Unambiguous determination of the commensurate antiferromagnetic structure of ${\mathrm{HoNi}}_2{\mathrm{B}}_2\mathrm{C}$ in the superconducting ground state

Crystal and magnetic structures of the intermetallic ${\mathrm{HoNi}}_2{\mathrm{B}}_2\mathrm{C}$ have been investigated using different single-crystal diffraction techniques at both room and low temperatures (superconducting phase). Combined refinement of the neutron and x-ray diffraction data performed at room temperature shows that the crystal structure of ${\mathrm{HoNi}}_2{\mathrm{B}}_2\mathrm{C}$ is well described in the tetragonal space group $I4/mmm$, in agreement with the literature. The magnetic ordering at low temperatures is accompanied by a structural phase transition to the orthorhombic space group $Fmmm$. However, only noncharacteristic orbits are occupied in this space group, and the symmetry reduction is entirely due to a lifting of the symmetry-induced restrictions on the anisotropic displacement parameters. Based on magnetic symmetry analysis, two types of commensurate antiferromagnetic structures with different magnetic space groups (MSGs) are found below $T_{\mathrm{N}}\approx 5.2$ K: MSG $P_{I}nnm$ (Ho spins are along the [100] direction of the parent structure) and MSG $C_{A}mca$ (Ho spins are along [110] of the parent structure). Both models agree well with conventional unpolarized neutron diffraction data at low temperature (2.4 K) with ordered magnetic moments of 9.16(9) ${\mu }_{\mathrm{B}}$/${\mathrm{Ho}}^{3+}$ atom. Spherical neutron polarimetry, on the other hand, allowed us to unambiguously solve the magnetic structure (MSG $C_{A}mca$) as well as to determine the population of two types of ${90}^{\circ }$ antiferromagnetic domains (0.45/0.55).

Figure 1

Perspective view of the ${\mathrm{HoNi}}_2{\mathrm{B}}_2\mathrm{C}$ crystal structure in SG $I4/mmm$. Blue and yellow planes perpendicular to the $c$ axis correspond to the layers of Ho/C and Ni, respectively. Ho-C, B-C, and Ni-B distances are drawn for clarity.

Figure 2

Selected fragment of the ${\mathrm{HoNi}}_2{\mathrm{B}}_2\mathrm{C}$ crystal structure at 2.4 K. Ho-C, B-C, and Ni-B distances are drawn, and their lengths are given in angstroms. B-Ni-B angles (in degrees) are also marked.

Figure 3

Top: Temperature dependences of the normalized integrated intensities of the magnetic Bragg reflections (111) and (100). Symbols correspond to unpolarized single-crystal neutron diffraction data. The solid line shows a fit to Eq. (1) in the vicinity (down to $0.85T_{\mathrm{N}}$, $T_{\mathrm{N}}\approx 5.2$ K) of the transition to the commensurate AFM state with ${\mathit{k}}_1=(0,0,1)$. The dashed line is an extension of the fit curve below $0.85T_{\mathrm{N}}$. Temperatures of SNP measurements (3.9 K) described in Sec. 3d and unpolarized neutron diffraction studies (2.4 K) described in Sec. 3c are indicated. Bottom: Temperature dependences of the normalized integrated intensities of the nuclear Bragg reflections (220), (200), and (0 0 12).

Figure 4

The possible $k$-maximal symmetries for a magnetic ordering of Ho with propagation vector $\mathit{k}=(0,0,1)$ on a paramagnetic phase with (left panel) SG $Fmmm$ and its corresponding gray group $Fmmm{1}^{\prime }$ and (right panel) SG $I4/mmm$ and its corresponding gray group $I4/mmm{1}^{\prime }$. Only the subgroups which allow nonzero magnetic moments are shown. Each magnetic space group label is shown together with the transformation from the corresponding parent unit cell basis $\{{\mathit{a}}_p,{\mathit{b}}_p,{\mathit{c}}_p\}$ to its standard setting. The allowed magnetic moment components on the Ho atoms which correspond to the AFM ordering are given at the bottom.

Figure 5

Magnetic structure of ${\mathrm{HoNi}}_2{\mathrm{B}}_2\mathrm{C}$ in (a) MSG $P_{I}nnm$ and (b) MSG $C_{A}mca$. Boron atoms are not shown for simplicity. Quality of the unpolarized neutron diffraction data refinement at 2.4 K in (c) MSG $P_{I}nnm$ and (d) MSG $C_{A}mca$. The experimental structure factors squared $F_{\mathrm{obs}}^2$ are plotted against the calculated ones $F_{\mathrm{calc}}^2$. The quality of the spherical neutron polarimetry data refinement at 3.9 K is shown in (e) MSG $P_{I}nnm$ and (f) MSG $C_{A}mca$. The experimental polarization matrix elements ${\mathcal{P}}_{\mathrm{obs}}^{ij}$ are plotted against the calculated ones ${\mathcal{P}}_{\mathrm{calc}}^{ij}$.