Magnetic structures of $R_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ intermetallic compounds: Evolution to ${\mathrm{Er}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ and ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$

The magnetic structures of ${\mathrm{Er}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ and ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ (monoclinic ${\mathrm{Dy}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$-type structure, $C2/m$ space group) have been studied by neutron powder diffraction, complemented by magnetization, specific heat measurements, and ${}^{166}\mathrm{Er}$ Mössbauer spectroscopy, over the temperature range 0.5 to 300 K. Their magnetic structures are compared with those of other $R_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ compounds. Antiferromagnetic ordering of the rare-earth sublattice is observed below the Néel temperatures of $T_N=4.8\left(2\right)\mathrm{K}$ and $T_N=2.6\left(3\right)\mathrm{K}$ for ${\mathrm{Er}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ and ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$, respectively. While ${\mathrm{Er}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ and ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ have the same crystal structure, they possess different magnetic structures compared with the other $R_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ ($R$ = Nd, Gd, Tb, Dy, and Ho) compounds. In particular, two different propagation vectors are observed below the Néel temperatures: $\mathbf{k}=[\frac{1}{2},\frac{1}{2},0]$ (for ${\mathrm{Er}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$) and $\mathbf{k}=[0.403\left(1\right),\frac{1}{2},0]$ (for ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$). For both compounds, the difference in propagation vectors is also accompanied by different orientations of the Er and Tm magnetic moments. Although the magnetic structures of ${\mathrm{Er}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ and ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ differ from those of the other $R_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ compounds, we have established that the two magnetic structures are closely related to each other. Our experimental and first-principles studies indicate that the evolution of the magnetic structures across the $R_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ series is a consequence of the complex interplay between the indirect exchange interaction and crystal field effects.

Figure 1

(Left panel) The dc susceptibility of ${\mathrm{Er}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ and ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ collected in field-cooled mode (FC; ${\mu }_{0}H$ = 0.5 T). The magnetic transition temperatures are marked by arrows. (Right panel) Zero field specific heat of ${\mathrm{Er}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ and ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$.

Figure 2

Rietveld refinements of the neutron diffraction patterns of ${\mathrm{Er}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ and ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ collected in the paramagnetic state. The vertical markers indicate Bragg reflections from the monoclinic ${\mathrm{Dy}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$-type structure with the difference between the experimental and calculated patterns given by the blue line.

Figure 3

Rietveld refinement of the neutron diffraction pattern of ${\mathrm{Er}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ collected at 1.4 K. The rows of Bragg markers, from top to bottom, represent ${\mathrm{Er}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ (nuclear) and ${\mathrm{Er}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ (magnetic), respectively. The difference between the experimental and calculated patterns is given by the blue line.

Figure 4

Rietveld refinement of the neutron diffraction pattern of ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ collected at 2.35 K. The rows of Bragg markers, from top to bottom, represent ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ (nuclear) and ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ (magnetic), respectively. The difference between the experimental and calculated patterns is given by the blue line.

Figure 5

Low angle region of neutron powder diffraction patterns of ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ obtained at 2.35, 1.95, 1.75, and 1.5 K (bottom to top).

Figure 6

Rietveld refinement of the neutron diffraction pattern of ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ at 1.5 K. The rows of Bragg markers, from top to bottom, represent ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ (nuclear) and ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ magnetic (first, third, and fifth harmonics), respectively. The difference between the experimental and calculated patterns is given by the blue line.

Figure 7

Magnetic structures of (a) ${\mathrm{Er}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ and (b) ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ at 1.5 K. The magnetic structure is projected onto the $a-c$ plane to better illustrate the square-wave modulation of the Tm magnetic moment along the $a$ axis. Each block corresponds to a nuclear unit cell. The Fe, Si, and C atoms are omitted for clarity.

Figure 8

${}^{166}\mathrm{Er}$ Mössbauer spectrum of ${\mathrm{Er}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ collected at 5 K.

Figure 9

Projected densities of states (pDOS) of the ${\mathrm{Er}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ compound based on the AFM1 order. The results were obtained using the PBE0+SO in wien2k, with magnetization along the [001] direction. The Fermi energy $E_F$ has been defined as the reference energy and is represented by dashed lines. (a) Total DOS and partial DOS of one erbium site (in blue). (b) pDOS for three individual Fe, Si, and C sites. In all cases, majority and minority spin components are shown.

Figure 10

Orientation of the ${}^{166}\mathrm{Er}$ EFG eigenaxes of ${\mathrm{Er}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ based on AFM1 order and using the PBE0 functional and including SO with magnetization direction along (a) $\left[100\right]$, (b) $\left[010\right]$, and (c) $\left[001\right]$. Representations of the electronic densities inside the Er sphere report the contribution to the density when only the $\left|L\right|=2$ (top), $\left|L\right|=4$ (middle), and their sum (bottom) are considered. A light (dark) color is used for positive (negative) values.

Figure 11

Local arrangement of the Tm atoms in ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ projected onto the $a-c$ plane.

Figure 12

Temperature dependences of the Tm-Tm interatomic distances $d_1, d_2$, and $d_3$ (Fig. 11) in ${\mathrm{Tm}}_2{\mathrm{Fe}}_2{\mathrm{Si}}_2\mathrm{C}$ ($T_N\sim 2.6\mathrm{K}$) as derived from the neutron diffraction refinements. Solid lines are guides to the eyes.