Atomic displacements and lattice distortion in the magnetic-field-induced charge-ordered state of ${\mathrm{SmRu}}_4{\mathrm{P}}_{12}$

Structural properties of ${\mathrm{SmRu}}_4{\mathrm{P}}_{12}$ in the anomalous magnetic ordered phase between $T^{*}\sim 14$ K and $T_{\text{N}}=16.5$ K in magnetic fields has been studied by x-ray diffraction. Atomic displacements of Ru and P, reflecting the field-induced charge order of the $p$ electrons, have been deduced by analyzing the intensities of the forbidden Bragg peaks, assuming a cubic space group $Pm\overline{3}$. Also, by utilizing a high-resolution x-ray diffraction experiment, we observed a splitting of fundamental Bragg peaks, clarifying that the unit cell in the magnetic ordered phase is rhombohedral elongated along the $\left[111\right]$ axis. The response of the rhombohedral domains to the magnetic field, which reflects the direction of the magnetic moment, is studied in detail.

Figure 1

Comparison of the observed intensities with the calculated intensities for the Bragg reflections in phase II at 15 K and 6 T $\parallel \left[\overline{1}\overline{1}2\right]$. The fundamental reflections and the forbidden reflections are shown by the circles and squares, respectively.

Figure 2

Comparison of the observed intensities with the calculated intensities for the Bragg reflections in phase II at 15 K and 6 T $\parallel \left[1\overline{1}0\right]$. The data are averaged for reversed fields at $\pm 6$ T. The fundamental reflections and the forbidden reflections are shown by the circles and squares, respectively.

Figure 3

Comparison of the observed intensities with the calculated intensities for the Bragg reflections in phase II at 15 K and 6 T $\parallel \left[001\right]$. The fundamental reflections and the forbidden reflections are shown by the circles and squares, respectively. The closed and open marks represent the $hk0$ and $h\overline{k}0$ reflections, respectively, which have equal calculated intensities. The indices of the $h\overline{k}0$ reflections are omitted.

Figure 4

(a) Schematic view of the atomic displacements in phase II in magnetic fields, assuming a cubic $Pm\overline{3}$ space group. The displacements are emphasized than the real shifts. (b) Top view of (a). Around Sm-$1a$, the cube of Ru expands and the angle $\varphi$ of P increases, whereas around Sm-$1b$ the atomic shifts are assumed opposite to those around Sm-$1a$.

Figure 5

Magnetic field dependence of the structure factor of the forbidden reflections in phase II at 15 K. The data points were deduced by taking the square root of the integrated intensities of the rocking scans. The data are normalized to unity at 5 T. The crosses represent the data for the $030$ reflection reported in Ref. [35].

Figure 6

Radial scan profiles along $\mathit{Q}=(8,8,8)+(h,h,h)$ at zero field (a) and at 5 T $\parallel \left[\overline{1}\overline{1}2\right]$ (b). X-ray energy is 10.7055 keV at $h=0$. Solid lines are the fits using asymmetric pseudo-Voigt functions.

Figure 7

Temperature dependence of the integrated intensity and the relative variation in the planar spacing ${(\mathrm{\Delta }d/d)}_{111}$ at zero field [(a) and (b)] and at 5 T [(c) and (d)], respectively, obtained from the fit of the data in Fig. 6. Open and closed circles correspond to peak A and B, respectively, in Fig. 6. Closed squares in (a) and (c) show the total intensity. The vertical arrows represent the HWHM of the peak profile at 20 K. Open squares in (b) are the calculated ${(\mathrm{\Delta }d/d)}_{111}$ assuming a rhombohedral distortion (see text).

Figure 8

Radial scan profiles along $\mathit{Q}=(10,10,0)+(h,h,0)$ at zero field (a) and at 5 T $\parallel \left[001\right]$ (b). X-ray energy is 10.9263 keV at $h=0$. Solid lines are the fits using asymmetric pseudo-Voigt functions.

Figure 9

Temperature dependence of the integrated intensity and the relative variation in the planar spacing ${(\mathrm{\Delta }d/d)}_{110}$ at zero field [(a) and (b)] and at 5 T [(c) and (d)], respectively, obtained from the fit of the data in Fig. 8. Open and closed circles correspond to peak A and B, respectively, in Fig. 8. Closed squares in (a) and (c) show the total intensity. Vertical arrows represent the HWHM of the peak profile at 20 K. Open squares in (b) are the calculated ${(\mathrm{\Delta }d/d)}_{110}$ assuming a rhombohedral distortion (see text).

Figure 10

(a) Radial scan profiles along (14, 0, 0)$+(h,0,0)$ at 5 T $\parallel \left[001\right]$. X-ray energy is 10.8189 keV at $h=0$. Solid lines are the fits using an asymmetric squared Lorentzian function. (b) and (c) show the temperature dependence of the integrated intensity and ${(\mathrm{\Delta }d/d)}_{010}$ obtained from the fit, respectively. Open square in (c) is the calculated ${(\mathrm{\Delta }d/d)}_{010}$ assuming a rhombohedral distortion (see text).

Figure 11

(a) Schematic of the rhombohedral domains and the AFM moments in a magnetic field along $\left[\overline{1}\overline{1}2\right]$. The angles between the magnetic field and the rhombohedral principal axis are also shown. (b) Magnetic phase diagram of ${\mathrm{SmRu}}_4{\mathrm{P}}_{12}$ for $H\parallel \left[\overline{1}\overline{1}2\right]$ obtained in the present work. The $\left[\overline{1}\overline{1}1\right]$ domain is favored in phase II and the $\left[111\right]$ domain in phase III, respectively.