Abstract
A numerical hopping model is developed to investigate the temperature dependence of electrical conductivity in a distribution of localized states. An exponential energy dependence of the density-of-states (DOS) distribution (N 0/E 0)exp(E/E 0) is assumed, together with a high DOS value (1 × 1018-1 × 1021cm−3 eV−1) at the Fermi level. The low-field dc conductivity is dominated by a subset of the localized states centred at a mean transport energy E t located well above E t; both E t and the distribution width (ΔE ≫ kT) increase with increasing temperature, in the range 50–500 K. It is found that the effective activation energy E act and apparent conductivity pre-factor σ0 both increase with increasing temperature. In any given temperature range, E act decreases with increasing DOS at E F, while σ0 decreases exponentially with increasing E 0. A linear relationship between log(σT 1/2) and T −1/4 is predicted up to high temperatures with a strong positive correlation between the pre-factor σ00 and the slope T1/4 0. This model is applied to electrical transport in amorphous carbon and carbon alloys where bonding and antibonding π orbitals produce highly localized states decoupled from extended a states. Unlike the classical variable-range hopping at the Fermi level, this model describes experimental data using physically acceptable values of N(EF), E0 and the localization radius 1/γ = 5 Å. Its predictions provide a framework to understand the increased conductivity brought about by ‘dopant’ atom incorporation.