Magneto-optic and transverse-transport properties of noncollinear antiferromagnets

Previous studies on the anomalous Hall effect in coplanar noncollinear antiferromagnets are revisited and extended to magneto-optic properties, namely magneto-optic Kerr effect (MOKE) and x-ray magnetic dichroism (XMCD). Starting from group-theoretical considerations the shape of the frequency-dependent conductivity tensor for various actual and hypothetical spin configurations in cubic and hexagonal ${\mathrm{Mn}}_{3}X$ compounds is determined. Calculated MOKE and x-ray dichroism spectra are used to confirm these findings and to give estimates of the size of the effects. For ${\mathrm{Mn}}_3\mathrm{IrPt}$ and ${\mathrm{Mn}}_3\mathrm{PtRh}$ alloys the concentration dependence of the anomalous and spin Hall conductivity is studied in addition.

Figure 1

Left: Cubic unit cell of ${\mathrm{Mn}}_3\mathrm{Ir}$ in the triangular antiferromagnetic structure. Ir atoms are colored in gold (light gray), Mn atoms in purple (dark gray). Black arrows indicate the direction of the magnetic moments on the Mn sites, the gray-shaded surface marks the (111) plane. Right: View along [111] on the kagome planes; Mn atoms and moments in alternating layers are colored in red and blue [57].

Figure 2

Anomalous Hall conductivity of ${\mathrm{Mn}}_3{\mathrm{Ir}}_{1-x}{\mathrm{Pt}}_x$ (top) and ${\mathrm{Mn}}_3{\mathrm{Rh}}_{1-x}{\mathrm{Pt}}_x$ (middle) as well as spin Hall conductivity of ${\mathrm{Mn}}_3{\mathrm{Rh}}_{1-x}{\mathrm{Pt}}_x$ alloys (bottom) as a function of concentration $x$. Open symbols are results excluding vertex corrections (NV) and filled symbols such including them (VC), see text for details. The intrinsic values of the pure compounds are shown in addition. All results were obtained using the Kubo-Středa formula.

Figure 3

Absorptive part of the optical conductivity tensor elements ${\sigma }_{\mathrm{xx}}^1, {\sigma }_{zz}^1$ (top) and ${\sigma }_{\mathrm{xy}}^2$ (middle) of ${\mathrm{Mn}}_3\mathrm{Ir}$ (full lines) together with the Kerr rotation angle ${\theta }_{\mathrm{K}}$ (bottom). In addition, corresponding results for ferromagnetic bcc-Fe are given (dashed lines).

Figure 4

Total absorption coefficient $\overline{\mu }$ at the ${\mathrm{L}}_{2,3}$ edge of a single Mn atom of noncollinear antiferromagnetic ${\mathrm{Mn}}_3\mathrm{Ir}$ (top) and corresponding circular dichroism spectra $\mathrm{\Delta }\mu$ (bottom), both for incidence ${\vec{q}}_{\mathrm{rad}}$ perpendicular (along [111]) and polar w.r.t. the magnetization direction.

Figure 5

Top: Differential polarization-averaged x-ray absorption spectra $\mathrm{\Delta }\overline{\mu }$ comparing incidence along high-symmetry directions in ${\mathrm{Mn}}_3\mathrm{Ir}$. Bottom: X-ray magnetic circular dichroism (XMCD) spectra $\mathrm{\Delta }\mu$ in ${\mathrm{Mn}}_3\mathrm{Ir}$ for incidence along the same directions as in the top panel.

Figure 6

Nonmagnetic unit cell of ${\mathrm{Mn}}_3\mathrm{Ge}$ in space group $P{6}_3/mmc$ (194). Mn atoms on the Wyckoff positions $6h$ are colored in magenta (dark gray) and Ge atoms (Wyckoff positions $2h$) are colored in light gray [57].

Figure 7

Antiferromagnetic structures of ${\mathrm{Mn}}_3\mathrm{Ge}, \mathrm{AFM}i$ with $i=0...19$, discussed in this contribution [57].

Figure 8

Top: Differential polarization-averaged x-ray absorption spectra $\mathrm{\Delta }\overline{\mu }$ comparing incidence along high-symmetry directions in ${\mathrm{Mn}}_3\mathrm{Ge}$ with AFM1 magnetic structure (see Fig. 7). Bottom: Corresponding x-ray magnetic circular dichroism (XMCD) spectra $\mathrm{\Delta }\mu$.

Figure 9

Top: Differential polarization-averaged x-ray absorption spectra $\mathrm{\Delta }\overline{\mu }$ comparing incidence along high-symmetry directions in ${\mathrm{Mn}}_3\mathrm{Ge}$ with AFM2 magnetic structure (see Fig. 7). Bottom: Corresponding x-ray magnetic circular dichroism (XMCD) spectra $\mathrm{\Delta }\mu$.

Figure 10

Top: Differential polarization-averaged x-ray absorption spectra $\mathrm{\Delta }\overline{\mu }$ comparing incidence along high-symmetry directions in ${\mathrm{Mn}}_3\mathrm{Ge}$ with AFM3 magnetic structure (see Fig. 7). Bottom: Corresponding x-ray magnetic circular dichroism (XMCD) spectra $\mathrm{\Delta }\mu$.

Figure 11

Top: Differential polarization-averaged x-ray absorption spectra $\mathrm{\Delta }\overline{\mu }$ comparing incidence along high-symmetry directions in ${\mathrm{Mn}}_3\mathrm{Ge}$ with AFM4 magnetic structure (see Fig. 7). Bottom: Corresponding x-ray magnetic circular dichroism (XMCD) spectra $\mathrm{\Delta }\mu$.