Formal analogy between multiple-trapping and polarization models: a physical picture for the Cole–Cole formula

Abstract

Dielectric properties of insulators can often be explained with a complex dielectric constant taking the well-known form proposed by Cole and Cole. The latter consists in correcting the simple Debye expression to account for anomalous dispersion in time required to line up dipoles in a dielectric submitted to a static field. Despite its success, the Cole–Cole formula has not received any convincing physical explanation. It introduces anomalous dispersion from a mathematical trick and is not based on a particular description of the polarization at the microscopic scale. The physical reasons for the dispersion are therefore obscure. Carrier transport is another field where anomalous dispersion occurs. In this case, however, numerous published works have contributed to draw some clear physical pictures for it, the most successful models involving hopping and multiple-trapping (MT) in the presence of an exponential distribution of traps. This work shows that the Cole–Cole formalism is formally equivalent to the MT model of highly dispersive transport when written under adequate conditions. This analogy provides a possible physical framework for the Cole–Cole formula, leading for instance to an expression of the characteristic time involved in this formula as a function of the Debye relaxation time.

Dielectric Engineering for Manipulating Exciton Transport in Semiconductor Monolayers

“…Such differences in the observed anomalous coefficient and the effective diffusivities across the different samples can be explained by a multitrapping and release (MTR) exciton transport model 45 shown in Figure 2f. In energetically disordered semiconductors under the low excitation density limit, transport becomes sub-diffusive due to the capture (temporary or until recombined) of mobile excitons at trap sites.…”

Mobility reduction and apparent activation energies produced by hopping transport in the presence of Coulombic defects

“…Properties of anomalous dispersive transport resulting from MT on an exponential trap distribution are also similar to that of hopping in an exponential bandtail. Both mechanisms give rise to currents obeying equation (11) with the same dispersion parameter α = k B T/ε c [2,23,24]. Discrepancies between these models are expected at short time and high field.…”

“…First, high field hopping transport is characterized by a field-dependent dispersion parameter which departs from the low field value. Second, multiple-trapping can yield a specific current behaviour at short time [23,24]. In all cases, discrimination between coulombic trap-limited MT and hopping can be done because of the modified Arrhenius law characterizing hopping, provided the effective temperature (1/T -1/T 0 ) -1 clearly differs from T. The crossing temperatures T 0 reported in figures 3 and 7 are very high and confusion would be possible at low T in the situation of an experimental data analysis.…”

“…Trap-limited mobilities also exhibit a Poole-Frenkel field dependence in the multipletrapping (MT) model of Gaussian transport (see [23,24] for recent treatments of MT for Gaussian and dispersive transport). In this situation, however, the field-assisted lowering of the activation energy is initially assumed and included in the thermal release frequency W from Coulombic traps (equation ( 14)).…”