Percolation and conduction in restricted geometries

Abstract

The finite-size scaling behaviour for percolation and conduction is studied in two-dimensional triangular-shaped random resistor networks at the percolation threshold. The numerical simulations are performed using an efficient star-triangle algorithm. The percolation exponents, linked to the critical behaviour at corners, are in good agreement with the conformal results. The conductivity exponent, t ′ = ζ/ν, is found to be independent of the shape of the system. Its value is very close to recent estimates for the surface and bulk conductivity exponents.

Influence of Nanotube Length on the Optical and Conductivity Properties of Thin Single-Wall Carbon Nanotube Networks

“…Interestingly, the universal percolation exponents t and s for the 820 nm and 210 nm tubes have similar values. These values are close to those theoretically expected for a large conducting 2D network, − where s = t ≈ 1.3. Kholodenko and Freed argue for the exact 2D exponent value, 23/18 ≈ 1.28.…”

“…Rather, they are found to be in accord with those predicted by GEM theory in 3D, s ≈ 0.8 and t ≈ 2.0. This indicates that an increasing fraction of SWCNT connections are being made in the third dimension in these short tube networks. ,, …”

Electronic Durability of Flexible Transparent Films from Type-Specific Single-Wall Carbon Nanotubes

“…The metallic data (Figure b) yield σ 0 = 3.3 × 10 –4 □ Ω –1 and h c = 12 ± 2 nm with a critical exponent of 0.92 ± 0.1, while the fit of the semiconducting data (Figure e) yields σ 0 = 7.7 x10 –5 □ Ω –1 and h c = 6 ± 2 nm with a critical exponent of 1.15 ± 0.1. Results from both film types are in agreement with accepted values of the critical exponent (α ≈ 1) for percolated 2D conducting networks. − Film thicknesses range from 9 to 72 nm, much less than the wavelength of light, and the transmittance as a function of sheet resistance, R s , is fit to the expression , T ( R S ) = true( 1 + 1 2 R normalS normalμ 0 normalε 0 normalσ normalo normalp normalσ normald normalc true) − 2 where ε 0 and μ 0 are the permittivity and permeability of free space, respectively, σ op is the optical conductivity, and σ dc is the direct current conductivity. The ratio σ op /σ dc quantifies the combined optical and electronic quality of the films.…”

Dielectric mixtures: electrical properties and modeling

“…In the case of disordered systems, their physical behavior can usually be described by a power law with a percolation threshold and a critical exponent depending on the geometry [299,300,209,301] and on the conduction of the system [287,288,291,290,302]. One should also not exclude the local field changes in the composite due to interaction of inclusions.…”