowires to calculate wire resistivities. Predicted experimental resistivities agreed within about 10%. The results further indicate that grain boundary scattering was the dominant scattering mechanism in scaled Ru interconnects.Comment: 4 pages. 2 figure

Temperature-Dependent Resistivity of Alternative Metal Thin Films

The temperature coefficients of the resistivity (TCR) of Cu, Ru, Co, Ir, and W thin films have been investigated as a function of film thickness below 10 nm. Ru, Co, and Ir show bulk-like TCR values that are rather independent of the thickness whereas the TCR of Cu increases strongly with decreasing thickness. Thin W films show negative TCR values, which can be linked to high disorder. The results are qualitatively consistent with a temperature-dependent semiclassical thin film resistivity model that takes into account phonon, surface, and grain boundary scattering.Comment: 11 pages, 4 figure

Resistivity scaling and electron relaxation times in metallic nanowires

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

First principles-based screening method for resistivity scaling of anisotropic metals

The resistivity scaling of metals is a crucial factor for further downscaling of interconnects in nanoelectronic devices that affects signal delay, heat production, and energy consumption. Here, we present a screening method for metals with highly anisotropic band structures near the Fermi level with the aim to select promising materials in terms of their electronic transport properties and their resistivity scaling at the nanoscale. For this, we consider a temperature-dependent transport tensor, based on band structures obtained from first principles. This transport tensor allows for a straightforward comparison between different anisotropic metals in nanostructures with different lattice orientations. By evaluating the temperature dependence of the tensor components, we also find strong deviations from the zero-temperature transport properties at standard operating temperature conditions around room temperature.Comment: 25 pages, 8 figure