Revealing hidden magnetoelectric multipoles using Compton scattering

Magnetoelectric multipoles, which are odd under both space-inversion $\mathcal{I}$ and time-reversal $\mathcal{T}$ symmetries, are fundamental in understanding and characterizing magnetoelectric materials. However, the detection of these magnetoelectric multipoles is often not straightforward as they remain “hidden” in conventional experiments in part since many magnetoelectrics exhibit combined $\mathcal{IT}$ symmetry. In this paper, we show that the antisymmetric Compton profile is a unique signature for all the magnetoelectric multipoles, since the asymmetric magnetization density of the magnetoelectric multipoles couples to space via spin-orbit coupling, resulting in an antisymmetric Compton profile. We develop the key physics of the antisymmetric Compton scattering using symmetry analysis and demonstrate it using explicit first-principles calculations for two well-known representative materials with magnetoelectric multipoles, insulating ${\mathrm{LiNiPO}}_4$ and metallic ${\mathrm{Mn}}_2\mathrm{Au}$. Our work emphasizes the crucial roles of the orientation of the spin moments, the spin-orbit coupling, and the band structure in generating the antisymmetric Compton profile in magnetoelectric materials.

Figure 1

Cartoons showing the atomic-site ($as$) and local-moment (loc) contributions to the ME multipole moments. The illustrations are for the $z$ component of the toroidal moment $t_z$. The magnetization texture on a sphere around an atom, as shown on the left, gives the $as$ contribution to $t_z$. The loc contribution from the local magnetic dipole moments is shown on the right.

Figure 2

Antisymmetric Compton profile in ${\mathrm{LiNiPO}}_4$. (a) The crystal and magnetic ($Pn{m}^{\prime }a$) structures of ${\mathrm{LiNiPO}}_4$. The arrows indicate the spin directions at the Ni sites. Representation of the allowed ME multipoles, toroidal moment $t_y$ and quadrupole moment $q_{xz}$, in this magnetic structure are shown at the right. (b) The symmetric ($J^s$) and the antisymmetric ($J^a$) parts of the Compton profile along the $y$ direction in the momentum space, computed for the magnetic structure, shown in (a). (c) The absence of the antisymmetric part for other momentum directions, $p_x$ and $p_z$. (d) The distribution of the antisymmetric part of the line integral of the electron momentum density along $p_z$ in the $p_x\text{−}{p}_y$ plane of the momentum space, indicating, further, the antisymmetry along $p_y$. The antisymmetric parts in all figures are magnified by a factor of ${10}^3$.

Figure 3

Role of SOC in the antisymmetric Compton profile. (a) Band structure along $k_y$ for the magnetic configuration in Fig. 2 both in the presence (left) and absence (right) of SOC. As seen from this plot, the bands are symmetric in the absence of SOC, while the presence of asymmetry is visible when SOC effects are included, indicating the crucial role of SOC in generating the asymmetry in the band dispersion. This is also reflected in the antisymmetric Compton profile $J^a\left(p_y\right)$ in (b). (b) The variation of $J^a\left(p_y\right)$ as a function of SOC, indicating strong dependence on the scaled SOC $\lambda ={\lambda }_c\times {\lambda }_r$, where ${\lambda }_r$ is the strength of the SOC in the real material and $\lambda$ is the SOC that we artificially enforce in the calculation. (c) The variation of $J^a\left(p_y\right)$ with the Hubbard $U$, indicating a weak dependence on $U$. The inset shows the small changes near the peak for a $U$ variation of 3–7 eV in intervals of 1 eV. (d) The variation of the magnitude of the computed atomic-site ME multipoles ${\tilde{t}}_y$ and ${\tilde{q}}_{xz}$ (in units of ${\mu }_B$) with $U$ (top panel). As in (c), both decrease with increasing $U$. The same variation for a different muffin-tin radius (2.6 a.u.) of Ni (bottom panel). Although the magnitudes of the multipoles depend on the muffin-tin radius, the trend in their variations with $U$ remains the same.

Figure 4

Manipulation of the magnetic structure and the corresponding antisymmetric Compton profile. (a) The variation of the antisymmetric Compton profile as a function of canting angle $\theta$ from $0^{\circ }$ to ${33}^{\circ }$. The arrows indicate the direction of increasing values of the canting angle $\theta$. The inset shows the corresponding variation in the magnitudes of the computed atomic-site ME toroidal moment ${\tilde{t}}_y$ and the quadrupole moment ${\tilde{q}}_{xz}$ in units of ${10}^{-3}{\mu }_B$ for the same values of $\theta$ as in the main plot. (b)–(d) The symmetric ($J^s$) and the antisymmetric ($J^a$) parts of the Compton profile along (b) the $p_{xyz}$ direction for the magnetic structure $P{n}^{\prime }{m}^{\prime }{a}^{\prime }$, (c) the $p_z$ direction for the magnetic space group $Pnm{a}^{\prime }$, and (d) the $p_x$ direction for the magnetic space group $P{n}^{\prime }ma$. In all cases the antisymmetric part of the Compton profile is magnified by a factor of ${10}^3$.

Figure 5

Schematic illustrations of the various symmetry-allowed magnetic structures corresponding to the magnetic space groups (a) $P{n}^{\prime }{m}^{\prime }{a}^{\prime }$, (b) $Pnm{a}^{\prime }$, and (c) $P{n}^{\prime }ma$. The arrows indicate the direction of the spin moments at the Ni atoms.

Figure 6

Antisymmetric Compton profile in ${\mathrm{Mn}}_2\mathrm{Au}$. (a) The crystal and magnetic structures of the ground state of ${\mathrm{Mn}}_2\mathrm{Au}$ corresponding to the Néel vector [110]. Arrows indicate the direction of the spin magnetic moments at the Mn sites. (b) The antisymmetric parts ($J^a$) of the Compton profile along $p_x$ and $p_y$, indicating equal and opposite contributions along these directions. This results in an asymmetry along $\left[1\overline{1}0\right]$ but an absence of asymmetry along [110]. (c) The computed antisymmetric Compton scattering profiles along the $\left[1\overline{1}0\right]$ and [110] directions reflect the presence and absence of asymmetry, respectively. The convolution of the computed antisymmetric profile along the $p_{1\overline{1}0}$ direction with a Gaussian function of 0.44 a.u. FWHM is also shown. (d) The magnetic structure for Néel vector $\left[1\overline{1}0\right]$. The corresponding antisymmetric Compton profile along (e) $p_x$ and $p_y$ and (f) $p_{110}$ and $p_{1\overline{1}0}$. In contrast to the Néel vector [110], here the antisymmetric profiles along $p_x$ and $p_y$ are the same in both magnitude and sign, leading to a vanishing antisymmetric profile along $p_{1\overline{1}0}$ but a nonzero $J^a$ along the [110] direction. All profiles are normalized, so that the total profile is normalized to 63 electrons, which is the total number of valence electrons per formula unit of ${\mathrm{Mn}}_2\mathrm{Au}$ in the calculation. The antisymmetric part of the Compton profile is magnified by a factor of ${10}^3$.

Figure 7

The asymmetry in the band structure of ${\mathrm{Mn}}_2\mathrm{Au}$, induced by the ME multipoles, for the Néel vectors [110] [(a)–(d)] and $\left[1\overline{1}0\right]$ [(e)–(h)]. The band structure along the (a) $k_x$, (b) $k_y$, (c) [110], and (d) $\left[1\overline{1}0\right]$ directions in the momentum space for the Néel vector [110] with the corresponding magnetic structure shown in Fig. 6. Comparison of (a) and (b) shows that the asymmetries of the bands are opposite along $k_x$ and $k_y$ for the Néel vector [110]. This results in an absence of asymmetry along the [110] direction in (c), while the asymmetry along the $\left[1\overline{1}0\right]$ direction is clearly visible in (d). Comparison with bands in the absence of SOC shows that the asymmetry is present only in the presence of SOC. Doubling the strength of the SOC further enhances this asymmetry, indicating the importance of SOC. (e)–(h) The same as in (a)–(d) but for the Néel vector $\left[1\overline{1}0\right]$. The corresponding magnetic structure is in Fig. 6. In this case, bands along (e) $k_x$ and (f) $k_y$ have the same asymmetry, leading to (g) asymmetric bands along the [110] direction and (h) absence of asymmetry along the $\left[1\overline{1}0\right]$ direction in the momentum space.