Spontaneous lattice distortion and crystal field effects in HoB4

TL;DR

HoB4 exhibits spontaneous lattice distortion tied to quadrupolar order in a Shastry–Sutherland lattice. The authors combine low-temperature PXRD, ultrasound-velocity measurements, and crystal-field (CEF) calculations to show a tetragonal-to-monoclinic transition around $T_{N2}$ with a large $C_{44}$ softening signaling ferroquadrupolar ordering and a quasi-degenerate Ho$^{3+}$ ground state. The monoclinic distortion accommodates the in-plane 2-in-2-out magnetic structure, linking spin, orbital, and lattice degrees of freedom via strong spin–orbit coupling. The work highlights spin-lattice-quadrupole coupling as a key mechanism controlling the ground state in HoB4 and related tetraborides, with implications for multipolar order in frustrated lattices.

Abstract

The tetraboride HoB4 crystallizes in a tetragonal structure (space group P4/mbm), with the Ho atoms realizing a Shastry-Sutherland lattice. It orders antiferromagnetically at TN1 = 7.1 K and undergoes further magnetic transition at TN2 = 5.7 K. The complex magnetic structures are attributed to competing order parameters of magnetic and quadrupolar origin with significant magnetoelastic coupling. Here, we investigate the response of the lattice of HoB4 across the antiferromagnetic phase transitions by using low-temperature powder x-ray diffraction and ultrasound-velocity measurements, supported by crystal electric field (CEF) calculations. Below TN2, the crystal structure of HoB4 changes to monoclinic (space group P21/b) as a macroscopic manifestation of the quadrupolar ordering. Between 300 and 3.5 K, the total distortion amplitude is 0.46~Å and the relative volume change is $3.5 \times 10^{-3}$. This structural phase transition is compatible with the huge softening of the modulus $C_{44}$ observed around TN2 due to ferroquadrupolar order. A lattice instability developing immediately below TN1 is seen consistently in x-ray and ultrasound data. CEF analysis suggests a quasi-degenerated ground state for the Ho$^{3+}$ ions in this system.

Figures (7)

• Figure 1: (a) Temperature evolution of PXRD patterns of HoB$_4$ for $300 \, \mathrm{K} \geq T \geq 3.5 \, \mathrm{K}$. The color scale is used to indicate the diffracted intensity (counts/s). PXRD patterns spanning the magnetic transitions collected upon (b) cooling and (c) heating. A clear structural distortion is seen below $T_{\rm N2}$. The double peaks for each family of reflections in the tetragonal setting (indexed at the top) are due to the presence of Cu K$\alpha_1$ and Cu K$\alpha_2$ wavelengths.
• Figure 2: Le Bail fitting to the PXRD pattern at 3.5 K treated with (a) the tetragonal space group $P4/mbm$ and (b) with the monoclinic space group $P2_1/b$.
• Figure 3: (a) Rietveld-refined PXRD pattern of HoB$_4$ at 3.5 K using the space group $P2_1/b$. Corresponding monoclinic crystal structure of HoB$_4$ at 3.5 K: view along (b) the $c$ axis and (d) the $a$ axis. The same projections for the tetragonal structure for comparison are shown in (c) and (e). Ho atoms are depicted in blue, whereas the B atoms are displayed in green. Labels for the different B positions in the monoclinic unit cell in (b) refer to Table I. Black dashed lines in the inset of (a) show that the Ho sublattice in the monoclinic phase preserves the geometry of a Shastry-Sutherland lattice.
• Figure 4: Temperature dependence of lattice parameters, angle, unit-cell volume, and volume change between 8 and 3.5 K in HoB$_4$ upon cooling. The broken lines are guides to the eye.
• Figure 5: Variation of the calculated crystal structure factor squared ($F^2$) for the reflections (140) and (330) versus the displacement parameter $n$ (defined in the text), when the atom specified (in the figure) is displaced in the monoclinic unit cell. The dashed lines are guides to the eye.