Heat capacity measurements on FeAs-based compounds: a thermodynamic probe of electronic and magnetic states

We report heat capacity measurements of the pnictide materials SmFeAsO$_{1-x}$F$_x$, NdFeAsO, LaFeAsO$_{1-x}$F$_x$ and LiFeAs. For SmFeAsO$_{1-x}$F$_x$, with x close to 0.1, we use 3 He measurements to demonstrate a transfer of entropy from the peak at TN to a previously unidentified ~2 K feature which grows with increasing doping. Our results on the Sm samples are compared with a similarly doped La sample to elucidate the crystal field levels of the Sm3+ ion at 0, 23, and 56 meV which lead to a Schottky-like anomaly, and also show that there is a significant increase in the Sommerfeld coefficient $\gamma$ when La is replaced by Sm or Nd. The lattice contribution to the heat capacity of the superconducting oxypnictides is found to vary negligibly with chemical substitution. We also present a heat capacity measurement of LiFeAs showing the feature at Tc, which is significantly rounded and much smaller than the BCS value.

Abstract. We report heat capacity measurements of the pnictide materials SmFeAsO 1−x Fx, NdFeAsO, LaFeAsO 1−x Fx and LiFeAs. For SmFeAsO 1−x Fx, with x close to 0.1, we use 3 He measurements to demonstrate a transfer of entropy from the peak at T N to a previously unidentified ∼ 2 K feature which grows with increasing doping. Our results on the Sm samples are compared with a similarly doped La sample to elucidate the crystal field levels of the Sm 3+ ion at 0, 23, and 56 meV which lead to a Schottky-like anomaly, and also show that there is a significant increase in the Sommerfeld coefficient γ when La is replaced by Sm or Nd. The lattice contribution to the heat capacity of the superconducting oxypnictides is found to vary negligibly with chemical substitution. We also present a heat capacity measurement of LiFeAs showing the feature at Tc, which is significantly rounded and much smaller than the BCS value.

Introduction
The discovery of superconductivity in pnictide compounds containing iron came as an enormous surprise to the condensed matter physics and materials chemistry community. This started with derivatives of LaFePO (T c = 4 K) [1], and then, attracting far greater attention, LaFeAsO 1−x F x (T c = 26 K) [2]. The LnFeAsO 1−x F x (1111) materials have provided a remarkable range of superconducting materials with T c rising quickly to 43 K in SmFeAsO 1−x F x [3], and subsequently 55 K in that compound and oxygen deficient SmFeAsO 1−δ [4,5]. Double layered FeAs-based (122) materials with the general formula AFe 2 As 2 (T c = 38 K) [6,7] and the single layered (111) and stoichiometric LiFeAs (T c ∼ 16K) [8,9] and NaFeAs (T c = 9 K) [10] also superconduct.
These new materials can be considered analogous to the cuprate high-T c superconductors but based on anti-PbO-type FeAs layers rather than planar CuO 2 layers and with more than one band relevant to the superconductivity. The superconducting properties and novel magnetism in these compounds have been a great spur to recent research. The tantalising prospect is that the analogy will offer insights that will identify common aspects of the superconducting mechanism in both types of material. Specific heat measurements are a powerful probe of magnetism and superconductivity since they probe the entropy involved with the electronic phase transitions, and provide a straightforward comparison with theoretical work. In the pnictide superconductors both Fe magnetism and superconductivity, and the magnetic behaviour of the rare-earth moments are amenable to investigation using this technique.
In this paper we present heat capacity results on three SmFeAsO 1−x F x samples in the doping range where the coexistence of magnetic order and superconductivity has been suggested, undoped SmFeAsO and NdFeAsO where magnetism is present without superconductivity, and superconducting samples of LaFeAsO 1−x F x and LiFeAs where there is no magnetic order. Following a review of previous work (section 2) and a description of the technique (section 3), we present our results. These are divided into the 4 He heat capacity results on the oxypnictide compounds in section 4 and the 3 He heat capacity results on the same materials in section 5. The results on LiFeAs are presented in section 6. We describe the experimental methodology in section 3 and discuss the results in section 7.

Previous heat capacity studies of pnictide compounds
A significant number of specific heat measurements have already been carried out on pnictide compounds although they have covered only a small selection of the compounds synthesized so far. SmFeAsO 1−x O x has already been studied at the dopings x = 0, 0.05, 0.07, 0.15, and 0.20 [11,12], away from the intermediate doping range we consider here, revealing a peak in the specific heat in the undoped sample at T SDW ∼ 130 K and λ anomalies near T N ∼ 4 K. In LaFeAs(O,F) [13,14,16] the structural (155 K) and magnetic transitions (143 K) are clearly resolved and have a small combined entropy of 0.032 R [14]. The Sommerfeld coefficient was found to be relatively small, γ = 3.7 mJmol −1 K −2 [14]. The low-temperature values of γ were found to vary in applied magnetic field as γ(H) ∝ √ H, suggestive of line nodes in the superconducting gap function [16]. In this issue McGuire et al. [15] examine the structural and magnetic transitions around 150 K in CeFeAsO, PrFeAsO, and NdFeAsO, finding clear separation between them and distinct heat capacity anomalies at each transition. LaFePO has also been studied in detail [17,18,19], γ ∼ 10 mJmol −1 K −2 and relatively small specific heat jumps have been measured at T c , ∆C p /γT c ∼ 0.6.
A smaller number of 122 materials have also been measured: BaNi 2 As 2 has a first-order phase transition at T o = 130 K and a Sommerfeld coefficient γ = 10.8 ± 0.1 mJmol −1 K −2 . A superconducting sample with T c = 0.68 K shows a very well defined specific heat jump at T c , with ∆C/γT c = 1.31. This is good evidence for bulk superconductivity, but further analysis was not pursued [20]. A more recent study of a Ba 0.6 K 0.4 Fe 2 As 2 single crystal [21] found a ∼ 1 K wide specific heat peak at T c = 34.6 K and probed the field dependence of the feature. Evidence for strong coupling was found and the κ values were over 100.
There have been relatively few predictions made concerning the heat capacity of the pnictide superconductors, but one is particularly relevant to our work here [22]. It has been suggested that SmFeAsO 1−x F x has a nematic electronic phase overlapping the superconducting phase, probably in the doping range considered here. The lowtemperature specific heat would change from C ∝ T β , β < 2 in the superconducting phase to C ∝ T 2/3 in the nematic phase, with a crossover to the conventional d = 3 scalings C ∝ T close enough to the nematic critical point.

Experimental details
The SmFeAsO 1−x F x samples were synthesized by conventional solid state reaction methods, as described in Refs [3,4]. Standard powder x-ray diffraction patterns were measured where all peaks could be indexed to the tetragonal ZrCuSiAs-type structure. DC resistivity and magnetisation measurements were made to determine the midpoint (10 % to 90 % width) of the resistive and diamagnetic transitions with T c = 10(7), 17 (8), and 25(8) K for x = 0.1, 0.12, and 0.13 respectively. These samples have previously been investigated using µSR [23], which suggested the coexistence of superconductivity and magnetism. LaFeAsO 0.9 F 0.1 (T c = 22 K) was synthesised by a method adapted from Ref. [2]; a mixture of La, La 2 O 3 , LaF 3 , F and As was heated at 600 • C for 12 hrs in an evacuated silica ampoule then thoroughly re-ground, pelletised and heated to 1150 • C for a further 48hrs in evacuated silica. The resulting material was examined by laboratory powder x-ray diffraction and found to contain LaFeAsO 0.9 F 0.1 (T c = 22 K) as the majority phase with small amounts of FeAs (< 5mol %) and LaOF (< 2.5mol %). The LiFeAs sample used in our experiments was synthesized by the method described in Ref. [8] and corresponds to Sample 2 described in that study, with T c = 12 K. While the diamagnetic shift in this sample is far smaller than ideal, or that of the T c = 16 K sample [8], µSR measurements on this sample demonstrated that the superconducting sample volume was > 80 % [24], suggesting that particle size effects may be affecting the bulk DC susceptibility measurement.
The heat capacity measurements were carried out using a Quantum Design Physical Properties Measurement System (PPMS) over the temperature range 0.4 to 300 K, with measurements below 2 K carried out using the standard 3 He insert. Samples were in the form of pressed powder pellets and were affixed to the measurement stage using Apiezon N-grease. The measurement technique employed by the PPMS is the two-tau relaxation method [25,26] where the temperature of the sample is measured while a heat pulse is applied and for an equal time afterwards while the sample cools. The time dependence of the temperature variation in each part of the measurement is described by the sum of two exponentials describing the heat capacity of the sample and the measurement stage, and the coupling between them. The effectiveness of this technique is discussed critically by Lashley et al. [27], and we can expect the absolute heat capacity values to be correct to within ±2 % for 5 ≤ T ≤ 300 K and ±5 % for T ≤ 5 K. These errors are comparable with the error in the sample mass.

4 He results on oxypnictides
The data for the oxypnictide materials SmFeAsO 0.9 F 0.1 , SmFeAsO 0.88 F 0.12 , SmFeAsO 0.87 F 0.13 , and LaFeAsO 0.9 F 0.1 from 2 to 200 K are shown in Figure 1. The lattice and electronic parts of the heat capacity data are fitted to the equation: where γ is the Sommerfeld coefficient, and C D and C E are Debye and Einstein functions respectively. We found that the data were well described by including only one Einstein mode, giving a more simple fitting function than was used by Tropeano et al. [12]. The total amplitude of the lattice modes is almost exactly value expected from the Dulong-Petit law. Fits were carried out over the temperature range 10 ≤ T /K ≤ 200 K as Apiezon N grease produces small heat capacity anomalies above 200 K. The parameters derived from fitting equation 1 to the data shown in Figure 1 are shown in Table 1. As is apparent from Figure 1, the lattice contributions are almost identical for the four samples, and the main differences are the change in the electronic term γ going from the La compound to the Sm compounds, and the λ anomalies at T Sm N (hereafter T N ) in each of the Sm samples. The difference between the γ values of the Sm and La compounds, ∼ 120 vs. 26 mJmol −1 K −2 , is quite remarkable. Our values of γ are comparable with those reported by Ding et al. [11] for samples with slightly different dopings. Similarly high values of γ have been observed in Nd (as we show in section 5 below) and Ce oxypnictides.
In Figure 2 the raw data near T N and the residues between the data in Figure 1 and the fits given in Table 1 near T c are shown. The peak in C/T near T N is clear for each of the samples and we can see the slow drop in T N with increasing doping. The height of the peak also seems to drop with increasing doping. Carrying out entropy subtractions suggested that the magnetic entropy was also dropping as the doping was increased and in the x = 0.1 and 0.12 samples an upturn in C/T was observed at the lowest temperatures, though as we will see in section 5 this is a hint of what is actually going on at low-temperatures. to the raw heat capacity data for the SmFeAsO 1−x Fx and LaFeAsO 0.9 F 0.1 samples described in Section 4. The quoted errors are 1σ from the fitting.  Table 1. Near the superconducting T c and T Fe N in the right-hand panel of Figure 2 the difference between the data and the fits is far smaller. It seems that the peak around T c in the La compound is well defined and of a similar size to those found previously, though somewhat broader. In the Sm compounds the temperatures of T c and T Fe N have been determined by other techniques [23]. The feature seems to grow with increasing doping and in the x = 0.1 sample must correspond to Fe ordering. In the x = 0.12 sample the anomaly starts at T c and continues up to T Fe N . The x = 0.13 sample has almost coincident T c and T Fe N and we observe a broad and less structured feature at the appropriate temperature.
Seeing that the entropy values extracted from the low-temperature Néel ordering of the Sm magnetic moments approximate much more closely to S = 1/2 than the S = 5/2 expected for free ions [28] we sought to investigate the possibility of a Schottky-like anomaly due to the crystal field levels being gradually populated with increasing temperature. This can be done by comparing the heat capacity of Sm and La-containing oxypnictides, but it would be challenging to attempt this on undoped samples since significant anomalies appear at the structural and magnetic phase transition(s) near 150 K. Instead, we used our measurements of the 10 % doped samples where the superconducting and magnetic anomalies above 10 K are very small. From the La compound we get an excellent estimate of the lattice contribution because there is no rare-earth moment, and we can easily adjust for the difference in the Sommerfeld coefficient in our fitting of the difference. We took a direct molar subtraction between data sets measured at common temperature points and fitted the difference, ∆C = C(Sm) − C(La), to the function: where ∆γ is the difference between the electronic specific heats, g i are the degeneracies of each of the levels, and ∆ 1 and ∆ 2 are the energy gaps (in K) of the S = 1/2 → S = 3/2 and S = 1/2 → S = 5/2 transitions respectively. The parameters derived from this analysis are ∆ 1 = 266(3) K = 22.92(3) meV and ∆ 2 = 650(50) K = 56(4) meV. The former value is in agreement with the estimates coming from the thermal activation of the µSR relaxation rate [29,30] and the latter value has previously been derived from magnetic susceptibility measurements (∆ = 520 K for x = 0 and ∆ = 620 K for x = 0.15) [28]. Sm containing materials are not amenable to investigation using inelastic neutron scattering, which is generally the most powerful probe of the crystal field levels in these materials. However, CeFeAsO 1−x F x has been investigated, and crystal field doublets at 0, 18.7, and 58.4 meV have been found for an x = 0.16 superconducting sample [31]. In the undoped sample three doublets in the paramagnetic phase split into six singlets in the antiferromagnetic phase. It is interesting to note the similarity between the crystal field levels in these two materials and the observation that the 18.7 meV mode has a strongly enhanced local magnetic susceptibility below T c , which is reminiscent of the increased µSR relaxation rates observed for the Sm compounds [29,30]. The other parameter extracted from this analysis is the difference in the Sommerfeld coefficients, ∆γ = 53(2) mJmol −1 K −2 , which is far lower than the values extracted from fitting the raw data. This would suggest that the true value of γ ∼ 80 mJmol −1 K −2 in the Sm oxypnictides -still significantly larger than for LaFeAs(O, F). (Ding et al. [11] found a similar value from some of their fitting.) It seems likely that failing to include the Schottky-like anomaly due to the crystal field levels in the fitting of the raw data leads to an erroneously high value of γ, as we found above and has been noted by other reports [11]. Observing the doping dependence of the crystal field levels in these materials is likely to prove a fruitful area of future study, particularly since the rare-earth moment fluctuations are evidently well coupled to the Fe fluctuations in the FeAs planes [29,31].

3 He results on oxypnictides
Our 3 He heat capacity results for SmFeAsO, SmFeAsO 0.9 F 0.1 , and NdFeAsO are plotted in Figure 3 after the subtraction of non-magnetic backgrounds. We made the assumption that the lattice contribution to the Sm samples would be identical and used the La lattice background for the Nd and corrected for the difference in the Sommerfeld coefficients, estimating a value for the Nd compound of ∼ 60 mJmol −1 K −2 . The Nd and Sm magnetic ordering leads to well defined λ anomalies at the Néel temperature of the rare-earth moments. We find T Nd N = 1.9 K in reasonable agreement with the value determined by neutron diffraction (T N = 1.96(3) K [32]). Other neutron diffraction measurements have shown that the Fe moments order at around 141 K with an ordered moment of 0.25(7) µ B [33]. We discuss the temperature variation of T Sm N with doping in section 7.
In the two Sm samples we see a further anomaly below the previously determined T N values at around 2 K. This feature grows significantly with increasing x as the feature at T N decreases in amplitude. We are unaware of any previous measurements on SmFeAsO 1−x F x below 2 K that have shown this feature, though our 4 He measurements show an upturn at low-temperature ( Figure 2) that implies something must be occurring, and the decrease in the height at T N has been reported elsewhere, if not discussed [11,12]. We note that if the superconducting state is becoming more robust with increasing doping there will be a drop in the electronic contribution to the specific heat below T c (> T N ). This would lead to a drop in the height of the magnetic features before the background is subtracted and a reduced estimate of the magnetic entropy after background subtraction.

LiFeAs
The LiFeAs sample (T c = 12 K) gives us an opportunity to observe the superconducting behaviour without the effects of magnetic ordering of either rareearth or iron moments. Since it superconducts without any non-stoichiometry we compare our results with those on LaFePO reported elsewhere [17,18,19]. Data on LiFeAs in zero-field and 12 T are shown in Figure 4 and we plot the difference between them in the inset. A clear peak is shown in the vicinity of T c although it is evidently broadened, as we would expect on the basis of susceptibility measurements [8]. (We also see a difference between the heat capacity traces above 20 K, which we cannot explain.) In the inset we plot a linear trend following the low-temperature behaviour of the peak in the residue and extending up to the midpoint of the transition to estimate the size of the specific heat anomaly in this compound. This has two inherent difficulties: the transition is relatively broad and may affect the curvature below T c to a greater extent than we have assumed, and more importantly, the upper critical field, B c2 , is far greater than our measurement field. On the basis of susceptibility measurements B c2 > 80 T [9], a result which is consistent with the value extracted from µSR measurements [24]. T c is therefore likely to have only been shifted by < 1 K by our measurement field and so the residue does not have a full subtraction of the superconducting part of the heat capacity. From the line plotted in the inset we can estimate the normalized specific heat jump at T c to be ∆C/γT c = 0.26, with γ ∼ 23 mJmol −1 K −2 . This is about 1/3 the value found for LaFePO, which has a much lower B c2 ∼ 3.5 T [19]. The small specific heat jump might suggest a small superconducting sample volume, which magnetic susceptibility measurements also suggested [8], but µSR measurements on a sample from the same batch gave a superconducting sample volume > 80 % [24]. On this basis the small shift in T c is more likely to have led to the small specific heat jump. Both higher applied magnetic fields and samples with a more sharply defined T c will be necessary to gain a more complete understanding of the thermodynamic properties of LiFeAs.

Discussion
Our results show a new feature in the heat capacity of SmFeAsO 1−x F x at around 2 K which appears to grow and shift to lower temperature with increasing doping. We are not aware of this feature being reported elsewhere, nor any predictions of such a feature. Since the entropy released at the T N peak seems to be dropping as this feature grows it seems likely that this 2 K feature is related to the Sm moments. The presence of the feature at low-temperature in the undoped sample suggests that it is not related to the nematic phase suggested by Xu et al. [22] if their phase diagram is appropriate. Alternatives may be successive ordering of components of the Sm magnetic moments or reorientations of the moments between states with a different entropy, which are altered by the doping-dependent orthorhombic to tetragonal structural distortion [34].  In NdFeAsO we have verified the T N value previously determined using neutron scattering [32]. In Table 2 we have collated the values of T N determined in this study and the magnetic entropy values extracted. Our undoped Sm-oxypnictide has a higher T N than has been suggested by previous studies [11,12]. It is not clear at this stage why this value is increased above previous measurements, although our value does sit closer to the trend of decreasing T N with increasing doping suggested by other measurements. Where we have 3 He data the magnetic entropy values are R ln 2 within the experimental error (we take this to be that of the low temperature accuracy of the cryostat) but for the SmFeAsO 0.88 F 0.12 and SmFeAsO 0.87 F 0.13 the data above 2 K suggest a significant amount of entropy is missing. Given the behaviour seen in the SmFeAsO 0.9 F 0.1 sample, it seems very likely that a similar feature is present below 2 K and accounts for this missing entropy. The entropy values we obtain for the rare-earth moment ordering suggest that both Sm and Nd are in a S = 1/2 state at low-temperature. In the Sm x = 0.1 sample we found that the higher temperature heat capacity includes a Schottky-like anomaly due to the higher crystal field levels at 23 and 56 meV above the ground state. These levels are similar to those observed for superconducting CeFeAsO 0.84 F 0.16 using inelastic neutron scattering (0, 18.7, and 58.4 meV) [31]. It will be interesting to discover the variation of these crystal field levels in other materials as this field of research progresses, particularly if the rareearth moments have any influence on the superconductivity. The heat capacity features associated with the superconducting transitions in the SmFeAsO 1−x F x are less well defined than those reported for superconducting samples without any magnetic transitions. The fit residues suggest that both magnetic and superconducting transitions contribute features, but while they are present at the temperatures suggested by other techniques [23], the entropy change is small and diffuse. The features in LaFeAsO 0.9 F 0.1 and LiFeAs are much better defined and are comparable with other reports on similar samples.

Conclusions
We have studied a series of Sm, Nd, and La-containing oxypnictides and investigated the effect that the rare-earth magnetic moments have on the magnetism and superconductivity. The λ anomalies at each Néel temperature give clear evidence for ordering of the rare-earth moments, confirming previous studies. We have extended those studies by investigating the form of the heat capacity in the Sm samples below 2 K, which provides evidence for a new feature occurring slightly below 2 K that grows with increasing doping. This provides a natural explanation for the previously observed drop in the size of the peak at T N ∼ 4 K with increasing doping. We also probed the crystal field levels of the Sm 3+ ions in SmFeAsO 0.9 F 0.1 and found they were consistent with both the R ln 2 entropy associated with magnetic ordering at low-temperature and also the free spin S = 5/2 value observed at high temperature. The energies of the levels appear to be similar to those found in superconducting CeFeAsO 0.84 F 0.16 using inelastic neutron scattering. In LaFeAsO 0.9 F 0.1 and LiFeAs we see distinct anomalies due to the superconducting transitions, whereas in the SmFeAsO 1−x F x samples magnetic contributions complicate the form of the data and the superconducting contribution is less distinct. There remain many avenues to be explored using heat capacity in this fast moving field, including the possibility of nematic electronic order when superconductivity has been suppressed in SmFeAsO 1−x F x and the trends of crystal field levels as the rare-earth ions are substituted and the oxygen doping is changed; but it is already clear that this fundemental thermodynamic technique, coupled with careful analysis, can provide much useful information on the properties of FeAs-based compounds. Table 2. Néel temperatures and magnetic entropy values for the samples investigated in this work. The methodology is discussed in Sections 4 and 5. Entropies extracted for the SmFeAsO 0.88 F 0.12 and SmFeAsO 0.87 F 0.13 are estimates due to the lack of 3 He data, calculated assuming a smooth trend down to T = 0. The error bars are based on the quoted accuracy of the calorimeter at low temperatures [26,27].

Doping (x)
T N (K) S mag /R