Local origin of the strong field-space anisotropy in the magnetic phase diagrams of ${\mathrm{Ce}}_{1-x}{\mathrm{La}}_x{\mathrm{B}}_6$ measured in a rotating magnetic field

Cubic $f$-electron compounds commonly exhibit highly anisotropic magnetic phase diagrams consisting of multiple long-range ordered phases. Field-driven metamagnetic transitions between them may depend not only on the magnitude, but also on the direction of the applied magnetic field. Examples of such behavior are plentiful among rare-earth borides, such as $R{\mathrm{B}}_6$ or $R{\mathrm{B}}_{12}$ ($R$ = rare earth). In this work, for example, we use torque magnetometry to measure anisotropic field-angular phase diagrams of La-doped cerium hexaborides, ${\mathrm{Ce}}_{1-x}{\mathrm{La}}_x{\mathrm{B}}_6$ ($x=0,0.18,0.28,0.5$). One expects that field-directional anisotropy of phase transitions must be impossible to understand without knowing the magnetic structures of the corresponding competing phases and being able to evaluate their precise thermodynamic energy balance. However, this task is usually beyond the reach of available theoretical approaches, because the ordered phases can be noncollinear, possess large magnetic unit cells, involve higher-order multipoles of $4f$ ions rather than simple dipoles, or just lack sufficient microscopic characterization. Here we demonstrate that the anisotropy under field rotation can be qualitatively understood on a much more basic level of theory, just by considering the crystal-electric-field scheme of a pair of rare-earth ions in the lattice, coupled by a single nearest-neighbor exchange interaction. Transitions between different crystal-field ground states, calculated using this minimal model for the parent compound ${\mathrm{CeB}}_6$, possess field-directional anisotropy that strikingly resembles the experimental phase diagrams. This implies that the anisotropy of phase transitions is of local origin and is easier to describe than the ordered phases themselves.

Figure 1

Magnetic-field dependence of the elastic neutron scattering intensity in pure ${\mathrm{CeB}}_6$ at ${\mathbf{q}}_1=\left\{\frac{1}{4}\frac{1}{4}0\right\}$ or $\left\{\frac{3}{4}\frac{3}{4}0\right\}, {\mathbf{q}}_2=\left\{\frac{1}{4}\frac{1}{4}\frac{1}{2}\right\}$, and ${\mathbf{q}}_{\text{AFQ}}=\left\{\frac{1}{2}\frac{1}{2}\frac{1}{2}\right\}$ for three different field directions: $\mathbf{B}\parallel \left[111\right]$, [112], and [001]. The data were measured in different experiments with different crystal orientations, therefore intensities are not comparable. The datasets are shifted vertically for clarity. Phase transitions are marked with arrows, and the domain selection transition is indicated with asterisks.

Figure 2

Design of the sample holder for cantilever magnetometry at NHMFL with a pair of cantilevers on opposite sides.

Figure 3

Sample holder is mounted in a cylindrical socket rotator [64]. The magnetic-field direction is vertical.

Figure 4

(a) Magnetic-field dependence of the torque, $\tau \left(B\right)$, for the magnetic-field direction $\mathbf{B}\parallel \left[112\right]$, measured on four samples of ${\mathrm{Ce}}_{1-x}{\mathrm{La}}_x{\mathrm{B}}_6$ ($x=0,0.18,0.28,0.50$) during upward and downward field sweeps, as indicated with arrows. (b) Magnetic-field derivative, $\partial \tau /\partial B$, of the data in panel (a). The curves in both panels are shifted vertically for clarity. Smooth pale lines in the background are empirical fits to the data. Vertical black arrows mark the estimated phase transition fields.

Figure 5

Field-angular magnetic phase diagrams of ${\mathrm{Ce}}_{1-x}{\mathrm{La}}_x{\mathrm{B}}_6$ (from top to bottom: $x=0,0.18,0.28,0.50$) as measured by torque magnetometry at $T=20\mathrm{mK}$. The field is rotated in the $\left[1\overline{1}0\right]$ plane. The field derivative of the magnetic torque, $\partial \tau /\partial B$, measured in field sweeps upon increasing the field, is plotted for every sample on the left as a color map. The corresponding field-angular phase diagrams in polar coordinates are presented on the right. The angular dependence for ${\mathrm{CeB}}_6$ at $T=1.4\mathrm{K}$ from Ref. [26] and the data for high-symmetry directions from our neutron-scattering measurements (Refs. [21, 53], and present work) as well as from Refs. [21, 29, 37, 38, 39, 41, 45, 49, 54, 58, 60, 61, 62] are overlaid for comparison.

Figure 6

Polar plot of the field-angular magnetic phase diagram of ${\mathrm{CeB}}_6$ measured at $T=20\mathrm{mK}$ in the $C_4$-symmetric field plane, $\mathbf{B}\perp \left[001\right]$, using torque magnetometry. The color map shows the field derivative of the magnetic torque, $\partial \tau /\partial B$, measured in field sweeps upon increasing the field. The data points show fitting results both for the upward (bottom-right sector) and downward (top-left sector) field sweeps, which are essentially identical within the experimental error. The data from our neutron-scattering measurements and from Refs. [21, 26, 37, 49, 53, 61] are overlaid for comparison (for the meaning of the symbols, see Fig. 5).

Figure 7

(a) Isosurfaces of the magnetization density in ${\mathrm{CeB}}_6$ from $\mathrm{DFT}+U/\mathrm{PBE}/\mathrm{PAW}$ calculations. (b), (d) Band structures and projected densities of states for ${\mathrm{CeB}}_6$ and ${\mathrm{LaB}}_6$ systems. (c) Natural spin density for the ground-state quartet calculated at the CASSCF level of theory.

Figure 8

(a) A schematic Zeeman diagram for the single-ion ${\widehat{\mathcal{H}}}_{\text{m1}}$ model in different orientations of $\mathbf{B}$ on a spherical grid. The shading covers an energy range of the Zeeman-split states due to the field-angular anisotropy of the $g$ factors. (b), (c) Anisotropy of the state magnetization $M_{D_i}\propto \partial E_{D_i}{\left(B\right)/\partial B|}_{B\to 0}$ for the $D_1$ and $D_2$ doublets (see Tables 2 and 3 in Appendix pp2), presented as color maps on a unit sphere and in a polar projection onto the 2D plane, respectively. (d),(e) Polar maps of the magnetization, $M(B,\theta )\propto \partial F/\partial B$, and the corresponding magnetic-field derivative of the torque, $\partial \tau /\partial B\propto {\partial }^{2}F/\partial \theta \partial B\propto \partial M(B,\theta )/\partial \theta$, respectively, as a function of magnetic-field strength $B$ from 0 to 5 T and angle $\theta$ in the $ab$ plane for $\mathbf{B}\perp \left[001\right]$.

Figure 9

(a) Superposed Zeeman diagrams and (b) a set of magnetization curves obtained in the two-site model (${\widehat{\mathcal{H}}}_{\text{m2}}$) for magnetic-field scans up to 5 T along different directions in the $\mathbf{B}\perp \left[1\overline{1}0\right]$ plane. The black arrows in panel (a) indicate ground-state level crossings responsible for the appearance of magnetization steps. (c) Calculated magnetic-field dependence of the total magnetization in the two-site model (${\widehat{\mathcal{H}}}_{\text{m2}}$) for the three high-symmetry directions in the cubic crystal. (d) Experimental magnetization curves, reproduced with permission from Ref. [26] for comparison. (e),(f) The total magnetization $M(B,\theta )$ and its angle derivative $\partial M(B,\theta )/\partial \theta$, which is equivalent to the experimentally measured magnetic-field derivative of the torque $\partial \tau /\partial B$, presented as polar maps as a function of magnetic field strength $B$ from 0 to 5 T and its in-plane direction $\theta$ for $\mathbf{B}\perp \left[001\right]$ and $\mathbf{B}\perp \left[1\overline{1}0\right]$, respectively.