Unravelling competing microscopic interactions at a phase boundary: A single-crystal study of the metastable antiferromagnetic pyrochlore ${\mathrm{Yb}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$

We report inelastic neutron scattering measurements from our newly synthesized single crystals of the structurally metastable antiferromagnetic pyrochlore ${\mathrm{Yb}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$. We determine the four symmetry-allowed nearest-neighbor anisotropic exchange parameters via fits to linear spin wave theory supplemented by fits of the high-temperature specific heat using the numerical linked-cluster expansion method. The exchange parameters so determined are strongly correlated to the values determined for the $g$-tensor components, as previously noted for the related Yb pyrochlore ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. To address this issue we directly determined the $g$ tensor from electron paramagnetic resonance of 1% Yb-doped ${\mathrm{Lu}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$, thus enabling an unambiguous determination of the exchange parameters. Our results show that ${\mathrm{Yb}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$ resides extremely close to the classical phase boundary between an antiferromagnetic ${\mathrm{\Gamma }}_5$ phase and a splayed ferromagnet phase. By juxtaposing our results with recent ones on ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, our work illustrates that the Yb pyrochlore oxides represent ideal systems for studying quantum magnets in close proximity to classical phase boundaries.

Figure 1

(a) and (b) Sections of classical phase diagram for the anisotropic exchange model [Eq. (1)] relevant for (a) ${\mathrm{Yb}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$ and (b) ${\mathrm{Yb}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. (c) Comparison of the specific heat of ${\mathrm{Yb}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$ from a representative single crystal [18] and the powder sample studied in Ref. [19]. (d) Optical images of representative single crystals, adapted from Ref. [18].

Figure 2

(a), (b), (d), and (e) Comparison of constant-energy slices (centered at energy $E$ with an energy dependent energy resolution function Appendix pp3) of the 3 T field polarized spin waves between ${\mathrm{Yb}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$ at $1.8\mathrm{K}$ (left) and linear spin wave theory using the best fit $J_1\text{−}{J}_4$ parameters within Eq. (1) (right). The overall intensity scale is consistent between panels, but arbitrary. Comparison between the (c) magnetic susceptibility and (f) specific heat and NLC calculations for the parameters listed in Table 1.

Figure 3

EPR spectra from a 50 mg collection of 1% Yb-doped ${\mathrm{Lu}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$ microcrystallites at different temperatures (arbitrary scale, offset for clarity). The $g$-tensor values are determined from the minimum and maximum of the broad EPR absorption $g_{\pm }=4.20\left(5\right)$, $g_z=1.93\left(2\right)$, which are shown with the green and blue vertical lines. An unidentified $g\approx 2$ impurity is present (indicated by the star).

Figure 4

(a) Constant energy transfer slice ($E=0.3$ meV) for ${\mathrm{Yb}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$ in zero field at $T=1.8\mathrm{K}$. “Rods” of scattering are visible along $\langle 111\rangle$ directions as well as diffuse scattering at $\left(\overline{2}\overline{2}0\right)$. (b) Static (equal time) structure factor calculated using NLC at $T=1.8\mathrm{K}$ using our best fit parameters from the field-polarized spin waves (see Table 1), which also shows rods of scattering and intensity at (220).

Figure 5

Magnetization vs magnetic field for a ${\mathrm{Yb}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$ single crystal. (a) Data taken at $T=10$ K for three high symmetry directions of pyrochlore lattice. (b) Data taken at $T=0.4$ K, with field along [110]. (c) Data taken at $T=2$ K. (d) Calculated single ion magnetization (using $g_z=1.93$, $g_{\pm }=4.20$) at $T=2$ K.

Figure 6

(a) MACS neutron sample mount with 28 coaligned single crystals of ${\mathrm{Yb}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$. (b) Example rocking scan taken at MACS across a (111) nuclear peak at $T\sim 6$ K. The peak is split into three peaks indicating a mosaic of $\le 5^{\circ }$.

Figure 7

Nominal temperature dependence of neutron scattering data taken on MACS. (a) and (b) A lack of additional intensity on expected AFM Bragg peak positions. A shift of the peaks is observed instead (resulting in a net-zero profile with some regions of negative and positive differences). (c) Overlay of the intensity along $[-1,-1,l]$ of the low energy excitations $E=0.3\mathrm{meV}$, at both temperatures (3 T data used as background subtraction).

Figure 8

${\mathrm{Yb}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$ INS data presented as a typical spin wave dispersion plot along $[h,h,-1]$. The dispersion is constructed from the combination of several constant energy slices through the $\left[hhl\right]$ plane.

Figure 9

(a) Zero-field INS spectrum of ${\mathrm{Yb}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$ at 60 mK, within the ordered state. All reciprocal space directions were integrated $\pm 0.1$ r.l.u. in their respective perpendicular components (for example, the cut along $\left[hh0\right]$ is integrated $\pm 0.1$ r.l.u. in $\left[00\mathrm{l}\right])$. The excitations are broad and featureless, despite the ordered ground state, similar to powder INS data reported in Ref. [44]. (b) comparison of (111) Bragg peak at 60 mK and 90 K showing added intensity from magnetic ordering. Plots were integrated from $\pm 0.1$ meV and $\pm 0.1$ r.l.u. in $\left[00l\right]$.

Figure 10

Illustration of the nonuniqueness of the $g$ factors obtained from the fit reported in Ref. [40]. Each set of $g$ factors represents a set of six crystal field parameters ($B_{kq}$) with transition energies and relative intensities within $1\%$ of the result computed using the best fit parameters of Ref. [40]. The color of each point show the variation of the Van Vleck contribution (${\chi }_0$) to the susceptibility. Several important limits are indicated: ${\mathrm{\Gamma }}_6$ doublet (octahedron cage), ${\mathrm{\Gamma }}_7$ doublet (cube cage), as well as pure $|\pm 1/2\rangle$, $|\pm 5/2\rangle$, and $|\pm 7/2\rangle$ doublets. The $g$ factors in the shaded region are not physical for a pure $J=7/2$ manifold in a $D_{3\mathrm{d}}$ crystal field [10].

Figure 11

Energy resolution functions at energy transfer $\mathrm{\Delta }E$ for specified energy transfers ${\left(\mathrm{\Delta }E\right)}_0$ used in the theoretical calculations to emulate the behavior of the experimental setup. Practically these functions can be modeled by a two-sided Gaussian, with energy dependent width for $\mathrm{\Delta }E<{\left(\mathrm{\Delta }E\right)}_0$.

Figure 12

Comparison of constant-energy slices (centered at energy $E$ with an energy dependent energy resolution function, see Fig. 11) of the 3 T field polarized spin waves between ${\mathrm{Yb}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$ at $1.8\mathrm{K}$ (left) and linear spin wave theory using the best fit exchange parameters within linear spin-wave theory (right). Overall intensity scale is consistent between panels, but arbitrary. For the 1.3 and $1.5\mathrm{meV}$ cuts we assume the energy resolution function is the same as the $1.1\mathrm{meV}$ case.