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Magnetic phase transitions and magnetic entropy in the XY antiferromagnetic pyrochlores (Er1−xYx)2Ti 2O7 John F. Niven, Michel B. Johnson, Alex Bourque, Patrick J. Murray, David D. James, Hanna A. Da?bkowska, Bruce D. Gaulin and Mary Anne White Proc. R. Soc. A 2014 470, 20140387, published 10 September 2014
Supplementary data
"Data Supplement" http://rspa.royalsocietypublishing.org/content/suppl/2014/09/09/rs pa.2014.0387.DC1.html
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Magnetic phase transitions and magnetic entropy in the XY antiferromagnetic pyrochlores (Er1−xYx)2Ti2O7 John F. Niven1 , Michel B. Johnson2 , Alex Bourque3 , Patrick J. Murray3 , David D. James3 ,
Research
Hanna A. D¸abkowska4 , Bruce D. Gaulin4,5 and
Cite this article: Niven JF, Johnson MB, Bourque A, Murray PJ, James DD, D¸abkowska HA, Gaulin BD, White MA. 2014 Magnetic phase transitions and magnetic entropy in the XY antiferromagnetic pyrochlores (Er1−x Yx )2 Ti2 O7 . Proc. R. Soc. A 470: 20140387. http://dx.doi.org/10.1098/rspa.2014.0387
Mary Anne White1,2,3
Received: 12 May 2014 Accepted: 8 August 2014
Subject Areas: physical chemistry, materials science Keywords: pyrochlore, antiferromagnetism, magnetic entropy, phase transition, heat capacity Author for correspondence: Mary Anne White e-mail: [email protected]
One contribution to a Special feature ‘New developments in the chemistry and physics of defects in solids’.
Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2014.0387 or via http://rspa.royalsocietypublishing.org.
1 Department of Physics and Atmospheric Science, 2 Institute for
Research in Materials, and 3 Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada B3H 4R2 4 Brockhouse Institute for Materials Research, and 5 Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1 We present the results of experimental determination of the heat capacity of the pyrochlore Er2 Ti2 O7 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 (Er1−x Yx )2 Ti2 O7 (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 Er2 Ti2 O7 and for (Er1−x Yx )2 Ti2 O7 , the magnetic entropy as a function of temperature evolves with two plateaus: the first at R ln 2, and the other at R ln 16. When a very high magnetic field is applied, the first plateau is washed out. The influence of dilution at low values is similar to the increase of magnetic field, as we show by examination of the critical temperature versus critical field curve in reduced terms.
2014 The Author(s) Published by the Royal Society. All rights reserved.
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2
Among magnetic materials, the pyrochlore family, of general formula A2 B2 O7 , is especially interesting. The overall cubic structure consists of interpenetrating A and B sublattices of vertexsharing 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 Er2 Ti2 O7 has several features that are unusual among the rare earth pyrochlores. In the absence of applied magnetic field, Er2 Ti2 O7 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 Er2 Ti2 O7 is known to occur by quantum disorder [2,4]. A simple theoretical model for Er2 Ti2 O7 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 ψ2 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 firstorder 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, Smag . 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 Er2 Ti2 O7 , 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, Er2 Ti2 O7 has the most reproducible heat
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1. Introduction
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Figure 1. The ψ2 ground state spin configuration for the XY antiferromagnet Er2 Ti2 O7 [2].
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(a) Synthesis and crystal growth Pure erbium titanate (Er2 Ti2 O7 ) and yttrium titanate (Y2 Ti2 O7 ) and solid solutions of the general formula (Er1−x Yx )2 Ti2 O7 were synthesized by the ball-milling-activated solid-state reaction of Er2 O3 (Alfa Aesar, 99.99%) and Y2 O3 (Sigma-Aldrich, 99.99%) with TiO2 (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 (Er1−x Yx )2 Ti2 O7 [15], using the same samples as investigated herein.
(b) Heat capacity The heat capacities of (Er1−x Yx )2 Ti2 O7 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) Er2 Ti2 O7 (i) Results We have determined the heat capacity of Er2 Ti2 O7 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 Er2 Ti2 O7 are presented in the electronic supplementary material.
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2. Experimental methods
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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 Er2 Ti2 O7 , and of (Er1−x Yx )2 Ti2 O7 in which Er3+ ions have been diluted with diamagnetic Y3+ , as functions of temperature, magnetic field and dilution. The magnetic dilution with Y3+ only perturbs the magnetic interactions as Er2 Ti2 O7 and Y2 Ti2 O7 are isostructural, and the ionic radius of six-coordinate Er3+ is 0.890 Å, whereas that of Y3+ 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 Er2 Ti2 O7 , and also as a function of field and of dilution?
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4
heating cooling
Cp(J K–1 mol–1)
10
5
0
0.4
0.6
0.8
1.0
1.2 T (K)
1.4
1.6
1.8
2.0
Figure 2. Experimental Cp values at zero field for Er2 Ti2 O7 expressed per mole of Er2 Ti2 O7 , on heating and cooling, showing that the transition is second order. (Online version in colour.)
Neutron scattering results [19] suggest that the low-temperature magnetic phase transition in Er2 Ti2 O7 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 Er2 Ti2 O7 as a function of temperature and of magnetic field. The total measured heat capacity of Er2 Ti2 O7 , 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, Cp , and at constant volume, CV . Thus, for Er2 Ti2 O7 , at all values of applied magnetic field, the total experimentally measured heat capacity can be expressed as expt
Cp
lattice,optic
= Clattice,acoustic + CV V
magnetic
+ (Cp − CV ) + CV
,
(3.1)
where all heat capacities here are expressed per mole based on the formula (Er1−x Yx )2 Ti2 O7 . 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 Er2 Ti2 O7 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 Er2 Ti2 O7 was
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90
60 50 40 30 20 10 0
20
40
60
80
100
T (K)
Figure 3. Experimental Cp values at zero field for Er2 Ti2 O7 compared with the optic, magnetic and (Cp − CV ) contributions, where their sum is shown as CV . Values are shown per mole of Er2 Ti2 O7 . (Online version in colour.)
calculated using the Debye model, with a Debye characteristic temperature of ΘD = 724 K, based on its Young’s modulus [20]. The optic contributions to heat capacity of Er2 Ti2 O7 were calculated using the Einstein model. Assignments and frequencies of some of the Raman and IR optic modes of Er2 Ti2 O7 have been published [21,22], and the remaining Raman and IR modes were extrapolated from those calculated for other pyrochlores of the general formula A2 Ti2 O7 (A = Sm, Gd, Yb) [23] (H. C. Gupta 2008, private communication) using trends with atomic mass of A. The optic modes of Er2 Ti2 O7 and their degeneracies are given in the electronic supplementary material. The Cp − CV term for Er2 Ti2 O7 was calculated from Cp − CV =
TVα 2 , βT
(3.2)
where T is the temperature, V is the molar volume, α is the thermal expansion coefficient and βT is the isothermal compressibility. The value of βT for Er2 Ti2 O7 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 Er2 Ti2 O7 has not been reported, so the thermal expansion coefficient of isostructural Tb2 Ti2 O7 [26] was used as an estimate. The lattice expansion of Y2 Ti2 O7 has been measured at 120 K, and shows this estimate to be reasonable [27]. Because Cp − CV 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 magnetic
. to the determination of CV The contributions of the terms to the heat capacity of Er2 Ti2 O7 in zero-field are shown in comparison with the experimental heat capacity in figure 3. Note the significant contribution of magnetic
both near the transition temperature and also at much higher temperatures. The magnetic heat capacity of Er2 Ti2 O7 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].
CV
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heat capacity (J K–1 mol–1)
80
5
CV Cp Cp–CV CVmagnetic CVacoustic CVoptic
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(iii) Magnetic contributions to the entropy
6
(i) Results We also have determined the heat capacities of single crystals in which Er3+ was diluted with Y3+ (general formula (Er1−x Yx )2 Ti2 O7 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 Er3+ ions with diamagnetic Y3+ ions causes the transition temperature to fall in zero applied field (figure 5 and the electronic supplementary material). The critical (percolation) concentration of Y3+ at zero field, beyond which there is no distinct TN , 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 Er3+ 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 Y3+ is more than 0.1. At a field of 3 T, the transition was suppressed for all samples examined. The dilution of Er3+ by Y3+ can be considered to be equivalent to the application of magnetic field on Er2 Ti2 O7 , as we discuss further below.
(ii) Magnetic contributions to the heat capacity A major goal of the (Er1−x Yx )2 Ti2 O7 study was to delineate the magnetic contribution to the heat capacity, and hence the entropy, as functions of Er3+ spin dilution by diamagnetic Y3+ , temperature and magnetic field. The approach here was similar to that taken for Er2 Ti2 O7 except that now contributions for Er2 Ti2 O7 and Y2 Ti2 O7 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, Cp ((Er1−x Yx )2 Ti2 O7 ) = (1 − x)Cp (Er2 Ti2 O7 ) + xCp (Y2 Ti2 O7 ),
(3.3)
where Cp (Y2 Ti2 O7 ) experimental data have been presented elsewhere [29]. We obtained the non-magnetic terms for Cp (Er2 Ti2 O7 ) as described above, and the remaining contribution, magnetic
CV
(Er3+ ), was then assessed via equation (3.1) by difference. magnetic
(Er3+ ) in (Er1−x Yx )2 Ti2 O7 can be seen from the zero-field data in figure 6. The trends of CV As Y3+ dilutes Er3+ , the transition temperature decreases (see also figure 5). When an external magnetic
magnetic field is applied to the (Er1−x Yx )2 Ti2 O7 solid solutions, the CV
(Er3+ ) behaviour
...................................................
(b) (Er1−x Yx )2 Ti2 O7
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The corresponding magnetic entropy of Er2 Ti2 O7 was assessed as a function temperature and magnetic field (figure 4). (To assess the entropy, the low-temperature heat capacity data for Er2 Ti2 O7 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 TN is a good match (figure 4a) 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 Er2 Ti2 O7 per Er3+ at zero field reaches the R ln 2 (=5.76 J K−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 4b). At higher temperatures, the magnetic entropy of Er2 Ti2 O7 continues to evolve, reaching an upper limit that is essentially independent of applied magnetic field (figure 4c). This limit is R ln 16 (=23.1 J K−1 mol−1 ), corresponding to the 16-fold degeneracy of the Er3+ 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 Er2 Ti2 O7 . However, the temperature at which R ln 16 is accomplished is rather low (ca 200 K) considering the theoretical prediction of crystal field energy levels for Er2 Ti2 O7 out to about 75 meV [28].
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(a)
0.8
7
R ln(2)
0.7
Smag R–1
0.4 0.3 0.2 0.1 0
Smag(J K–1 (mol Er3+)–1)
(b)
1 TN
9 8 7 6 5 4 3 2 1
10
T (K)
0T 0.5 T 1T 1.25 T 1.5 T 1.75 T
2T 2.5 T 3T 5T 7T 9T
R ln(2)
0
5
10 T (K)
15
20
(c)
20 15 10 5
3 H (T)
5
200 7
100 9 0
)
1
(K
0
T
Smag(J K–1 (mol Er3+)–1)
25
Figure 4. Magnetic entropy of Er2 Ti2 O7 per Er3+ as a function of temperature and applied magnetic field: (a) in zero field in comparison with models for above TN [7] with 9 or 12 terms used in the series expansion, in comparison with the experimentally derived data; (b) in various magnetic fields applied along the [110] direction at low temperature; and (c) over a wider temperature range. (Online version in colour.) is similar to that observed in Er2 Ti2 O7 . 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 (Er1−x Yx )2 Ti2 O7 solid solutions. (See the electronic supplementary material for magnetic
plots of CV
(Er3+ ) at no applied field and at 1 and 3 T.)
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partial-sum-12 partial-sum-9 data
0.5
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0.6
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1.5
8
0 field
TN (K) 0.5
0
0.1
0.2
0.3 0.4 0.5 0.6 x = mole fraction Y 3+
0.7
0.8
Figure 5. Néel temperature for (Er1−x Yx )2 Ti2 O7 as a function of mole fraction of Y3+ (= x), in zero field and a 1 T field applied along the [110] direction.
x=0 x = 0.031 x = 0.085 x = 0.183 x = 0.471
Cmagnetic (J K–1 (mol Er3+)–1)
12 10 8 6 4 2
0
0.5
1.0
1.5 T (K)
2.0
2.5
3.0
Figure 6. Magnetic heat capacity of (Er1−x Yx )2 Ti2 O7 in zero applied field. (Online version in colour.)
(iii) Magnetic contributions to the entropy The magnetic entropies of the Er3+ ions in (Er1−x Yx )2 Ti2 O7 solid solutions were calculated from magnetic
magnetic
(Er3+ ). Unfortunately, it was not possible to reliably extrapolate CV (Er3+ ) to T = 0 K CV 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 (Er1−x Yx )2 Ti2 O7 , whether above or below the critical field, Smag (Er3+ ) first plateaus at R ln 2, and then at R ln 16, corresponding to the entropy associated with the isolated doublet state and 16-fold degeneracy of the Er3+ 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 Er3+ with diamagnetic Y3+ , 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.
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1.0
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1T
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(a)
9 ...................................................
20
R ln(16)
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Smag (J K–1 (mol Er3+)–1)
25
x=0 x = 0.031 x = 0.085
15
10 R ln(2) 5
0 (b)
Smag (J K–1 (mol Er3+)–1)
25
20
R ln(16) x=0 x = 0.031 x = 0.085
15
10 R ln(2) 5
0 (c)
Smag (J K–1 (mol Er3+)–1)
25
R ln(16)
20
x=0 x = 0.031 x = 0.085 x = 0.183
15
10 R ln(2) 5
0
10
20
30
40 50 T (K)
60
70
80
Figure 7. Magnetic entropy per mole of Er3+ of (Er1−x Yx )2 Ti2 O7 in (a) zero field, (b) 1 T field and (c) 3 T field. The magnetic field was applied along the [110] direction. (Online version in colour.)
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0.8
10
3T 1T
D (meV)
0.5 D (meV)
x=0
2.0 1.5 1.0 0.5
0.4
0
0.3
2
4
6
8
10
H (T)
0.2 0.1 0
0.1
0.2
0.3 x
0.4
0.5
0.6
Figure 8. The splitting of the ground state Kramers doublet, , of (Er1−x Yx )2 Ti2 O7 as a function of magnetic dilution (x) and magnetic field. Note the linear relationship between and field for Er2 Ti2 O7 (see inset). Typical uncertainty in is shown by the error bar. (Online version in colour.)
Tr = TN / TN (B = 0)
1.5 x=0 x = 0.031 x = 0.085 1.0
0.5
0
0.5
1.0
Br = B/Bc(TN = 0 K)
Figure 9. Reduced transition temperature as a function of reduced magnetic field for (Er1−x Yx )2 Ti2 O7 , showing that samples with different Y3+ doping levels exhibit the same relative behaviour as Er2 Ti2 O7 . (Online version in colour.)
Above the critical field, the magnetic contribution to the magnetic heat capacity of pure Er2 Ti2 O7 and (Er1−x Yx )2 Ti2 O7 solid solutions was fitted to a Schottky function magnetic
CV
= cR
kT
2
e−/kT , (1 + e−/kT )2
(3.4)
where is the gap associated with the splitting of the ground state Kramers doublet and c is a numerical coefficient. The results (figure 8) show that the corresponding splitting in the two-level system increases both with Y3+ 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 (Er1−x Yx )2 Ti2 O7 as a function of reduced Néel
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0.6
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We have shown that the antiferromagnetic to paramagnetic phase transition behaviour of Er2 Ti2 O7 and Y3+ -doped Er2 Ti2 O7 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 Er2 Ti2 O7 above TN 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 Er2 Ti2 O7 , the magnetic entropy as a function of temperature evolves with two plateaus: the first at R ln 2, and the other at R ln 16, at about 10 and 200 K, respectively. When a very high magnetic field is applied, the first plateau is washed out. With dilution of Er3+ with diamagnetic Y3+ , 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 Er2 Ti2 O7 , in (Er1−x Yx )2 Ti2 O7 solid solutions, the magnetic entropy per Er3+ evolves in two steps, with a plateau at R ln 2 and another at R ln 16. Above the critical field, the splitting of the ground state Kramers doublet in (Er1−x Yx )2 Ti2 O7 increases both with dilution and with magnetic field. 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.
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.
References 1. Gardner JS, Gingras MJP, Greedan JE. 2010 Magnetic pyrochlore oxides. Rev. Mod. Phys. 82, 53–107. (doi:10.1103/revmodphys.82.53) 2. Poole A, Wills AS, Lelièvre-Berna E. 2007 Magnetic ordering in the XY pyrochlore antiferromagnet Er2 Ti2 O7 : a spherical neutron polarimetry study. J. Phys. Condensed Matter 19, 452201. (doi:10.1088/0953-8984/19/45/452201) 3. Blöte HWJ, Wielinga RF, Huiskamp WJ. 1969 Heat-capacity measurements on rare-earth double oxide R2 M2 O7 . Physica 43, 549–569. (doi:10.1016/0031-8914(69)90187-6) 4. Ross KA, Qiu Y, Copley JD, Dabkowska HA, Gaulin BD. 2014 Order by disorder spin wave gap in the XY pyrochlore magnet Er2 Ti2 O7 . Phys. Rev. Lett. 112, 057201. (doi:10.1103/ PhysRevLett.112.057201)
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4. Conclusion
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temperature and reduced magnetic field, as follows. We define the reduced Néel temperature (Tr ) as the Néel temperature of that composition at a given field (TN ) relative to the Néel temperature of that system extrapolated to zero field (TN (B = 0)): Tr = TN /TN (B = 0). Similarly, the reduced field (Br ) is the actual field (B) relative to the critical field at which TN = 0 K (Bc (T = 0 K)): Br = B/Bc (TN = 0 K). Our data for pure Er2 Ti2 O7 provide a smooth curve and the data for (Er1−x Yx )2 Ti2 O7 , 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 (Er1−x Yx )2 Ti2 O7 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).
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5. Savary L, Ross KA, Gaulin BD, Ruff JPC, Balents L. 2012 Order by quantum disorder in Er2 Ti2 O7 . Phys. Rev. Lett. 109, 167201. (doi:10.1103/PhysRevLett.109.167201) 6. Zhitomirsky ME, Gvozdikova MV, Holdsworth PCW, Moessner R. 2012 Quantum order by disorder and accidental soft mode in Er2 Ti2 O7 . Phys. Rev. Lett. 109, 077204. (doi:10.1103/ PhysRevLett.109.077204) 7. Oitmaa J, Singh RRP, Javanparast B, Day AGR, Bagheri BV, Gingras MJP. 2013 Phase transition and thermal order-by-disorder in the pyrochlore antiferromagnetic Er2 Ti2 O7 : a high-temperature series expansion study. Phys. Rev. B 88, 220404. (doi:10.1103/PhysRevB. 88.220404) 8. Jacobs P. 2013 Thermodynamics, p. 91. London, UK: Imperial College Press. 9. Ruff JPC et al. 2008 Spin waves and quantum criticality in the frustrated XY pyrochlore antiferromagnet Er2 Ti2 O7 . Phys. Rev. Lett. 101, 147205. (doi:10.1103/PhysRevLett.101. 147205) 10. Sosin SS, Prozorova LA, Lees MR, Balakrishnan G, Petrenko OA. 2010 Magnetic excitations in the XY-pyrochlore antiferromagnet Er2 Ti2 O7 . Phys Rev. B 82, 094428. (doi:10.1103/ PhysRevB.82.094428) 11. Dalmas de Réotier P et al. 2012 Magnetic order, magnetic correlations, and spin dynamics in the pyrochlore antiferromagnet Er2 Ti2 O7 . Phys. Rev. B 86, 104424. (doi:10.1103/ PhysRevB.86.104424) 12. Shannon RD. 1976 Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Crystallogr. A 32, 751–767. (doi:10.1107/ S0567739476001551) 13. Gardner JS, Gaulin BD, Paul DMcK. 1998 Single crystal growth by the floating-zone method of a geometrically frustrated pyrochlore antiferromagnet, Tb2 Ti2 O7 . J. Cryst. Growth 191, 740–745. (doi:10.1016/S0022-0248(98)00382-0) 14. Da¸bkowska HA, D´zbkowski AB. 2010 Crystal growth of oxides by optical floating zone technique. Experimental approach to defects determination. In Springer handbook of crystal growth, defects and characterization (eds G Dhanaraj, K Byrappa, V Prasad, M Dudley), pp. 367–391. Berlin, Germany: Springer. 15. Bryan C, Whitman CA, Johnson MB, Niven JF, Murray P, Bourque A, D´zbkowska HA, Gaulin BD, White MA. 2012 Thermal conductivity of (Er1−x Yx )2 Ti2 O7 pyrochlore oxide solid solutions. Phys. Rev. B 86, 054303. (doi:10.1103/PhysRevB.86.054303) 16. Kennedy CA, Stancescu M, Marriott RA, White MA. 2007 Recommendations for accurate heat capacity measurements using a quantum design physical property measurement system. Cryogenics 47, 107–112. (doi:10.1016/j.cryogenics.2006.10.001) 17. Hwang JS, Lin KJ, Tien C. 1997 Measurement of heat capacity by fitting the whole temperature response of a heat-pulse calorimeter. Rev. Sci. Instrum. 68, 94–101. (doi:10.1063/1.1147722) 18. Lashley JC et al. 2003 Critical examination of heat capacity measurements made on a quantum design physical property measurement system. Cryogenics 43, 369–378. (doi:10.1016/S0011-2275(03)00092-4) 19. Champion JDM et al. 2003 Er2 Ti2 O7 : evidence of quantum order by disorder in a frustrated antiferromagnetic. Phys. Rev. B 68, 020401. (doi:10.1103/PhysRevB.68.020401) 20. van Dijk MP, de Vries KJ, Burggraaf AJ. 1983 Oxygen ion and mixed conductivity in compounds with the fluorite and pyrochlore structure. Solid State Ionics 9&10, 913–920. (doi:10.1016/0167-2738(83)90110-8) 21. Maczka ˛ M, Hanuza J, Hermanowicz K, Fuentes AF, Matsuhira K, Hiroi Z. 2008 Temperaturedependent Raman scattering studies of the geometrically frustrated pyrochlores Dy2 Ti2 O7 , Gd2 Ti2 O7 and Er2 Ti2 O7 . J. Raman Spec. 39, 537–544. (doi:10.1002/jrs.1875) 22. Martos M, Julián-López B, Cordoncillo E, Escribano P. 2009 Structural and spectroscopic study of a new pink chromium-free Er2 (Ti,Zr)2 O7 ceramic pigment. J. Am. Ceram. Soc. 92, 2987–2992. (doi:10.1111/j.1551-2916.2009.03335.x) 23. Gupta HC, Brown S, Rani N, Gohel VB. 2001 Lattice dynamic investigation of the zone center wavenumbers of the cubic A2 Ti2 O7 pyrochlores. J. Raman Spec. 32, 41–44. (doi:10.1002/10974555(200101)32:13.0.CO;2-R) 24. Scott PR, Midgley A, Musaev O, Muthu DVS, Singh S, Suryanarayanan R, Revcolevschi A, Sood AK, Kruger MB. 2011 High-pressure synchrotron X-ray diffraction study of the pyrochlores: HO2 Ti2 O7 , Y2 Ti2 O7 and Tb2 Ti2 O7 . High Pressure Res. 31, 219–227. (doi:10.1080/ 08957959.2010.548333)
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25. Knop O, Brisse F, Castelliz L. 1965 Determination of the crystal stucture of erbium titanate, Er2 Ti2 O7 , by X-ray and neutron diffraction. Can. J. Chem. 43, 2812–2826. (doi:10.1139/v65-392) 26. Saha S et al. 2009 Low-temperature and high-pressure Raman and x-ray studies of pyrochlore Tb2 Ti2 O7 : phonon anomalies and possible phase transition. Phys. Rev. B 79, 134112. (doi:10.1103/PhysRevB.79.134112) 27. Ruff JPC, Gaulin BD, Castellan JP, Rule KC, Clancy JP, Rodriguez J, Dabkowska HA. 2007 Structural fluctuations in the spin-liquid state of Tb2 Ti2 O7 . Phys. Rev. Lett. 99, 237202. (doi:10.1103/PhysRevLett.99.237202) 28. Bertin A, Chapuis Y, Dalmas de Réotier P, Yoauanc A. 2012 Crystal field effect in the R2 Ti2 O7 pyrochlore compounds. J. Phys. Condensed Matter 24, 246003. (doi:10.1088/09538984/24/25/256003) 29. Johnson MB, James DD, Bourque A, Dabkowska HA, Gaulin BD, White MA. 2009 Thermal properties of the pyrochlore, Y2 Ti2 O7 . J. Solid State Chem. 182, 725–729. (doi:10.1016/j. jssc.2008.12.027) 30. Gaulin BD, Gardner JS. 2004 Chapter 8: experimental studies of frustrated pyrochlore antiferromagnets. In Frustrated spin systems (ed. HT Diep), pp. 457–489. Singapore: World Scientific.
Magnetic phase transitions and magnetic entropy in the XY antiferromagnetic pyrochlores (Er1−xYx)2Ti 2O7 John F. Niven, Michel B. Johnson, Alex Bourque, Patrick J. Murray, David D. James, Hanna A. Da?bkowska, Bruce D. Gaulin and Mary Anne White Proc. R. Soc. A 2014 470, 20140387, published 10 September 2014
Supplementary data
"Data Supplement" http://rspa.royalsocietypublishing.org/content/suppl/2014/09/09/rs pa.2014.0387.DC1.html
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Magnetic phase transitions and magnetic entropy in the XY antiferromagnetic pyrochlores (Er1−xYx)2Ti2O7 John F. Niven1 , Michel B. Johnson2 , Alex Bourque3 , Patrick J. Murray3 , David D. James3 ,
Research
Hanna A. D¸abkowska4 , Bruce D. Gaulin4,5 and
Cite this article: Niven JF, Johnson MB, Bourque A, Murray PJ, James DD, D¸abkowska HA, Gaulin BD, White MA. 2014 Magnetic phase transitions and magnetic entropy in the XY antiferromagnetic pyrochlores (Er1−x Yx )2 Ti2 O7 . Proc. R. Soc. A 470: 20140387. http://dx.doi.org/10.1098/rspa.2014.0387
Mary Anne White1,2,3
Received: 12 May 2014 Accepted: 8 August 2014
Subject Areas: physical chemistry, materials science Keywords: pyrochlore, antiferromagnetism, magnetic entropy, phase transition, heat capacity Author for correspondence: Mary Anne White e-mail: [email protected]
One contribution to a Special feature ‘New developments in the chemistry and physics of defects in solids’.
Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2014.0387 or via http://rspa.royalsocietypublishing.org.
1 Department of Physics and Atmospheric Science, 2 Institute for
Research in Materials, and 3 Department of Chemistry, Dalhousie University, Halifax, Nova Scotia, Canada B3H 4R2 4 Brockhouse Institute for Materials Research, and 5 Department of Physics and Astronomy, McMaster University, Hamilton, Ontario, Canada L8S 4M1 We present the results of experimental determination of the heat capacity of the pyrochlore Er2 Ti2 O7 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 (Er1−x Yx )2 Ti2 O7 (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 Er2 Ti2 O7 and for (Er1−x Yx )2 Ti2 O7 , the magnetic entropy as a function of temperature evolves with two plateaus: the first at R ln 2, and the other at R ln 16. When a very high magnetic field is applied, the first plateau is washed out. The influence of dilution at low values is similar to the increase of magnetic field, as we show by examination of the critical temperature versus critical field curve in reduced terms.
2014 The Author(s) Published by the Royal Society. All rights reserved.
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2
Among magnetic materials, the pyrochlore family, of general formula A2 B2 O7 , is especially interesting. The overall cubic structure consists of interpenetrating A and B sublattices of vertexsharing 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 Er2 Ti2 O7 has several features that are unusual among the rare earth pyrochlores. In the absence of applied magnetic field, Er2 Ti2 O7 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 Er2 Ti2 O7 is known to occur by quantum disorder [2,4]. A simple theoretical model for Er2 Ti2 O7 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 ψ2 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 firstorder 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, Smag . 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 Er2 Ti2 O7 , 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, Er2 Ti2 O7 has the most reproducible heat
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1. Introduction
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Figure 1. The ψ2 ground state spin configuration for the XY antiferromagnet Er2 Ti2 O7 [2].
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(a) Synthesis and crystal growth Pure erbium titanate (Er2 Ti2 O7 ) and yttrium titanate (Y2 Ti2 O7 ) and solid solutions of the general formula (Er1−x Yx )2 Ti2 O7 were synthesized by the ball-milling-activated solid-state reaction of Er2 O3 (Alfa Aesar, 99.99%) and Y2 O3 (Sigma-Aldrich, 99.99%) with TiO2 (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 (Er1−x Yx )2 Ti2 O7 [15], using the same samples as investigated herein.
(b) Heat capacity The heat capacities of (Er1−x Yx )2 Ti2 O7 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) Er2 Ti2 O7 (i) Results We have determined the heat capacity of Er2 Ti2 O7 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 Er2 Ti2 O7 are presented in the electronic supplementary material.
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2. Experimental methods
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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 Er2 Ti2 O7 , and of (Er1−x Yx )2 Ti2 O7 in which Er3+ ions have been diluted with diamagnetic Y3+ , as functions of temperature, magnetic field and dilution. The magnetic dilution with Y3+ only perturbs the magnetic interactions as Er2 Ti2 O7 and Y2 Ti2 O7 are isostructural, and the ionic radius of six-coordinate Er3+ is 0.890 Å, whereas that of Y3+ 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 Er2 Ti2 O7 , and also as a function of field and of dilution?
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4
heating cooling
Cp(J K–1 mol–1)
10
5
0
0.4
0.6
0.8
1.0
1.2 T (K)
1.4
1.6
1.8
2.0
Figure 2. Experimental Cp values at zero field for Er2 Ti2 O7 expressed per mole of Er2 Ti2 O7 , on heating and cooling, showing that the transition is second order. (Online version in colour.)
Neutron scattering results [19] suggest that the low-temperature magnetic phase transition in Er2 Ti2 O7 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 Er2 Ti2 O7 as a function of temperature and of magnetic field. The total measured heat capacity of Er2 Ti2 O7 , 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, Cp , and at constant volume, CV . Thus, for Er2 Ti2 O7 , at all values of applied magnetic field, the total experimentally measured heat capacity can be expressed as expt
Cp
lattice,optic
= Clattice,acoustic + CV V
magnetic
+ (Cp − CV ) + CV
,
(3.1)
where all heat capacities here are expressed per mole based on the formula (Er1−x Yx )2 Ti2 O7 . 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 Er2 Ti2 O7 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 Er2 Ti2 O7 was
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90
60 50 40 30 20 10 0
20
40
60
80
100
T (K)
Figure 3. Experimental Cp values at zero field for Er2 Ti2 O7 compared with the optic, magnetic and (Cp − CV ) contributions, where their sum is shown as CV . Values are shown per mole of Er2 Ti2 O7 . (Online version in colour.)
calculated using the Debye model, with a Debye characteristic temperature of ΘD = 724 K, based on its Young’s modulus [20]. The optic contributions to heat capacity of Er2 Ti2 O7 were calculated using the Einstein model. Assignments and frequencies of some of the Raman and IR optic modes of Er2 Ti2 O7 have been published [21,22], and the remaining Raman and IR modes were extrapolated from those calculated for other pyrochlores of the general formula A2 Ti2 O7 (A = Sm, Gd, Yb) [23] (H. C. Gupta 2008, private communication) using trends with atomic mass of A. The optic modes of Er2 Ti2 O7 and their degeneracies are given in the electronic supplementary material. The Cp − CV term for Er2 Ti2 O7 was calculated from Cp − CV =
TVα 2 , βT
(3.2)
where T is the temperature, V is the molar volume, α is the thermal expansion coefficient and βT is the isothermal compressibility. The value of βT for Er2 Ti2 O7 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 Er2 Ti2 O7 has not been reported, so the thermal expansion coefficient of isostructural Tb2 Ti2 O7 [26] was used as an estimate. The lattice expansion of Y2 Ti2 O7 has been measured at 120 K, and shows this estimate to be reasonable [27]. Because Cp − CV 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 magnetic
. to the determination of CV The contributions of the terms to the heat capacity of Er2 Ti2 O7 in zero-field are shown in comparison with the experimental heat capacity in figure 3. Note the significant contribution of magnetic
both near the transition temperature and also at much higher temperatures. The magnetic heat capacity of Er2 Ti2 O7 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].
CV
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heat capacity (J K–1 mol–1)
80
5
CV Cp Cp–CV CVmagnetic CVacoustic CVoptic
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(iii) Magnetic contributions to the entropy
6
(i) Results We also have determined the heat capacities of single crystals in which Er3+ was diluted with Y3+ (general formula (Er1−x Yx )2 Ti2 O7 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 Er3+ ions with diamagnetic Y3+ ions causes the transition temperature to fall in zero applied field (figure 5 and the electronic supplementary material). The critical (percolation) concentration of Y3+ at zero field, beyond which there is no distinct TN , 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 Er3+ 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 Y3+ is more than 0.1. At a field of 3 T, the transition was suppressed for all samples examined. The dilution of Er3+ by Y3+ can be considered to be equivalent to the application of magnetic field on Er2 Ti2 O7 , as we discuss further below.
(ii) Magnetic contributions to the heat capacity A major goal of the (Er1−x Yx )2 Ti2 O7 study was to delineate the magnetic contribution to the heat capacity, and hence the entropy, as functions of Er3+ spin dilution by diamagnetic Y3+ , temperature and magnetic field. The approach here was similar to that taken for Er2 Ti2 O7 except that now contributions for Er2 Ti2 O7 and Y2 Ti2 O7 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, Cp ((Er1−x Yx )2 Ti2 O7 ) = (1 − x)Cp (Er2 Ti2 O7 ) + xCp (Y2 Ti2 O7 ),
(3.3)
where Cp (Y2 Ti2 O7 ) experimental data have been presented elsewhere [29]. We obtained the non-magnetic terms for Cp (Er2 Ti2 O7 ) as described above, and the remaining contribution, magnetic
CV
(Er3+ ), was then assessed via equation (3.1) by difference. magnetic
(Er3+ ) in (Er1−x Yx )2 Ti2 O7 can be seen from the zero-field data in figure 6. The trends of CV As Y3+ dilutes Er3+ , the transition temperature decreases (see also figure 5). When an external magnetic
magnetic field is applied to the (Er1−x Yx )2 Ti2 O7 solid solutions, the CV
(Er3+ ) behaviour
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(b) (Er1−x Yx )2 Ti2 O7
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The corresponding magnetic entropy of Er2 Ti2 O7 was assessed as a function temperature and magnetic field (figure 4). (To assess the entropy, the low-temperature heat capacity data for Er2 Ti2 O7 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 TN is a good match (figure 4a) 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 Er2 Ti2 O7 per Er3+ at zero field reaches the R ln 2 (=5.76 J K−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 4b). At higher temperatures, the magnetic entropy of Er2 Ti2 O7 continues to evolve, reaching an upper limit that is essentially independent of applied magnetic field (figure 4c). This limit is R ln 16 (=23.1 J K−1 mol−1 ), corresponding to the 16-fold degeneracy of the Er3+ 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 Er2 Ti2 O7 . However, the temperature at which R ln 16 is accomplished is rather low (ca 200 K) considering the theoretical prediction of crystal field energy levels for Er2 Ti2 O7 out to about 75 meV [28].
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(a)
0.8
7
R ln(2)
0.7
Smag R–1
0.4 0.3 0.2 0.1 0
Smag(J K–1 (mol Er3+)–1)
(b)
1 TN
9 8 7 6 5 4 3 2 1
10
T (K)
0T 0.5 T 1T 1.25 T 1.5 T 1.75 T
2T 2.5 T 3T 5T 7T 9T
R ln(2)
0
5
10 T (K)
15
20
(c)
20 15 10 5
3 H (T)
5
200 7
100 9 0
)
1
(K
0
T
Smag(J K–1 (mol Er3+)–1)
25
Figure 4. Magnetic entropy of Er2 Ti2 O7 per Er3+ as a function of temperature and applied magnetic field: (a) in zero field in comparison with models for above TN [7] with 9 or 12 terms used in the series expansion, in comparison with the experimentally derived data; (b) in various magnetic fields applied along the [110] direction at low temperature; and (c) over a wider temperature range. (Online version in colour.) is similar to that observed in Er2 Ti2 O7 . 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 (Er1−x Yx )2 Ti2 O7 solid solutions. (See the electronic supplementary material for magnetic
plots of CV
(Er3+ ) at no applied field and at 1 and 3 T.)
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partial-sum-12 partial-sum-9 data
0.5
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0.6
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1.5
8
0 field
TN (K) 0.5
0
0.1
0.2
0.3 0.4 0.5 0.6 x = mole fraction Y 3+
0.7
0.8
Figure 5. Néel temperature for (Er1−x Yx )2 Ti2 O7 as a function of mole fraction of Y3+ (= x), in zero field and a 1 T field applied along the [110] direction.
x=0 x = 0.031 x = 0.085 x = 0.183 x = 0.471
Cmagnetic (J K–1 (mol Er3+)–1)
12 10 8 6 4 2
0
0.5
1.0
1.5 T (K)
2.0
2.5
3.0
Figure 6. Magnetic heat capacity of (Er1−x Yx )2 Ti2 O7 in zero applied field. (Online version in colour.)
(iii) Magnetic contributions to the entropy The magnetic entropies of the Er3+ ions in (Er1−x Yx )2 Ti2 O7 solid solutions were calculated from magnetic
magnetic
(Er3+ ). Unfortunately, it was not possible to reliably extrapolate CV (Er3+ ) to T = 0 K CV 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 (Er1−x Yx )2 Ti2 O7 , whether above or below the critical field, Smag (Er3+ ) first plateaus at R ln 2, and then at R ln 16, corresponding to the entropy associated with the isolated doublet state and 16-fold degeneracy of the Er3+ 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 Er3+ with diamagnetic Y3+ , 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.
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1T
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(a)
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20
R ln(16)
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Smag (J K–1 (mol Er3+)–1)
25
x=0 x = 0.031 x = 0.085
15
10 R ln(2) 5
0 (b)
Smag (J K–1 (mol Er3+)–1)
25
20
R ln(16) x=0 x = 0.031 x = 0.085
15
10 R ln(2) 5
0 (c)
Smag (J K–1 (mol Er3+)–1)
25
R ln(16)
20
x=0 x = 0.031 x = 0.085 x = 0.183
15
10 R ln(2) 5
0
10
20
30
40 50 T (K)
60
70
80
Figure 7. Magnetic entropy per mole of Er3+ of (Er1−x Yx )2 Ti2 O7 in (a) zero field, (b) 1 T field and (c) 3 T field. The magnetic field was applied along the [110] direction. (Online version in colour.)
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0.8
10
3T 1T
D (meV)
0.5 D (meV)
x=0
2.0 1.5 1.0 0.5
0.4
0
0.3
2
4
6
8
10
H (T)
0.2 0.1 0
0.1
0.2
0.3 x
0.4
0.5
0.6
Figure 8. The splitting of the ground state Kramers doublet, , of (Er1−x Yx )2 Ti2 O7 as a function of magnetic dilution (x) and magnetic field. Note the linear relationship between and field for Er2 Ti2 O7 (see inset). Typical uncertainty in is shown by the error bar. (Online version in colour.)
Tr = TN / TN (B = 0)
1.5 x=0 x = 0.031 x = 0.085 1.0
0.5
0
0.5
1.0
Br = B/Bc(TN = 0 K)
Figure 9. Reduced transition temperature as a function of reduced magnetic field for (Er1−x Yx )2 Ti2 O7 , showing that samples with different Y3+ doping levels exhibit the same relative behaviour as Er2 Ti2 O7 . (Online version in colour.)
Above the critical field, the magnetic contribution to the magnetic heat capacity of pure Er2 Ti2 O7 and (Er1−x Yx )2 Ti2 O7 solid solutions was fitted to a Schottky function magnetic
CV
= cR
kT
2
e−/kT , (1 + e−/kT )2
(3.4)
where is the gap associated with the splitting of the ground state Kramers doublet and c is a numerical coefficient. The results (figure 8) show that the corresponding splitting in the two-level system increases both with Y3+ 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 (Er1−x Yx )2 Ti2 O7 as a function of reduced Néel
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We have shown that the antiferromagnetic to paramagnetic phase transition behaviour of Er2 Ti2 O7 and Y3+ -doped Er2 Ti2 O7 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 Er2 Ti2 O7 above TN 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 Er2 Ti2 O7 , the magnetic entropy as a function of temperature evolves with two plateaus: the first at R ln 2, and the other at R ln 16, at about 10 and 200 K, respectively. When a very high magnetic field is applied, the first plateau is washed out. With dilution of Er3+ with diamagnetic Y3+ , 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 Er2 Ti2 O7 , in (Er1−x Yx )2 Ti2 O7 solid solutions, the magnetic entropy per Er3+ evolves in two steps, with a plateau at R ln 2 and another at R ln 16. Above the critical field, the splitting of the ground state Kramers doublet in (Er1−x Yx )2 Ti2 O7 increases both with dilution and with magnetic field. 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.
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.
References 1. Gardner JS, Gingras MJP, Greedan JE. 2010 Magnetic pyrochlore oxides. Rev. Mod. Phys. 82, 53–107. (doi:10.1103/revmodphys.82.53) 2. Poole A, Wills AS, Lelièvre-Berna E. 2007 Magnetic ordering in the XY pyrochlore antiferromagnet Er2 Ti2 O7 : a spherical neutron polarimetry study. J. Phys. Condensed Matter 19, 452201. (doi:10.1088/0953-8984/19/45/452201) 3. Blöte HWJ, Wielinga RF, Huiskamp WJ. 1969 Heat-capacity measurements on rare-earth double oxide R2 M2 O7 . Physica 43, 549–569. (doi:10.1016/0031-8914(69)90187-6) 4. Ross KA, Qiu Y, Copley JD, Dabkowska HA, Gaulin BD. 2014 Order by disorder spin wave gap in the XY pyrochlore magnet Er2 Ti2 O7 . Phys. Rev. Lett. 112, 057201. (doi:10.1103/ PhysRevLett.112.057201)
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4. Conclusion
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temperature and reduced magnetic field, as follows. We define the reduced Néel temperature (Tr ) as the Néel temperature of that composition at a given field (TN ) relative to the Néel temperature of that system extrapolated to zero field (TN (B = 0)): Tr = TN /TN (B = 0). Similarly, the reduced field (Br ) is the actual field (B) relative to the critical field at which TN = 0 K (Bc (T = 0 K)): Br = B/Bc (TN = 0 K). Our data for pure Er2 Ti2 O7 provide a smooth curve and the data for (Er1−x Yx )2 Ti2 O7 , 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 (Er1−x Yx )2 Ti2 O7 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).
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