Monte Carlo Simulation of a Growing Pb-Film on Cu (100) and (111) Surfaces
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the lower three substrate layers were fixed in their position and only the uppermost atom layer was movable. The whole algorithm of the present simulation can be summarized as follows: (1) The start of a new simulation cycle is characterized by generating a new adatom at a random position on the growth front. (2) A virtual displacement is given to one substrate atom in the first substrate layer or one film atom using a random number generator to obtain a move in continuos space. (3) The difference in the potential energy between the initial and the virtually moved film configuration, AE, is calculated by the applied potential. (4) If AE is negative, which means the total energy of the system is lowered, the new position of the virtually moved atom is accepted and the displacement becomes real. (5) If AE is positive, the Boltzmann factor exp(AE/kbT) is calculated from the change of energy associated with the virtual displacement. It represents a barrier which the total system has to overcome to move the observed atom. This decision is done by the generation of another random number R. Only if R is less than the Boltzmann factor the virtual displacement is accepted. (6) The continuos repetition of points (2) to (5) is performed for every movable film or substrate atom for r Monte Carlo steps. (7) Starting with point (1) a new adatom is deposited. The described algorithm was also used to calculate the total internal energy of a system with already deposited film atoms. In this case the film atoms were placed at well-defined positions at the substrate to form a Pb-overlayer with a superstructure as observed in STM [3]. Then the simulation procedure was applied to enable the reconstruction of the Pb film and the upper substrate layer. Using T = 1K, only a virtual displacement with an energy change less than zero was allowed. Hence the total system achieved its energy minimum after a few simulation cycles. Because of ambiguities in the interpretation of the STM results on the development of a surface alloy on the Cu (100) [3] surface the described method was used to calculate the difference in the total energy of a deposited Pb-overlayer forming a superstructure with and without incorporated Cu atoms. Comparing the two cases it was necessary to fix the number of all atoms in the film and especially in the substrate. Hence all substrate atoms incorporated in the Pboverlayer representing a surface alloy had to be attached at the substrate backside for the non surface alloy case. This procedure made it possible to compare the total energy of both systems. Two different types of potentials have been applied to simulate the interaction of the deposited film atoms and the substrate atoms: the simple two body potential (Morse-potential) and a recently developed embedded atom potential (EAM) of the Johnson-type [5]. The Morse potential
(D(r1j) = D.{ exp [-2cc (rij -ro)I-2. exp [-al (rij -r 0)]} D .... dissociation energy of a dimer (potential minimum) a .... determines shape of potential interaction type Cu-Cu Pb-Pb Cu-Pb (geo
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