Better mobilities through better theories

Humanity could really do with some new energy materials. If these are to be more efficient than what we have already, they are likely to be more complex; if they are to be cheaper, then they are likely to be more disordered. Solid-state theory, mostly dating from the 1950s and 1960s, is really best at predicting the behaviour of perfect infinite crystals at zero temperature. So let's make some better theories!

Semiconductors are required for a number of applications: they often form one of the electrodes of a battery, they are used as the active material in solar cells, LEDs and solid-state lasers, low-thermal conductivity semiconductors can be used as thermoelectrics to generate electricity from temperature difference.
A key (and highly technologically relevant) technical feature of a semiconductor is the charge-carrier mobility. This is a phenomenological quantity, it is a product of a competition between processes. It is not a direct ground state property of the material, and is strongly temperature dependent. Most theories of semiconductor charge carrier mobility have an empirical parameter (often an effective scattering time, in a Drude like model of mobility). This means that while relative mobility can be predicted within one material class, absolute predictions of mobility are lacking.

The situation is not entirely hopeless though. As electronic structure techniques are getting more sophisticated, it is possible to calculate more subtle aspects of a material, such as the electron phonon coupling (This electron-phonon coupling is between the nuclear and electronic degrees of freedom, which is adiabatically separated and ignored once you apply the Born-Oppenheimer approximation.)

There are a number of different avenues to explore, depending on the interests of the student, and the existence of new experimental data to explain. The model polaron Lagrangian could be extended (while retaining analytic solution) to attempt to increase the accuracy of the approximation, perhaps by extending the Gaussian form to a set of correlated Gaussian processes. The effective Lagrangian could be extended to provide greater material specific detail. Diagrammatic Monte-Carlo can provide direct evaluation of the effective Lagrangian, and a code for this could be written to compare to the Feynman variational results. Further response functions of the polaron variation state could be computed, which enable comparison to experimental observables. For instance, the optical absorption of the polaron state could be compared to transient absorption measurements on these materials; the frequency dependent mobility could be calculated and compared to Terahertz and microwave conductivity measurements.
There are theoretical models built on a compatible path integral basis to describe disorder that exists in non-crystalline systems, but so far these have not been interfaced to polaron mobility theories.

As well as immediate method development, there is a large scope to apply these codes to material groups and classes. This requires the characterisation of materials, either by recourse to the material databases of synthetic data derived from electronic structure calculations, such as the Materials Project, and through the use of standard electronic structure packages such as VASP and Gauss
The central aim is to develop methods which can offer fully predictive temperature-dependent mobilities for a wide variety of systems of technical interest, and thereby offer design clues for, and methods to computationally screen, new materials.
Though this project is envisaged as theoretical and computational, it will require a deep familiarisation and contact with experimental methods, material specific properties, and interaction with experimental collaborators and peers who are measuring mobilities and response functions in these materials.