Isotropic parallel antiferromagnetism in the magnetic field induced charge-ordered state of $\mathrm{Sm}{\mathrm{Ru}}_4{\mathrm{P}}_{12}$ caused by $p-f$ hybridization

Nature of the field-induced charge-ordered phase (phase II) of $\mathrm{Sm}{\mathrm{Ru}}_4{\mathrm{P}}_{12}$ has been investigated by resonant x-ray diffraction (RXD) and polarized neutron diffraction (PND), focusing on the relationship between the atomic displacements and the antiferromagnetic (AFM) moments of Sm. From the analysis of the interference between the nonresonant Thomson scattering and the resonant magnetic scattering, combined with the spectral function obtained from x-ray magnetic circular dichroism, it is shown that the AFM moment of Sm prefers to be parallel to the field (${\mathit{m}}_{\text{AF}}\parallel \mathit{H}$), giving rise to large and small moment sites around which the ${\mathrm{P}}_{12}$ and Ru cage contract and expand, respectively. This is associated with the formation of the staggered ordering of the ${\mathrm{\Gamma }}_7$-like and ${\mathrm{\Gamma }}_8$-like crystal-field states, providing a strong piece of evidence for the charge order. PND was also performed to obtain complementary and unambiguous conclusion. In addition, isotropic and continuous nature of phase II is demonstrated by the field-direction invariance of the interference spectrum in RXD. Crucial role of the $p-f$ hybridization is shown by resonant soft x-ray diffraction at the P $K$ edge ($1s\leftrightarrow 3p$), where we detected a resonance due to the spin polarized $3p$ orbitals reflecting the AFM order of Sm.

Figure 1

X-ray energy dependences of the intensity of the 333 Bragg diffraction at 2 K (phase III) in magnetic fields of 0 and $\pm 6$ T along $\left[1\overline{1}0\right]$ after absorption correction.

Figure 2

(a) X-ray energy dependences of the intensity of the 333 Bragg diffraction for the $\pi -{\pi }^{\prime }$ channel at 15 K in magnetic fields of $\pm 6$ T along $\left[1\overline{1}0\right]$. Solid lines are the fits with Eq. (1), using the magnetic spectral function ${\alpha }^{\left(1\right)}\left(\omega \right)$ obtained from XMCD. See text.

Figure 3

Two possible models of atomic displacements in the $\left[\overline{1}11\right]$ AFM domain for the field applied along $\left[1\overline{1}0\right]$. We call the Sm atom with its moment oriented to $\left[\overline{1}11\right]$ at zero field as Sm-1 and the other one as Sm-2. In a magnetic field, a ferromagnetic component ${\mathit{m}}_{\text{F}}$ is induced, resulting in a smaller moment for Sm-1 and a larger moment for Sm-2 ($m_1<m_2$). In (a), the ${\mathrm{P}}_{12}$ cage and the Ru cube expand around the small moment site of Sm-1, whereas in (b) they expand around the large moment site of Sm-2. Only the Ru atoms are shown.

Figure 4

(a) Temperature dependences of the 333 Bragg-diffraction intensity for $\pi -{\pi }^{\prime }$ at $E=6.712$ keV ($E2$) in magnetic fields of 0 and $\pm 6$ T. (b) Temperature dependences of the average and difference intensities for the data at $\pm 6$ T and the nonresonant intensity at 6.680 keV measured in the $\pi -{\pi }^{\prime }$ channel. (c) Temperature dependences of the intensity at 0 T and the averaged intensity at $\pm 6$ T for $\pi -{\sigma }^{\prime }$. The lines are guides for the eye.

Figure 5

Temperature dependences of the crystal and magnetic structure factors for $\pi -{\pi }^{\prime }$ (top) and $\pi -{\sigma }^{\prime }$ (bottom), which are deduced from the intensity in Fig. 4. The solid lines are guides for the eye.

Figure 6

Energy spectrum of the 333 Bragg diffraction at $\pm 6$ T in phase II measured as a function of the field direction, as represented by the azimuthal angle $\psi$.

Figure 7

(a) Intensities and (b) flipping ratios of the 111, 114, and 330 Bragg diffraction of polarized neutrons for up and down spin states in phase II at 5 T $\parallel \left[1\overline{1}0\right]$ and 14 K. The background has been subtracted. The flipping ratios are compared with the calculations. F and AF represent the ferro- and antiferromagnetic Bragg points, respectively.

Figure 8

Temperature dependence of the 111 antiferromagnetic Bragg intensities for up (solid circle) and down (open circle) spin polarizations. The background has been subtracted. The solid and broken lines are guides for the eye.

Figure 9

Results of resonant soft x-ray diffraction at the P $K$ edge using a $\pi$-polarized incident beam without analyzing the final polarization. (a) X-ray energy dependences of the 010 Bragg intensity at 14 K in phase II in magnetic fields applied along the [001] axis. BG represents the background signal due to the fluorescence. (b) Temperature dependence of the 010 intensity at resonance in magnetic fields of 0 and $\pm 6$ T. The background has been subtracted. (c) X-ray energy dependences of the 111 Bragg intensity at 9 K in phase III in magnetic fields of 0 and $\pm 6$ T along the $\left[11\overline{2}\right]$ axis. (d) Comparison of the energy spectrum of the 010 and 111 Bragg intensities at 9 K at zero field after background subtraction.

Figure 10

(a) XMCD (filled circles) and XAS (dashed line) of $\mathrm{Sm}{\mathrm{Ru}}_4{\mathrm{P}}_{12}$ at 15 K and 7 T in phase II around the Sm $L_3$ edge. (b) Real and imaginary parts of the magnetic dipolar spectral function $f_{\text{m}}\left(\omega \right)$ in an arbitrary unit. Solid lines are the fits using Eq. (5).

Figure 11

(a) A model structure representing the relationship between the atomic displacements and the magnetic moments of Sm ions in magnetic fields applied along [110] in phase II. The surrounding cage of ${\mathrm{P}}_{12}$ and Ru expand (contract) around the small (large) moment site of Sm-1 (Sm-2) where the AFM moment has an antiparallel (parallel) component along the applied field. (b) The AFM ordered state at zero field with equal moments at Sm-1 and Sm-2 and equal volumes of the surrounding cage. (c) The same as (a) but for the reversed field direction. The magnetic moment of Sm-1 becomes larger than that of Sm-2. (d), (e), (f) Schematic of the charge density of the $p$ electrons, which is represented by the size of the circle, and the electronic state of the $f$ electrons, corresponding to the situation of (a), (b), (c), respectively. The indices of 7 and 8 represent the ${\mathrm{\Gamma }}_7$-like and ${\mathrm{\Gamma }}_8$-like states, respectively.