Abstract
Electrons in a non-centrosymmetric crystal experience effective Zeeman fields in momentum space, known as spin-orbit coupling(SOC) fields. Unlike usual Zeeman fields, these SOC fields pin the spins of electrons with opposite momentum to opposite directions. In this thesis, we present theoretical studies on two types of SOC fields, the Rashba SOC and the Ising SOC, with particular emphasis on their roles in creating novel superconducting states and Berry phase effects. In the first part of the thesis, we show that in superconductors with usual s-wave interactions, Rashba and Ising SOC fields can generate unconventional electron pairing. In particular, we discuss the transport properties of these novel Cooper pairs and explain how they can be used to create Majorana fermions. The second part of the thesis is devoted to the finding of a new type of valley Hall effect, called the spin valley Hall effect, which originates from combined effects of Rashba and Ising SOCs in transitionmetal dichalcogendies(TMDs). Experimental schemes to detect the novel spin valley Hall effects are proposed.