Competing interactions in dysprosium garnets and generalized magnetic phase diagram of $\mathrm{S}=\frac{1}{2}$ spins on a hyperkagome network

The anisotropy and magnetic ground state of hyperkagome dysprosium gallium garnet ${\mathrm{Dy}}_3{\mathrm{Ga}}_5{\mathrm{O}}_{12}$ are investigated, along with that of its closest structural analog and archetypal Ising multiaxis antiferromagnet, dysprosium aluminum garnet ${\mathrm{Dy}}_3{\mathrm{Al}}_5{\mathrm{O}}_{12}$, using a combination of neutron scattering techniques, including polarized neutron powder diffraction. Results show a dramatic change from an Ising-like anisotropy in ${\mathrm{Dy}}_3{\mathrm{Al}}_5{\mathrm{O}}_{12}$, to a quasiplanar one in ${\mathrm{Dy}}_3{\mathrm{Ga}}_5{\mathrm{O}}_{12}$. According to a point charge modeling, this is due to small variations of the oxygen positions surrounding ${\mathrm{Dy}}^{3+}$ ions. The magnetic ground state of ${\mathrm{Dy}}_3{\mathrm{Ga}}_5{\mathrm{O}}_{12}$ is investigated for the first time and is found to be similar to that of ${\mathrm{Dy}}_3{\mathrm{Al}}_5{\mathrm{O}}_{12}$, yet with a much lower $T_{\mathrm{N}}$. Mean-field calculations show that the dipolar interaction favors distinct magnetic ground states, depending on the anisotropy tensor: a multiaxis antiferromagnetic state is favored in the case of strong Ising-like anisotropy, like in ${\mathrm{Dy}}_3{\mathrm{Al}}_5{\mathrm{O}}_{12}$, whereas a complex ferrimagnetic state is stabilized in the case of planar anisotropy. The ${\mathrm{Dy}}_3{\mathrm{Ga}}_5{\mathrm{O}}_{12}$ crystal field parameters locate the latter close to the boundary between those two ground states, which, alongside competition between dipolar and a small but finite magnetic exchange, may explain its low $T_{\mathrm{N}}$. To widen the scope of these experimental results, we performed mean-field calculations to generate the magnetic phase diagram of an effective anisotropic pseudospin $S=1/2$, characterized by general $g_{xx}$, $g_{yy}$, and $g_{zz}$ Landé factors. A very rich magnetic phase diagram, encompassing complex phases, likely disordered, is evidenced when magnetic anisotropy departs from the strong Ising case. With magnetic anisotropy being controllable through appropriate tuning of the rare-earth oxygen environment, these results emphasize the potential of rare-earth hyperkagome networks for the exploration of new magnetic phases.

Figure 1

Intertwined hyperkagome networks (in blue and orange) of rare-earth triangles in garnets $R_3{A}_5{\mathrm{O}}_{12}$ ($A$ = Al, Ga). The networks are not connected to each other within the nearest-neighbor distance. The magnetic ordering shown corresponds to the $AFA$ one [8] ($Ia\overline{3}{d}^{\prime }$), with three magnetic sublattices having moments parallel or antiparallel to the cubic axis ($a$, $b$, $c$ in yellow, blue and red, respectively), for each hyperkagome network. First-neighbor spins are orthogonal to each other. In the grey area, local anisotropy axes ($X,Y,Z$) are drawn as thin arrows (see text).

Figure 2

(a) DyAG flipping difference ($y_{+}-y_{-}$) diffraction pattern collected at 5 K in 5 T. The measured (calculated) 2D pattern is shown in the top (bottom) panel. (b) Projection on the Bragg angle $2\theta$. Reflections positions are marked by vertical ticks. The black line shows the difference between the experimental points (blue) and the model (orange line). $\gamma$ is the azimuthal angle and $\nu$ is the elevation angle in the laboratory coordinate system $\left(xyz\right)$, where $\mathit{x}\parallel {\mathit{k}}_i$, $\mathit{z}\parallel \mathit{B}$.

Figure 3

(a) DyGG flipping difference ($y_{+}-y_{-}$) diffraction pattern collected 5 K and 5 T. (b) Projection on the Bragg angle $2\theta$. Notations are the same as for Fig. 2.

Figure 4

Magnetization components ${\mu }_{\left(110\right)}$, ${\mu }_{\left(1\overline{1}0\right)}$ and ${\mu }_{\left(001\right)}$ versus temperature for DyAG (a) in 6 T, and for DyGG (b) in 5 T. The insets show magnetic ellipsoids at 5 K. In both panels, dotted lines show susceptibility calculations using the point charge model, and solid lines represent calculations using the CEF model (CEF1 and CEF2 models for DyGG are differentiated by thin and thick lines, respectively). All calculations are based on the parameters given in Table 1.

Figure 5

(a) Constant $\text{Q}=2.1$ ${\AA }^{-1}$ cut of the $S(Q,\omega )$ of DyAG at 5 K (blue square symbols). In red is shown the neutron cross section calculated for the CEF parameters of Table 1. Green lines show the excitation levels determined by optical spectroscopy in Refs. [21, 28]. (b) Corresponding temperature evolution of constant $\text{Q}=1.5$ ${\AA }^{-1}$ scans.

Figure 6

(a) Constant Q = 2.1 ${\AA }^{-1}$ cut of the $S(Q,\omega )$ of DyGG at 5 K (blue square symbols). In red is shown the neutron cross section calculated for the CEF1 (dotted line) and CEF2 (continuous line) parameters of Table 1. Green lines show the excitation levels determined by optical spectroscopy in Refs. [21, 28, 38]. (b) Corresponding temperature evolution of constant Q = 2.1 ${\AA }^{-1}$ scans. The inset focuses on the T evolution of the higher energy levels.

Figure 7

Evolution of the main ellipsoid axes with the oxygen $y$ position $(0.03,y,0.65)$ with effective charge $-2e$ calculated by the point charge model at 5 K in 5 T. The top inserts show the different types of anisotropies obtained for various $y$ coordinates (shown as dots on the graph, of the same color scheme as the inserts). The bottom inserts illustrate the $y$ evolution of the oxygen environment of ${\mathrm{Dy}}^{3+}$ ion.

Figure 8

Rietveld refinement of DyAG at 1.5 K, using the magnetic space group $Ia\overline{3}{d}^{\prime }$ (experimental: empty red circles, calculated: black line, Bragg positions (crystal $+$ magnetic contributions): green ticks. The difference between the experimental and calculated profiles is displayed at the bottom of the graph as a blue continuous line). The corresponding evolution of the ${\mathrm{Dy}}^{3+}$ magnetic moment with temperature is shown in the inset. The $\mathbf{*}$ symbol indicates a non-magnetic impurity of unknown origin.

Figure 9

Rietveld refinement of DyGG at 32 mK, using the magnetic space group $Ia\overline{3}{d}^{\prime }$ (Experimental: empty red circles, calculated: black line, Bragg positions (crystal $+$ magnetic contributions): green ticks. The difference between the experimental and calculated profiles is displayed at the bottom of the graph as a blue continuous line). The evolution of the ${\mathrm{Dy}}^{3+}$ magnetic moment with temperature is shown in the inset.

Figure 10

($\mathcal{J}$, $x_{\mathrm{dip}}$) magnetic phase diagram calculated for DyGG crystal field parameters CEF1 and CEF2 (see Table 1). $AFA$, $FC$ and $(X0Z,0YZ)$ labels are defined in the text. The dotted line and the arrow close to the boundary illustrate the fact that a finite $\mathcal{J}\approx 0.02\mathrm{K}$ leads to a position in the phase diagram consistent with a reduced $T_{\mathrm{N}}$ value, because of competition effects between different phases.

Figure 11

General magnetic phase diagram calculated for a pseudospin $1/2$ ($\mathcal{J}=1$ and $x_{\mathrm{dip}}=100$ fixed) on a hyperkagome lattice, as a function of Landé factors $g_{xx}$, $g_{yy}$, and $g_{zz}$. $AFA$ (purple), $FC$ (light blue), $LS$ (light grey), and $AFAy$ (orange) label different types of magnetic ordering, as described in the text. The points corresponding to the $g$ values extracted from the CEF of DyAG and the CEF1 and CEF2 schemes of DyGG are indicated as red, dark green and light green spheres, respectively.

Figure 12

Evolution with $\mathcal{J}$ of the sum $\mathrm{\Sigma }$ (see text for definition) of the three moments on any given first-neighbor triangle of the hyperkagome lattice in the $FC$ phase. Calculations are performed for $x_{\mathrm{dip}}=100$.

Figure 13

DyAG (left) and DyGG (right) INS experimental spectra vs calculations, as obtained from the ${\chi }^2$ minimization described in the text. The data correspond to constant $Q$ scans as a function of temperature. See the main text for corresponding experimental details.

Figure 14

Calculated magnetic specific heat for DyAG (left) and DyGG (right).

Figure 15

Powder averaged magnetization vs magnetic field $H$ for DyAG (left) and DyGG (right) at different temperatures. The black lines show the calculations as described in the text.

Figure 16

Variation with $g_{xx}$, $g_{yy}$, and $g_{zz}$ of the $m_x$, $m_y$, and $m_z$ components written in the local bases of phases $AFA$ and $FC$ (in the latter, $m_x$, $m_y$ correspond to the 16e site, $m_z$ to the 8b site). Calculations are performed calculated for a pseudospin $1/2$ with $\mathcal{J}=1$ and $x_{\mathrm{dip}}=100$ fixed.

Figure 17

Evolution with $\mathcal{J}$ (from $\mathcal{J}=2$ to $\mathcal{J}=2.75$) of the magnetic phase diagrams as a function of $g_{xx}$, $g_{yy}$, and $g_{zz}$. Same color code as Fig. 11. Calculations are performed for a fixed $x_{\mathrm{dip}}$ value = 100.