Physical properties of a candidate quantum spin-ice system Pr2Hf2O7

TL;DR

This study investigates the physical properties of Pr2Hf2O7, revealing its potential as a quantum spin-ice candidate through structural, magnetic, and neutron scattering analyses.

Contribution

The paper provides a comprehensive characterization of Pr2Hf2O7, including crystal field analysis and magnetic behavior, establishing its suitability as a quantum spin-ice system.

Findings

No long-range magnetic order down to 90 mK

Presence of slow spin dynamics and spin freezing

Crystal field analysis confirms Ising anisotropy

Abstract

Physical properties of a pyrohafnate compound Pr2Hf2O7 have been investigated by ac magnetic susceptibility \chi_ ac(T), dc magnetic susceptibility \chi(T), isothermal magnetization M(H) and heat capacity C_p(T) 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 (Fd-3m) 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_ac(T) 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 Pr^3+. The crystal field parameters and ground state wavefunction of Pr^3+ have been determined. The Ising anisotropic nature of magnetic ground state is inferred from the INS as well as \chi(T) and M(H) data. Together these properties make Pr2Hf2O7 a candidate compound for quantum spin-ice behavior.

Tables

Table 1. Table 1: Observed and calculated crystal-field transition energies ( E 𝐸 E ) and integrated intensities ( I 𝐼 I ) within the ground state multiplet H 4 3 superscript subscript 𝐻 4 3 {}^{3}H_{4} of Pr 2 Hf 2 O 7 at 10 K. The intensities are relative to the highest observed peak. The best-fit crystal field parameters B q k superscript subscript 𝐵 𝑞 𝑘 B_{q}^{k} are: B 0 2 = 33.8 subscript superscript 𝐵 2 0 33.8 B^{2}_{0}=33.8  meV, B 0 4 = 426.0 subscript superscript 𝐵 4 0 426.0 B^{4}_{0}=426.0  meV, B 3 4 = 192.7 subscript superscript 𝐵 4 3 192.7 B^{4}_{3}=192.7  meV, B 0 6 = 165.3 subscript superscript 𝐵 6 0 165.3 B^{6}_{0}=165.3  meV, B 3 6 = − 111.8 subscript superscript 𝐵 6 3 111.8 B^{6}_{3}=-111.8  meV, and B 6 6 = 178.6 subscript superscript 𝐵 6 6 178.6 B^{6}_{6}=178.6  meV. The goodness-of-fit parameter χ 2 = 3.12 superscript 𝜒 2 3.12 \chi^{2}=3.12 .

Levels	$E_{obs}$ (meV)	$E_{cal}$ (meV)	$I_{obs}$	$I_{cal}$
${\Gamma }_3^{+}$	0	0
${\Gamma }_1^{+}$	9.1(1)	8.82	-	-
${\Gamma }_3^{+}$	56.5(2)	56.63	0.42(10)	0.55
${\Gamma }_1^{+}$	82.2(3)	82.11	0.39(8)	0.47
${\Gamma }_3^{+}$	94.5(3)	94.56	1.0	1.0
${\Gamma }_2^{+}$	109.5(4)	109.46	0.33(6)	0.38

Table 2. Table 2: Parameters obtained from the analysis of magnetic susceptibility of Pr 2 Hf 2 O 7 by modified Curie-Weiss law.

$H$ direction	${\chi }_0$	$C$	${\theta }_p$
	(emu/mol Pr)	(emu K/mol Pr)	(K)
$\left[100\right]$	$4.9\left(1\right)\times {10}^{-3}$	0.79(1)	$-0.81\left(4\right)$
$\left[110\right]$	$4.8\left(1\right)\times {10}^{-3}$	0.79(1)	$-0.93\left(5\right)$
$\left[111\right]$	$4.9\left(1\right)\times {10}^{-3}$	0.79(1)	$-0.86\left(5\right)$

Equations

$$⟨M⟩=\frac{(k_{\mathrm{B}}T{)}^2}{g_{\parallel }{\mu }_{\mathrm{B}}{H}^{2}S}{\int }_0^{g_{\parallel }{\mu }_{\mathrm{B}}HS/k_{\mathrm{B}}T}x\tanh (x)dx$$

$$\begin{array}{rl}{H}_{\mathrm{CEF}}=& B_0^2{C}_0^2+B_0^4{C}_0^4+B_3^4(C_{-3}^4+C_3^4)\\ & +B_0^6{C}_0^6+B_3^6(C_{-3}^6+C_3^6)+B_6^6(C_{-6}^6+C_6^6),\end{array}$$

$$\begin{array}{rl}{\Gamma }_3^{+}=& 0.823{|}^3{H}_4,\pm 4⟩\pm 0.513{|}^3{H}_4,\pm 1⟩-0.136{|}^3{H}_4,\pm 2⟩\\ & +0.139{|}^1{G}_4,\pm 4⟩\pm 0.087{|}^1{G}_4,\pm 1⟩\\ & \mp 0.058{|}^3{H}_5,\pm 4⟩\mp 0.050{|}^3{H}_5,\pm 2⟩\end{array}$$

$$\begin{array}{rl}{\Gamma }_1^{+}=& 0.173{|}^3{H}_4,3⟩+0.950{|}^3{H}_4,0⟩-0.173{|}^3{H}_4,-3⟩\\ & +0.162{|}^1{G}_4,0⟩+0.038{|}^3{F}_2,0⟩.\end{array}$$

Full text

I INTRODUCTION

The rare earth pyrochlore oxides $R_{2}B_{2}$O7 ($R$ is a trivalent rare earth and $B$ a tetravalent transition metal or Ge, Sn, Pb) containing magnetic networks of corner-sharing tetrahedra are well known for their exotic ground state (arising from the combined effect of crystal field anisotropy and dipolar and exchange interactions) and have been an interesting topic of research for the past few decades Gardner2010 ; Castelnovo2012 ; Gingras2014 ; Ramirez1999 ; Siddharthan1999 ; Hertog2000 ; Bramwell2001 . The research interests in these systems were invigorated after the discovery of spin-ice behavior in Dy2Ti2O7 and Ho2Ti2O7 Ramirez1999 ; Siddharthan1999 ; Hertog2000 ; Bramwell2001 . The crystal field induced Ising anisotropic ground state (where the spins are forced to point along the local $\langle 111\rangle$ direction) of these classical spin-ice materials along with ferromagnetic dipolar interactions give rise to a ‘two-in/two-out’ configuration (the so called ‘ice rule’) of rare earth moments Harris1997 . The two-in/two-out spin configuration of these classical spin-ices is equivalent to the arrangement of protons in water ice, thereby they also possess low-temperature Pauling residual entropy $(1/2)R\ln(3/2)$ Ramirez1999 . The fundamental excitations at the heart of spin-ice physics are the emergent magnetic monopoles arising from the creation and anihilation of ‘three-in/one-out’ or ‘one-in/three-out’ spin configurations Castelnovo2008 . The experimental observation of Dirac strings via neutron scattering have provided evidence of these magnetic monopoles Castelnovo2008 ; Morris2009 ; Fennell2009 . The monopole dynamics provides further insights to the emergent magnetic phases in spin-ice materials Castelnovo2008 ; Fennell2009 ; Morris2009 ; Jaubert2009 ; Bramwell2009 .

The titanate pyrochlores Tb2Ti2O7 and Yb2Ti2O7 also present interesting magnetic ground states. Tb2Ti2O7 is a spin-liquid system that shows a cooperative critical paramagnetic ground state Gardner1999 ; Gardner2001 ; Molavian2007 . Yb2Ti2O7 exhibits the characteristic features of quantum spin-ice (QSI) behavior Ross2011 ; Pan2016 . The ground state of Yb2Ti2O7 is however very much sample dependent ranging from ferromagnetic order to disorder. A Higgs transition from a magnetic Coulomb liquid state to a ferromagnetic state (the Higgs phase of magnetic monopoles) has been suggested for ferromagnetically ordered Yb2Ti2O7 Chang2012 . The quantum fluctuations play an important role in quantum spin-ice materials. While in a classical spin-ice system, because of strong magnetic anisotropy, only Ising interactions exist, in a quantum spin-ice system the magnetic anisotropy is not so strong and significant non-Ising terms exist in the Hamiltonian, therefore quantum fluctuations strongly dominate the spin dynamics in QSI systems. In order to have strong quantum fluctuations the dipolar interaction has to be small which one would naively expect to be the case with the materials having rare earth ions with smaller moments. This condition is fulfilled for Yb3+, and because of the smaller dipolar interaction the ground state of Yb2Ti2O7 is mainly dominated by exchange interactions Ross2011 .

Pr3+ and Ce3+ have much smaller magnetic moments compared to Dy3+ and Ho3+, therefore the pyrochlore systems containing Pr and Ce is expected to fullfill the condition of smaller dipolar interaction. As the dipolar interaction is proportional to the square of magnetic moment, the dipolar interaction in Pr-pyrochlores (say Pr2Sn2O7) will be only $\sim 9\%$ of that in Dy2Ti2O7. Indeed the Pr-pyrochlores Pr2Sn2O7 Zhou2008 and Pr2Zr2O7 Kimura2013 , and Ce-pyrochlore Ce2Sn2O7 Sibille2015 have been found to show the characteristic features of a quantum spin-ice/spin-liquid system. Pr2Ir2O7 on the other hand shows a metallic spin-liquid ground state Nakatsuji2006 that is situated close to a quantum critical point Tokiwa2014 . Of particular interest are the Pr-based pyrochlores which have a non-Kramers doublet as ground state. Lee et al. Lee2012 used a gauge mean-field theoretical approach to suggest that the QSI phase in a system with a non-Kramers doublet ground state [such as in Pr${}_{2}B_{2}$O7 ($B$ = Zr, Sn, Ir)] is more stable than in a system with a Kramers doublet ground state.

The hafnate pyrochlore Pr2Hf2O7 is another potential Pr-based system for the study of possible quantum spin-ice behavior. Pr2Hf2O7 is reported to form in a cubic pyrochlore ($Fd\bar{3}m$) structure Karthik2012 . A recent muon spin relaxation study on Pr2Hf2O7 by Foronda et al. Foronda2015 found muon-induced anisotropic local distortion for which they proposed a splitting of the non-Kramers doublet of Pr3+. Extending our work on pyrohafnates Anand2015 here we report the physical properties of Pr2Hf2O7 which indeed seems to be a potential candidate for quantum spin-ice behavior. Our investigations using various complementary techniques exclude long-range magnetic ordering down to 90 mK. The ac magnetic susceptibility $\chi_{\rm ac}(T)$ shows a rather broad frequency dependent anomaly near 0.2 K at 160 Hz. The frequency dependent behavior of this anomaly at such a low temperature is a reminiscence of spin-ice dynamics. With a $J$ value of 4, Pr3+ has a $(2J+1=9)$-fold degenerate ground state which under the action of crystal electric field (CEF) can split into a combination of singlets and doublets. In order to determine whether the CEF-split ground state of Pr3+ is consistent with spin-ice physics we performed an inelastic neutron scattering measurement and found that Pr2Hf2O7 indeed has a well isolated Ising anisotropic doublet ground state. The CEF ground state of Pr2Hf2O7 thus meets the requirements for quantum spin-ice behavior. A recent work by Sibille et al. Sibille2016 also reports a quantum spin-ice behavior in Pr2Hf2O7.

II EXPERIMENTAL DETAILS

The polycrystalline Pr2Hf2O7 sample was prepared at the Crystal Lab, Helmholtz-Zentrum Berlin (HZB) by solid-state reaction method using the high purity materials: Pr6O11 (99.99%, Alfa Aesar) and HfO2 (99.95%, Alfa Aesar). A single phase sample was obtained by firing the stoichiometric mixture of Pr6O11 and HfO2 at 1300 ∘C for 50 h followed by three more successive grindings, pelletizings and firings at 1500 ∘C for 50 h. The polycrystalline La2Hf2O7 sample was prepared using La2O3 (99.999%, Alfa Aesar) and HfO2 as detailed in Ref. Anand2015 . The phase purity and quality of the samples were checked by room temperature powder x-ray diffraction using the Bruker-D8 laboratory-based x-ray diffractometer.

A large single crystal of Pr2Hf2O7 was successfully grown by the Floating-zone technique in a four mirror type high-temperature optical floating zone furnace (Crystal Systems corp. FZ-T-12000-X-VII-VPO-PC) equipped with Xenon arc lamps, in flowing argon atmosphere. The quality of the single crystal was checked by Laue diffraction. The single crystal was also oriented by the Laue method and cut into small pieces for magnetic measurements.

The magnetic measurements down to 2 K and up to 14 T were performed using a Quantum Design superconducting quantum interference device vibrating sample magnetometer (SQUID-VSM) and VSM option of a Quantum Design physical properties measurement system (PPMS) at Mag Lab, HZB. The heat capacity measurements down to 1.8 K were performed by the adiabatic-relaxation technique using the PPMS at Mag Lab, HZB. The ac susceptibility measurements down to 90 mK were performed at Helmholtz-Zentrum Dresden-Rossendorf using compensated coil-pair susceptometer built in a He-3/4 dilution refrigerator using a SR830 lock-in amplifier. We also made an attempt to measure the resistivity but the resistance was found to be beyond the measurement range of PPMS indicating an insulating behavior. The resistance measurement using a multimeter further confirmed the insulating ground state of Pr2Hf2O7.

The high-resolution synchrotron x-ray diffraction measurements were performed on the MS-beamline Willmott2013 at Paul Scherrer Institute (PSI), Switzerland. The sample was very finely ground and filled into a thin capillary (0.3 mm in diameter). In order to minimize the profile shape dependence on capillary diameter and reduce the effect of preferred orientation the sample was rotated continuously. X-rays of energy 25 keV were used for the measurement. The exact wavelength of the x-rays (0.4957 Å) and the instrument profile parameters were determined by measurements on the standard LaB6 powder (NIST) under the identical experimental condition.

The inelastic neutron scattering measurements were performed on the high neutron flux beamline MERLIN at the ISIS facility of the Rutherford Appleton Laboratory, Didcot, U.K. Bewley2006 ; Bewley2009 . About 20 g powder samples each of Pr2Hf2O7 and La2Hf2O7 were used for the INS measurements. The powder samples were mounted inside a thin-walled aluminum can using a thin wrapper of aluminum foil in a cylindrical geometry with diameter approximately 40 mm and height 40 mm, which was then mounted into a closed-cycle refrigerator using He-4 exchange gas. The measurements were carried out at 10 K, 100 K and 300 K using neutrons of incident energies $E_{i}=18.9$ meV, 40.1 meV, 86.3 meV, 136.0 meV and 195.0 meV.

III Crystallography

The room temperature high-resolution synchrotron powder XRD pattern of polycrystalline Pr2Hf2O7 is shown in Fig. 1 along with the structural Rietveld refinement profile. For Rietveld refinement we used the software FullProf Rodriguez1993 . The refinement revealed no impurity phase and confirmed the reported ${\rm Eu_{2}Zr_{2}O_{7}}$-type face-centered cubic (space group $Fd\bar{3}m$) pyrochlore structure. In this structure both Pr and Hf form individual three-dimensional networks of corner-shared tetrahedra. The Wyckoff positions (atomic coordinates) of Pr, Hf, O1 and O2 are 16d (1/2,1/2,1/2), 16c (0,0,0), 48f ($x_{\rm O1}$,1/8,1/8) and 8b (3/8,3/8,3/8), respectively. The lattice parameter $a=10.6728(1)$ Å and the $x$-coordinate of O1 $x_{\rm O1}=0.3351(4)$ obtained from the refinement agree very well with the reported values Karthik2012 . The agreement factors for the refinement are $\chi^{2}=10.1$, $R_{\rm p}=2.91\%$ and $R_{\rm wp}=3.66\%$. We also checked for possible Pr/Hf site mixing and refined the occupancies of atoms which did not make any noticeable improvement in the fit or agreement factors. From our synchrotron data we anticipate that Pr/Hf site mixing and/or oxygen deficiency, if present, are less than 0.5%. This suggests a well-ordered pyrochlore structure for Pr2Hf2O7 consistent with the previous report Karthik2012 and with the prediction based on the ratio of the cation radii $r_{\rm Pr}/r_{\rm Hf}\approx 1.59$ which is well within the range of 1.36–1.71 required for a stable ordered pyrochlore phase Gardner2010 .

IV DC Magnetic Susceptibility and Magnetization

The zero-field-cooled (ZFC) dc magnetic susceptibility $\chi(T)$ data of polycrystalline Pr2Hf2O7 are shown in Fig. 2. It is clear from Fig. 2 that at $T\geq 2$ K the $\chi(T)$ data show no anomaly related to magnetic phase transition. As can be seen from the upper inset of Fig. 2, the magnitude of $\chi$ at low-$T$ is rather large. The ZFC and field-cooled (FC) $\chi(T)$ data (not shown) do not show any thermal hysteresis above 2 K. The $\chi(T)$ data follow modified Curie-Weiss behavior, $\chi(T)=\chi_{0}+C/(T-\theta_{\rm p})$ (lower inset of Fig. 2). A fit of ZFC $\chi^{-1}(T)$ data (measured in magnetic field $H=0.1$ T) in $12~{}{\rm K}\leq T\leq 25$ K gives $\chi_{0}=5.02(6)\times 10^{-3}$ emu/mol Pr, $C=0.81(1)$ emu K/mol Pr and $\theta_{\rm p}=-0.01(2)$ K. Upon correcting the $\chi(T)$ data for demagnetization effects (approximating the sample by a sphere) we obtain $\chi_{0}=5.02(6)\times 10^{-3}$ emu/mol Pr, $C=0.81(1)$ emu K/mol Pr and $\theta_{\rm p}=+0.06(2)$ K. The positive $\theta_{\rm p}$ suggests the effective interaction in Pr2Hf2O7 to be ferromagnetic in nature. The $C$ value yields an effective moment of $\mu_{\rm eff}\approx 2.54\,\mu_{\rm B}$/Pr which is significantly lower than the expected paramagnetic state value of $\mu_{\rm eff}=g_{J}\sqrt{J(J+1)}=3.58\,\mu_{\rm B}$/Pr for free Pr3+ ions ($g_{J}=4/5$ and $J=4$). The reduced value of $\mu_{\rm eff}$ reflects the Ising anisotropic nature of magnetic ground state. A fit of the $H=1.0$ T $\chi^{-1}(T)$ data in $50~{}{\rm K}\leq T\leq 300$ K yields $\mu_{\rm eff}\approx 2.91\,\mu_{\rm B}$/Pr which is also smaller than the free ion value and can be attributed to crystal field effect.

The isothermal magnetization $M(H)$ data of polycrystalline Pr2Hf2O7 at five different temperatures between 2 K and 50 K are shown in Fig. 3. For $T\geq 20$ K the $M(H)$ isotherms are almost linear in $H$ and become nonlinear at $T\leq 10$ K. The nonlinearity can be attributed to the saturation behavior of magnetization which is best reflected in $M(H)$ curve at 2 K. At 2 K, the near saturation value of magnetization is $M\approx 1.15\,\mu_{\rm B}$/Pr at 5 T which is only about 36% of the theoretical saturation magnetization $M_{\rm sat}=g_{J}J\,\mu_{\rm B}=3.2\,\mu_{\rm B}$/Pr for free Pr3+ ions. However, the observed $M$ is consistent with the local $\langle 111\rangle$ Ising anisotropic behavior. We fitted the $M(H)$ data by Bramwell2000 ; Xu2015

[TABLE]

that describes the $H$ dependence of powder-averaged $M$ in the paramagnetic state of an Ising pyrochlore within effective spin $S=1/2$ model. Here $g_{\parallel}$ is the longitudinal $g$-factor and the transverse $g$-factor $g_{\perp}$ is assumed to be zero. The fit of $M(H)$ data at 2 K (shown by solid red line in Fig. 3) gives $g_{\parallel}=4.78(1)$, accordingly the paramagnetic state moment is expected to be $m=g_{\parallel}S\,\mu_{\rm B}\approx 2.4\,\mu_{\rm B}$/Pr which is very close to the $\mu_{\rm eff}\approx 2.54\,\mu_{\rm B}$/Pr obtained above from $\chi(T)$. Further it is seen that while Eq. (1) predicts a saturation tendency at high $H$ (see the fit in Fig. 3), the $M(H)$ data show a weak increase, this departure is clearly noticed at $H>6$ T. Such a behavior of $M(H)$ hints for the presence of spin fluctuations/non-Ising contribution.

The nearest neighbor dipole-dipole interaction $D_{\rm nn}$ corresponding to our Ising ground state moment $\mu_{\rm eff}=2.54\,\mu_{\rm B}$/Pr can be estimated using the relation $D_{\rm nn}=({5}/{3})({\mu_{0}}/{4\pi}){\mu_{\rm eff}^{2}}/{r_{\rm nn}^{3}}$ Bramwell2001 ; Hertog2000 , $\mu_{0}$ being the magnetic permeability of free space and $r_{\rm nn}=(a/4)\surd 2$ the nearest neighbor distance. For Pr2Hf2O7 the lattice parameter $a=10.6389(1)$ Å, therefore $D_{\rm nn}\approx 0.12$ K. An estimate of effective magnetic interaction $J_{\rm eff}$ can be obtained from the value of Weiss temperature, $\theta_{\rm p}=zJ_{\rm eff}S(S+1)/3k_{B}$ ($z=6$ is the number of the nearest neighbor) which for $\theta_{\rm p}=+0.06$ K and $S=1/2$ (effective spin) yields $J_{\rm eff}=+0.04$ K (in our notation a positive sign indicates ferromagnetic interaction and a negative sign the antiferromagnetic interaction). The effective interaction $J_{\rm eff}=J_{\rm nn}+D_{\rm nn}$ where $J_{\rm nn}$ is the nearest neighbor exchange interaction which for $J_{\rm eff}=+0.04$ K and $D_{\rm nn}\approx+0.12$ K would imply $J_{\rm nn}\approx-0.08$ K. As expected, the exchange interaction is thus antiferromagnetic in nature. A more realistic estimate of $J_{\rm nn}$ can be obtained from heat capacity $C_{\rm p}(T)$. The height or the temperature of the peak of $C_{\rm p}(T)$ has been found to give very reasonable estimate of $J_{\rm nn}$ Gardner2010 ; Hertog2000 ; Melko2004 which does not suffer from the issue of choice of fitting range of temperature that causes a little variation in the value of $\theta_{\rm p}$. Sibille et al. Sibille2016 reported a value of $C_{\rm p}\approx 3.2$ J/mol K near the peak around 1.8 K. The $C_{\rm peak}\approx 3.2$ J/mol K corresponds to $J_{\rm nn}/D_{\rm nn}\approx 0.7$ in the phase diagram of dipolar spin ice Hertog2000 which for $D_{\rm nn}\approx+0.12$ K yields $J_{\rm nn}\approx-0.08$ K. We thus see that both $\theta_{\rm p}$ and $C_{\rm peak}$ give a very consistent estimate of the antiferromagnetic exchange interaction in Pr2Hf2O7.

V AC Magnetic Susceptibility

The ac magnetic susceptibility $\chi_{\rm ac}$ of polycrystalline Pr2Hf2O7 measured at frequencies 16 Hz, 160 Hz and 1.6 kHz is shown in Fig. 4(a). Both real $\chi^{\prime}$ and imaginary $\chi^{\prime\prime}$ parts of $\chi_{\rm ac}(T)$ show broad peaks. The peak position depends on the frequency, with increasing frequency the peak position shifts to higher temperatures. At 16 Hz $\chi^{\prime}(T)$ shows a peak near 0.16 K which shifts to 0.18 K at 160 Hz and 0.28 K at 1.6 kHz. The frequency dependent ac susceptibility clearly indicates very slow spin dynamics and is reminiscent of spin-ice behavior. This kind of frequency dependent $\chi_{\rm ac}(T)$ associated with spin-freezing has also been observed in the classical spin-ice systems Dy2Ti2O7 Matsuhira2001 and Ho2Ti2O7 Matsuhira2000 and dynamic spin-ice systems Pr2Sn2O7 Matsuhira2004 and Pr2Zr2O7 Kimura2013 . Sibille et al. [29] also found a similar frequency dependent peak in ac susceptibility of Pr2Hf2O7. Through the $\chi_{\rm ac}$ measurements at several frequencies they suggested cooperative spin-dynamics in this compound.

We have determined the single-spin relaxation time $\tau_{\rm spin}$ according to the Casimir-du Pré relation in the paramagnetic limit Casimir1938 , $\tau_{\rm spin}=\chi^{\prime\prime}/\omega\chi^{\prime}$, the $T$ dependence of which is shown in Fig. 4(b). For comparison, $\tau_{\rm spin}(T)$ of Ho2Ti2O7 Thomas is also shown in Fig. 4(b). The single-spin relaxation time is a sensitive and direct measure of the spin dynamics of magnetic materials, in particular in the vicinity of the spin-freezing temperature. The $\tau_{\rm spin}(T)$ of Pr3+ magnetic moments of Pr2Hf2O7 scales pretty well with that of Ho3+ in Ho2Ti2O7 by multiplying with the ratio of their spin-freezing temperatures. We thus see that $\tau_{\rm spin}(T)$ of Pr2Hf2O7 follows qualitatively and quantitatively the temperature dependent single-spin relaxation time observed for the spin-ice compound Ho2Ti2O7.

We also see a distinct thermal relaxation behavior in Pr2Hf2O7. However, the ac susceptibility of the sample has a reproducible temperature dependence and serves as an internal sample thermometer. Susceptibility data have been taken during stepwise heating up and cooling of the mixing chamber of the used He-3/4 fridge over long periods. At very low temperatures, the temperature has been kept constant for days [relaxed susceptibility-temperature data points are shown in Fig. 4(a)].

The sample temperature $T_{\rm sample}$ deduced from the $\chi_{\rm ac}$ data is shown in Fig. 4(c) as a function of time $t$. The $T_{\rm sample}(t)$ represents the thermal relaxation in polycrystalline Pr2Hf2O7. The $T_{\rm sample}(t)$ data in Fig. 4(c) show a very slow thermal relaxation in Pr2Hf2O7. We fit the $T_{\rm sample}(t)$ data by $T_{\rm sample}(t)=T_{0}+\Delta T\exp(-t/\tau_{\rm sample})$, the fits are shown by orange solid lines in Fig. 4(c). The mean thermal relaxation time $\tau_{\rm sample}(T)$ obtained from the fit is shown in Fig. 4(d). The $\tau_{\rm sample}$ is very large and increases very rapidly as $T$ decreases, following a power law behavior, $\tau_{\rm sample}(T)=R_{\rm thermal}(T)C_{\rm sample}(T)\sim 1/T^{\alpha}$ with $\alpha=4.5$. The observed thermal relaxation time of the sample exceeds the typical time windows of standard heat-capacity measurements. As the thermal coupling resistance $R_{\rm thermal}(T)$ is expected to follow a $1/T^{3}$ power law due to phononic transport in the low-temperature limit, the sample relaxation time of Pr2Hf2O7 allows for an estimate of the temperature dependence of the heat capacity, $C_{\rm sample}(T)\sim 1/T^{1.5}$ in the investigated temperature interval (0.1–0.25 K). This increase of heat capacity with decreasing temperature at low-$T$ indicates that, in contrast to observations in classical spin-ice materials, the spin entropy drops in the temperature range of spin freezing. In consequence, the ice entropy might not be conserved in Pr2Hf2O7. We interpret this observation as a hint for the occurence of quantum fluctuations in the low-temperature limit.

VI Heat Capacity

The $C_{\rm p}(T)$ data of polycrystalline Pr2Hf2O7 are shown in Fig. 5 for 1.8 K $\leq T\leq$ 300 K. As can be seen from Fig. 5 the $C_{\rm p}(T)$ data do not show any anomaly at $T\geq 1.8$ K, though an upturn feature is seen at $T\leq 10$ K which is shown in detail in the upper inset of Fig. 5. The magnetic contribution to heat capacity $C_{\rm mag}(T)$ estimated after subtracting the lattice contribution (equivalent to formula mass and unit cell volume corrected $C_{\rm p}(T)$ of La2Hf2O7) is shown in the lower inset of Fig. 5. A broad Schottky type peak near 50 K is seen in $C_{\rm mag}(T)$ data. The $T$ dependence of $C_{\rm mag}$ is described by the crystal field. The crystal field contribution to heat capacity $C_{\rm CEF}(T)$ calculated according to the CEF level scheme deduced from the analysis of INS data (Sec. VII), shown by solid red curve in the lower inset of Fig. 5, agrees well with the $C_{\rm mag}(T)$.

The room temperature value of $C_{\rm p}\approx 228$ J/mol K is substantially lower than the expected high-$T$ limit Dulong-Petit value, reflecting the large value of Debye temperature $\Theta_{\rm D}$. We estimate $\Theta_{\rm D}$ by fitting the $C_{\rm p}(T)$ data by a combination of the Debye and Einstein models of lattice heat capacity added to the fixed crystal field contribution $C_{\rm CEF}(T)$. The fit of $C_{\rm p}(T)$ data by CEF+Debye+Einstein models Anand2015a is shown by the solid red curve in Fig. 5 which gives $\Theta_{\rm D}=790(7)$ K and Einstein temperature $\Theta_{\rm E}=158(2)$ K with 66% weight to Debye term and 34% to Einstein term. The obtained $\Theta_{\rm D}$ is similar to that of the related pyrohafnates La2Hf2O7 ($\Theta_{\rm D}=792(5)$ K) and Nd2Hf2O7 ($\Theta_{\rm D}=785(6)$ K) Anand2015 .

VII Inelastic Neutron Scattering and Crystal Field Excitations

The bulk properties measurement discussed above, particularly the ac susceptibility data hint for spin-ice like behavior. Therefore in order to check if the ground state is consistent with the physics of spin-ice which requires a doublet as ground state we determine the crystal field states using inelastic neutron scattering. Figure 6 shows the color-coded contour maps of the INS response from powder Pr2Hf2O7 and La2Hf2O7 showing energy transfer $E$ versus wave vector $Q$ for neutrons of incident energy $E_{i}=40.1$ meV, 86.3 meV and 136.0 meV at 10 K. A comparison of the contour maps of Pr2Hf2O7 and La2Hf2O7 clearly show five dispersionless excitations in Pr2Hf2O7 at around 9.1 meV, 56.5 meV, 82.2 meV, 94.5 meV and 109.5 meV at low-$Q$. Since La2Hf2O7 does not show strong phonon scattering at low-$Q$, these low-$Q$ excitations of Pr2Hf2O7 seems to have magnetic origin due to the crystal field excitations from Pr3+.

With a $J$ value of 4, the ground state (GS) of Pr3+ possesses nine-fold degeneracy, when this ${}^{3}H_{4}$ GS multiplet is subject to a CEF interaction with $D_{3d}$ symmetry (point symmetry $\bar{3}m$) due to the pyrochlore structure, it splits into three doublets and three singlets, designated by the symmetry decompositions 3$\Gamma_{3}^{+}+2\Gamma_{1}^{+}+\Gamma_{2}^{+}$ in the irreducible representations of $D_{3d}$. Accordingly there ought to be five CEF excitations from the GS multiplet which we clearly see in the INS spectra at 10 K. For a $D_{3d}$ symmetry with $z$ axis along the local cubic $\langle$111$\rangle$ direction, the CEF Hamiltonian is given by

[TABLE]

where $B_{q}^{k}$ are the crystal field parameters and $C_{q}^{k}$ the tensor operators cfbook .

The $Q$-integrated one-dimensional energy cuts of the INS response from the low-$Q$ region (where the phonon contributions are relatively weaker) were made to obtain the scattering intensity $S(Q,\omega)$. The magnetic scattering $S_{\rm M}(Q,\omega)$ from Pr2Hf2O7 was obtained by subtracting the phonon contribution using the INS data of nonmagnetic reference La2Hf2O7. The plots of $S_{\rm M}(Q,\omega)$ versus $E$ are shown in Fig. 7 where $S_{\rm M}(Q,\omega)=S(Q,\omega)_{\rm Pr_{2}Hf_{2}O_{7}}-\alpha\,S(Q,\omega)_{\rm La_{2}Hf_{2}O_{7}}$ with $\alpha=0.8$ being the ratio of the neutron scattering cross sections of Pr2Hf2O7 and La2Hf2O7. The five CEF excitations near 9.1 meV, 56.5 meV, 82.2 meV, 94.5 meV and 109.5 meV are clearly visible in the $S_{\rm M}(Q,\omega)$ plots in Fig. 7.

The $S(Q,\omega)$ of La2Hf2O7 does not account well for the phonon background of Pr2Hf2O7, particularly at low $E$ the phonon contributions to the INS response of the two compounds are quite different. This aspect of the INS data is evident in the $\phi$-integrated energy cuts $S(\langle\phi\rangle,\omega)$ shown in Fig. 8(a). For a better estimate of the phonon background we adopted the approach suggested by Boothroyd et al. Boothroyd2003 . Accordingly the low-angle phonon background is estimated from the high-angle scattering of Pr2Hf2O7 itself where the magnetic scattering is negligibly small. For this the high-angle Pr2Hf2O7 spectrum was scaled by a factor equal to the ratio of the high- to low-angle La2Hf2O7 spectra. The scaled phonon background estimated this way from the Pr2Hf2O7 data in $119^{\circ}\leq\phi\leq 121^{\circ}$ is shown in Fig. 8(a). A substantial difference between the phonon background from the La2Hf2O7 spectrum and scaled high-angle Pr2Hf2O7 spectrum is obvious from Fig. 8(a). The $\phi$-integrated magnetic scattering $S_{\rm M}(\langle\phi\rangle,\omega)$ estimated using this scaled background is shown in Fig. 8(b). The corresponding $Q$ is obtained using the relation $\hbar^{2}Q^{2}/2m_{\rm n}=E_{i}+E_{f}-2(E_{i}E_{f})^{1/2}\cos\phi$, where $E_{f}=E_{i}-E$ and $m_{\rm n}$ is the rest mass of neutron.

In order to obtain quantitative information of the CEF states we analyzed the magnetic INS data using the software SPECTRE SPECTRE which allowed us to use the intermediate-coupling free ion basis states for diagonalizing the $H_{\rm CEF}$. We employed the complete set of 91 intermediate coupling basis states following the approach used for the related Pr-based system Pr2Sn2O7 Princep2013 . For least-square fitting of the observed INS excitations (energies and relative intensities) the CEF parameters of Pr2Sn2O7 were used as the starting parameters. The results of our analysis of INS data are summarized in Table 1. The fits of INS data are shown in Figs. 7(a), 7(b) and 8(b). We used a Lorentzian shape peak function for fitting the magnetic INS specta. Further details of our analysis can be found in our recent paper on Nd2Zr2O7 Xu2015 where we employed a similar approach for analyzing the INS data by the crystal field model.

The CEF parameters $B_{q}^{k}$ obtained from the analysis are: $B^{2}_{0}=33.8$ meV, $B^{4}_{0}=426.0$ meV, $B^{4}_{3}=192.7$ meV, $B^{6}_{0}=165.3$ meV, $B^{6}_{3}=-111.8$ meV, and $B^{6}_{6}=178.6$ meV. The CEF analysis of INS data reveal the ground state to be a doublet, first excited state a singlet at 8.82 meV, second excited state a doublet at 56.63 meV, third excited state a singlet at 82.11 meV, fourth excited state a doublet at 94.56 meV and fifth excited state a singlet at 109.46 meV. The CEF level scheme of Pr2Hf2O7 is found to be very similar to that of Pr2Sn2O7 Princep2013 and Pr2Zr2O7 Kimura2013 .

The wavefunction of the ground state non-Kramers doublet is found to be

[TABLE]

A significant mixing of $|^{3}H_{4},\pm 4\rangle$ with $|^{3}H_{4},m_{J}\neq\pm 4\rangle$ terms as well as with ${}^{1}G_{4}$ and ${}^{3}H_{5}$ is found in the ground state. Based on the CEF ground state wavefunction we estimate $g_{\parallel}=4.80$ and $g_{\perp}=0$ which are consistent with the values deduced from the magnetic measurements. The ground state wave function gives the magnetic moment of $2.4~{}\,\mu_{\rm B}$/Pr which is close to the effective moment obtained from the Curie-Weiss analysis of magnetic susceptibility.

The crystal field parameters obtained from the analysis of INS data are able to reproduce the powder dc susceptibility as well as the magnetic contribution to the heat capacity. The magnetic susceptibility calculated from these CEF parameters is shown in Fig. 2, which show an excellent agreement with the experimental data and thus supports the obtained CEF states and parameters. The anisotropic CEF suceptibility yields $\chi_{\parallel}/\chi_{\perp}\sim 45$ at 10 K where $\chi_{\parallel}$ is the susceptibility parallel to $\langle 111\rangle$ and $\chi_{\perp}$ is perpendicular to $\langle 111\rangle$. This value of $\chi_{\parallel}/\chi_{\perp}$ is very close to that of Pr2Sn2O7 for which $\chi_{\parallel}/\chi_{\perp}\sim 60$ at 10 K Princep2013 . The $\chi_{\perp}$ is purely Van-Vleck like therefore $\chi_{\parallel}/\chi_{\perp}$ increases very rapidly as $T\rightarrow 0$, e.g. at 3.5 K $\chi_{\parallel}/\chi_{\perp}\sim 120$. The $C_{\rm CEF}(T)$ calculated from the CEF parameters is shown in the lower inset of Fig. 5 (solid red line), in good agreement with $C_{\rm mag}(T)$ data, thus further supporting the deduced CEF level scheme. Our CEF level scheme (CEF parameters, eigenvalues and wavefunction) are in good agreement with the recent results by Sibille et al. Sibille2016 , though the values of observed CEF excitations and deduced $B_{q}^{k}$ parameters are slightly different.

VIII Single crystal Pr2Hf2O7

The ZFC $\chi(T)$ of single crystal Pr2Hf2O7 is shown in Fig. 9 for the $H$ applied along the three crystallographic directions [100], [110] and [111]. Consistent with the $\chi(T)$ of the polycrystalline sample the $\chi(T)$ of the single crystal does not show any anomaly. The $\chi(T)$ for $H$ along [100], [110] and [111] show very similar $T$ dependence with similar magnitude. In order to estimate the $\theta_{\rm p}$ and $\mu_{\rm eff}$ the $\chi(T)$ data at $12~{}{\rm K}\leq T\leq 25$ K were analyzed by modified Curie-Weiss behavior, the representative fit for [111] direction is shown in the inset of Fig. 9). The fit parameters are listed in Table 2. The $C$ values yield $\mu_{\rm eff}\approx 2.51\,\mu_{\rm B}$/Pr for the single crystal which, as expected, is very similar to that obtained for the polycrystalline sample.

The isothermal $M(H)$ data of single crystal Pr2Hf2O7 measured at 2 K are shown in Fig. 10. The $M(H)$ data reflect the Ising anisotropy. For a system with $\langle 111\rangle$ Ising anisotropy, the saturation magnetizations $M_{s}$ are given by Fukazawa2002 : $g^{\prime}J^{\prime}\mu_{\rm B}(1/\sqrt{3})$ for [100] (two-in/two-out), $g^{\prime}J^{\prime}\mu_{\rm B}(\sqrt{2/3}\times 2)/4$ for [110] (one-in/one-out, two free) and $g^{\prime}J^{\prime}\mu_{\rm B}(1+1/3\times 3)/4$ for [111] (three-in/one-out). The observed magnetizations at $H=14$ T are $M_{[100]}=1.46\,\mu_{\rm B}$/Pr, $M_{[110]}=1.21\,\mu_{\rm B}$/Pr and $M_{[111]}=1.34\,\mu_{\rm B}$/Pr. The observed anisotropic $M$ gives the ratio $M_{[100]}/M_{[110]}$ = 1.21 and $M_{[100]}/M_{[111]}$ = 1.09 which are in line with Ising anisotropy but somewhat smaller than the theoretically expected ratios of $\sqrt{2}$ and $2/\sqrt{3}$, respectively. With $g^{\prime}J^{\prime}\mu_{\rm B}\approx 2.51\,\mu_{\rm B}$, the effective moment, one would estimate $M_{[100]}=1.45\,\mu_{\rm B}$/Pr, $M_{[110]}=1.02\,\mu_{\rm B}$/Pr and $M_{[111]}=1.26\,\mu_{\rm B}$/Pr. Thus we see that while the observed $M_{[100]}$ and $M_{[111]}$ are close to their expected values, the observed $M_{[110]}$ is significantly higher than the expected value. The deviations from the Ising picture possibly indicates the presence of additional non-Ising terms in the Hamiltonian. Further we also notice that the $M$ is not saturated even at 14 T field which could be due to the mixing of ground state multiplet and/or non-Ising contributions.

IX Summary and Conclusions

We have synthesized polycrystalline and single crystal samples of the pyrohafnate compound Pr2Hf2O7 and investigated their structural and physical properties by means of synchrotron x-ray diffraction, dc magnetic susceptibility, isothermal magnetization, ac susceptibility, heat capacity and inelastic neutron scattering measurments. The high-resolution synchrotron x-ray diffraction revealed the well-ordered pyrochlore ($Fd\bar{3}m$) structure with Pr/Hf site mixing and/or oxygen deficiency (if present) at a level lower than 0.5%. Our ac susceptibility measurments down to 90 mK did not reveal any evidence for a magnetic transition to a long range ordered phase. The $\chi_{\rm ac}(T)$ is found to show a frequency dependent broad peak, a characteristic feature of spin-freezing in spin-ice systems, indicating slow spin dynamics in Pr2Hf2O7 at low temperature. The single-spin relaxation time for Pr2Hf2O7 is found to be very similar to that of the spin-ice Ho2Ti2O7.

The dc magnetic susceptibility and magnetization data show the Ising anisotropic features commonly observed in 227 pyrochlore spin-ice systems. The single crystal magnetic data clearly reveal this Ising anisotropic behavior, which is most obvious from the field dependence of directional dependent isothermal magnetization data. The inelastic neutron scattering data reveal five well defined magnetic excitations which are well accounted for by a model based on crystal electric field, accordingly we have determined the crystal field states of Pr3+. The CEF ground state is a well isolated non-Kramers doublet. The CEF suceptibility yields $\chi_{\parallel}/\chi_{\perp}\sim 120$ at 3.5 K revealing an Ising type anisotropy. The ground state wavefunction shows significant mixing of $|^{3}H_{4},\pm 4\rangle$ state with other multiplets. Such a mixing can cause quantum fluctuations. The single crystal $M(H)$ data reflect the presence of non-Ising contribution which are very favorable for quantum spin-ice behavior.

The key observations from our investigations are i) a weakly ferromagnetic interaction as inferred from the Weiss temperature, ii) crystal field ground state is a doublet well-separated from the first excited state, iii) low-$T$ magnetism described by an effective pseudospin-1/2 model, iv) anisotropic g-tensor, v) Ising-type anisotropy, and vi) slow spin dynamics. All these indicate that the magnetic ground state of Pr2Hf2O7 is consistent with spin-ice physics. In addition, there is some indication for the presence of a non-Ising contribution that would imply a dynamic spin-ice behavior instead of classical spin-ice behavior. The magnetic properties of Pr2Hf2O7 are very similar to those of the dynamic spin-ice systems Pr2Sn2O7 Zhou2008 ; Princep2013 and Pr2Zr2O7 Kimura2013 . Recent investigations by Sibille et al. [29] also concludes in favor of a quantum spin-ice behavior in Pr2Hf2O7 where they also implied low energy inelastic neutron scattering in addtion to ac susceptibility for the study of spin dynamics in this compound. Further investigations are required to explore the quantum spin-ice ground state of Pr2Hf2O7.

Acknowledgements.

We acknowledge Helmholtz Gemeinschaft for funding via the Helmholtz Virtual Institute (Project No. VH-VI-521) and DFG through Research Training Group GRK 1621 and SFB 1143. We also acknowledge the support of the HLD-HZDR, a member of the European Magnetic Field Laboratory (EMFL).