## Abstract

We present the results of experimental determination of the heat capacity of the pyrochlore Er_{2}Ti_{2}O_{7} as a function of temperature (0.35–300 K) and magnetic field (up to 9 T), and for magnetically diluted solid solutions of the general formula (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7} (*x*≤0.471). On either doping or increase of magnetic field, or both, the Néel temperature first shifts to lower temperature until a critical point above which there is no well-defined transition but a Schottky-like anomaly associated with the splitting of the ground state Kramers doublet. By taking into account details of the lattice contribution to the heat capacity, we accurately isolate the magnetic contribution to the heat capacity and hence to the entropy. For pure Er_{2}Ti_{2}O_{7} and for (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7}, the magnetic entropy as a function of temperature evolves with two plateaus: the first at

## 1. Introduction

Among magnetic materials, the pyrochlore family, of general formula *A*_{2}*B*_{2}O_{7}, is especially interesting. The overall cubic structure consists of interpenetrating *A* and *B* sublattices of vertex-sharing tetrahedra. Either *A* or *B* or both can be paramagnetic ions, giving rise to phenomena such as spin glasses, spin liquids and disordered spin ices [1].

The pyrochlore Er_{2}Ti_{2}O_{7} has several features that are unusual among the rare earth pyrochlores. In the absence of applied magnetic field, Er_{2}Ti_{2}O_{7} orders antiferromagnetically into a non-coplanar state, shown schematically in figure 1 [2], with a Néel temperature of 1.2 K [3].

The ground state selection of the magnetically ordered state in Er_{2}Ti_{2}O_{7} is known to occur by quantum disorder [2,4]. A simple theoretical model for Er_{2}Ti_{2}O_{7} based on local *XY*-like spins coupled by near-neighbour antiferromagnetic exchange and long-range dipole interactions on the pyrochlore lattice led to expectations of a Palmer–Chalker non-co-linear-ordered state, related to, but different from, the *ψ*_{2} state which is actually observed. Recent neutron scattering experiments established a detailed spin Hamiltonian based on anisotropic exchange for this system. The actual ground state selected could then be understood in terms of this new anisotropic exchange Hamiltonian and ground state selection via order by disorder. This mechanism selects ordered states entropically, rather than energetically, on the basis of which ground state possesses a higher density of low-energy fluctuations [5,6]. Although neutron scattering experiments strongly suggest that the transition is second order, several numerical simulations predicted a first-order transition [1]. However, very recent theoretical studies which now include anisotropic exchange [7] show that, when dipolar interactions beyond nearest neighbour are included, the transition is indeed predicted to be second order.

One of the most important criteria to understand ordering in a solid is its corresponding entropy, and the evolution of entropy with temperature. Indeed, entropy was the first thermodynamic property for which *ab initio* calculations were carried out, by Sackur and Tetrode, about a century ago [8]. For magnetic systems, the important property is the magnetic entropy, *S*_{mag}. Unfortunately, the magnetic contribution to the entropy is usually difficult to delineate quantitatively from experimental information owing to uncertainty of the phononic contributions to the total entropy. However, for Er_{2}Ti_{2}O_{7}, we have the unusual circumstance of a well-known structure and high-quality data to assess the phononic contributions to the heat capacity and entropy. Furthermore, of all the rare earth titanates, Er_{2}Ti_{2}O_{7} has the most reproducible heat capacity, based on different samples measured in different laboratories [9–11], thereby allowing quantitative assessment of the magnetic entropy.

Here, we investigate the heat capacity and corresponding magnetic entropy of Er_{2}Ti_{2}O_{7}, and of (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7} in which Er^{3+} ions have been diluted with diamagnetic Y^{3+}, as functions of temperature, magnetic field and dilution. The magnetic dilution with Y^{3+} only perturbs the magnetic interactions as Er_{2}Ti_{2}O_{7} and Y_{2}Ti_{2}O_{7} are isostructural, and the ionic radius of six-coordinate Er^{3+} is 0.890 Å, whereas that of Y^{3+} is 0.900 Å [12].

The questions addressed here are as follows. How does the antiferromagnetic to paramagnetic phase transition change with magnetic field and with dilution? How does the magnetic entropy evolve as a function of temperature in pure Er_{2}Ti_{2}O_{7}, and also as a function of field and of dilution?

## 2. Experimental methods

### (a) Synthesis and crystal growth

Pure erbium titanate (Er_{2}Ti_{2}O_{7}) and yttrium titanate (Y_{2}Ti_{2}O_{7}) and solid solutions of the general formula (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7} were synthesized by the ball-milling-activated solid-state reaction of Er_{2}O_{3} (Alfa Aesar, 99.99%) and Y_{2}O_{3} (Sigma-Aldrich, 99.99%) with TiO_{2} (Sigma-Aldrich, 99.99%). Large single crystals were grown using the optical floating-zone method 13,14]. Synthesis and crystal growth details are given in our previous publication that describes thermal conductivity of single crystals of the general formula (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7} [15], using the same samples as investigated herein.

### (b) Heat capacity

The heat capacities of (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7} single crystals (*x*=0, 0.031, 0.085, 0.183, 0.471) were measured via relaxation calorimetry using a Physical Property Measurement System (PPMS; Quantum Design, San Diego, CA) with a ^{3}He cryogenic system [16]. The calorimeter consisted of an alumina sample platform with a thin-film RuO resistive heater and a Cernox sensor (Lake Shore Cryogenics), both mounted under the platform. A known amount of power was applied to the heater and the sensor monitored the temperature response. When the temperature increase reached a set value, typically 2% of the system temperature, the power was stopped and the temperature decay monitored over time. The two-tau data analysis method was used to calculate heat capacity from the temperature relaxation profile [17].

Heat capacity was measured from 0.35 to 300 K with an external magnetic field ranging from 0 to 9 T, all under vacuum (*p*<10^{−4} Torr). The crystals were aligned using X-ray diffraction such that all in-field measurements were carried out with the [110] lattice vector parallel to the applied magnetic field. To ensure good thermal contact between the samples and platform, the bottom side of each sample was polished, and a thin layer of Apiezon *N* high-vacuum grease was applied.

Sample masses ranged from 2.5 to 3.0 mg for ^{3}He measurements (0.35 *K*<*T*<10 K) and from 8 to 24 mg for ^{4}He measurements (10 *K* <*T*<300 K). Smaller samples were used during ^{3}He measurements to avoid long relaxation times, which have been found to reduce the accuracy of heat capacity measurements [16]. Uncertainty in the measured heat capacity is less than 1% for 5 *K*<*T*<300 K, and less than 5% for *T*<5 K [16,18].

## 3. Results and discussion

### (a) Er_{2}Ti_{2}O_{7}

#### (i) Results

We have determined the heat capacity of Er_{2}Ti_{2}O_{7} in single crystal form over a wide range of temperatures (0.3–300 K) and 12 values of applied magnetic field up to 9 T. The experimental heat capacity data for Er_{2}Ti_{2}O_{7} are presented in the electronic supplementary material.

Neutron scattering results [19] suggest that the low-temperature magnetic phase transition in Er_{2}Ti_{2}O_{7} is second order, as does the shape of the heat capacity anomaly at low magnetic fields [9]. However, relaxation calorimetry can determine heat capacity as the temperature of the sample platform is ramped up or down, allowing definitive determination of the transition order. The absence of hysteresis (figure 2) definitively showed the second-order nature of the transition. The zero-field data are in extremely good agreement with our previous studies 9] and with results from other laboratories over their overlapping temperature range (0.35–20 K) [10,11], showing a zero-field Néel temperature of 1.21 K.

#### (ii) Magnetic contributions to the heat capacity

The goal here was to delineate the magnetic contribution to the heat capacity, and hence the magnetic entropy, of Er_{2}Ti_{2}O_{7} as a function of temperature and of magnetic field. The total measured heat capacity of Er_{2}Ti_{2}O_{7}, which was determined at constant pressure, can be separated into three components: heat capacity at constant volume owing to lattice vibrations (acoustic and optic); heat capacity at constant volume owing to the material's magnetic spins; and the difference between heat capacity at constant pressure, *C*_{p}, and at constant volume, *C*_{V}. Thus, for Er_{2}Ti_{2}O_{7}, at all values of applied magnetic field, the total experimentally measured heat capacity can be expressed as
_{1−x}Y_{x})_{2}Ti_{2}O_{7}. At very low temperatures (*T*<0.3 K), there also can be a heat capacity contribution from the nuclear spin degeneracy of the system, but this contribution is insignificant in the temperature range studied here [11]. Note that equation (3.1) provides a more accurate determination of the magnetic contribution than subtraction of the heat capacity of an isostructural diamagnetic material, even if scaled in some way, as is common in this field. It is rare that sufficient information is available to accurately extract the magnetic heat capacity via equation (3.1), but here we do have such information and it is especially useful at higher temperatures (*T*>20 K) where the magnetic and lattice contributions are comparable (see below).

In the Er_{2}Ti_{2}O_{7} lattice, there are two formula units per unit cell, giving three acoustic and 63 optic degrees of freedom. The acoustic contribution to the lattice heat capacity of Er_{2}Ti_{2}O_{7} was calculated using the Debye model, with a Debye characteristic temperature of *Θ*_{D}=724 K, based on its Young's modulus [20].

The optic contributions to heat capacity of Er_{2}Ti_{2}O_{7} were calculated using the Einstein model. Assignments and frequencies of some of the Raman and IR optic modes of Er_{2}Ti_{2}O_{7} have been published 21,22], and the remaining Raman and IR modes were extrapolated from those calculated for other pyrochlores of the general formula *A*_{2}Ti_{2}O_{7} (*A*=Sm, Gd, Yb) [23] (H. C. Gupta 2008, private communication) using trends with atomic mass of *A*. The optic modes of Er_{2}Ti_{2}O_{7} and their degeneracies are given in the electronic supplementary material.

The *C*_{p}−*C*_{V} term for Er_{2}Ti_{2}O_{7} was calculated from
*T* is the temperature, *V* is the molar volume, *α* is the thermal expansion coefficient and *β*_{T} is the isothermal compressibility. The value of *β*_{T} for Er_{2}Ti_{2}O_{7} is from indirect measurement [20] but is in good agreement with other pyrochlores [24]. The unit cell volume is from X-ray and neutron diffraction [25]. The coefficient of thermal expansion for Er_{2}Ti_{2}O_{7} has not been reported, so the thermal expansion coefficient of isostructural Tb_{2}Ti_{2}O_{7} [26] was used as an estimate. The lattice expansion of Y_{2}Ti_{2}O_{7} has been measured at 120 K, and shows this estimate to be reasonable [27]. Because *C*_{p}−*C*_{V} is a small contribution to the overall heat capacity (approx. 1% at 300 K and less at lower temperatures), the uncertainty in *α* does not introduce significant error to the determination of

The contributions of the terms to the heat capacity of Er_{2}Ti_{2}O_{7} in zero-field are shown in comparison with the experimental heat capacity in figure 3. Note the significant contribution of

The magnetic heat capacity of Er_{2}Ti_{2}O_{7} at low temperature as a function of 12 different values of external magnetic field up to 9 T shows (see the electronic supplementary material) that, when a magnetic field is applied along the [110] lattice vector, the transition peak first becomes depressed and the Néel temperature is lowered, until the critical field of between 1.5 and 1.75 T, above which the phase transition peak broadens into a Schottky-like anomaly, in accord with our earlier studies [9] and other recent work [11].

#### (iii) Magnetic contributions to the entropy

The corresponding magnetic entropy of Er_{2}Ti_{2}O_{7} was assessed as a function temperature and magnetic field (figure 4). (To assess the entropy, the low-temperature heat capacity data for Er_{2}Ti_{2}O_{7} were extrapolated to *T*=0 K without introduction of much uncertainty as most of the magnetic entropy develops at higher temperatures.) At zero applied field, the evolution of the magnetic entropy above *T*_{N} is a good match (figure 4*a*) to recent calculations based on series expansion of the Hamiltonian [7], but only if more than 10 terms are used in the expansion. The magnetic entropy of Er_{2}Ti_{2}O_{7} per Er^{3+} at zero field reaches the ^{−1} mol^{−1}) limit by 8 K, corresponding to an isolated doublet state [11]. However, as the field increases, more thermal energy is required to achieve this same magnetic entropy (figure 4*b*).

At higher temperatures, the magnetic entropy of Er_{2}Ti_{2}O_{7} continues to evolve, reaching an upper limit that is essentially independent of applied magnetic field (figure 4*c*). This limit is ^{−1} mol^{−1}), corresponding to the 16-fold degeneracy of the Er^{3+} ground state in the crystal field of the lattice. The accomplishment of this full magnetic entropy indicates that there is no residual magnetic entropy at *T*=0 K in Er_{2}Ti_{2}O_{7}. However, the temperature at which *ca* 200 K) considering the theoretical prediction of crystal field energy levels for Er_{2}Ti_{2}O_{7} out to about 75 meV [28].

### (b) (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7}

#### (i) Results

We also have determined the heat capacities of single crystals in which Er^{3+} was diluted with Y^{3+} (general formula (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7} with *x*=0.031, 0.085, 0.183, 0.471) as a function of temperature and in magnetic field applied along the [110] direction. The full experimental heat capacity datasets are presented in the electronic supplementary material.

Dilution of the Er^{3+} ions with diamagnetic Y^{3+} ions causes the transition temperature to fall in zero applied field (figure 5 and the electronic supplementary material). The critical (percolation) concentration of Y^{3+} at zero field, beyond which there is no distinct *T*_{N}, is about *x*=0.6.

The lower Néel temperature on dilution results from the system requiring less thermal energy to transition from the antiferromagnetic to the paramagnetic phase as the Er^{3+} ions are diluted. The same trend is found at a 1 T field except that the transition is suppressed entirely when the mole fraction of Y^{3+} is more than 0.1. At a field of 3 T, the transition was suppressed for all samples examined. The dilution of Er^{3+} by Y^{3+} can be considered to be equivalent to the application of magnetic field on Er_{2}Ti_{2}O_{7}, as we discuss further below.

#### (ii) Magnetic contributions to the heat capacity

A major goal of the (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7} study was to delineate the magnetic contribution to the heat capacity, and hence the entropy, as functions of Er^{3+} spin dilution by diamagnetic Y^{3+}, temperature and magnetic field.

The approach here was similar to that taken for Er_{2}Ti_{2}O_{7} except that now contributions for Er_{2}Ti_{2}O_{7} and Y_{2}Ti_{2}O_{7} must both be taken into account. Because the parent compounds are isostructural, it is an excellent approximation that the heat capacity of the solid solution can be expressed by the rule of mixtures,
*C*_{p}(Y_{2}Ti_{2}O_{7}) experimental data have been presented elsewhere [29]. We obtained the non-magnetic terms for *C*_{p}(Er_{2}Ti_{2}O_{7}) as described above, and the remaining contribution,

The trends of _{1−x}Y_{x})_{2}Ti_{2}O_{7} can be seen from the zero-field data in figure 6. As Y^{3+} dilutes Er^{3+}, the transition temperature decreases (see also figure 5). When an external magnetic field is applied to the (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7} solid solutions, the _{2}Ti_{2}O_{7}. In a field of 1 T, the magnetic phase transition peaks are depressed and the Néel temperature is decreased relative to the zero-field results. This is because the magnetic spins become partially aligned in the direction of the magnetic field and the system requires less thermal energy to disorder the magnetic spins, i.e. to undergo the antiferromagnetic–paramagnetic phase transition. For *x*=0.183 and 0.471, the transition peaks are completely depressed at 1 T. In a 3 T magnetic field, the phase transition peaks are completely depressed for all (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7} solid solutions. (See the electronic supplementary material for plots of

#### (iii) Magnetic contributions to the entropy

The magnetic entropies of the Er^{3+} ions in (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7} solid solutions were calculated from *T*=0 K for *x*>0.085. However, for all *x*≤0.085 samples, the results show very similar behaviour for all values of *x* (figure 7). As with pure (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7}, whether above or below the critical field, *S*_{mag}(Er^{3+}) first plateaus at ^{3+} ground state in the crystal field of the lattice, respectively.

### (c) Influence of dilution and field

The phase transition and magnetic entropy results presented above indicate a strong similarity between dilution of Er^{3+} with diamagnetic Y^{3+}, and application of a magnetic field. In both cases, the perturbation first causes a decrease in Néel temperature and then, past a certain critical value, the transition is depressed and replaced by a Schottky-like heat capacity anomaly in the magnetic heat capacity.

Above the critical field, the magnetic contribution to the magnetic heat capacity of pure Er_{2}Ti_{2}O_{7} and (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7} solid solutions was fitted to a Schottky function
*c* is a numerical coefficient. The results (figure 8) show that the corresponding splitting in the two-level system increases both with Y^{3+} doping and with applied magnetic field.

The similarity of the effect of doping and magnetic field led us to undertake an examination of the universality of the magnetic behaviour of (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7} as a function of reduced Néel temperature and reduced magnetic field, as follows. We define the reduced Néel temperature (*T*_{r}) as the Néel temperature of that composition at a given field (*T*_{N}) relative to the Néel temperature of that system extrapolated to zero field (*T*_{N}(*B*=0)): *T*_{r}=*T*_{N}/*T*_{N}(*B*=0). Similarly, the reduced field (*B*_{r}) is the actual field (*B*) relative to the critical field at which *T*_{N}=0 K (*B*_{c}(*T*=0 K)): *B*_{r}=*B*/*B*_{c}(*T*_{N}=0 K). Our data for pure Er_{2}Ti_{2}O_{7} provide a smooth curve and the data for (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7}, i.e. for the samples and magnetic fields which allow a transition, fall on the same curve (figure 9). The fact that the critical temperature versus critical field curve (in reduced terms) is independent of diamagnetic doping at the magnetic site shows that the quenched disorder so introduced is weak, and these systems are well removed from any percolation limit. The (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7} system is manifestly three dimensional, and therefore this is as one might expect, because three-dimensional percolation thresholds are typically near 0.6 [30], whereas our most highly doped sample that shows a distinct transition at more than one field investigated here contains almost an order of magnitude less disorder (*x*=0.085).

## 4. Conclusion

We have shown that the antiferromagnetic to paramagnetic phase transition behaviour of Er_{2}Ti_{2}O_{7} and Y^{3+}-doped Er_{2}Ti_{2}O_{7} is very similar. On either doping or increase of magnetic field, or both, the Néel temperature first shifts to lower temperature until a critical point above which there is no well-defined transition.

Accurate quantitative analysis of the magnetic heat capacity shows that the magnetic contributions to the entropy of Er_{2}Ti_{2}O_{7} above *T*_{N} in zero field require at least 10 terms in the series expansion of the Hamiltonian for a good representation, highlighting the importance of subtle magnetic interactions in this system. For pure Er_{2}Ti_{2}O_{7}, the magnetic entropy as a function of temperature evolves with two plateaus: the first at

With dilution of Er^{3+} with diamagnetic Y^{3+}, the Néel temperature decreases. The critical temperature versus critical field curve (in reduced terms) is independent of diamagnetic doping at the magnetic site, implying that the quenched disorder introduced by doping is weak, and the system is far removed from any percolation limit. Interestingly, the influence of dilution is similar to the increase of magnetic field. As for pure Er_{2}Ti_{2}O_{7}, in (Er_{1−x}Y_{x})_{2}Ti_{2}O_{7} solid solutions, the magnetic entropy per Er^{3+} evolves in two steps, with a plateau at _{1−x}Y_{x})_{2}Ti_{2}O_{7} increases both with dilution and with magnetic field.

## Funding statement

This work was supported by NSERC of Canada, and facilities used at McMaster's Brockhouse Materials Research Institute and Dalhousie's Institute for Research in Materials were supported by the Canada Foundation for Innovation and NSERC. M.A.W. dedicates this paper to the memory of Professor Patrick Jacobs, with gratitude for introducing her to thermodynamics.

## Acknowledgements

We gratefully acknowledge the assistance of Dr Antoni Dabkowski and Professor T. S. Cameron in aligning the crystals, and Lauren Bilinsky for preliminary data analysis.

## Footnotes

One contribution to a Special feature ‘New developments in the chemistry and physics of defects in solids’.

- Received May 12, 2014.
- Accepted August 8, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.