Dynamics of a Charge Carrier Driven by Oscillating Fields in Materials with Impurities

In conductive materials and semiconductors, a charge carrier under the effects of an electric field will suffer collisions due to thermal fluctuations and impurities in the lattice, altering their trajectory. The electronic properties of these materials depend on the nature and frequency of these collisions; thus, they must be accounted for in any model dealing with electrical conduction. Tracking all collisions individually, while it may be possible within certain limits, forces the model to a large degree of approximation. This work introduces a Monte Carlo-based methodology to electrical transport in Ohmic materials that consists of two parts, the utilization of probability distribution functions (PDFs) for a set of collisions (coarse grain), as opposed to solving the transport equations for individual collisions and the use of homotopies to parameterize PDFs what produces a continuous set of PDFs once a relatively small number of them are explicitly parameterized. With the current approach, simulation times are from a few hundred to a few thousand times smaller than explicitly solving the transport equations. Average collision times are generated from distributions for a set of n collisions (the grain size), and from there, transport properties are calculated. Simulations were used to solve equations of motion based on the Drude’s Model of electrical conductivity. The results of the simulations are then used to generate probability distributions for various combinations of input parameters in order to coarse-grain the transport model. Grain sizes of n=5 and n=50 were considered. A homotopy on start time was first created by evaluating select distribution parameters across a half cycle. An excellent agreement non-coarse grained model was obtained.The electric field was then incorporated into the model parameterization leading to a PDF that, via a homotopy, can generate average collision time for any initial position of the carrier under any electric field within a continuous range). Results were validated using the non-coarse grained simulation under conditions not used for the parametrization for up to 500,000 collisions, with current density values being above 98.9% accurate. The goal of this work was to build a homotopy or mapping that, given some input parameters, could output some transport properties to aid experimental studies. The material of choice for this work was an ideal ohmic conductor with a mean free path of 4.3× 10−9m