Simulation of the Nucleation and Growth of Si Grains Inside Au Thin Films
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(1)
(2)
(3)
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(j1)
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(2)
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between the resistors in the network and increasing gradually the
number of broken connections, it is possible to calculate the equivalent resistance as a function of interrupted connections and, for
,-example,
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to predict the disap-
(0)
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FIG. 1 - Random resist network used in the simulation of the twodimensional crystallization process. Node (ij) illustrates process of nucleation of crystals due to the transformation of Au nodes into Si nodes.
pearance of the last long-range connectivity (percolation path). A two-dimensional random resistor network will be used to represent the Au thin film. The appearance of the Si nuclei (embryos) will be represented by substituting the Au resistors with Si ones. This procedure is equivalent to the one where the interconnections are interrupted in the percolation theory, because the Si resistors are orders of magnitude more resistive than the Au ones. From the experimental point of view, the substitution
step corresponds to the nucleation of the Si grains at the interface and the rapid growth of them reaching the Au surface [3,4]. Fig. 1 shows a random resistor network of [(i+l) 0(j+1)] order. Each node corresponding to 4 (four) Au resistors. In this figure node (i ;j) illustrates the presence of a Si nucleus. As the crystallization in the Au/a-Si:H bilayer system occurs in two-dimensions, the random resistor network will represent any Au plane parallel to the surface. Considering the surface plane, the simulation results will directly reproduce micrographies obtained by scanning electron microscopy (SEM). Concerning electrical measurements, it will be indifferent if the random resistor network simulates the surface plane or any interior Au plane. The overall crystallization process, taking into account nucleation, growth and impingement of neighbors grains (the formation of a continuous metallic network), will be simulated by the evolution of the random network. Each iteration of the simulation program results in a new network with new substitutions (new embryos or grown grains) and a new equivalent resistance value. Plotting the resistor network and the calculated equivalent resistance as a function of the number of iterations, permits the study of the crystallization process as a function of time and to compare with experiments. The simulation of physical mechanisms as heterogeneous nucleation, growth rate reaction-limited and the impingement of adjacent nuclei will be described in the next section. SIMULATION PROGRAM The computational program starts by generating a resistor matrix of [(i+l) 01+)] order and associating to each node (corresponding to 4 resistors) a random number between 0 and 1. In the first iteration, the program begins the nucleation procedure. All nodes which random numbers are between 0 and b are transformed into Si nodes. The parameter b varying between 0 and 1 is 168
related to the formation frequency y' of the embryos. For example, when b is equal to 0.1, around 10 % of the n
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