(a) and (b) Two-dimensional sketch of a network model of a system of sintered nanograins with (a) small and (b) large thickness of the depletion zone λ. The conducting core region and the depletion zone are symbolized by the light blue and the dark blue shading, respectively. (c) Magnification of a single grain contact with the geometrical neck diameter . The (smaller) diameter of the conducting neck is the diameter of the light blue channel.
(a) and (b) Sketch of the two possibilities that may lead to a breakdown of the conductance when λ increases: (a) The bonds between large grains may become interrupted ( ) while the cores are still large and therefore conducting. This leads to . (b) The cores of small grains may become too small to host free electrons ( ) while the bonds are still conducting, leading to . (c) Two-dimensional sketch for the construction of the relative free surface s for the special case that next-nearest neighbors just touch each other. One easily recognizes that .
(a) Sketch of the theoretical prediction for the normalized mobility versus Nr for (a) and (b) . The theoretical curve for (Eq. (7) ) is indicated by the dashed curve and the cut-off value (Eq. (9) ) by the arrow. The expected experimental values in both cases are shown by the red circles. Note the jump in the experimental values in case (a).
Numerical simulations for the site percolation threshold : fraction of percolating systems versus occupation probability. The straight line represents the site percolation threshold for an ideal 2d film and the interrupted lines approximate 3d films of , and 31 monolayers.
Critical surface gas density versus the mean grain diameter for systems with (a) N = 1 and (b) N = 5 layers. In each figure, the full (red) line represents Eq. (7) with slope , independent of . The dashed lines show the theoretical predictions of Eq. (9) that depend on while the symbols show the corresponding numerical simulations. The values for (which here depend on N and ) are: (pink) circles for N = 1 and for N = 5, (blue) pluses for N = 1 and for N = 5, (green) crosses for N = 1 and for N = 5 and (red) squares for N = 1 and for N = 5.
Mobility μ (that corresponds to the diffusion constant Ds ) versus surface density Nr for various mean grain-sizes for fixed number of layers N = 11 and occupation probability leading to the coordination number . Open and filled symbols represent the numerical simulations of the monodisperse and the polydisperse systems, respectively. The dashed lines represent the theoretical prediction of Eq. (8) . As in Fig. 3(a) , where is indicated by the arrows and by the beginnings of the dashed curves.
Mobility μ versus surface density Nr for fixed mean grain size and number of layers N = 11 and varying occupation probability . The meanings of the open (filled) symbols, dashed lines, and the arrows are the same as in Fig. 6 . Note that both cases and occur in this figure.
Mobility μ versus surface density Nr for fixed mean grain size nm and occupation probability and varying number of layers N. The meanings of the open (filled) symbols, dashed lines, and the arrows are the same as in Fig. 6 .
Characteristics for two different systems with the parameters nm, N = 11, and (a) and nm, N = 3, and (b). The grain sizes are taken from Eq. (12) with (red circles), (green diamonds), and (blue triangles). While for system (a) the occupation probability is much larger than the critical density , for system (b) is very close to (see Fig. 4 ). The dashed line shows the theoretical prediction for monodisperse systems (see Eq. (6) ), the arrows mark according to Eq. (9) . Inset: Fraction of conducting grains (full circles) and fraction of conducting bonds (crosses) of the total number of grains and bonds of the whole structure for and for both systems (a) and (b).
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