  We investigate properties of exotic pairing states in a three-dimensional
Fermi gas with three-dimensional spin-orbit coupling and an effective Zeeman
field. The interplay of spin-orbit coupling, effective Zeeman field and pairing
can lead to first-order phase transitions between different phases, and to
interesting nodal superfluid states with gapless surfaces in the momentum
space. We then demonstrate that pairing states with zero center-of-mass
momentum are unstable against finite center-of-mass momentum
Fulde-Ferrell-Larkin-Ovchinnikov (FFLO) states, with the center-of-mass
momentum of the pairs opposite to the direction of the effective Zeeman field.
Unlike conventional FFLO states, these FFLO states are induced by the
coexistence of spin-orbit coupling and Fermi surface deformation, and have
intriguing features like first-order transitions between different FFLO states,
nodal FFLO states with gapless surfaces in momentum space, and exotic fully
gapped FFLO states. With the recent theoretical proposals for realizing
three-dimensional spin-orbit coupling in ultracold atom gases, our work is
helpful for the future experimental studies, and provides valuable information
for the general understanding of pairing physics in spin-orbit coupled
fermionic systems.
