Abstract

We report an extension of the theory of Sheng, Xing and Wang (SXW) (Sheng L, Xing D Y and Wang Z D 1995 Phys. Rev. B 51 7325), which permits the calculation of size effects from the statistical properties that characterize the surface on a microscopic scale, for samples in which the average height-height autocorrelation function (ACF) is described either by a Gaussian or by an exponential. We also report measurements of the topography of a gold film deposited,sited on a mica substrate using a scanning tunnelling microscope (STM) on a gold sample 70 nm thick deposited under ultrahigh vacuum on a mica substrate preheated to 300 degrees C. From the STM images we compute the average ACF which characterizes the surface of the film on the scale of 10 nm x 10 nm, and determine by least-squares fitting the r.m.s. amplitude delta and the lateral correlation length xi corresponding to a Gaussian and to an exponential that best represent the ACF data. Using the modified SXW (mSXW) theory and a Gaussian and an exponential representation of the ACF data, we calculate the quantum reflectivity R characterizing the interaction between the electrons and the surface, and the decrease in conductivity her attributable to electron-surface scattering, for mean free paths 2.5 nm less than or equal to l less than or equal to 1000 nm. We compare the predictions of the classical Fuchs-Sondheimer (FS) model for the average quantum reflectivity R = (R), calculated with the mSXW model, with the predictions of the quantum theory, using both the Gaussian and the exponential representation of the ACE We find that Delta sigma predicted by FS theory for R = (R) exceeds that predicted by the quantum mSXW theory, by an amount that increases with increasing l. This discrepancy can be traced to the angular dependence of the quantum reflectivity R[cos(theta)]. We also find that the decrease in conductivity Delta sigma predicted by mSXW theory for a Gaussian representation of the data is larger than that predicted for an exponential representation of the same ACF data. We attribute this to the fact that the reflectivity R is determined by the Fourier transform of, the ACF, and the Gaussian and the exponential that best represent the ACF data exhibit Fourier transforms that are similar in the regions where k xi; similar to 1, but are different in the regions where k xi 1 and k xi > 1 (k: wave vector).