Physical properties of the candidate quantum spin-ice system ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$

Physical properties of a pyrohafnate compound ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ have been investigated by ac magnetic susceptibility ${\chi }_{\mathrm{ac}}\left(T\right)$, dc magnetic susceptibility $\chi \left(T\right)$, isothermal magnetization $M\left(H\right)$, and heat-capacity $C_{\mathrm{p}}\left(T\right)$ measurements on polycrystalline as well as single-crystal samples combined with high-resolution synchrotron x-ray diffraction (XRD) for structural characterization and inelastic neutron scattering (INS) to determine the crystal-field energy-level scheme and wave functions. Synchrotron XRD data confirm the ordered cubic pyrochlore $\left(Fd\overline{3}m\right)$ structure without any noticeable site mixing or oxygen deficiency. No clear evidence of long-range magnetic ordering is observed down to 90 mK, however the ${\chi }_{\mathrm{ac}}\left(T\right)$ evinces slow spin dynamics revealed by a frequency dependent broad peak associated with spin freezing. The INS data reveal the expected five well-defined magnetic excitations due to crystal-field splitting of the $J=4$ ground-state multiplet of the ${\mathrm{Pr}}^{3+}$. The crystal-field parameters and ground-state wave function of ${\mathrm{Pr}}^{3+}$ have been determined. The Ising anisotropic nature of the magnetic ground state is inferred from the INS as well as $\chi \left(T\right)$ and $M\left(H\right)$ data. Together these properties make ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ a candidate compound for quantum spin-ice behavior.

Figure 1

Synchrotron powder x-ray-diffraction pattern of polycrystalline ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ recorded at room temperature. The solid line through the experimental points is the Rietveld refinement profile calculated for the ${\mathrm{Eu}}_2{\mathrm{Zr}}_2{\mathrm{O}}_7$-type face-centered-cubic (space group $Fd\overline{3}m$) pyrochlore structure. The short vertical bars mark the Bragg peaks positions and the lowermost curve represents the difference between the experimental and calculated intensities. Inset: An expanded scale view showing the details of the refinement in a small angle range at high $2\theta$.

Figure 2

Zero-field-cooled magnetic susceptibility $\chi$ of polycrystalline ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ plotted as inverse magnetic susceptibility ${\chi }^{-1}$ as a function of temperature $T$ for $2\mathrm{K}\le T\le 300$ K measured in magnetic field $H=1.0$ T. The solid curve is the crystal-field susceptibility corresponding to the crystal-field parameters obtained from the analysis of inelastic neutron-scattering data. Upper inset: Low-$T \chi \left(T\right)$ for $H=0.1$ and 1.0 T and $2\mathrm{K}\le T\le 25$ K. Lower inset: ${\chi }^{-1}\left(T\right)$ for $H=0.1$ T and $2\mathrm{K}\le T\le 25$ K. The solid red line is the fit of the ${\chi }^{-1}\left(T\right)$ data by the modified Curie-Weiss law fitted over $12\mathrm{K}\le T\le 25$ K and the dashed line is an extrapolation.

Figure 3

Isothermal magnetization $M$ of polycrystalline ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ as a function of applied magnetic field $H$ for $0\le H\le 7$ T measured at the indicated temperatures. The solid curve is the fit of 2-K $M\left(H\right)$ data by Eq. (1) with an effective longitudinal $g$ factor $g_{\parallel }=4.78\left(1\right)$.

Figure 4

(a) Real ${\chi }^{\prime }$ and imaginary ${\chi }^{\prime \prime }$ parts of zero-field-cooled ac magnetic susceptibility ${\chi }_{\mathrm{ac}}$ of polycrystalline ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ as a function of temperature $T$ for $0.09\mathrm{K}\le T\le 0.5$ K measured at indicated frequencies and ac magnetic fields (in zero dc field). (b) Single-spin relaxation time ${\tau }_{\mathrm{spin}}\left(T\right)={\chi }^{\prime \prime }\left(T\right)/\omega {\chi }^{\prime }\left(T\right)$ of ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$. Also shown is the ${\tau }_{\mathrm{spin}}\left(T\right)$ for ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$ [41]. The open circles are the data points of ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ normalized and scaled to the spin-freezing temperature of ${\mathrm{Ho}}_2{\mathrm{Ti}}_2{\mathrm{O}}_7$. (c) Sample temperature $T_{\mathrm{sample}}$ as a function of time $t$ showing the thermal relaxation in polycrystalline ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$. The orange solid lines are the fits of the $T_{\mathrm{sample}}\left(t\right)$ data. (d) Thermal relaxation time ${\tau }_{\mathrm{sample}}\left(T\right)$ obtained from the fitting of $T_{\mathrm{sample}}\left(t\right)$ in (c).

Figure 5

Heat capacity $C_{\mathrm{p}}$ of polycrystalline ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ as a function of temperature $T$ for 1.8 K $\le T\le$ 300 K measured in zero field. The solid curve is the fit by Debye + Einstein models of lattice heat capacity plus crystal-field contribution as deduced from inelastic neutron-scattering (INS) data. Upper inset: Expanded view of low-$T C_{\mathrm{p}}\left(T\right)$ data over $1.8\mathrm{K}\le T\le 15$ K. Lower inset: Magnetic contribution to heat capacity $C_{\mathrm{mag}}\left(T\right)$ of ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$. The solid curve represents the crystal-field contribution to heat capacity according to the CEF level scheme deduced from the INS data.

Figure 6

Inelastic neutron-scattering response, a color-coded map of the intensity, energy transfer $\left(E\right)$ vs momentum transfer $\left(Q\right)$ for ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ (upper panel) and ${\mathrm{La}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ (lower panel) measured at 10 K with the incident energy $E_i=40$, 86, and 136 meV. The arrows mark the position of CEF excitations in ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$. The corresponding energy positions in ${\mathrm{La}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ are also marked.

Figure 7

$Q$-integrated inelastic magnetic scattering intensity $S_{\mathrm{M}}(Q,\omega )$ vs energy transfer $E$ of ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ integrated over (a) momentum $Q$ range (1.25–1.75) ${\AA }^{-1}$ for $E_i=18.9$ meV, and (b) $Q$ range (4–5) ${\AA }^{-1}$ for $E_i=136.0$ meV, showing all five transitions within the ground-state multiplet ${}^3{H}_4$ of ${\mathrm{Pr}}^{3+}$ at 10 K. The solid lines are the fits of the data according to the crystal-field model discussed in the text.

Figure 8

(a) $\varphi$-integrated inelastic scattering intensity $S(\langle \varphi \rangle ,\omega )$ vs energy transfer $E$ of ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ and ${\mathrm{La}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ for $E_i=136.0$ meV integrated over scattering angle $9^{\circ }\le \varphi \le {11}^{\circ }$ together with the scaled phonon background $S(\langle \varphi \rangle ,\omega )$ from $\langle \varphi \rangle$ range ${119}^{\circ }\le \varphi \le {121}^{\circ }$ of ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$. (b) Magnetic scattering intensity $S_{\mathrm{M}}(\langle \varphi \rangle ,\omega )$ vs $E$ of ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ for $\langle \varphi \rangle$ range $9^{\circ }\le \varphi \le {11}^{\circ }$, showing four transitions within the ground-state multiplet ${}^3{H}_4$ of ${\mathrm{Pr}}^{3+}$ at 10 K. The solid line is the fit of the data according to the crystal-field model discussed in text. The apparent peak near 45 meV is an artifact arising from phonon contribution.

Figure 9

Zero-field-cooled magnetic susceptibility $\chi$ of single-crystal ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ as a function of temperature $T$ for $2\mathrm{K}\le T\le 50$ K for the three crystallographic directions [100], [110], and [111], measured in magnetic field $H=0.1$ T. Inset: Inverse susceptibility ${\chi }^{-1}\left(T\right)$ for $H=0.1$ T and $2\mathrm{K}\le T\le 25$ K. The solid red line is the fit of the ${\chi }^{-1}\left(T\right)$ data by the modified Curie-Weiss law in $12\mathrm{K}\le T\le 25$ K and the dashed line is an extrapolation.

Figure 10

Isothermal magnetization $M$ of single-crystal ${\mathrm{Pr}}_2{\mathrm{Hf}}_2{\mathrm{O}}_7$ as a function of applied magnetic field $H$ for $0\le H\le 14$ T measured at 2 K for the three crystallographic directions [100], [110], and [111].