Surface-induced resistivity of thin metallic films bounded by a rough fractal surface

Abstract

We have extended the modified formalism of Sheng, Xing, and Wang [J. Phys.: Condens. Matter 11 L299 (1999)] to allow the calculation of the conductivity of a thin metallic film bounded by a rough fractal surface. We utilized the so-called k-correlation model proposed by Palasantzas and Barnas [Phys. Rev. B 48, 14 472 (1993); 56, 7726 (1997)], to describe the height-height autocorrelation function corresponding to a self-affine roughness. This extension permits the calculation of the conductivity of the film as a function of the r.m.s. roughness amplitude \ensuremath{\delta}, of the lateral correlation length \ensuremath{\xi}, of the mean free path in the bulk l, and of the roughness exponent H. We found that the degree of surface irregularity, represented by the roughness exponent H characterizing the surface, does influence the conductivity of the film, as first discovered by Palasantzas and Barnas. However, this influence manifests itself for large bulk mean free paths $l\ensuremath{\approx}1000\mathrm{nm}$ and for large correlation lengths $\ensuremath{\xi}\ensuremath{\approx}5\mathrm{nm},$ in which case the conductivity of the film for $H=1$ exceeds by about 30% the conductivity for $H=0.2,$ an effect which is smaller than that reported by Palasantzas and Barnas. For correlation lengths \ensuremath{\xi} below 1 nm and mean free paths $l\ensuremath{\approx}100\mathrm{nm},$ the influence of the roughness exponent H on the conductivity is reduced to below 10%, and for smaller mean free paths and correlation lengths the conductivity becomes insensitive to H. We also found that Mathiessen's rule is severily violated in the case of thin metallic films. The resistivity of the film coincides roughly with the surface-limited resistivity only in the case of ultrathin films $t<5\mathrm{nm}.$ For thicker films $100\mathrm{nm}>t>5\mathrm{nm},$ the resistivity of the film exceeds by some 20 to 30 % the value dictated by Mathiessen's rule. And conversely, the apparent surface-induced resistivity estimated assuming the validity of Mathiessen's rule, exceeds by nearly one order of magnitude the true surface-induced resistivity, except in the case of ultrathin films $t<5\mathrm{nm}.$