Optical Probe for Anomalous Hall Resonance in Ferromagnets with Spin Chirality

We have investigated the infrared optical Hall conductivity, ${\sigma }_{xy}(\omega )$ for band-filling-controlled ferromagnetic crystals of ${\mathrm{Nd}}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$, revealing the dynamical properties of their anomalous Hall effect (AHE). A resonant structure and its systematic filling dependence were observed in the Hall conductivity spectra in the midinfrared region (typically at 0.1 eV), while similar effects were not discerned in the diagonal (longitudinal or ordinary) conductivity spectra. This property of ${\sigma }_{xy}(\omega )$ provides crucial and essential information to understand the microscopic mechanism of AHE including its dc limit. Specifically, the interband transition at the magnetic-monopole-like band-anticrossing point, which is split by spin chirality, is the dominant source in AHE.

Figure 1

(a) Crystal structure of pyrochlore type ${\mathrm{Nd}}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$ and (b) schematic configuration of the Nd and Mo spins. Temperature dependence of (c) resistivity and (d) Hall conductivity in a magnetic field of 0.2 T applied along [100] for ${\mathrm{Nd}}_2({\mathrm{Mo}}_{1-x}{\mathrm{Nb}}_x{)}_2{\mathrm{O}}_7$ and $({\mathrm{Nd}}_{1-y}{\mathrm{Ca}}_y{)}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$.

Figure 2

The reflectivity (a), Kerr rotation (b), and Kerr ellipticity (c) spectra over the photon-energy range of 0.1–2.5 eV in ${\mathrm{Nd}}_2({\mathrm{Mo}}_{1-x}{\mathrm{Nb}}_x{)}_2{\mathrm{O}}_7$ and $({\mathrm{Nd}}_{1-y}{\mathrm{Ca}}_y{)}_2{\mathrm{Mo}}_2{\mathrm{O}}_7$ at 10 K. The real part of the diagonal optical conductivity spectra ${\sigma }_{xx}^{\prime }(\omega )$, and the real (e) and imaginary (f) parts of the Hall conductivity ${\tilde{\sigma }}_{xy}(\omega )$ in the infrared region. For ${\sigma }_{xx}^{\prime }(\omega )$, the spiky structures due to the optical phonons (0.02–0.06 eV) are removed from the figure for clarity. The dots on the vertical axis indicate the dc values. Note ${\sigma }_{xy}^{\prime \prime }(\omega )$ should go to zero in the dc limit by definition.

Figure 3

(a) The schematic view of the two-band model and its parameters $E_F$, $\mu$, and $m$ representing the Fermi energy, the distance of the band-anticrossing point from $E_F$, and half of the energy gap, respectively. (b) Hole ($y$) and electron ($x$) doping (band-filling change $\delta =x-y$) dependence of the dc anomalous Hall conductivity ${\sigma }_{xy}(\mathrm{dc})$, the peak magnitude of the real part of the optical Hall conductivity in the midinfrared region ${\sigma }_{xy}^{\prime }(E_{\mathrm{MIR}})$, and the estimated magnetization of the Mo spin ($M_{\mathrm{Mo}}$) at 10 K. (c) Doping dependence of the peak position $E_{\mathrm{MIR}}$ in ${\sigma }_{xy}^{\prime }(\omega )$ spectra located in the midinfrared region. The line shows the peak position shift corresponding to the slope $E_{\mathrm{MIR}}/\delta$ of $1\mathrm{eV}·\mathrm{Mo}$. Comparison between the experimental spectra (solid lines) and the two-band model calculations (broken lines) for (d) the real and (e) the imaginary part of ${\tilde{\sigma }}_{xy}(\omega )$ for $x=0$ and 0.06.