Relationship between the surface morphology and the height distribution curve in thermal evaporated Au thin films

We present the formal deduction of an analytic expression for the height distribution curves of scanning tunneling microscopy (STM) images obtained from Au polycrystalline thin films with a wide range of thicknesses (60-1800 nm). That expression is a function of statistical parameters measured from the minimalist surface elements of the images and can be employed when these parameters satisfy several statistical conditions. We show that the required experimental parameters fulfil these conditions in a reasonable way and we present the numerical synthesis of experimental height distribution curves. The interface width calculated from the synthesized curves is compared with the root mean square height of the images, and as a consequence the ≈1/3 scaling exponent is confirmed and the complex origin of the interface width in a competitive columnar growth is shown. The description of the minimalist parameters set is completed with two facts: all they present a similar scaling exponent and, following the deduced expression, the experimental height distribution curves collapse in a unique distribution curve for all the thicknesses. Several parameters are proposed as a clear alternative to the usual interface width and to the morphologic slope.