Monopole-based formalism for the diagonal magnetoelectric response

We develop the formalism of the macroscopic monopolization—that is, the magnetoelectric monopole moment per unit volume—in periodic solids, and discuss its relationship to the magnetoelectric effect. For the series of lithium transition metal phosphate compounds, we use first-principles density functional theory to calculate the contributions to the macroscopic monopolization from the global distribution of magnetic moments within the unit cell, as well as from the distribution of magnetization around the atomic sites. We find one example within the series (LiMnPO${}_4$) that shows a macroscopic monopolization corresponding to a ferromonopolar ordering consistent with its diagonal magnetoelectric response. The other members of the series (Li$M$PO${}_4$, with $M=\text{Co}$, Fe, and Ni) have zero net monopolization but have antiferromonopolar orderings that should lead to $q$-dependent diagonal magnetoelectric effects.

Figure 1

Diagonal [(a) and (b)] and off-diagonal [(c) and (d)] magnetoelectric responses of monopolar and toroidal spin arrangements (from Ref. 11).

Figure 2

Representation of (left to right) positive and negative magnetoelectric monopoles, the $z$ component of the toroidal moment and the $z^2$ component of the quadrupole moment.

Figure 3

Representative arrangements of local magnetic moments (shown by arrows) that have monopolar contributions. The arrangements in (a) and (b) (point group $4m^{\prime }$) are purely monopolar and have equal and opposite monopoles. (c) (point group $m^{\prime }{m}^{\prime }$) consists of the sum of a monopolar contribution (of size half that of (b) and a quadrupolar contribution; the decomposition is shown in the lower panel.

Figure 4

Calculation of the monopolization for two different one-dimensional antiferromagnetic periodic arrangements of magnetic moments. Our choice of unit cell is indicated by the shaded area in each case. (a) A nonmonopolar state, which is space-inversion symmetric with respect to each moment site. (b) A monopolar state.

Figure 5

Allowed values of the monopolization for the antiferromagnetic chain of Fig. 4 as a function of displacement $\lambda$ from the nontoroidal case ($\lambda =0$). The cartoons at the bottom indicate the corresponding positions of the magnetic moments within the unit cell.

Figure 6

Effect on the magnetic moment configuration of Fig. 4b (middle panel) of a reversal of all magnetic moments (lower panel) and of a reversal of the noncentrosymmetric distortion $d$ (upper panel). Note that the upper and lower final states are identical, with the moments in the upper and lower panels translated by half a unit cell relative to each other.

Figure 7

Structure of the lithium transition metal phosphates. The 1–4 labeling of the transition metal atoms is consistent with their labeling in Tables  and .

Figure 8

Calculated magnetization density isosurface for LiNiPO${}_4$. The blue and red surfaces correspond to positive or negative density, respectively.