Figure 1

(a) Structure of a ${\mathrm{GaFeO}}_3$ crystal with a space group of $Pc{2}_{1}n$. An electric polarization appears along the screw $b$ axis, which causes the breaking of inversion symmetry. The magnetic structure is ferrimagnetic with a magnetic point group of $m^{\prime }{2}^{\prime }m$ below $T_C$ ($\sim 205\mathrm{K}$). The $c$ axis is a magnetic easy axis and a saturated moment is about $0.6{\mu }_B/\mathrm{Fe}$. (b) Experimental configuration for the observation of x-ray nonreciprocal directional dichroism (XNDD). A synchrotron x-ray beam was monochromated by a pair of Si(111) crystals. The x-ray beam is injected onto the $(\overline{1}00)$ plane, while an alternating magnetic field is applied along the $c$ axis in Voigt configuration, with an amplitude of 50 mT and a frequency of 10 Hz. Ionization chambers were used for the detection of an incident intensity ($I_0$) and a transmission ($I_1$). XNDD was observed as the first-harmonic component of the transmission [$\Delta I_1(f)$] responding to the magnetic field with a lock-in amplifier.Reuse & Permissions

Figure 2

Spectra of (a) the x-ray absorption $\mu t$ and (b) the XNDD $\Delta \mu t$ of a ${\mathrm{GaFeO}}_3$ crystal at 50 K for $E^{\omega }\parallel b$ (red) and $E^{\omega }\parallel c$ (blue). $\Delta \mu t$ is defined as the difference of absorption coefficients when a magnetic field is applied parallel ($H+$) and antiparallel ($H-$) to the $c$ axis. In (b) the spectrum of the second-harmonic component of the magnetic-field modulation for $E^{\omega }\parallel c$ is also shown with open circles.Reuse & Permissions

Figure 3

Temperature dependence of (a) the XNDD spectra in the pre-edge, and (b) the peak height of XNDD spectrum for $E^{\omega }\parallel b$ (red) and $E^{\omega }\parallel c$ (blue). The temperature dependence of the magnetization is also shown in (b) for comparison, which was measured in a static magnetic field of 50 mT along the $c$ axis after a zero field cooling process (a solid line) and in the field cooling run (a broken line). The magnitude of XNDD as detected by a modulation field of 50 mT shows a similar temperature dependence of the magnetization taken after the zero-field cooling. At low temperatures, the magnetization cannot be reversed by the ac magnetic field because the coercive force of ${\mathrm{GaFeO}}_3$ exceeds 50 mT below 40 K.Reuse & Permissions

Figure 4

Processes relevant to the microscopic origin of the XNDD effect in ${\mathrm{GaFeO}}_3$ for (a) $E^{\omega }\parallel b$ and (b) $E^{\omega }\parallel c$. In the vicinity of Fe $K$ absorption edge, both the electric dipole ($E1$) process from Fe $1s$ to Fe $4p$ and the electric quadrupole ($E2$) process from Fe $1s$ to Fe $3d$ appear. For $E1$–$E2$ interference, spin-orbit coupling and breaking symmetry of $b$ mirror is crucial. The $A$–$G$ indices of $3d$ and $4p$ levels correspond with the $A$–$G$ peaks of XNDD spectra shown in Fig. 2a, respectively. See text.Reuse & Permissions