The influence of steps on the island distribution function in thin solid film formation

The gamma distribution has been shown to be the simplest function for describing the universal size distribution function of InAs quantum dots on GaAs(001) substrate (Fanfoni et al 2007 Phys. Rev. B 75 245312), although there is no reason to believe that it could not be applied to any film growth where the principal feeding of the islands occurs through surface diffusion as proposed, after numerical simulations in the two-dimensional case, by Mulheran and Blackman (1995 Phil. Mag. Lett. 72 55). As far as InAs/GaAs(001) quantum dots are concerned, by fitting the data of Ebiko et al (1999 Phys. Rev. B 60 8234) and Krzyzewski et al (2002 Phys. Rev. B 66 201302(R)) to the gamma function, beta = 4.5 and 2.9 are the respective returned values. The latter value appears anomalous because for a Poissonian distribution of points (centers of islands), beta = 3.5-3.6. Moreover, the greater the spatial correlation the greater the beta value. We prove that the presence of steps alters the distribution of nucleation centers throughout the substrate in such a way that its variance increases with respect to the substrate without steps and, as a consequence, beta decreases.