INTERATOMIC SPIN-ORBIT COUPLING: A MECHANISM FOR SPIN- SPIRAL-CAUSED FERROELECTRICITY T. A. Kaplan and S. D. Mahanti Michigan State University APS March, 2007

We had considered a simple model that gave the effect. The model involves inter-atomic spin-orbit coupling, and considers just ‘direct hopping’ between magnetic atoms a and b. O indicates the mid-point. An initial crude estimate suggested realism, perhaps. Very recently: discovered an additional, physically real contribution that appreciably reduces the size; now probably too small. But precise model physically correct; still gives qualitative insight; further, interatomic s.o. coupling might be important in superexchange. For our simplified model we show: Introduction b a O

For O a crystal center of inversion, and the direction of the spiral wave vector along the line including a and b, the ferroelectric polarization, in agreement with Katsura, Nagaosa, Balatzky (KNB). This unchanged for a Cr-Cr pair in, where O not a ctr. of inv., but there are mirror planes through O. However, direction of does not satisfy this in general. does not change sign when one of a n.n. pair of magnetic ions changes from 1/2-filled shell. (Interest via Co-Cr vs Cr-Cr pairs in.)

The Model Start with pair of atoms, e.g. hydrogens, or ions. For Cr’s, each ion: (3d) in orbitals. The atoms have different spin directions. Spin states quantized along z):, and a b y x z ab O

Co Cr O From Menyuk review, 1970 Cobalt chromite

Spin orbit coupling: Interatomic matrix elements: ptpt a b Cr-Cr a b H-H s THE CALCULATION

The wave function to 1 st order (in ) gives the electron density to 1 st order x y For spiral, contribution to dipole moment d from spin z- component same for each bond, ferroelectricity (similar to KNB).

So only the z-component of m enters: We considered V having the same symmetry as that of a n.n. pair of Cr’s in the spinel, namely, V(x,y,z)=V(-x,y, z)=V(x,-y, z). Origin at O (O not a center of inversion). This symmetry leads to the vanishing of certain of these quantities. E.g. for the H-H model, one set of such quantities is This vanishes for = x and z (by y-integral), leaving only ; , overlap bond charge ferroelectric polarization in y direction. + -

This direction given by found by KNB who assumed O is a center of inversion. We find conclusion unchanged for O not a coi., but with symmetry of Cr-Cr pair in. Same conclusion for pair of ions. However, for general, direction of not that of above. (It will also have a component along ).

Sign change of under ½-filled shell? Imagine one atom having d-levels in a crystal field such that there are just two low-lying orbitals. Case I: 1 electron at b; Case II: 3 electrons at b; 1 electron at a in both cases. Put p d2 d1 a s p d2 d1 “up” “down” b 1-electron energy-level scheme Can show that

Contribution to the charge density from electron hopping from a to b: Case I, 1 electron on b: (Sign difference from.) Case II, 3 electrons on b: Intra-atomic exchange J < , 1-electron energy difference; required for stability of model ground state. Then sign of d same in cases I and II. Argument shows how for filled shell.