Quantification of local Ising magnetism in rare-earth pyrogermanates ${\mathrm{Er}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$ and ${\mathrm{Yb}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$

Recently, a local-Ising-type magnetic order was inferred from neutron diffraction of the antiferromagnetic ${\mathrm{Er}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$ (pg-ErGO) with an applied magnetic field. Here, we use neutron spectroscopy to investigate the energetics of pg-ErGO and the isostructural ${\mathrm{Yb}}_2{\mathrm{Ge}}_2{\mathrm{O}}_7$ (pg-YbGO) to evaluate the adequacy of the local-Ising description. To begin, we generate a model of the magnetic structure of pg-YbGO using neutron diffraction and find a net ferromagnetic moment. While pg-ErGO and pg-YbGO have highly symmetric crystal structures ($P{4}_1{2}_{1}2$ tetragonal space group 92) with only one trivalent rare-earth magnetic site, the point symmetry of the rare-earth site is low with only a single symmetry element (point group ${\mathrm{C}}_1$). For both compounds, the energy scale of the first excited state is large compared to the magnetic ordering temperature, suggesting Ising character. The ground-state Kramer's doublet of both compounds is dominated by the maximal $m_{\mathrm{J}}$ component. However, the low point group symmetry of the rare-earth site leads to finite mixing of all other $m_{\mathrm{J}}$'s, which suggests potential deviations from Ising behavior. Moreover, quasielastic scattering is observed deep in the ordered state of pg-ErGO and pg-YbGO that may be due to non-Ising behavior. The dominant magnetic interaction in both compounds is found to be magnetostatic by considering the magnetic excitations in the ordered state. From consideration of these data, the pg-YbGO is more Ising-like than pg-ErGO. Also, quantum multicritical points are anticipated with applied magnetic field in both compounds.

Figure 1

Structure of pg-ErGO. (a) Crystal structure of one unit cell, where the erbium sites are represented by a large green circle, germanium is shown as the intermediately sized pink circle encapsulated by its coordinating tetrahedron, and oxygen is illustrated as the smallest red circle. (b) The local coordination sphere of erbium. (c) The magnetic structure is visualized with black arrows, and the nearest-neighbor erbium bond distances are shown as green (3.5743 Å), red (3.5961 Å), and blue (3.8014 Å). These images build on renderings by VESTA [25].

Figure 2

Magnetic structures of pg-ErGO and pg-YbGO. The data are shown in comparison with the best-fit models of the two smallest $R_{wp} {\mathrm{\Gamma }}_2$ and ${\mathrm{\Gamma }}_3$ and the residuals for (a) pg-YbGO and (b) pg-ErGO. Tick marks identify reciprocal lattice vectors of (001), (100), (002), (101), (110), (102), and (111). The overall scale is the same, with different arbitrary offsets added to remove paramagnetic oversubtraction. Residuals are shown below data and models. The magnetic structures associated with the modeled intensities are shown for (c) pg-YbGO ${\mathrm{\Gamma }}_2$, (d) pg-YbGO ${\mathrm{\Gamma }}_3$, (e) pg-ErGO ${\mathrm{\Gamma }}_2$, and (f) pg-ErGO ${\mathrm{\Gamma }}_3$. The magnetic structure images build on renderings by VESTA [25].

Figure 3

High-energy neutron spectroscopy of pg-ErGO single-ion effects. Data are from SEQUOIA with $E_{\mathrm{i}}=160\mathrm{meV}$ and $T=5\mathrm{K}$. (a) An intensity map is shown with the isobar values in units of mb/sr/meV/Er. (b) An averaging of momentum transfers between $Q=[2,4]{\AA }^{-1}$ with fits to resolution limited Gaussians.

Figure 4

High-energy neutron spectroscopy of pg-YbGO single-ion effects. Data are from SEQUOIA with $E_{\mathrm{i}}=160\mathrm{meV}$ and $T=5\mathrm{K}$. The isobar intensity values are in units of mb/sr/meV/Er.

Figure 5

Low-energy neutron spectroscopy of pg-ErGO in the ordered state. Data are from CNCS with $E_{\mathrm{i}}=1.55\mathrm{meV}$. The isobar intensity values are in units of mb/sr/meV/Er. The data are shown on a logarithmic intensity scale.

Figure 6

Low-energy neutron spectroscopy of magnetic ordering effects in pg-ErGO. Data are from CNCS with $E_{\mathrm{i}}=1.55\mathrm{meV}$. Momentum dependence is averaged in the range from $0.2–0.8{\AA }^{-1}$. The data displayed as x symbols were not included in the fitting of inelastic scattering. Lines are fits to Eq. (1).

Figure 7

Low-energy neutron spectroscopy of pg-ErGO in the ordered state. Data are from CNCS with $E_{\mathrm{i}}=1.55\mathrm{meV}$. The isobar intensity values are in units of mb/sr/meV/Yb. The data are shown on a logarithmic intensity scale.

Figure 8

Low-energy neutron spectroscopy of magnetic ordering effects in pg-YbGO. Data are from CNCS with $E_{\mathrm{i}}=1.55\mathrm{meV}$. Momentum dependence is averaged in the range from $0.2–0.8{\AA }^{-1}$. Lines are fits to Eq. (2).

Figure 9

Neutron powder diffraction of pg-YbGO. Temperatures of (a) 10 K and (b) $<1\mathrm{K}$ are shown. The Al and Cu contributions are from the equipment holding the sample.