Revisiting spin ice physics in the ferromagnetic Ising pyrochlore ${\mathrm{Pr}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$

Pyrochlore materials are characterized by their hallmark network of corner-sharing rare-earth tetrahedra, which can produce a wide array of complex magnetic ground states. Ferromagnetic Ising pyrochlores often obey the “two-in-two-out” spin ice rules, which can lead to a highly degenerate spin structure. Large moment systems, such as ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ and ${\mathrm{Dy}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$, tend to host a classical spin ice state with low-temperature spin freezing and emergent magnetic monopoles. Systems with smaller effective moments, such as ${\mathrm{Pr}}^{3+}$-based pyrochlores, have been proposed as excellent candidates for hosting a “quantum spin ice” characterized by entanglement and a slew of exotic quasiparticle excitations. However, experimental evidence for a quantum spin ice state has remained elusive. Here, we show that the low-temperature magnetic properties of ${\mathrm{Pr}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ satisfy several important criteria for continued consideration as a quantum spin ice. We find that ${\mathrm{Pr}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ exhibits two distinct spin-correlation time scales of $\tau \ge {10}^{-4}$ and $\sim {10}^{-10}$ s in the spin ice regime. Our comprehensive bulk characterization and neutron scattering measurements enable us to map out the magnetic field-temperature phase diagram, producing results consistent with expectations for a ferromagnetic Ising pyrochlore. We identify key hallmarks of spin ice physics and show that the application of small magnetic fields (${\mu }_0{H}_c\sim 0.5$ T) suppresses the spin ice state and induces a field-polarized, ordered spin-ice phase. Together, our work clarifies the current state of ${\mathrm{Pr}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ and encourages future studies aimed at exploring the potential for a quantum spin ice ground state in this system.

Figure 1

(a) Crystal structure of ${\mathrm{Pr}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ shown in two perspectives. The left image highlights the rare-earth sublattice of corner-sharing tetrahedra. The right image depicts a slice through the structure, highlighting the edge-sharing network of Pr–O polyhedra and Sn–O octahedra. (b) Measured, calculated (Rietveld), and difference room temperature XRD profiles of polycrystalline ${\mathrm{Pr}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ collected at room temperature. Samples show excellent crystallinity with no detectable impurity phases.

Figure 2

(a) Field dependence of the isothermal magnetization data with the best fit to the powder-averaged $J{}_{\mathrm{eff}}=\frac{1}{2}$ Ising model ($g_{\perp }=0$) superimposed on the 1.8 K data. (b) dc (VSM) magnetization measurements as a function of magnetic field. The derivative $dM/dH$ was used to extract the characteristic field for the onset of the field-induced order. (c) Temperature dependence of the dc susceptibility and the ac susceptibility at selected frequencies and measurement conditions. The inset shows the Curie-Weiss analysis (fit from 3 K to 20 K) of the dc susceptibility data, which suggests the presence of weak net ferromagnetic exchange. (d) Frequency dependence of the real part of the ac susceptibility ${\chi }^{\prime }$ for various temperatures in the absence of an external dc field (left, points), fits to the model described in the text (left, lines), and extracted spin correlation times as a function of temperature (right). Filled and hollow points show results obtained from ${\chi }^{\prime }$ data normalized to dc data over different temperature ranges. (e) Temperature dependence of the imaginary part of the ac susceptibility ${\chi }^{″}$ for various frequencies in the absence of an external dc field (left) and the corresponding Arrhenius fit of the freezing transition (right), which yields an activation energy $E_a=2.6$ K. (f) Field dependence of the freezing transition in ${\chi }^{″}$ using an intermediate driving frequency $f=1$ kHz.

Figure 3

(a) Heat capacity data for polycrystalline ${\mathrm{Pr}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ at selected magnetic fields. The lattice contribution from nonmagnetic ${\mathrm{La}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ is indicated by a solid red curve. (b) The lattice-subtracted specific heat as a function of field demonstrates a low-temperature nuclear Schottky anomaly and a higher temperature peak. (c) The same data shown in (b), but now presented using an $x$-axis log scale to emphasize the two distinct features observed for a given field. (d) Modeling of the nuclear Schottky anomaly using two different models: (1) an analytic form from Kimura et al. [24] for ${\mathrm{Pr}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$ and (2) a generalized statistical-mechanical model provided by Gopal et al. [61]. (e) The magnetic component of $C_{p,m}/T$ after removal of the nuclear component. (f) Integration of the magnetic heat capacity provides insight into magnetic ground state. Regardless of the choice of nuclear Schottky model, at zero field we recover the full Pauling entropy, consistent with a spin-ice ground state. At the highest magnetic fields measured here, we nearly recover the full $Rln\left(2\right)$ expected for a well-isolated crystal field ground state doublet.

Figure 4

(a) Measured, calculated (Rietveld), and difference neutron powder diffraction profiles ($T=1.6$ K, ${\lambda }_{\mathrm{n}}=2.41$ Å) of polycrystalline ${\mathrm{Pr}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ in both the absence and presence of an external dc magnetic field. Contributions from the Bragg reflections of the sample container were fit using a Pawley refinement. Magnetic intensity appears in the 4 T data at $\mathit{k}=\mathit{0}$ Bragg positions. (b) The parent crystal structure of ${\mathrm{Pr}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ (left) and the field-induced ordered spin ice magnetic structure (right) showing only the Pr atoms for clarity.

Figure 5

Field-temperature phase diagram of ${\mathrm{Pr}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ as determined by heat capacity (DR $C_{\text{p}}$ and ${}^4\mathrm{He}{C}_{\text{p}}$), ac susceptibility (DR ${\chi }_{\text{ac}}$), and dc susceptibility (${}^4\mathrm{He}{\chi }_{\text{dc}}$). Regions include a high-temperature paramagnetic (PM) state, field-polarized phase (FPP), spin ice (SI), and partial spin freezing (SF). The inset depicts an enlarged version of the low-temperature, low-field portion of the phase diagram.

Figure 6

(a) Comparison of the $Q$ dependence of the energy-integrated magnetic scattering for polycrystalline ${\mathrm{Pr}}_2{\mathrm{Sn}}_2{\mathrm{O}}_7$ at various temperatures with the corresponding reverse Monte Carlo (RMC) fits as described in the text. Percentages denote the portion of tetrahedra that exhibit a two-in-two-out spin configuration. (b) Restricting the available parameter space to lie exclusively in the spin ice manifold for the RMC refinement cannot completely account for the differential magnetic cross section at 0.05 K. Instead, the model requires the addition of a $Q$-independent vertical offset.

Figure 7

(a) Intensity vs energy transfer $E$ integrated over a $Q$ range of $[0.25,0.95]{\AA }^{-1}$, measured after zero-field cooling at base temperature [$T=0.20$ K for Expt. 1 (red circles); $T=0.02$ K for Expt. 2 (black circles)]. The shaded gray region indicates the elastic portion of the spectrum. (b) Intensity vs momentum transfer $Q$ for energy-integrated data (empty circles) and inelastic data (filled squares). Data from experiments 1, 2, and 3 are labeled on the plots. For Expt. 1 and Expt. 2, energy integration ranges are $[-0.17,0.37]$ meV and [0.014,0.37] meV for energy-integrated and inelastic data, respectively. For Expt. 3 (D7 data), energy integration extends to $\sim 3.5$ meV. The shaded gray regions denote $Q$ regions affected by oversubtracted Bragg scattering. (c) Intensity vs $E$ measured in zero field at 0.5 K after zero-field cooling (Expt. 1, red circles) and at 0.5 K in zero field after a 0.02 K measurement in a 9 T field (Expt. 2, black circles), showing a strong dependence on magnetic-field history. (d) Intensity vs $Q$ for energy-integrated data (empty circles) and inelastic data (filled squares). Red symbols show data collected at 0.5 K after zero-field cooling (Expt. 1) and black symbols show data collected at 0.5 K in zero field after a 0.02 K measurement in a 9 T field. (e) Same as (a), except at 4.2 K. (f) Same as (b), except at 4.2 K and with integration range $[-0.37,0.37]$ meV for the energy-integrated data.