Hopping model for charge transport in amorphous carbon

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 x 10(18)-1 x 10(21) cm(-3) eV(-1)) at the Fermi level. The low-field de conductivity is dominated by a subset of the localized states centred at a mean transport energy E-t located well above E-F; both E-t and the distribution width (DeltaE much greater than 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 sigma (0) both increase with increasing temperature. In any given temperature range, E-act decreases with increasing DOS at E-F, while sigma (0) decreases exponentially with increasing E-0. A linear relationship between log(sigmaT(1/2)) and T-1/4 is predicted up to high temperatures with a strong positive correlation between the pre-factor sigma (00) and the slope T-0(1/4). This model is applied to electrical transport in amorphous carbon and carbon alloys where bonding and antibonding pi orbitals produce highly localized states decoupled from extended sigma states. Unlike the classical variable-range hopping at the Fermi level, this model describes experimental data using physically acceptable values of N(E-F), E-0 and the localization radius 1/gamma = 5 Angstrom. Its predictions provide a framework to understand the increased conductivity brought about by 'dopant' atom incorporation.