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

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.