Unravelling the complex magnetic structure of multiferroic pyroxene NaFeGe2O6: A combined experimental and theoretical study.

Magnetic order and the underlying magnetic model of the multiferroic pyroxene NaFeGe2O6 are systematically investigated by neutron powder diffraction, thermodynamic measurements, density-functional bandstructure calculations, and Monte-Carlo simulations. Upon cooling, NaFeGe2O6 first reveals one-dimensional spin-spin correlations in the paramagnetic state below about 50 K, revealed by magnetic diffuse scattering. The sinusoidal spin-density wave with spins along the a-direction sets in at 13 K, followed by the cycloidal configuration with spins lying in the (ac) plane below 11.6 K. Microscopically, the strongest magnetic coupling runs along the structural chains, J1 ' 12 K, which is likely related to the one-dimensional spin-spin correlations. The interchain couplings J2 ' 3:8K and J3 ' 2:1K are energetically well balanced and compete, thus giving rise to the incommensurate order in sharp contrast to other transition-metal pyroxenes, where one type of the interchain couplings prevails. The magnetic model of NaFeGe2O6 is further completed by the weak single-ion anisotropy along the a-direction. Our results resolve the earlier controversies regarding the magnetic order in NaFeGe2O6 and establish relevant symmetries of the magnetic structures. These results, combined with symmetry analysis, enable us to identify the possible mechanisms of the magnetoelectric coupling in this compound. We also elucidate microscopic conditions for the formation of incommensurate magnetic order in pyroxenes.