Dynamical magnetic charges and linear magnetoelectricity

Magnetoelectric (ME) materials are of fundamental interest and have been investigated for their broad potential for technological applications. The search for, and eventually the theoretical design of, materials with large ME couplings present challenging issues. First-principles methods have only recently been developed to calculate the full ME response tensor $\alpha$ including both electronic and ionic (i.e., lattice-mediated) contributions. The latter is proportional to both the Born dynamical electric charge $Z^{\mathrm{e}}$ and its analog, the dynamical magnetic charge $Z^{\mathrm{m}}$. Here we present a theoretical study of the magnetic charge $Z^{\mathrm{m}}$ and the mechanisms that could enhance it. Using first-principles density functional methods, we calculate the atomic $Z^{\mathrm{m}}$ tensors in ${\mathrm{Cr}}_2{\mathrm{O}}_3$, a prototypical magnetoelectric, and in KITPite, a fictitious material that has previously been reported to show a strong ME response arising from exchange striction effects. Our results confirm that in ${\mathrm{Cr}}_2{\mathrm{O}}_3$, the $Z^{\mathrm{m}}$ values and resulting ME responses arise only from spin-orbit coupling (SOC) and are therefore rather weak. In KITPite, by contrast, the exchange striction acting on the noncollinear spin structure induces large $Z^{\mathrm{m}}$ values that persist even when SOC is completely absent.

Figure 1

Sketch showing how the six lattice-mediated responses indicated by solid circles (orange) are each built up from the four elementary tensors indicated by open circles: the Born charge $Z^{\mathrm{e}}$ (yellow), magnetic charge $Z^{\mathrm{m}}$ (blue), internal strain $\Lambda$ (green), and force-constant inverse $K^{-1}$ (magenta). Each lattice-mediated response is given by the product of the three elementary tensors connected to it, as indicated explicitly in Eqs. (9)–(14).

Figure 2

Structure of ${\mathrm{Cr}}_2{\mathrm{O}}_3$. (a) In the rhombohedral primitive cell, four Cr atoms align along the rhombohedral axis with AFM magnetic moments shown by (blue) arrows. (b) Each Cr atom is at the center of a distorted oxygen octahedron.

Figure 3

Symmetry pattern of the Born and magnetic charge tensors for (a) the Cr atom in ${\mathrm{Cr}}_2{\mathrm{O}}_3$, (b) the O atom in ${\mathrm{Cr}}_2{\mathrm{O}}_3$, and the ${\mathrm{O}}^2$ atom in ${\mathrm{CaAlMn}}_3{\mathrm{O}}_7$, (c) the Ca, Al, and ${\mathrm{O}}^1$ atoms in ${\mathrm{CaAlMn}}_3{\mathrm{O}}_7$, and (d) the Mn and ${\mathrm{O}}^3$ atoms in ${\mathrm{CaAlMn}}_3{\mathrm{O}}_7$. The elements indicated by an asterisk vanish in the absence of SOC for $Z^{\mathrm{m}}$ in ${\mathrm{CaAlMn}}_3{\mathrm{O}}_7$.

Figure 4

Planar view of the ${\mathrm{CaAlMn}}_3{\mathrm{O}}_7$ (KITPite) structure. The broad arrows (blue) on the Mn atoms represent the magnetic moment directions in the absence of electric or magnetic fields. Small (black) arrows indicate the atomic forces induced by an external magnetic field applied in the $\widehat{y}$ direction.