Superconductivity is a macroscopic quantum phenomenon at low temperatures, characterized by the complete absence of electrical resistance and the expulsion of magnetic flux fields. The foundational Bardeen, Cooper, and Schrieffer (BCS) theory provides the analysis of the microscopic mechanism, attributing the origin of superconductivity to the formation of Cooper pairs through an attractive electron-phonon interaction.

In recent decades, the discovery of superconductors beyond BCS theory has sparked explorations into unconventional pairing states. This thesis explores unconventional superconductors that intrinsically have extra degrees of freedom, more specifically, multi-band unconventional superconducting systems with spin-orbit coupling. We focus on analytical approaches to simplify the multi-band system into the minimized version and peel off the redundancy to better understand unconventional pairing states.

Identifying the leading pairing state in unconventional superconductors is a long-standing problem, which motivates us to investigate a handy method to analyze a multi-band superconducting system like a single-band one. We construct a general connection between the fluctuating order and the pairing state, and by using this connection, we find a quick route to qualitatively locating the leading pairing state. We use j = 3/2 superconductor as an example to show that it is feasible for multi-band superconductors. Further, we use a complete calculation of the linearized gap equation to show the leading pairing state of YPtBi.

The consequences of multi-band physics in a superconducting system are poorly understood. Impurity scattering seems to be a useful tool for exploring unconventional pairing states. Thus, we focus on the impurity-induced bound states of a Rashba superconducting bilayer. We study Yu-Shiba-Rusinov states from unconventional s-wave pairing states encountering impurities. We use the superconducting fitness to quantify the inter-band pairing and classify the generalized variety of pair-breaking impurities in the multi-band system.

Further, we analytically investigate the real-space Green’s function of the superconducting bilayer model. To effectively use impurity to identify pairing symmetries in unconventional superconductors with spin-orbit coupling, we apply real-space Green’s function to study the behavior of bound states with different impurities. In the single-impurity regime, we find the results of the local density of states are nearly similar with different classifications of

impurities. In the two-impurity regime, we find that the non-magnetic sublattice-localized impurities can identify the A1g state, and the magnetic and non-magnetic cases can quantitatively identify odd-parity states A1u and A2u. We also find the significant Rashba spin-orbit coupling effects in real space that can cause the anisotropic local density of

states distributions for all pairing states.