Electron-hole diffusion in disordered superconductors

In a superconductor, the excitations at energies above the gap can be understood as a superposition of electrons and holes. In this work, we study their diffusive behavior in disordered superconductors in an electron-hole basis. That is, we treat electron and hole diffusion as well as electron-hole conversion processes. We refer to this concept as electron-hole diffusion. We develop a formalism based on semiclassical Green's functions in Nambu space that allows us to treat disorder in superconductors, and use said formalism to study diffusive propagation in conventional superconductors. We focus on different properties that relate to electron-hole diffusion in order to understand more in depth the properties of disordered conventional superconductors and their possible applications. We show that the speed of propagation in disordered superconductors is given by the energy dependent group velocity $v_g=v_F\sqrt{E^2-\Delta^2}/E$ and determine that the conditions for the diffusive regime to take place in the superconducting state differ from those in metals. In superconductors there exist two energy scales that determine the onset of the diffusive regime. The first energy scale is given, similar to the normal metal case, by the inverse of the scattering time. The second energy scale does not depend on disorder strength, but instead on the energy carried by the diffusing particle and the strength of the superconducting gap. Two regimes can be defined depending on which energy scale dominates, and a novel energy scale $\varepsilon_*$, that separates these two regimes, emerges. We later study thermal conductivity in superconductors, putting special emphasis in the particular behavior of the weak localization correction. We show that the behavior of the weak localization is temperature dependent. This dependence varies in the two energy regimes defined by $\varepsilon_*$. We discuss its behavior in the different regimes, and highlight the case of a dirty superconductor ($\tau_e \Delta \ll 1$), where we theorize that the novel energy scale $\varepsilon_*$, given in this case by $\varepsilon_*=\sqrt{\Delta/\tau_e}$, could be experimentally measured. We discuss as well the use of disordered superconductors in the field of quantum computation. We build on a proposal where a disordered superconductor is used as a way to extend the exchange interaction between solid-state spin qubits. In the setup, the exchange interaction is possible via virtual propagation through the superconductor at energies below the superconducting gap. We discuss the viability of the setup under different experimental conditions. We show that the effects of external magnetic fields or spin-orbit (SO) coupling in the superconductor decrease the coupling range. We also highlight however the role of the geometry of the superconductor, which has a very strong impact on the coupling range with gains of over an order of magnitude from a 2D film to a quasi-1D strip. We estimate that for superconductors with weak SO coupling (e.g., aluminum), exchange rates of up to $100\,$MHz in the presence of external magnetic fields of up to $100$mT could be achieved over distances of over $1\mu\text{m}$.Finally, we study the density of states anomaly in disordered conventional superconductors. We focus on the two-dimensional case. For energies larger than the superconducting gap we obtain a logarithmic correction in $\tau_e E$ with the leading order correction due to superconductivity proportional to $\Delta^2/E^2$. For energies close to the gap, the behavior of the DOS anomaly is divergent. However, as opposed to the logarithmic divergence encountered in the normal metal state when approaching the Fermi energy, in the superconducting case this divergence is stronger and proportional to $\sqrt{\Delta/(E-\Delta)}$. This divergence shows as a decrease of the superconducting density of states peak as disorder increases