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

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 the unit vector connecting sites $i$ and $j$ and $\alpha =x,y$, due partially to magnetoelectric coupling effects, are shown to be responsible for the spin-flop helical phase in CuFeO${}_2$ at low magnetic field and temperature. Usual biquadratic {\it symmetric} exchange, likely due to magnetoelastic coupling, is found to support the stability of axial magnetic states at higher fields in this nearly-Heisenberg like stacked triangular antiferromagnet. A model Hamiltonian which also includes substantial interplane and higher-neighbor intraplane exchange interactions, reproduces the unique series of observed commensurate and incommensurate periodicity phases with increasing applied magnetic field in this highly frustrated system. The magnetic field-temperature phase diagram is discussed in terms of a Landau-type free energy.