Interatomic spin-orbit coupling: mechanism for spin-spiral-caused ferroelectricity

There are two general classes of mechanisms that have been proposed for spin-spiral caused ferroelectricity, one based on ionic displacements as primary cause, the other on charge distortion without ionic displacements. Here we discuss the latter$^{1,2}$. The mechanism proposed here is illustrated by a model where a pair of ions a and b each have low-lying s- and excited p- states with a prescribed spin state $\chi_a$ for the a-site states, similarly for the b-site, and there are 2 electrons; interatomic spin orbit coupling resides in inter-ion hopping due to s-p matrix elements of the spin-orbit coupling operator $\propto \nabla V \times \mathbf{p}\cdot\mathbf{s}$, $V, \mathbf{p}, \mathbf{s}$ are 1- electron potential, momentum, spin, respectively. Assuming the symmetry of a nearest-neighbor pair of cubic- spinel B-sites (there's no center of inversion (coi)) we find an electric dipole moment in the direction $\mathbf{r}_ {ab}\times(\mathbf{S}_a\times\mathbf{S}_b)$, as was found when there is a coi$^{1,2}$. For the spins in a chain parallel to the spiral wave vector in CoCr$_2$O$_{4}$, direction [110]$^3$, this results in ferroelectricity, as observed$^{4}$; the spins in a [1,-1,0]-directed chain, give an antiferroelectric component. Extension to a pair of Cr$^{3+}$ ions will be discussed.\\ 1. H. Katsura et al,Phys. Rev. Lett.\textbf{95},057205 (2005)\\ 2. T. A. Kaplan and S. D. Mahanti, cond-mat/0608227, 10 Aug 2006 \\ 3. N. Menyuk et al, J. Phys. (Paris) \textbf{25}, 528 (1964)\\ 4. Y. Yamasaki et al, Phys. Rev. Lett. \textbf{96}, 207204 (2006)