Long range order in the dipolar XY antiferromagnet Er$_2$Sn$_2$O$_7$

TL;DR

This study demonstrates a second order transition to a long-range Palmer-Chalker antiferromagnetic state in Er$_2$Sn$_2$O$_7$, confirming its status as a dipolar XY pyrochlore antiferromagnet with complex fluctuation dynamics.

Contribution

First experimental observation of a second order transition to Palmer-Chalker order in Er$_2$Sn$_2$O$_7$, clarifying its magnetic ground state and the role of fluctuations.

Findings

Observation of a second order transition at 108 mK.

Detection of a gap opening in magnetic excitations.

Confirmation of dipolar XY pyrochlore antiferromagnet behavior.

Abstract

Er$_2$Sn$_2$O$_7$ remains a puzzling case among the extensively studied frustrated compounds of the rare-earth pyrochlore family. Indeed, while a first order transition towards a long-range antiferromagnetic state with the so-called Palmer-Chalker structure is theoretically predicted, it has not been observed yet, leaving the issue, as to whether it is a spin-liquid candidate, open. We report on neutron scattering and magnetization measurements which evidence a second order transition towards this Palmer-Chalker ordered state around 108 mK. Extreme care was taken to ensure a proper thermalization of the sample, which has proved to be crucial to successfully observe the magnetic Bragg peaks. At the transition, a gap opens in the excitations, superimposed on a strong quasielastic signal. The exchange parameters, refined from a spin wave analysis in applied magnetic field, confirm that Er$_2$Sn$_2$O$_7$ is a realization of the dipolar XY pyrochlore antiferromagnet. The proximity of competing phases and the strong XY anisotropy of the Er$^{3+}$ magnetic moment might be at the origin of enhanced fluctuations, leading to the unexpected nature of the transition, the low ordering temperature, and the observed multi-scale dynamics.

Tables

Table 1. TABLE S 1: Coordinates, written in the cubic F ​ d ​ 3 ¯ ​ m 𝐹 𝑑 ¯ 3 𝑚 Fd\bar{3}m structure of the pyrochlore lattice, of the site dependent local 𝒂 i subscript 𝒂 𝑖 {\boldsymbol{a}_{i}} and 𝒃 i subscript 𝒃 𝑖 {\boldsymbol{b}_{i}} vectors spanning the local XY anisotropy planes. The CEF axes 𝒛 i subscript 𝒛 𝑖 {\boldsymbol{z}_{i}} of the rare earth is perpendicular to those planes.

Site	1	2	3	4
CEF axis ${𝒛}_i$	$(1,1,\overline{1})$	$(\overline{1},\overline{1},\overline{1})$	$(\overline{1},1,1)$	$(1,\overline{1},1)$
Coordinates	$(1/4,1/4,1/2)$	$(0,0,1/2)$	$(0,1/4,3/4)$	$(1/4,0,3/4)$
${𝒂}_i$	$(\overline{1},\overline{1},\overline{2})$	$(1,1,\overline{2})$	$(1,\overline{1},2)$	$(\overline{1},1,2)$
${𝒃}_i$	$(\overline{1},1,0)$	$(1,\overline{1},0)$	$(1,1,0)$	$(\overline{1},\overline{1},0)$

Equations

$${\mathbf{J}}_i\tilde{\mathcal{J}}{\mathbf{J}}_j=$$

$$+$$

$${\mathbf{S}}_i={\mathbf{z}}_i$$

$${\mathbf{S}}_i=\cos \varphi {\mathbf{a}}_i+\sin \varphi {\mathbf{b}}_i.$$

$${\mathbf{S}}_{1,4}$$

$${\mathbf{S}}_{2,3}$$

$${\mathbf{S}}_{1,2}$$

$${\mathbf{S}}_{3,4}$$

Full text

Frustration in magnetism is usually characterized by the inability of a system to condense into an ordered state, even well below the temperature range of the magnetic interactions Lacroix . This reflects the presence, at the classical level, of a large ground state degeneracy, which prevents the system from choosing a unique ground state. Nevertheless, the system may eventually order owing to the presence of additional terms in the Hamiltonian, like second neighbor or Dzyaloshinski-Moriya interactions, or owing to “order by disorder” phenomena Villain80 ; Shender82 , which lift the degeneracy and stabilize a unique ordered state. Conversely, fluctuations, originating for example from the proximity of competing phases, can hinder magnetic ordering, resulting in an unconventional correlated state with exotic excitations.

The pyrochlore oxide Er2Sn2O7 appears to belong to this category. In this compound, the Er3+ magnetic moments reside on the vertices of a lattice made of corner sharing tetrahedra, and are confined by a strong XY anisotropy within local planes, perpendicular to the $\langle 111\rangle$ axes. The magnetic interactions are found to be governed by dipolar interactions, in addition to a quasi-isotropic antiferromagnetic exchange tensor Guitteny13 . Mean field calculations show that, in the $T=0$ phase diagram, this interaction tensor locates Er2Sn2O7 in an antiferromagnetic phase, called “Palmer-Chalker” phase Palmer00 (see Figure 1), close to the boundary with another ordered phase, the so-called $\psi_{2}$ phase, which is realized in the related compound Er2Ti2O7 Poole07 . The predicted Néel temperature is about $T_{\rm N}^{\rm MF}\approx 1.3$ K. Monte-Carlo simulations Yan13 have pointed out that fluctuations tend to lower the ordering temperature, with $T_{\rm N}^{\rm MC}\approx 200$ mK.

Experimentally, no phase transition has been detected in Er2Sn2O7 down to 20 mK in muon spin relaxation measurements Lago05 and down to about 100 mK in magnetization Matsuhira02 ; Guitteny13 and neutron scattering experiments Sarte11 ; Bramwell04 . Yet Palmer-Chalker like short-range correlations have been reported below 5 K, taking the form of a broad diffuse quasielastic signal in neutron scattering measurements Sarte11 ; Guitteny13 . In this letter, using neutron diffraction and magnetization measurements, we show that Er2Sn2O7 does order in the Palmer-Chalker state at a Néel temperature $T_{\rm N}\approx 108$ mK. This long range ordering is characterized by magnetic Bragg peaks which develop on top of the broad diffuse scattering. The latter disappears progressively as the ordered magnetic moment increases, indicating a second order transition. Concomitantly, the slow dynamics previously observed in ac susceptibility above the magnetic transition persist at low temperature and coexist with the Palmer-Chalker ordering. In addition, on entering the ordered phase, inelastic neutron scattering (INS) experiments reveal new features in the spin excitation spectrum stemming from the magnetic Bragg peaks. Using INS measurements performed in applied magnetic field, we determine the exchange parameters of a model Hamiltonian, confirming previous estimations.

Magnetization and ac susceptibility measurements were performed on a SQUID magnetometer equipped with a dilution refrigerator developed at the Institut Néel Paulsen01 . Neutron diffraction experiments were performed on the G4.1 diffractometer (LLB-Orphée facility) with $\lambda=2.426$ Å. INS measurements were carried out on the triple axis spectrometer 4F2 using a final wavevector $k_{f}=1.15$ Å*-1*. The energy resolution was about 70 $\mu$eV. The sample was a pure polycrystalline Er2Sn2O7 compound, synthesized by a solid state reaction from a stoichiometric mixture of Er2O3 (99.9%) and SnO2 (99.996%). The powder was ground and heated for 6–8 h four times from 1400 to 1450*∘*C in air, cooled down to room temperature, and reground after each calcination.

An important issue regarding the measurements at very low temperature concerns the thermalization of the powder sample. For magnetization measurements, a few milligrams of Er2Sn2O7 were mixed with apiezon N grease in a copper pouch, to improve the thermal contact and reduce the thermalization time. For neutron scattering experiments, a dedicated vanadium cell was used in the dilution fridge. The cell was filled with He gas up to 40 bars. During the experiments, it became obvious that a non negligible heating was induced by the sample’s activation in the neutron beam. This effect was all the more important when the incident flux was large. Special care was then taken to minimize this effect for a better temperature control. For INS measurements, the neutron flux was reduced with a lead attenuator. For diffraction measurements, short counting times (of about 30 min) were programmed, alternating with deactivation (hence cooling) periods of 1 h. This thermalization issue probably explains why the transition had not been reported in previous neutron scattering measurements.

Zero field cooled - field cooled (ZFC-FC) magnetization measurements show an antiferromagnetic transition at $T_{\rm N}=108\pm 5$ mK (see inset of Figure 2). It manifests as a peak in the ZFC curve, while the FC magnetization sharply increases at the transition. This effect, although less pronounced, is similar to what is observed in Er2Ti2O7 Petrenko11 and could be due to the polarization of uncompensated magnetic moments at domain boundaries. This magnetic transition is also evidenced by a peak at $T_{\rm N}$ in the real part of the ac susceptibility (see top of Figure 2). A frequency dependent signal, a signature of slow dynamics, comes on top of this peak, as a bump which moves towards high temperature when the frequency increases. It is associated with a peak in the imaginary part of the ac susceptibility, which follows the same thermal activated law above and below the Néel temperature (see bottom inset of Figure 2), showing that the slow dynamics is not affected by the transition.

Neutron diffraction confirms the presence of antiferromagnetic ordering at very low temperature, as shown in Figure 3. Intensity increases on existing crystalline Bragg peak positions, indicating a ${\bf k}={\bf 0}$ propagation vector. The main magnetic peaks appear at $1.07$ and $1.22$ Å*-1*, corresponding to the $Q=(111)$ and $(002)$ wavevectors, and emerge from the diffuse scattering signal characteristic of short-range Palmer-Chalker correlations Sarte11 ; Guitteny13 . The latter has almost disappeared at the lowest temperature ($T=68$ mK), indicating that most of the magnetic moment is ordered (see Figure 3(a)). Rietveld refinements show that, among the possible irreducible representations authorized by the ${\bf k}={\bf 0}$ propagation vector Wills06 ; Poole07 , neutron intensities can only be properly modeled by the $\Gamma_{7}$-Palmer-Chalker representation supmat . At the lowest temperature, the ordered moment is $3.1~{}\mu_{\rm B}$, i.e. about 80 % of the magnetic moment of the Er3+ ground state doublet, estimated at $3.8~{}\mu_{\rm B}$ Guitteny13 (see Figure 3(b)). The remaining Er3+ moments are embedded in short-range only Palmer-Chalker correlations, as reflected by the persistence of a weak diffuse signal. This ordered moment value is consistent with what is expected in a conventional second order phase transition at the corresponding $T/T_{\rm N}$ ratio. The second order nature is further confirmed by the gradual increase of the ordered moment below $T_{\rm N}$ (see inset of Figure 3(b)).

We now focus on the excitation spectrum associated with this magnetic ordering. As previously reported, at temperatures as high as 10 K, the INS spectrum includes a strong quasielastic signal Sarte11 . Its intensity is stronger around $Q_{o}=1.1$ Å*-1*, which corresponds to the position of the maximum intensity of the Palmer-Chalker diffuse scattering (see Figure 4(a)). On cooling, the intensity increases, while the width decreases, corresponding to a slowing down of the fluctuations, reaching a characteristic time of about $10^{-11}$ s close to $T_{\rm N}$. This quasielastic signal persists down to temperatures below $T_{\rm N}$ (see Figure 4(b)). Nevertheless, because of self-heating in the neutron beam, the effective temperature actually reached by the sample at the lowest temperature of the dilution fridge ($T_{\rm min}=70$ mK), is estimated to be 105 mK from the intensity of the magnetic Bragg peaks, thus just below $T_{\rm N}$ (corresponding to an ordered moment of $0.8~{}\mu_{\rm B}$). Additional features, stemming from the Bragg peaks at $Q\approx 1$ and $1.2$ Å*-1*, also arise in the ordered regime (see Figure 4(b)). In the cut shown in Figure 4(d), this additional signal manifests as a broad band in the $0.2\leq\omega\leq 0.5$ meV range, superimposed on the quasielastic response.

To further analyze these results, we have refined the exchange parameters previously estimated from magnetization measurements Guitteny13 . We have used the well-documented procedure Savary12 ; Ross11 , which consists in applying a magnetic field to drive the ground state towards a field polarized state and analyzing the spin excitations in terms of conventional spin waves. Owing to the polycrystalline nature of the Er2Sn2O7 sample, the magnetic field is applied simultaneously in all crystallographic directions, leading to an average excitation spectrum. INS measurements were performed as a function of magnetic field at 1.5 K. Above about $\mu_{o}H=$ 1.5 T, the response is no longer quasielastic-like, and a dispersive spectrum is observed, with the opening of a gap (see Figure 5(a)). Its value is $\Delta=0.26\pm 0.04$ meV at $Q_{o}=1.1$ Å*-1* for a field of $\mu_{o}H=2$ T. $\Delta$ increases roughly linearly further increasing the field, as illustrated in Figure 5(c).

To determine the exchange parameters, we consider the following Hamiltonian: ${\cal H}={\cal H}_{\rm CEF}+\frac{1}{2}\sum_{<i,j>}{\bf J}_{i}{\cal\tilde{J}}{\bf J}_{j}+g_{\rm J}\mu_{\rm B}{\bf H}.{\bf J}_{i}$, where $\bf{J}$ is the Er3+  magnetic moment, ${\cal H}_{\rm CEF}$ is the crystal electric field (CEF) Hamiltonian, $\bf{H}$ denotes the magnetic field, $g_{\rm J}$ is the effective $g$ factor, and ${\cal\tilde{J}}$ is the anisotropic exchange tensor, which incorporates the dipolar interaction truncated to its nearest neighbors. It is written in the $(\bf{a}_{ij},\bf{b}_{ij},\bf{c}_{ij})$ frame linked with Er3+-Er3+$\langle ij\rangle$ nearest neighbors bonds Malkin10 :

[TABLE]

where $D_{\rm nn}=\dfrac{\mu_{o}}{4\pi}\dfrac{(g_{\rm J}\mu_{\rm B})^{2}}{r^{3}_{\rm nn}}=0.022$ K is the pseudo-dipolar contribution (with $r_{\rm nn}$ the nearest neighbor distance in the pyrochlore lattice), ${\cal J}_{a}$, ${\cal J}_{b}$, ${\cal J}_{c}$ are effective exchange parameters, and ${\cal J}_{4}$ corresponds to the antisymmetric Dzyaloshinski-Moriya interaction. This model takes into account the specific CEF scheme which strongly confines the spins within the local XY planes Guitteny13 . The powder average of the spin excitation spectrum $S(Q,\omega)_{H}$ is then calculated for a given field ${\bf H}$ using the random phase approximation (RPA).

To compare with the polycrystalline experimental data shown in Figure 5(a), the average over all field directions has to be performed. As a first approximation, we consider that the powder spectrum can be described by averaging over the high symmetry directions of the system, $\langle 001\rangle$, $\langle 110\rangle$ and $\langle 111\rangle$, taking into account their multiplicity: $\overline{S(Q,\omega)}=\frac{1}{13}(3S(Q,\omega)_{{\bf H}\parallel[001]}+6S(Q,\omega)_{{\bf H}\parallel[110]}+4S(Q,\omega)_{{\bf H}\parallel[111]})$. Assuming $J_{4}=0$, since the Dzyaloshinski-Moriya is expected to be small compared to symmetric exchange, the best agreement is obtained with ${\cal J}_{a}={\cal J}_{b}=0.03\pm 0.003$ K, ${\cal J}_{c}=0.05\pm 0.01~{}$K, thus consistent with the uncertainty range given in Ref. Guitteny13, . Calculations reproduce the strong intensity close to $Q_{o}$ and the presence of a gap of about 0.25 meV (see Figure 5(b)). They also account for the field dependence of the gap at $Q_{o}$ above 1.5 T, as shown in Figure 5(c) by the blue open circles. RPA calculations performed with these exchange parameters at $T=0$ in zero field, predict a gapped flat mode at 0.2 meV, together with excitations dispersing up to 0.4 meV (see Figure 4(c)). While the value of the gap is consistent with the measured inelastic contribution, the predicted dispersion is hardly distinguishable in the data of Figure 4(b).

It is instructive to compare the Er2Sn2O7 behavior with results obtained on Gd2Sn2O7. The latter, in which the Gd3+ magnetic moment is almost isotropic, is known as the archetype of the dipolar Heisenberg pyrochlore antiferromagnet. In spite of very different anisotropies in both compounds, calculations predict identical behaviors, for thermodynamic and dynamical properties: the transition towards the Palmer-Chalker phase is expected to be first order Cepas05 ; Yan13 , while a spin wave spectrum, similar to Figure 4(c), should develop at low temperature, consisting in dispersive branches on top of a gapped flat mode DelMaestro04 ; DelMaestro07 ; Sosin09 . Gd2Sn2O7 indeed follows these predictions Bonville03 ; Stewart08 , and the opening of the gap occurs well above $T_{\rm N}$ Sosin08 . In Er2Sn2O7, the scenario appears more complex: the temperature dependence of the ordered magnetic moment points out a second order transition. The lack of clear dispersion in the experimental INS spectrum can be attributed to the proximity of $T_{\rm N}$, so that conventional spin waves are expected to develop at lower temperature. However, the strong XY character of the system, which can lead to unconventional excitations as lines of vortices Savit1978 , insufficiently captured by the RPA, might induce a peculiar $Q$ dependence of the spectrum.

The strong ratio between dipolar and exchange interactions in Er2Sn2O7, which is about $0.5-0.7$ (against 0.15 in Gd2Sn2O7 DelMaestro07 ) also appears crucial. Preliminary calculations with Er2Sn2O7 parameters at $T=0$ and ${\bf Q}=(1,1,1)$ show that accounting for the long range part of the dipolar interaction beyond the truncation considered above and in Ref. Guitteny13, ; Yan13, , slightly reduces the gap value, but does not affect the ground state. It was however established in Heisenberg systems that, at finite temperature, long range dipolar interactions tend to weaken the Palmer-Chalker state stability, due to the proximity of an unconventional state with a ${\bf k}=(1/2,1/2,1/2)$ propagation vector Enjalran03 ; Cepas04 (proposed to be the magnetic state stabilized in Gd2Ti2O7 Champion01 ). The proximity of Er2Sn2O7 with such a state, in addition to the proximity with the $\psi_{2}$ state evoked above, might reinforce the fluctuations in the Palmer-Chalker state at finite temperature and explain the low ordering temperature, the unexpected second order of the transition as well as the structure of the spectrum close to the transition. Note that the very low temperature considered here could suggest a role of quantum fluctuations, but they have been shown not to destabilize the Palmer-Chalker state much DelMaestro04 ; Yan13 .

Finally, as reported in numerous frustrated magnets Gardner10 , Er2Sn2O7 hosts a complex spin dynamics where several time scales coexist, even in the long-range order regime. In addition to the fast fluctuations observed by inelastic neutron scattering, slow dynamics are observed in ac susceptibility above $T_{\rm N}$. They persist below $T_{\rm N}$, coexisting both with the magnetic ordering along with magnetic moments fluctuating at the muon time scale Lago05 . The characteristic energy barrier of this slow dynamics, $E=0.9$ K ($=0.078$ meV), is smaller than the gap of the Palmer-Chalker state. It might be associated with the spins at the boundary between the six existing domains, as previously reported in kagome and pyrochlore systems Lhotel11 ; Tardif15 .

We have shown that the pyrochlore Er2Sn2O7 orders in the Palmer-Chalker state at about 100 mK. As confirmed by the analysis of the exchange couplings, it is the realization of the dipolar XY pyrochlore antiferromagnet. The absence of a first order transition along with multi-scale dynamics are signatures of an unconventional magnetic state. Further calculations, accounting for the XY character and for the long range dipolar interactions at finite temperature are needed to understand the role of fluctuations and may enlighten the existence of exotic excitations. The proximity of Er2Sn2O7 with competing phases might also enhance fluctuations, as recently proposed in the context of another pyrochlore compound Yb2Ti2O7 Robert15 ; Jaubert15 . In that perspective, our study points out the importance of multiphase competitions in the novel magnetic states emerging in frustrated systems.