Electrical conductivity of crack-template-based transparent conducting films: mean-field approximation, effective medium theory, and simulation

In our work, crack-template-based transparent conducting films were modeled as networks corresponding to the edges of a two-dimensional Poisson--Voronoi diagram. Two types of networks were considered: the original one, in which the conductivity of each edge was inversely proportional to its length, and the effective one, where all edges had the same conductivity obtained from the effective medium theory. The mean field approximation was used for analytical evaluation of the electrical conductivity. Direct numerical calculations for the Poisson--Voronoi diagram showed that the mean field approximation overestimated the conductivity of the original network by approximately 13\%, and of the effective network by 79\%. In addition, a hexagonal network with an edge conductivity distribution corresponding to the Poisson--Voronoi diagram was studied: for it, the predictions of the effective medium theory turned out to be more accurate than for the Poisson--Voronoi diagram, which was explained by the greater structural homogeneity of the periodic hexagonal lattice. Our results showed that when modeling crack-template-based transparent conducting films, especially in the case of hierarchical cracks with variable width (where the resistance was not simply proportional to the length), the application of the mean field approximation could potentially lead to significant errors.