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Biquadratic {\it antisymmetric} exchange terms of the form $ - [C_{ij}e^{\alpha}_{ij}({\bf s}_i\times{\bf s}_j)_z]^2$, where ${\bf e}_{ij}$ is theunit vector connecting sites $i$ and $j$ and $\alpha = x,y$, due partially tomagnetoelectric coupling effects, are shown to be responsible for the spin-flophelical phase in CuFeO$_2$ at low magnetic field and temperature. Usualbiquadratic {\it symmetric} exchange, likely due to magnetoelastic coupling, isfound to support the stability of axial magnetic states at higher fields inthis nearly-Heisenberg like stacked triangular antiferromagnet. A modelHamiltonian which also includes substantial interplane and higher-neighborintraplane exchange interactions, reproduces the unique series of observedcommensurate and incommensurate periodicity phases with increasing appliedmagnetic field in this highly frustrated system. The magnetic field-temperaturephase diagram is discussed in terms of a Landau-type free energy.

Biquadratic antisymmetric exchange and the magnetic phase diagram of magnetoelectric CuFeO$_2$