Indirect exchange interaction in two-dimensional materials with quartic dispersion

We investigate the indirect magnetic exchange interaction between two magnetic moments in a two-dimensional semiconductor with quartic dispersion, featuring a singularity at the band edge. We obtain the Green's functions analytically to calculate the magnetic exchange interaction at zero temperature. We show that the singularity in the density of states (DOS) for quartic dispersion gives rise to an enhancement in the amplitude of the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction as the Fermi energy is swept toward the band edge. Furthermore, a region of finite exchange interaction arises, with a range increasing as the Fermi energy approaches the band edge. The results lay the possibility of an electrical/chemical control over the exchange interactions.