Hello, my name is Bradley Martin and I am a PhD student work of my supervisor, Jarvis Frost, in the Experimental Solidestate Group at Imperial College London, where I am researching how to develop better theories for better mobilities. Why is mobility important? Well, mobility affects whether a material will perform well as say a solar panel, an LED or a battery. It is a very important property of any semiconductor material. So why better theories? Well, because materials are complicated and mobility is determined by complex quantum processes, mobility is commonly calculated using semi-classical models such as the drood model, but we want to include more material properties and treat mobility quantumly. So where do we begin? A good start and point was provided by Herbert Frouelik in the 1950s, who developed a Hamiltonian that describes a single conduction band electron interacting with the polarization field in a material. The strength of the interaction is characterized by the dimensionless alpha parameter, which can be determined from material properties. This scenario can be pictures as an electron repellent negative ions than attracts impositive ions, the self-induced potential arises, which acts back on the electron and modifies its properties. This can also be viewed as an electron dress with a virtual cloud of phonons that move with the electron through the material, collectively known as a polaron. Frouelik's model has no known exact solution. Therefore, we see an approximate solution, so we copy Richard Feynman and use path integrals. These focus on all the possible trajectories that the polaron can take. The result is that the infinite number of phonons can instead be replaced with an effective potential. This potential looks like an attractive cool-on potential between the electron and itself at an earlier time. The caveat is that we are now working in an imaginary time, which replaces troublesome oscillations of exponential decays, as what quantum physics falls statistical physics. From this, we can get a variational principle that gives the n upper bounds to the free energy of the polaron by swapping out the effective cool-on potential for an effective harmonic potential. Now back to mobility. We start by finding the best harmonic approximation to the polaron, which looks like the electron coupled by spring to a fictitious particle standing in for every phonon. The mobility is given by the zero-frequency limit of the real conductivity of the electron, with a quantum part of the responses entirely contained in a memory function that we calculate with the harmonic model. The result are non-drewed like oscillations that alter the polaront mobility. I have extended Feynman's model to include multiple phonon modes, which replaces the single spring approximation by one where the electron is coupled by springs to many fictitious particles with different vibrational frequencies and masses. This allows us to more accurately model the polaron quantum response to give better predictions for the conductivity spectra of materials. Thank you for watching and please check out our code over on GitHub.