Mechanism of Type-II Multiferroicity in Pure and Al-Doped CuFeO_{2}

Type-II multiferroicity, where electric polarization is induced by specific spin patterns, is crucial in fundamental physics and advanced spintronics. However, the spin model and magnetoelectric coupling mechanisms in prototypical type-II multiferroic CuFeO_{2} and Al-doped CuFeO_{2} remain unclear. Here, by considering both spin and alloy degrees of freedom, we develop a magnetic cluster expansion method, which considers all symmetry allowed interactions. Applying such method, we not only obtain realistic spin model that can correctly reproduce observations for both CuFeO_{2} and CuFe_{1-x}Al_{x}O_{2}, but also revisit well-known theories of the original spin-current (SC) model and p-d hybridization model. Specifically, we find that (i) a previously overlooked biquadratic interaction is critical to reproduce the ↑↑↓↓ ground state and excited states of CuFeO_{2}; (ii) the combination of absent biquadratic interaction and increased magnetic frustration around Al dopants stabilizes the proper screw state; and (iii) it is the generalized spin-current (GSC) model that can correctly characterize the multiferroicity of CuFeO_{2}. These findings have broader implications for understanding novel magnetoelectric couplings in, e.g., monolayer multiferroic NiI_{2}.