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Cooper instability associated with superconductivity in the two-dimensionalsemi-Dirac semimetals is attentively studied in the presence of attractiveCooper-pairing interaction, which is the projection of an attractivefermion-fermion interaction. Performing the standard renormalization groupanalysis shows that the Cooper theorem is violated at zero chemical potentialbut instead Cooper instability can be generated only if the absolute strengthof fermion-fermion coupling exceeds certain critical value and transfermomentum is restricted to a confined region, which is determined by the initialconditions. Rather, the Cooper theorem would be instantly restored once afinite chemical potential is introduced and thus a chemical potential-tunedphase transition is expected. Additionally, we briefly examine the effects ofimpurity scatterings on the Cooper instability at zero chemical potential,which in principle are harmful to Cooper instability although they can enhancethe density of states of systems. Furthermore, the influence of competitionbetween a finite chemical potential and impurities upon the Cooper instabilityis also simply investigated. These results are expected to provide instructiveclues for exploring unconventional superconductors in the kinds of semimetals.

Cooper instability generated by attractive fermion-fermion interaction in the two-dimensional semi-Dirac semimetals