Modelling Size-Selected Growth of Nanodots by Using Reaction Kinetic Approach
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Modelling Size-Selected Growth of Nanodots by Using Reaction Kinetic Approach Kirsi Nevalainen, Marko Rusanen, and Ismo T. Koponen Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, Espoo, 02015, Finland
ABSTRACT The size selection of nanodots during the growth is modelled by using the reaction kinetic model with reaction rates for dot size dependent attachment and detachment processes, related to the free energy of dots. Long-lived metastable state is found near the minimum of free energy, but the kinetics of the growth causes the peak to overshoot the location of minimum of the energy. The size distribution of dots is shown to be nearly Gaussian with minor skewness originated from kinetics. However, it is argued that this skewness is inherently related to the effect of overshooting, and thus to the formation of the long lived metastable state with size selected nanodots. Based on RKM, a simple continuum model is derived to describe the size selection and narrowing of the size distribution. INTRODUCTION In growth of nanoclusters and dots on surfaces, the size-selection during the growth is one of the most interesting phenomena [1-3], which gives promises to realize fully selfassembled growth of regular sized nanodot arrays. The spontaneous size-selection occurs in 2D Stranski-Krastanov growth of nanodots [4], where the energy minimum is due to elastic misfit strain energy [2,4-5] and in the 3D Volmer-Weber growth where it emerges from surface energy [3]. In most cases self-assembled growth is observed in heteroepitaxy of semiconductor nanodots, but sometimes also in metallic dots [3]. Another possibility is to ascribe the sizeselection to the reaction kinetics of growth, and relate the size selection to size dependent reaction rates including the surface energy contribution [4,6,7]. The results of these studies support the view that the size-selection is a generic feature of growth, related to the existence of long lived metastable state of growth where the thermodynamically driven growth is balanced by kinetic effects. Due to the kinetics, the distribution peak and the average size overshoot the free energy minimum location [4-5]. In this work, we describe the size-selective growth in terms of a reaction kinetic model (RKM) with self consistent reaction rates for size dependent attachment and detachment processes. The RKM is solved numerically by using particle coalescence PCM simulation method. The results of the RKM are compared with the continuum model, obtained first reducing the RKM to Becker-Döring model, used then to derive the Fokker-Planck equation for growth (see [5]). We show that the size distributions are skewed, and that the skewness is inherently related to the kinetic effect of overshooting the energy minimum.
THE REACTION KINETIC MODEL The growth of the nanodots is described by the reaction kinetic model (RKM), which includes only adatom attachment and detachment processes A1 + As ↔ A1+s with dots of size s, and with the reaction rates being σs and γs, respect
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