Bond electronic polarization induced by spin

We study theoretically the electric polarization induced by noncollinear spin configurations in the limit of strong Hund coupling. We employ a model of two magnetic ions sandwiching an oxygen ion. It is shown that there appears a longitudinal polarization $P_x$ along the bond, which is roughly proportional to ${\left(m_r^x\right)}^2-{\left(m_l^x\right)}^2$ where $m_{r\left(l\right)}^x$ is the $x$ component of the spin orientation vector at right (left) magnetic ion. A numerical study of the model Hamiltonian yields both longitudinal and transverse electric dipole moments. The transverse polarization is shown to have a nonuniform as well as a uniform component, with the latter being consistent with the previous theory. The longitudinal polarization is nonuniform and oscillating with the period half that of the spin order, but the local magnitude is typically much larger than the uniform transverse polarization and may be detected by x-ray/neutron scattering experiments.

Figure 1

Arrangement of two magnetic ions and an oxygen ion in the M-O-M cluster. ${\widehat{m}}_l$ and ${\widehat{m}}_r$ are the magnetic moment orientations of the magnetic sites bridged to the intervening oxygen ion. Polarization $P_x$ along the cluster’s axis is found (see text for details).

Figure 2

The direction of longitudinal polarization $P_x=\mathbf{P}\bullet \widehat{x}$ in the ground state wave function plotted for $(m_l^x,m_r^x)$, $-1⩽m_l^x,m_r^x⩽1$. +(−) refers to $P_x>(<)0$. $P_x$ is indeterminate along two dotted lines ($m_r^x=m_l^x$ and $m_r^x=-m_l^x$) due to the level crossing occuring along these lines.

Figure 3

The calculated polarization $P_x$ and $P_y$ from exact diagonalization of the full M-O-M cluster Hamiltonian with a single hole. Dotted white lines indicate $P_x$ or $P_y$ equal to zero. $P_x$ vanishes not only when the spins are collinear, but also when ${\varphi }_l+{\varphi }_r=\pm \pi$, as discussed in Sec. 2. The overall directional dependence for $P_x$ is consistent with $\cos 2{\varphi }_r-\cos 2{\varphi }_l$. The sharp change in the sign of $P_x$ along ${\varphi }_r={\varphi }_l\pm \pi$ lines is due to the energy level crossing in the ground state. $P_y$ turns to zero along the lines (indicated as dotted white lines) where the spins become collinear. The parameters used in this calculation are $(U∕V,E_p∕V,\lambda ∕V)=(10,30,0.3)$.

Figure 4

(Color online) Induced polarization $P_x$ and $P_y$ in a unit of $L$ for the superexchange case with two holes in the three-atom cluster. The parameters used in this calculation are the same as in Fig. 3.

Figure 5

(a)–(c) Fourier components $A$, $B_1$, and $B_2$ in the polarization $P_x∕L$ and $P_y∕L$ using the fitting formula, Eq. (23). (d) Fourier components with $U∕V=10$ and varying $\lambda ∕V$. The inset shows very small $\lambda ∕V$ results.

Figure 6

(Color online) Spiral spin structure (red arrows) with an angle shift of $\pi ∕4$ between neighboring magnetic (M) ions and the induced dipole moments (parallel or antiparallel to black arrows) at oxygen (O) sites with half the periodicity. The uniform component is not shown since it is much smaller than the nonuniform part.