Size effects and charge transport in metals: Quantum theory of the resistivity of nanometric metallic structures arising from electron scattering by grain boundaries and by rough surfaces

Abstract

We discuss recent progress regarding size effects and their incidence upon the coefficients describing charge transport (resistivity, magnetoresistance, and Hall effect) induced by electron scattering from disordered grain boundaries and from rough surfaces on metallic nanostructures; we review recent measurements of the magneto transport coefficients that elucidate the electron scattering mechanisms at work. We review as well theoretical developments regarding quantum transport theories that allow calculating the increase in resistivity induced by electron-rough surface scattering (in the absence of grain boundaries) from first principles—from the parameters that describe the surface roughness that can be measured with a Scanning Tunnelling Microscope (STM). We evaluate the predicting power of the quantum version of the Fuchs-Sondheimer theory and of the model proposed by Calecki, abandoning the method of parameter fitting used for decades, but comparing instead theoretical predictions with resistivity measured in thin films where surface roughness has also been measured with a STM, and where electron-grain boundary scattering can be neglected. We also review the theory of Mayadas and Shatzkes (MS) [Phys. Rev. B 1, 1382 (1970)] used for decades, and discuss its severe conceptual difficulties that arise out of the fact that: (i) MS employed plane waves to describe the electronic states within the metal sample having periodic grain boundaries, rather than the Bloch states known since the thirties to be the solutions of the Schrödinger equation describing electrons propagating through a Krönig-Penney [Proc. R. Soc. London Ser. A 130, 499 (1931)] periodic potential; (ii) MS ignored the fact that the wave functions describing electrons propagating through a 1-D disordered potential are expected to decay exponentially with increasing distance, a fact known since the work of Anderson [Phys. Rev. 109, 1492 (1958)] in 1958 for which he was awarded the Nobel Prize in 1977; (iii) The current in the sample should be proportional to TN, the probability that an electron traverses N consecutive (disordered) grains found along a mean free path; MS assumed that TN = 1. We review unpublished details of a quantum transport theory based upon a model of diffusive transport and Kubo's linear response formalism recently published [Arenas et al., Appl. Surf. Sci. 329, 184 (2015)], which permits estimating the increase in resistivity of a metallic specimen (over the bulk resistivity) under the combined effects of electron scattering by phonons, impurities, disordered grain boundaries, and rough surfaces limiting the sample. We evaluate the predicting power of both the MS theory and of the new quantum model on samples where the temperature dependence of the resistivity has been measured between 4 K and 300 K, and where surface roughness and grain size distribution has been measured on each sample via independent experiments. We find that the quantum theory does exhibit a predicting power, whereas the predicting power of the MS model as well as the significance and reliability of its fitting parameters seems questionable. We explore the power of the new theory by comparing, for the first time, the resistivity predicted and measured on nanometric Cu wires of (approximately) rectangular cross section employed in building integrated circuits, based upon a quantum description of electron motion.