Unraveling 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 ${\mathrm{NaFeGe}}₂{\mathrm{O}}₆$ are systematically investigated by neutron powder diffraction, thermodynamic measurements, density-functional band-structure calculations, and Monte Carlo simulations. Upon cooling, ${\mathrm{NaFeGe}}₂{\mathrm{O}}₆$ first reveals one-dimensional spin-spin correlations in the paramagnetic state below about 50 K, uncovered 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, ${J}₁\ensuremath{\simeq}12\phantom{\rule{0.16em}{0ex}}\mathrm{K}$, which is likely related to the one-dimensional spin-spin correlations. The interchain couplings ${J}₂\ensuremath{\simeq}3.8\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ and ${J}₃\ensuremath{\simeq}2.1\phantom{\rule{0.16em}{0ex}}\mathrm{K}$ are energetically well balanced and compete, thus giving rise to the incommensurate order, in sharp contrast to other transition-metal pyroxenes, in which one type of the interchain couplings prevails. The magnetic model of ${\mathrm{NaFeGe}}₂{\mathrm{O}}₆$ is further completed by the weak single-ion anisotropy along the $a$ direction. Our results resolve the earlier controversies regarding the magnetic order in ${\mathrm{NaFeGe}}₂{\mathrm{O}}₆$ 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.