Magnetic structure of the magnetoelectric material ${\mathrm{Ca}}_2{\mathrm{CoSi}}_2{\mathrm{O}}_7$

Detailed investigation of ${\mathrm{Ca}}_2{\mathrm{CoSi}}_2{\mathrm{O}}_7$ was performed in its low-temperature magnetoelectric state combining neutron diffraction with magnetization measurements on single crystals. The crystal and magnetic structures well below the antiferromagnetic transition temperature of $T_{\text{N}}\approx 5.7$ K were determined using neutron diffraction. Neutron diffraction data imply no structural phase transition from 10 K down to 2.5 K and are well described within the orthorhombic space group $P{2}_1{2}_{1}2$ with a $3\times 3\times 1$ supercell compared with the high-temperature unmodulated state (tetragonal space group $P\overline{4}{2}_{1}m$). We found that in zero magnetic field the magnetic space group is $P{2}_1{2}_1^{\prime }{2}^{\prime }$ with antiferromagnetic order along the [100] or [010] axes for two types of ${90}^{\circ }$ twin domains, while neighboring spins along the [001] axis are ordered ferromagnetically. A noncollinear spin arrangement due to small canting within the $ab$ plane is allowed by symmetry and leads to the existence of the tiny spontaneous magnetization below $T_{\text{N}}$. The ordered moment with a magnitude of about 2.8 ${\mu }_{\text{B}}/{\mathrm{Co}}^{2+}$ at 2.5 K lies in the $ab$ plane. Distinct differences between the magnetic structure of ${\mathrm{Ca}}_2{\mathrm{CoSi}}_2{\mathrm{O}}_7$ as compared to those of ${\mathrm{Ba}}_2{\mathrm{CoGe}}_2{\mathrm{O}}_7$ and ${\mathrm{Sr}}_2{\mathrm{CoSi}}_2{\mathrm{O}}_7$ are discussed.

Figure 1

Comparison of the atomic shift between the low-temperature commensurately modulated lock-in state (SG $P{2}_1{2}_{1}2$) and the high-temperature unmodulated phase (SG $P\overline{4}{2}_{1}m$), according to our neutron diffraction data at 2.5 K and x-ray diffraction at about 500 K (Ref. [14]), respectively. The small gray square corresponds to the unit cell of the high-temperature unmodulated phase with SG $P\overline{4}{2}_{1}m$ and the large red one to the $3\times 3\times 1$ supercell of SG $P{2}_1{2}_{1}2$. Five inequivalent cobalt sites are numbered. Silicon atoms are not shown for simplicity; Their shifts are small and comparable with those of cobalt.

Figure 2

The possible $k$-maximal symmetries for a magnetic ordering of Co with propagation vector $\mathit{k}=\mathit{0}$ on a paramagnetic phase with space group $P{2}_1{2}_{1}2$ and its corresponding gray group $P{2}_1{2}_1{21}^{\prime }$. The magnetic space group label is shown together with the transformation from the parent $P{2}_1{2}_1{21}^{\prime }$ unit cell basis $\{{\mathit{a}}_p,{\mathit{b}}_p,{\mathit{c}}_p\}$ to its standard setting. The possible antiferromagnetic (AF) and ferromagnetic (F) components along the crystallographic directions $a$, $b$, and $c$ are given at the bottom separately for the Co1 and Co2-5 sites.

Figure 3

Quality of the neutron diffraction data refinement at 2.5 K. The observed structure factors squared ($F_{\mathrm{obs}}^2$) are plotted against the calculated ones ($F_{\mathrm{calc}}^2$). Eight representative Bragg reflections were selected to illustrate the fit quality considering (a) the pure nuclear structure or possible magnetic models (b) $M1$, (c) $M2$, and (d) $M3/M4$. The simplified (e) $M3/M4$ model with only 2 refined parameters is also presented for comparison. A fit of (f) the full data set is given in the unrestricted $M3/M4$ model (14 refined parameters) to evaluate the quality of the final simultaneous refinement of ${\mathrm{Ca}}_2{\mathrm{CoSi}}_2{\mathrm{O}}_7$ crystal and magnetic structures.

Figure 4

Magnetic structure in the full $M3/M4$ (14 parameters) and its simplified (2 parameters) models, according to our neutron diffraction data at 2.5 K. The small unit cell of the high-temperature unmodulated phase with SG $P\overline{4}{2}_{1}m$ is plotted in gray color and the large $3\times 3\times 1$ supercell of SG $P{2}_1{2}_{1}2$ is shown in red. Five inequivalent cobalt sites are numbered (see also Fig. 1 for comparison).

Figure 5

Upper panel: Temperature dependences of the normalised integrated intensity of the magnetic Bragg reflection (300). Symbols correspond to the unpolarized single-crystal neutron diffraction data measured by two instruments: HEiDi and TASP. The solid line shows a fit to Eq. (1). Lower panel: Temperature dependence of the normalized Co magnetic moment. The experimental data from the single-crystal neutron diffraction measurements are shown by circles. The solid line is a result of a modified molecular field model [Eq. (3)]. The dashed line is shown to illustrate the deviation of $\mu \left(T\right)$ from the conventional molecular field model [Eq. (2)].

Figure 6

Field dependences of the magnetization $M$ at 2 K (blue), 4 K (green), and 6 K (red). The field is applied along the [110] (upper panel) and [001] (lower panel) directions. There is no hysteresis for $B\parallel \left[001\right]$ indicating no detectable spontaneous (zero-field) canting of the magnetic moments along the $c$ axis.

Figure 7

Angular dependences of the magnetization $M$ at 2 K. $\varphi$ is the angle between the field and the [100] direction in the $ab$ plane. The corresponding crystallographic directions are indicated on the top of the figure.

Figure 8

Field dependences of the magnetization $M$ at 2 K. Upper panel: the whole measured region. Lower panel: magnified selected part to emphasize the difference between the curves in the intermediate field range. The field is applied along the [100] (red), [110] (green), and [001] (blue) directions. Curves correspond to the measurements with a Bitter magnet, while symbols in the bottom panel represent the MPMS data. The magnetization anomaly at around 11 T for $\mathit{B}\parallel \left[110\right]$, marked by an arrow, corresponds to the sign change of the ferroelectric polarization [13].