Anomalous charged-particle transport in organic and soft-condensed matter

Abstract

A general phase-space kinetic model for non-equilibrium charged particle transport through combined localised and delocalised states is presented that accounts for scattering, trapping/detrapping and recombination loss processes in organic and soft-condensed matter. The model takes the form of a generalised Boltzmann equation, for which an analytical solution is found in Fourier-Laplace space. A Chapman-Enskog-type perturbative solution technique is also applied, confirming the analytical results and highlighting the emergence of a density gradient series representation in the weak-gradient hydrodynamic regime. This representation validates Fick's law for this model, providing expressions for the flux transport coefficients of drift velocity and diffusion. By applying Fick's law, a generalised diffusion equation with a unique global time operator is shown to arise that coincides with both the standard diffusion equation and the Caputo fractional diffusion equation in the respective limits of normal and dispersive transport. A subordination transformation is used to efficiently solve the generalised diffusion equation by mapping from the solution of a corresponding classical diffusion equation. From the aforementioned density gradient expansion, we extend Fick's law to consider also the third-order transport coefficient of skewness. This extension is in turn applied to yield a corresponding generalised advection-diffusion-skewness equation. Negative skewness is observed and a physical interpretation is provided in terms of the processes of trapping and detrapping. By analogy with the generalised Einstein relation, a relationship between skewness, diffusion, mobility and temperature is also formed. The phase-space model is generalised further by introducing energy-dependence in the collision, trapping and loss frequencies. The solution of this resulting model is explored indirectly through balance equations for particle continuity, momentum and energy. Generalised Einstein relations (GER) are developed that enable the anisotropic nature of diffusion to be determined in terms of the measured field-dependence of the mobility. Interesting phenomena such as negative differential conductivity (NDC) and recombination heating/cooling are shown to arise from recombination loss processes and the localised and delocalised nature of transport. Fractional generalisations of the GER and mobility are also explored. Finally, a planar organic semiconductor device simulation is presented that makes use of the aforementioned generalised advection-diffusion equation to account for the trapping and detrapping of charge carriers. In this simulation, we use Poisson's equation to account for space-charge effects and Kirchhoff's circuit laws to account for RC effects. These considerations allow for a variety of charge transport experiments to be simulated in a planar geometry, including time of flight (TOF), charge extraction by linearly increasing voltage (CELIV) and resistance-dependent photovoltage (RPV) experiments. The simulation is used to explore a proposed experimental technique for the characterisation of the recombination coefficient, as well as to study what effects traps would have on the measured current.