Mathematical modelling of disorder in organic solar cells

Matt will work on mathematical methods to take into account the effects of disorder in solar cells. A particular emphasis will be on organic solar cells as well as hybrid solar cells containing organic components. These cells have crucial advantages over alternatives such as silicon as they can be produced directly from a solution that is cooled down. This production process means that disorder plays a larger role than in other solar cells, and the aim of this project will be systematically treat effects arising from this disorder.

Existing mathematical approaches to solar cells involve modelling the classical motion of charge carriers using differential equations. These equations describe the rate rate of change for the concentration of conducting electrons and holes as governed by diffusion, drift, and well as generation and recombination of charge carriers. The equations can also be adapted to include excitons (bound states formed out of electrons and holes). A further model for organic solar cells is the Gaussian disorder model. This model is more comprehensive than the differential equations, and explicitly takes account of quantum effects as well as statistical mechanics. In this model, the possible states of the charge carriers are localised at randomly distributed sites. The energies of these states follow a Gaussian distribution, and jump rates between sites are chosen in line with ideas from Statistical Mechanics. Finally, the solar cells could be described using the full machinery of many-particle quantum mechanics, with a second quantised Hamiltonian taking into account the motion of charge carriers in the solar cell, interaction with light, etc.

Two directions to be followed in this project are
1. The development an improved version of the drift diffusion equations that incorporates effects from the Gaussian disorder model, using a continuum limit.
2. Implementing disorder averages in the second quantised model of organic solar cells. This will make use of methods used to perform disorder averages in condensed matter theory, such as nonlinear sigma models.

These lines of research are based on preliminary results, including work of the supervisor and in Matt's MSc dissertation, where he considered second quantised models for solar cells based on e.g. the Hubbard model and the Jaynes-Cummings model. The balance between these topics will be adapted to the progress of the project.

This work will allow for a transfer of advanced methods developed in Mathematical Physics to photovoltaics, which is mostly dominated by experimental work and more phenomenological approaches. It will contribute to the development of clean energy, with an obvious environmental impact. It is connected to the EPSRC research areas of Mathematical Physics (in the Mathematical Sciences theme), Solar Technology (in the Energy theme) and Condensed Matter: Electronic Structure (in the Physical Sciences theme).