The epitaxial growth of nanometer-thick metallic thin films is in certain situations, such as in case of Pb-films, affected strongly by the electronic structure dependent total energy of the film [1]. This so-called quantum size effect (QSE) is related to the quantum well states originating from the spatial confinement of electrons in the nanometer-thick films. Recently a clear indication of the QSE in thin Pb-films has been observed in a form of a roughening transition, which has an oscillating dependence on temperature. The oscillations can be directly related to QSE on the film stability, where films with odd number of layers are more stable than films with even number of layers [1]. Thus films with these thicknesses are easier to grow in smooth epitaxial growth mode.
In this work, we present a simple phenomenological model, which describes the QSE on thin film growth and reproduces the experimentally observed stability of films with odd number of layers. The model is a modification of the simultaneous multilayer growth model, which takes into account the mass redistribution by interlayer processes [2,3]. In the model, mass redistribution is described in terms of macroscopic mass currents, which depend on film morphology and are related to the microscopic events of atoms on step edges of the layers. It is shown that very basic and simple processes of step crossing are enough to reproduce the qualitative features of growth of stable layers and the simultaneous bilayer growth associated with formation of stable layers. Because of its simplicity, the results of the model are only qualitative, but they suggest that the underlying processes needed to reproduce the basic generic features of the smooth epitaxial growth caused by the QSE are quite simple. Moreover, the results of the model can be used to estimate the relative frequencies of the occurrence of atomistic processes needed to maintain the observed simultaneous bilayer growth and roughening oscillations.
[1] F. Calleja, M.C.G. Passeggi, J.J. Hinarejos, A.L. Vásquez de Parga, and R. Miranda, Phys. Rev. Lett. 97, 186104 (2006).
[2] B. Voigtländer and T. Weber, Phys. Rev. B 54 (1996) 7709.
[3] Q. Fu and T. Wagner, Phys. Rev. Lett. 90, 2003, 106105
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