  A two-component quasi-two-dimensional superconductor with
Dzyaloshinsky-Moriya interaction is studied based on the Ginzburg-Landau and
Bogoliubov-de Gennes theories. Under external in-plane magnetic fields, the
order parameter of the superconducting state is a type of the Fulde-Ferrell
state with a finite momentum of Cooper pairs due to the Dzyaloshinsky-Moriya
interaction. It is shown that the superconducting diode effect can emerge when
a supercurrent flows parallel to the external magnetic field, characteristic of
chiral crystals. In the Bogoliubov-de Gennes theory, phase diagrams associated
with the transition of the Cooper-pair momentum and the Josephson phase between
spin-singlet and spin-triplet Cooper pairs are derived, and a close
relationship with the diode quality factor is demonstrated. Implications of
critical currents in the aspect of thermodynamics are also discussed. Based on
such an argument, it is argued that the first-order phase transition in terms
of Cooper-pair momentum and the coexistence of phases with different
Cooper-pair momentum and Josephson phase can occur. The argument also implies
the issue with the definition of critical currents calculated from the extremes
of the supercurrent when metastable states exist. Comments on purely
two-dimensional superconductors are also given.
