  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 $\uparrow\uparrow\downarrow\downarrow$ 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$_2$.
