Slow spin tunneling in the paramagnetic phase of the pyrochlore Nd2Sn2O7

TL;DR

This study reveals extremely slow, temperature-independent spin dynamics in Nd2Sn2O7's paramagnetic phase, suggesting a double spin-flip tunneling mechanism, contrasting with faster high-temperature relaxation processes.

Contribution

It demonstrates the existence of microsecond-range, slow spin tunneling dynamics in Nd2Sn2O7's paramagnetic phase, extending understanding of spin relaxation mechanisms in pyrochlore compounds.

Findings

Spin dynamics are extremely slow and temperature independent between 1.7 and 7 K.

The relaxation mechanism likely involves double spin-flip tunneling.

Field distribution at the muon site increases as temperature decreases.

Abstract

The insulating pyrochlore compound Nd2Sn2O7 has been shown to undergo a second order magnetic phase transition at Tc ~ 0.91 K to a noncoplanar all-in--all-out magnetic structure of the Nd3+ magnetic moments. An anomalously slow paramagnetic spin dynamics has been evidenced from neutron backscattering and muon spin relaxation (muSR). In the case of muSR this has been revealed through the strong effect of a 50 mT longitudinal field on the spin-lattice relaxation rate. Here, motivated by a recent successful work performed for Yb2Ti2O7 and Yb2Sn2O7, analyzing the shape of the muSR longitudinal polarization function, we substantiate the existence of extremely slow paramagnetic spin dynamics in the microsecond time range for Nd2Sn2O7. Between 1.7 and 7 K, this time scale is temperature independent. This suggests a double spin-flip tunneling relaxation mechanism to be at play, probably involving spin substructures such as tetrahedra. Unexpectedly, the standard deviation of the field distribution at the muon site increases as the system is cooled. This exotic spin dynamics is in sharp contrast with the dynamics above 100 K which is driven by the Orbach relaxation mechanism involving single Nd3+ magnetic moments.

Equations

$$A^{\mathrm{LF}}(t)=A_0^{\mathrm{LF}}\left[(1-F_{\mathrm{bg}})P_{Z,\mathrm{NSO}}(t)+F_{\mathrm{bg}}{P}_{Z,\mathrm{bg}}(t)\right],$$

Full text

I Introduction

Exotic magnetic fluctuations and correlations are expected to be observed for geometrically frustrated magnetic materials Moessner and Ramirez (2006); Gardner et al. (2010); Balents (2010); Lacroix et al. (2011); Gingras and McClarty (2014). Probably, the best documented experimental example is given by the ordered state of the pyrochlore insulator compound Tb2Sn2O7 for which signatures of unconventional fluctuations have been found from muon spin relaxation ($\mu$SR), different types of neutron scattering techniques and specific heat data Dalmas de Réotier et al. (2006); Bert et al. (2006); Chapuis et al. (2007); Rule et al. (2007); Mirebeau et al. (2008); Rule et al. (2009); Bonville (2010); Dalmas de Réotier et al. (2016). The existence of unexpected short-range correlations has been pointed out for the ordered state of the pyrochlore insulators Er2Ti2O7 Ruff et al. (2008), Yb2Ti2O7 Bonville et al. (2003); Ross et al. (2011a); Maisuradze et al. (2015), Yb2Sn2O7 Maisuradze et al. (2015) and the triangular system La2Ca2MnO7 Dalmas de Réotier et al. (2015). Exotic fluctuations have been discovered for cooperative paramagnets such as the spin-1/2 kagome lattice herbertsmithite ZnCu3(OH)6Cl2 Fu et al. (2015), the pyrochlore insulator Tb2Ti2O7 Ueland et al. (2006) and the triangular system NiGa2S4 Yaouanc et al. (2008); Nambu et al. (2015). A large range of fluctuation rates is usually observed, as documented in Ref. Nambu et al., 2015. However, the number of compounds which display magnetic ordering at low temperature and unconventional paramagnetic fluctuations is still restricted. Recently, the normal spinel CdHo2S4 Yaouanc et al. (2015), and the pyrochlore insulators Yb2Ti2O7, Yb2Sn2O7, Nd2Sn2O7, Nd2Zr2O7, and Er2Ti2O7 have been shown to belong to this family of compounds Maisuradze et al. (2015); Bertin et al. (2015); Xu et al. (2016); Orendáč et al. (2016).

Anomalously slow paramagnetic fluctuations with a correlation time roughly in the 100 ns time range have been reported for the cubic pyrochlore stannate Nd2Sn2O7 Bertin et al. (2015). They were unraveled through the study of the influence of an external magnetic field of 50 mT on the $\mu$SR spin-lattice relaxation rate. It has been shown recently that the study of the shape of the $\mu$SR longitudinal polarization function can be a very effective method to detect anomalously slow paramagnetic fluctuations Maisuradze et al. (2015). Motivated by this result, we have performed a low field $\mu$SR study of the paramagnetic state of Nd2Sn2O7 to further characterize its spin dynamics. Here we report on this detailed study.

Nd2Sn2O7 crystallizes in the pyrochlore structure in which the rare earth ions form a network of corner sharing regular tetrahedra. It exhibits a magnetic phase transition at $T_{\rm c}\approx 0.91$ K to a so-called all-in–all-out magnetic structure: the magnetic moment of an ion is collinear to the direction linking the corner at which the ion sits to the tetrahedron center and all four moments of a given tetrahedron point either inwards or outwards. A $\mu$SR spontaneous field has been observed, consistent with the lack of the divergence-free part of the Helmholtz decomposition of the magnetic-moment field for such a magnetic structure Brooks-Bartlett et al. (2014). In this frame, the long-range order is associated with the divergence-full component of the field. Persistent spin dynamics below $T_{\rm c}$ and anomalously slow paramagnetic fluctuations up to $\approx 30\,T_{\rm c}$ have been detected.

We shall first describe the experimental and data analysis methods in Sec. II. Our experimental results will be presented in Sec. III and discussed in Sec. IV. Finally, in Sec. V conclusions are gathered.

II Experimental and data analysis

The measurements were performed with a powder sample previously used for a $\mu$SR work Bertin et al. (2015). The asymmetry spectra were recorded at the MuSR and EMU spectrometers of the ISIS facility (Rutherford Appleton Laboratory). The sample was mixed with a small fraction of General Electric varnish and deposited to a silver plate.

The longitudinal-field geometry was adopted, for which an external field ${\bf B}_{\rm ext}$, if any, is set along the initial muon beam polarization ${\bf S}_{\mu}$ Yaouanc and Dalmas de Réotier (2011). By definition, ${\bf B}_{\rm ext}$ is applied along the $Z$ axis of the laboratory reference frame.

The measured longitudinal-field (LF) asymmetry spectra will be analyzed with the two-component model

[TABLE]

where $A^{\rm LF}_{0}$ is the initial $\mu$SR asymmetry. The longitudinal polarization function $P_{Z,{\rm NSO}}(t)$ describes the evolution of the $Z$ component of the muon spin ensemble Yaouanc and Dalmas de Réotier (2011). $F_{\rm bg}$ is the fraction of muons stopped outside of the sample, mainly in the silver backing plate and possibly in the cryostat window or walls. A weakly damped background function is required to account for the spectra at low fields. We find that $P_{Z,{\rm bg}}(t)$ can be taken as the so-called static Kubo-Toyabe (KT) function Hayano et al. (1979).

The simplest possible model for $P_{Z,{\rm NSO}}(t)$ is also the KT model. It depends on the mean field $B_{\rm LF}$ and the standard deviation $\Delta$ of the field distribution at the muon site assumed to be Gaussian and isotropic. Spin dynamics is described with the strong collision model Kehr et al. (1978) which requires the introduction of a field fluctuation frequency, $\nu_{\rm c}=1/\tau_{\rm c}$. Here $\tau_{\rm c}$ is the characteristic decay time of the correlation function of the field experienced by the muons. We have $P_{Z,{\rm NSO}}(t)=P_{Z,{\rm KT}}(t,B_{\rm LF},\Delta,\nu_{\rm c})$ Yaouanc and Dalmas de Réotier (2011).

III Experimental results

Examples of asymmetry spectra recorded with the MuSR spectrometer at four temperatures and the related fitting curves are presented in Fig. 1.

In contrast to Yb2Sn2O7 and Yb2Ti2O7 Maisuradze et al. (2015), the asymmetry monotonically decays with time at all fields for the four temperatures. So the signature of quasi-static fluctuations is not so obvious graphically as it was for the ytterbium compounds. However, a careful inspection of the spectra reveals that some of them depart from an exponential or stretched-exponential decay, i.e. the spin dynamics they probe is quasi-static. For instance, consider the spectra at 2.1 K. A break in the slope around $0.5\ \mu{\rm s}$ is found for the spectra with $B_{\rm ext}=5$ mT and nearby values.

The curves in Fig. 1 result from fits with Eq. 1. The fits were performed with common values for $F_{\rm bg}$ and the field width $\delta_{\rm bg}$ associated with $P_{Z,{\rm bg}}(t)$. Parameters $\Delta$ and $\nu_{c}$ tend to increase with $B_{\rm ext}$, in a manner reminiscent to what was observed for Yb2Ti2O7 and Yb2Sn2O7 Maisuradze et al. (2015). The overall quality of the fits is extremely good, with the exception of the low field spectra at the lowest temperature. We will return to this point at the end of Sect. III.

To ascertain the relevance of the observed deviations from the dynamical KT model as the sample is cooled down, we have extended the zero-field measurements to two lower temperatures. An example is presented in Fig. 2.

The dashed line clearly shows that the conventional Kubo-Toyabe function no longer provides an adequate fit to the data. Instead, the full line provides a remarkable fit. Here the dynamical KT function is replaced by an equal-weight sum of two polarization functions for which the KT Gaussian field distribution is extended to account for short-range correlations (SRC) Yaouanc et al. (2013a); Maisuradze et al. (2015). Alternative fits with either a weighted sum of two dynamical KT functions or a single SRC function still provide reasonable descriptions of the data (not shown). We cannot select the suitable model with the available information. However, the value of $\nu_{\rm c}$ extracted from the three models is the same within statistical uncertainties.

For a better determination of the observed deviations from the conventional KT model seen in ZF, we have performed ZF measurements with high statistics — about $80\times 10^{6}$ decay positrons recorded per spectrum — at the EMU spectrometer. The data were recorded at 7 K and below where the spectra substantially deviate from an exponential-like relaxation. The results are shown in Fig. 3.

A successful combined fit of the eight spectra has been done with a weighted sum of two dynamical KT functions, i.e. $P_{Z,{\rm NSO}}(t)=\sum_{i=1}^{2}F_{i}\,P_{Z,{\rm KT}}(t,B_{\rm LF}=0,\Delta_{i},\nu_{c})$ with $F_{1}+F_{2}=1$. This model requires a minimum number of free parameters. For the background we have $F_{\rm bg}=0.366\,(1)$ and $\delta_{\rm bg}=0.041\,(1)$ mT. These values are in the expected range for the background signal. The relative weight of the first component is $F_{1}=0.62\,(1)$. We have assumed $\nu_{\rm c}$ to be the same for the two relaxation functions and a common ratio for the two standard deviations for the eight spectra. The value for the ratio $\Delta_{2}/\Delta_{1}=2.11\,(2)$ was set from a prior fit with independent values of $\Delta_{1}$ and $\Delta_{2}$. The thermal dependences of the remaining free parameters, namely $\nu_{\rm c}$, $\Delta_{1}$ and $A^{\rm LF}_{0}$, are displayed in Fig. 4.

Within experimental uncertainties, $A^{\rm LF}_{0}$ is temperature independent, as it should. We observe a pronounced increase of $\Delta_{1}$ as the sample is cooled down, and this parameter is much smaller than one would have expected for the standard deviation of a field distribution arising from Nd3+ magnetic moments. The spin dynamics characterized by $\nu_{\rm c}$ is anomalously slow. Based on the Curie-Weiss temperature $\Theta_{\rm CW}=-0.32\,(1)$ K Bertin et al. (2015) of Nd2Sn2O7, we would have expected $\nu_{\rm c}$ of order $k_{\rm B}|\Theta_{\rm CW}|/\hbar\approx 4\times 10^{10}$ s*-1*, i.e. more than four orders of magnitude larger than observed. Here $k_{\rm B}$ and $\hbar$ are the Boltzmann and Dirac constants, respectively. The extremely small $\nu_{\rm c}$ value previously roughly inferred from the anomalous field dependence of the $\mu$SR relaxation rate Bertin et al. (2015) is therefore confirmed. The temperature independence of $\nu_{\rm c}$ (Fig. 4) is another remarkable feature. On the approach of a phase transition, but still outside the critical regime, it is expected that $\nu_{c}(T)\propto(T-T_{\rm c})$, $T_{\rm c}$ being a Curie or Néel temperature Moriya (1962); de Gennes and Villain (1960), at odds from our result.

The two-component model just used for the high statistics spectra allows us to satisfactorily fit the whole set of data shown in Fig. 1, including the low field spectra at 2.1 K. Taking the example of the zero-field spectrum recorded at 2.1 K, the confidence parameter $\chi^{2}$ changes from 1.33 to 0.97 for the one and two-component model, respectively. Figure 5 displays $\Delta_{1}(B_{\rm ext})$. The field width $\Delta_{1}$ increases with $B_{\rm ext}$, especially at low temperature similarly to what was observed in the ytterbium titanate and stannate Maisuradze et al. (2015).

Going back to the thermal dependence of the high statistics spectra, the thermal behavior of the two parameters characterizing the muon spin relaxation is exotic. The signature of the approach to the second-order phase transition is seen in $\Delta_{1}(T)$ rather than in $\nu_{\rm c}(T)$. This is further discussed in the next section, together with the $\Delta_{1}$ and $\nu_{\rm c}$ values. Before that, it is interesting to notice that these results are partly reminiscent of those recently obtained for Yb2Ti2O7 and Yb2Sn2O7 Maisuradze et al. (2015). These compounds display splayed ferromagnetic ground states Yaouanc et al. (2013b, 2016), completely different from the antiferromagnetic ground state of Nd2Sn2O7. In addition, they are characterized by first order magnetic phase transitions, again in contrast to Nd2Sn2O7. This suggests that the exotic paramagnetic fluctuations we observe are independent of the type of magnetic ordering at low temperature. They rather seem to reflect the frustrated magnetic interactions present in these rare-earth pyrochlore compounds.

IV Discussion

The existence of two relaxation channels has recently been inferred from ac susceptibility measurements at low temperature for Er2Ti2O7 Orendáč et al. (2016): an Orbach relaxation with a remarkable small activation energy and an attempt time much longer than expected, and a temperature independent relaxation. Here we shall discuss our results in relation to the interpretation of the Er2Ti2O7 susceptibility data. We shall first examine the possible reason for the absence of the first relaxation mechanism in the $\mu$SR data and consider the $\Delta_{1}$ and $\nu_{\rm c}$ parameters in turn.

The traditional Orbach process involves two real phonons with an excited crystal-electric-field (CEF) level as intermediate Orbach (1961). The spin-lattice relaxation measured by $\mu$SR above $\approx 100$ K in Nd2Sn2O7 is well accounted for with an activation energy $\Delta_{\rm CEF}$ = 39.8 meV, i.e. the energy of the third excited CEF energy level Bertin (2015); see Fig. 6. Similarly, an Orbach process explains the spin dynamics of the geometrically frustrated garnet Yb3Ga5O12 above 100 K Dalmas de Réotier et al. (2003) and also for Yb2Ti2O7 Dalmas de Réotier et al. (2004a) and Nd2Zr2O7 Xu et al. (2016). The scheme of the CEF levels of Nd3+ ions in Nd2Sn2O7 cannot explain our experimental findings at low temperature since the first excited CEF doublet is located at 26 meV above the ground state Bertin (2015).

The same energy misfit exists for Er2Ti2O7. For this compound it has been advocated that the four spins sitting at the corner of a tetrahedron rather than single spins are the building blocks which generate the relaxation measured with ac susceptibility Orendáč et al. (2016). This picture explains qualitatively the measured activation energy, and therefore justifies the existence of an Orbach relaxation, although with an attempt time much longer than expected. Its renormalization has been attributed to the bottleneck effect Abragam and Bleaney (1970). To understand the experimental data for Er2Ti2O7 a second temperature independent relaxation had to be added. This thermal behavior suggests a spin tunneling relaxation mechanism. The energy level scheme for a tetrahedron might support it through double spin-flip relaxation processes Bloembergen et al. (1959).

A priori the same picture should apply to Nd2Sn2O7. However, while the temperature independent relaxation channel has been found, an Orbach relaxation has not been detected. Recall that when two relaxation mechanisms with vastly different fluctuation frequencies are present, only the mechanism falling into the technique time window should drive the relaxation. An Orbach relaxation with an attempt time in the range of a second and an activation energy of $\approx 1$ meV as for Er2Ti2O7 Orendáč et al. (2016) cannot contribute to the $\mu$SR relaxation. We propose this is also the case for Nd2Sn2O7.

So we are left with the discussion of the characteristic parameters $\Delta_{1}$ and $\nu_{\rm c}$ for the remaining relaxation channel. The standard deviation $\Delta_{1}$ is much smaller than expected if there were no correlation between the spins. Indeed, following the methodology developed by van Vleck in the absence of correlations Van Vleck (1948), a field width of the order of a few hundred milliteslas is computed for different possible muon sites. Going further than this qualitative remark would require to model the static wavevector-dependent susceptibility and to know the muon site Dalmas de Réotier et al. (2004b). Concerning the fluctuation rate, we measure $\nu_{\rm c}\simeq 1~{}\mu{\rm s}^{-1}$ while for Er2Ti2O7 the value derived from ac susceptibility is approximately four orders of magnitude smaller. Since there is no detailed prediction available for the double spin-flip relaxation process, we cannot give a quantitative discussion and compare the results for Nd2Sn2O7 and Er2Ti2O7 in terms of their basic physical properties.

V Conclusions

The reported $\mu$SR measurements in the low temperature range of the paramagnetic phase of Nd2Sn2O7 point to the presence of anomalously slow spin tunneling dynamics. The magnetic fluctuation frequency is unexpectedly almost temperature independent within a relatively broad temperature range, while the standard field distribution at the muon site is anomalously small and reduces with increasing temperature.

These results suggest to focus attention on coupled spins within tetrahedra rather than to ionic spins. This is in sharp contrast to the dynamics above 100 K which only involves ionic spins.

Similarly, tetrahedra spin structures should also influence the low temperature properties of other pyrochlore compounds such as Yb2Sn2O7 and Yb2Ti2O7 for which a detailed study has recently been published Maisuradze et al. (2015). Interestingly, the physical properties of tetrahedra of spins may depend on characteristics of the fictitious spins 1/2. The time reversal symmetry property of the ground state doublets, i.e. Kramers vs non Kramers, and their spatial symmetry, i.e. vector-like or multipole-like, are expected to play key roles Curnoe (2007); Onoda and Tanaka (2010, 2011, 2011); Ross et al. (2011b); Huang et al. (2014). Simple theoretical methods are available for the computation of the generalized susceptibility for such reduced Hilbert space systems Becker et al. (1977); Richards (1984); Yaouanc (2009); Yaouanc and Dalmas de Réotier (2011). An exhaustive theoretical study of that susceptibility for different types of fictitious spins 1/2 on a tetrahedron might be of great help for the interpretation of a large spectrum of data ranging from specific heat and ac susceptibility to neutron scattering and $\mu$SR.

Acknowledgements.

This research project has been partially supported by the European Commission under the 7th Framework Programme through the ‘Research Infrastructures’ action of the ‘Capacities’ Programme, Contract No: CP-CSA_INFRA-2008-1.1.1 Number 226507-NMI3. We thank STFC for the provision of muon beamtime at the ISIS Facility, UK. PDR thanks ISIS for financial support.