Generalized balance equations for charged particle transport via localized and delocalized states: Mobility, generalized Einstein relations, and fractional transport

Abstract

A generalised phase-space kinetic Boltzmann equation for highly non-equilibrium charged particle transport via localised and delocalised states is used to develop continuity, momentum and energy balance equations, accounting explicitly for scattering, trapping/detrapping and recombination loss processes. Analytic expressions detail the effect of these microscopic processes on the mobility and diffusivity. 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 and recombination heating/cooling are shown to arise from recombination loss processes and the localised and delocalised nature of transport. Fractional transport emerges naturally within this framework through the appropriate choice of divergent mean waiting time distributions for localised states, and fractional generalisations of the GER and mobility are presented. Signature impacts on time-of-flight current transients of recombination loss processes via both localised and delocalised states are presented.

Third-order transport coefficients for localised and delocalised charged-particle transport

“… 24 . This would allow for the derivation of a skewness analogue of Einstein’s relation that would also take into account the field dependence of mobility 24 . This may also shed light on the recent results of Petrović et al .…”

“…( 1 ) has an enhanced diffusivity in the direction of the field due to trapping and detrapping. This enhancement manifests as the following generalised Einstein relation 24 …”

“…We previously reported 22 – 24 the development of a phase-space kinetic model wherein charged particles scatter due to collisions, enter and leave traps and undergo recombination. In this model, free particles are described by a phase-space distribution function f ( t , r , v ), defined by the generalised Boltzmann equation where E is the applied electric field and particles have charge e , mass m and number density n ( t , r ) ≡ ∫ d v f ( t , r , v ).…”

“…In the following, we describe charged particle transport using a full phase-space kinetic model as defined by a generalised Boltzmann equation with a corresponding trapping and detrapping operator. In our previous papers 22 – 24 , we introduced and studied such a generalised Boltzmann equation, deriving lower-order transport coefficients (up to diffusion) and generalisations of the Einstein relation. We will extend the results of these papers to determine the skewness tensor.…”

Analysis of a generalised Boltzmann equation for anomalous diffusion under time-dependent fields

“…It was shown that with particular choices of the trapping operator the fractional diffusion equation could be obtained from appropriate integrations and, when applied to organic semiconductor transport, the appropriate short and longtime behaviour was obtained, validating the approach [13,14]. This model was subsequently extended to include energy-dependent collision, trapping and loss rates, analysed using momentum transfer theory and used to generalise classical relations such as the Wannier and Einstein relations [16,17].…”