Rattling Behavior in a Simple Perovskite NaWO₃

Abstract

Rattling phenomena have been observed in materials characterized by a large cage structure but not in a simple ABO₃-type perovskite because the size mismatch, if it exists, can be relieved by octahedral rotations. Here, we demonstrate that a stoichiometric perovskite oxide NaWO₃, prepared under high pressure, exhibits anharmonic phonon modes associated with low-energy rattling vibrations, leading to suppressed thermal conductivity. The structural analysis and the comparison with the ideal perovskite KWO₃ without rattling behavior reveal that the presence of two crystallographic Na1 (2a) and Na2 (6b) sites in NaWO₃ (space group Im$\overline{3}$) accompanied by three in-phase WO₆ octahedral (a⁺a⁺a⁺) rotations generates an open space Δ ~ 0.5 Å for the latter site, which is comparable with those of well-known cage compounds of clathrates and filled skutterudites. The observed rattling in NaWO₃ is distinct from a quadruple perovskite AA′₃B₄O₁₂ (A, A′: transition metals) where the A (2a) site with lower multiplicity is the rattler. The present finding offers a general guide to induce rattling of atoms in pristine ABO₃ perovskites.

Graphical Abstract

INTRODUCTION

Inorganic solids with a network of polyhedral cages that host guest cations show interesting properties associated with characteristic phonon modes.^1–4 When guest cations are smaller relative to the space inside the cage, a local and anharmonic vibration, or “rattling”, occurs, leading to strong scattering of long wavelength acoustic phonons in clathrates (e.g., Sr₈Ga₁₆Ge₃₀),^5,6 superconductivity with strong electron–phonon coupling in β-pyrochlore IKOs₂O₆,^7–9 and heavy Fermion states in filled skutterudites (e.g., SmOs₄Sb₁₂).^10–13 It has been widely believed that the rattling vibration suppresses the lattice thermal conductivity, κ_L, resulting in the enhancement of thermoelectric efficiency.^14–16 Quasi-harmonic interaction between the cage and guest cation is also proposed.¹⁷ These features have stimulated numerous studies toward exotic physical properties coupled with rattling vibrational modes, such as high efficiency thermoelectric materials with low κ_L.

Perovskite ABO₃ is the most studied system in materials science because of the richness in chemical and physical properties including superconductivity,¹⁸ colossal magneto-resistivity,¹⁹ and ferroelectricity.²⁰ The diversity of properties arises from the ability of perovskite compounds to take various cations at the A and B sites,²¹ along with the substitution of oxide site by other anions.²² In addition, octahedral rotation of various patterns causes the deviation of ∠B–O–B angle from 180° in the ideal perovskite, which influences orbital overlap, superexchange interactions, and electron hopping between neighboring B sites. Here, the relief of size mismatch is the origin of octahedral rotations: when the Goldschmidt tolerance factor is smaller than 1, BO₆ octahedra rotate to satisfy the necessary the A site cation environment, leading to structural distortion from a cubic perovskite (space group: Pm$\overline{3}$m).²¹ Octahedral rotations match almost any A site sizes, leading to dense structures and therefore leaving little “open space” for rattling to occur. Recently, Tanaka et al. reported several quadruple perovskite oxides, ACu₃V₄O₁₂ (A = Mn, Cu).^23,24 In general, the A′ ions in AA′₃B₄O₁₂ are Jahn–Teller active transition metals (e.g., Cu²⁺ and Mn³⁺), and extensive BO₆ octahedral tilting permits square-planar A′O₄ coordination.²⁵ The resultant A site is 12-fold coordinated typically occupied by large alkaline earth metals (e.g., La), but when it is occupied by small transition metal ions, a rattling phenomenon emerges, as seen from a large atomic displacement parameter (ADP) and Einstein-like specific heat.^23,24

In this paper, we report the high-pressure synthesis of a stoichiometric tungsten bronze Na_xWO₃ (x = 1). Na_xWO₃ has been extensively studied for decades, and a perovskite structure with a⁺a⁺a⁺ octahedral rotation (space group: Im$\overline{3}$) is found in a high concentration regime (x ≥ 0.8).^26–29 However, there remains an uncertainty about its Na composition, as will be shown later. The present study follows our recent work on K_xWO₃, where the high-pressure method expanded the x amount, forming a stoichiometric tetragonal phase K_0.6WO₃ and a cubic (perovskite) one KWO₃.³⁰ Quite unexpectedly, we observed rattling behavior in this pristine perovskite NaWO₃ but not in KWO₃. We discuss the origin of the rattling phenomenon in NWO₃ by considering crystal structures and physical properties, in comparison with the 1:3 ordered quadruple perovskite AA′₃B₄O₁₂. A general guideline to the rattling phenomenon in perovskite is presented.

EXPERIMENTAL PROCEDURE

Polycrystalline samples of Na_xWO₃ (x = 0.5, 0.6, 0.75, 0.8, 1.0) were prepared using a high-pressure (HP) technique. Stoichiometric mixtures of Na₂WO₄, WO₂ (99%, Rare Metallic), and WO₃ (99.999%, Rare Metallic) were ground in a mortar for 30–60 min and pressed into a pellet. The pellets were sealed in a platinum capsule, inserted in a graphite tube heater, and enclosed in a pyrophyllite cube. These procedures were carried out in a N₂-filled groove box. The Na₂WO₄ precursor was prepared by heating Na₂WO₄·2H₂O (99.8%), Alfa Aesar) in air at 400 °C for 24 h; a complete loss of water was checked by infrared spectroscopy.³¹ The HP reactions were performed at 1000 °C for 30 min under pressures of 5–8 GPa. The samples were characterized through powder X-ray diffraction (XRD) with Cu Kα radiation at room temperature (RT). Synchrotron powder X-ray diffraction (SXRD) measurements were performed at RT on the BL02B2 beamline at SPring-8. The wavelengths employed were λ = 0.42044 Å for x = 0.5, 0.6, 0.75, 1.0 and λ = 0.41914 Å for x = 0.8. A glass capillary with a 0.2 mm inner diameter was used. The SXRD patterns were analyzed by the Rietveld refinement method using the program JANA2006.^32,33 Reaction procedures were similar to those followed for the synthesis of Li_0.5WO₃, with stoichiometric mixtures of Li₂WO₄ (99%, Kojundo Chemical), WO₂ (99%, Rare Metallic), and WO₃ (99.999%, Rare Metallic). Neutron diffraction (ND) measurements (λ = 1.5397 Å) for Li_0.5WO₃ were carried out using the high-resolution powder diffractometer BT-1 at the NIST Center for Neutron Research. The sample was loaded into a vanadium cell. The collected neutron data were analyzed by the program JANA2006.

Specific heat, C_p, was measured with a commercial calorimeter (Quantum Design PPMS) in a temperature range from 200 to 2 K. Thermal conductivity, κ, was measured by a standard four-contact method at temperatures between 300 and 100 K. Electrical resistivity, ρ, was measured with a four-probe method using a commercial apparatus (Quantum Design PPMS) equipped with an adiabatic demagnetization refrigerator. We used rectangular-shaped samples cut out from the pellets. Gold wires were attached to the samples with silver paste. Magnetization was measured with a SQUID magneto-meter (Quantum Design, MPMS).

RESULTS AND DISCUSSION

Figure 1a shows powder XRD patterns for a series of Na_xWO₃ at RT, which are indexed with a tetragonal supercell of a perovskite structure (2a_p × 2a_p × 2c_p) for x ≤ 0.75 and a cubic supercell (2a_p × 2a_p × 2a_p) for x ≥ 0.8, as indicated previously.^28,29 No impurity phases were detected for all samples. In Figure 1b, we plotted the x dependence of the reduced cell parameters, a_p, along with those of previous reports, where the samples were prepared by solid-state reaction at ambient pressure.^26–28 The results for x ≤ 0.8 agree with the previous results. Whereas the reported data show a deviation from the Vegard law above x = 0.85, our samples follow a linear evolution in the entire range up to x = 1. Extrapolating the proposed relation, a_p = 0.0819x + 3.7846,²⁸ to x = 1 gives a_p = 3.8665 Å, which is fairly consistent with the experimental value of 3.86596(1) Å. This fact strongly suggests that previous studies overestimated the Na content for x > 0.85.^26–28 Deviation from the nominal values in earlier works may be due to Na evaporation. The use of high pressure prevented such undesirable evaporation. Higher compressibility of alkali metal may can also help Na ions be incorporated in the perovskite lattice.

Figure 1.

(a) Powder XRD patterns of Na_xWO₃ measured at RT. (b) Pseudocubic (normalized) lattice parameters, a_p, plotted as a function of x, where data from previous studies^26–28 are included. The solid line represents the proposed relationship, a_p(x) = 0.0819x + 3.7846.²⁸

A Rietveld refinement for the x =1 data assuming the Im$\overline{3}$ structure confirmed the full stoichiometry of our specimen, with reasonable reliability factors of R_wp= 7.37% and R_p = 5.70% and goodness-of-fit of 2.22 (Figure 2a and Table 1). Rietveld analyses for other Na_xWO₃ samples gave results consistent with nominal Na compositions (Figure S1 and Table S1). It is remarkable that ADP of the Na2 site for NaWO₃ (x = 1) is 2.23(15) × 10⁻² Ų. This value corresponding to the isotropic root-mean-square displacement (rmsd) of 0.15(1) Å implies unusually large thermal vibration of sodium ions at the Na2 (6b) site. No anomaly is observed for the Na1 (2a) site (ADP = 7 × 10⁻³ Ų).

Figure 2.

(a) Rietveld refinement of SXRD of NaWO₃ at RT. Red crosses, green solid line, and blue solid line represent observed, calculated, and difference intensities, respectively. The green ticks are the position of Bragg peaks. The inset shows the (103) reflection. (b) Crystal structure of NaWO₃ (Im$\overline{3}$) with octahedral rotation of a⁺a⁺a⁺. Blue, yellow, gray, and red spheres denote Na1 (2a), Na2 (6b), W (8c), and O (24g), respectively. The dotted lines show a 2a_p × 2a_p × 2a_p cell. (c) Two nonequivalent cuboctahedral units of NaO12 around Na1 (yellow) and Na2 (blue). (d) Coordination geometry around Na1 (top) and Na2 (bottom).

Table 1.

Refined Structural Parameters of Na_xWO₃ (x = 1.0) at RT (Space Group: Im$\overline{3}$ (No. 204), a = 7.731934(8) Å)

atom	site	x	y	z	Uiso (×10−2 Å2)
Na	2a	0	0	0	0.7(2)
Na	6b	0	1/2	1/2	2.23(15)
W	8c	1/4	1/4	1/4	0.238(2)
O	24g	0	0.263(2)	0.230(2)	0.77(7)

Together with our previous report,³⁰ we have two isoelectric (d¹) compounds, NaWO₃ (Im$\overline{3}$) with a⁺a⁺a⁺ rotations and KWO₃ (Pm$\overline{3}$m) with a⁰a⁰a⁰ rotations. This offers an ideal opportunity to clarify the effect of WO₆ octahedral rotation on physical properties. First, let us present the temperature variation of resistivity, ρ, and magnetic susceptibility, χ (=M/H), in Figure 3. Both compounds exhibit metallic temperature dependence and almost the same ρ values in a whole temperature range examined (Figure 3a), where no signature of phase transition is observed. This behavior is paralleled with T-independent χ (Figure 3b), and hence, both compounds can be attributed to a Pauli paramagnetic metal. The Curie-type increases in χ at low temperatures below 20 K come from a small amount of free impurity spins (0.01–0.1%).

Figure 3.

(a) Temperature dependence of electrical resistivity, ρ, for 0.1 K < T < 200 K in NaWO₃ and 0.6 K < T < 200 K in KWO₃. The low-T region is magnified in the inset. (b) Temperature dependence of magnetic susceptibility, χ, for NaWO₃ and KWO₃ in the zero-field-and field-cooling process.

On the contrary, heat capacities of the two compounds behave differently, as shown in Figure 4a (C_P/T versus T). This difference can be highlighted for the C_P/T³–T plot (Figure 4b). A broad peak is observed at 30 K only for NaWO₃, suggesting the presence of a low-energy optical phonon. A similar peak in the C_P/T³ versus T curve has been known to appear in cage compounds having large ADPs:^9,34,35 these two features, namely, C_P/T³ peak and large ADPs, are hallmarks of local vibrational mode (or rattling) associated with guest cations inside cages.^3,4 Thus, the observed broad peak of NaWO₃ is most likely attributed to Na2 (6b) ions with the large ADP.

Figure 4.

(a) Temperature dependence of (a) C_P/T and (b) C_P/T ³ for NaWO₃ and KWO₃. Lines are fitting results (see the text for detail). (c) C_P/T ³ versus T plot for Na_0.8WO₃ and Li_0.5WO₃.

The C_P/T³ peak of cage compounds is typically characterized by Einstein specific heat within the “harmonic” approximation.^9,34,35 To do this, we analyzed our raw data of NaWO₃ by a combination of Debye and Einstein specific heats, along with an electronic specific heat of C_e(T) = γT (where γ is the electronic specific heat constant), which is given by

$$C_{\text{P}}(T)=\sum _i{f}_{\text{D},i}{\text{C}}_{\text{D},i}(T)+\sum _j{f}_{\text{E},j}{\text{C}}_{\text{E},j}(T)+{\text{C}}_e(T)$$ (1)

where f_D,i and f_E,j represent the number of the Debye and Einstein modes for the ith and jth contributions, respectively, all of which were determined by fitting. The sum of these is nothing but the total number of phonon modes in the unit cell, that is, ${\sum }_i{f}_{\text{D},i}{\sum }_j{f}_{\text{E},j}=15.C_{\text{D},i}(T)$ and C_E,j(T) denote, respectively, Debye and Einstein specific heat, which are expressed as follows:

$$C_{\text{D},i}(T)=3R{\left(\frac{T}{{\Theta }_{\text{D},i}}\right)}^3{\int }_0^{{\Theta }_{\text{D},i}/T}\frac{e^x{x}^4\text{d}x}{{\left(e^x-1\right)}^2}$$ (2)

$$C_{\text{E},j}(T)=R\frac{e^{{\Theta }_{\text{E},j}/T}}{{\left(e^{{\Theta }_{{\text{E}}_j}/T}-1\right)}^2}{\left(\frac{{\Theta }_{\text{E},j}}{T}\right)}^2$$ (3)

Here, R is the gas constant and ${\Theta }_{\text{D},i}$ and ${\Theta }_{\text{E},j}$ are Debye and Einstein temperatures, respectively, that were also used as fitting parameters. The least-squares fitting was carried out using the data in the temperature range of 2 K < T < 200 K. A reasonable agreement was obtained for NaWO₃ when we assume two Debye and one Einstein modes (i = 2, j = 1) and for KWO₃ with two Debye modes only (i = 2, j = 0) (see Figure 4b and Table 2). It is clear that the additional Einstein term can reproduce the main feature of the C_P/T³ peak in NaWO₃, suggesting rattling-like local vibrational mode. It is noteworthy that the introduction of Na site deficiency (i.e., Na_0.8WO₃) substantially suppresses the C_P/T³ peak, as shown in Figure 4c. This implies that the effect of lattice defect and disorder is not the origin of the peak, as opposed to random or disordered systems such as those containing Bi or Pb with 6s lone pair electrons.^36,37

Table 2.

Parameters Obtained by Fitting the Specific Heat Data of NaWO₃ to Equation 1, Where a Model with Two Debye and One Einstein Mode Was Used^a

mode	${\Theta }_{\text{D},i}{\Theta }_{\text{E},j}(\text{K})$	fD,i, fE,j
D1	962	6.62
D2	371	6.74
E1	151	1.64

^aThe terms of the Debye and Einstein modes are D_i (i = 1, 2) and E₁. The γ value was obtained as γ = 3.34 mJ/K².

The observed rattling phenomenon in NaWO₃ is unprecedented because the simple ABO₃ perovskite can be described by cubic closed-packed arrangement of AO₃ layers with B cations occupied at the interstitial octahedral site, and if the A cation is smaller, BO₆ octahedral rotation takes place to adjust and optimize the local coordination environment around A. In NaWO₃, however, the a⁺a⁺a⁺ rotation generates nonequivalent Na sites, Na1 (2a) and Na2 (6b) (Figure 2b,c). This necessarily leads to a situation that both sites cannot be simultaneously optimized in terms of the local environment, yielding relatively large guest-free-space Δ around the Na2 ion. The value of Δ can be roughly estimated by taking the difference between the cage distance d_cage (from the center) and ionic radius of a guest cation: Δ = d_cage – r(Na⁺). Our structural analysis indicates that Na1 has 12 equidistant Na1–O bonds (2.701(18) Å), whereas Na2 has four short (2.552(17) Å), four medium (2.778(17) Å), and four long (2.916(17) Å) bonds (Figure 2d). The available space of Na1 and Na2 is estimated by calculating d_cage = d – r(O^2–),^{9,24,34,35,38} where d is the Na–O bond length and r(O^2–) refers to the ionic radius of O^2– (=1.4 Å).³⁹ This yielded d_cage = 1.301 Å for Na1 and 1.152, 1.378, and 1.516 Å for Na2. Subsequently, using r(Na⁺) ~ 1 Å,³³ we obtained Δ ~ 0.3 Å for Na1O₁₂ and Δ ~ 0.5 Å for Na2O₁₂. The Δ value of Na2O₁₂ is parallel with those in the known cage compounds: Δ = 0.5–0.9 Å for filled skutterudites,^35,38 Δ = 0.3–0.4 Δ for A-site-ordered quadruple perovskites,^23,24 and Δ = 1–3 Å for clathrates and β-pyrochlores.^3,4,9,34 Together with the large rmsd of 0.15 Å, the large value of Δ for Na2 led us to conclude that the Na2 site is responsible for rattling local vibration.

A comparison with the A-site-ordered quadruple perovskite CuCu₃V₄O₁₂ further revealed that the coefficient of the Einstein specific heat, f_E,1 = 1.64 per the NaWO₃ formula, is larger than 0.27 per the “CuVO₃” formula (i.e., 1/4 × CuCu₃V₄O₁₂).^24,33 This result is fairly consistent with the difference in the rattling site: CuCu₃V₄O₁₂ has a rattler at the A site (2a) with the 2-fold multiplicity, whereas NaWO₃ has a rattler at the A′ site (6b) with 6-fold multiplicity. Another prominent feature that differentiates NaWO₃ from CuCu₃V₄O₁₂ is the degree of octahedral tilting. Although both compounds are isostructural, the octahedral (a⁺a⁺a⁺) tilting in CuCu₃V₄O₁₂ is very extensive and gives the ∠V–O–V bridging angle of ~140°, which forces the A′ site transition metal to adopt the square-planar coordination. As a result, the A site can be a rattler. In contrast, the ∠W–O–W angle for NaWO₃ is ~169°, meaning that moderate octahedral rotation makes the A′ site a rattler.

One can generalize a strategy to find rattling phenomena in the pristine ABO₃ perovskite. Among 15 tilting systems in perovskite,^26,40–42 four systems (a⁺a⁺a⁺, a⁰b⁺b⁺, a⁰b⁺c^–, and a⁺a⁺c^–) have nonequivalent A sites. Examples include Li_xWO₃ (x ≤ 0.5) and Li_0.2ReO₃ with a⁺a⁺a⁺,⁴³ NaTaO₃, NaNbO₃, SrZrO₃, and SrHfO₃ with a⁰b⁺c^–,^41,44 and NH₄MnCl₃ with a⁰b⁺c^–.⁴¹ Unfortunately, most of them exist as an intermediate phase (e.g., 803–893 K for NaTaO₃). As far as we are aware, Li_0.5WO₃ with a⁺a⁺a⁺ is stable at RT,³³ so we prepared the phase-pure Li_0.5WO₃ under high pressure and measured C_P. As expected, a C_P/T³ peak is found at 20 IK, which can be fitted by introducing the Einstein specific heat with ${\Theta }_{\text{E}}=98\text{K}$ (Figure 4c and Figure S4 and Table S4). Extensive research on rattling compounds has shown that ${\Theta }_{\text{E}}$ decreases with increasing the cage size.^3,4,34,35,38 In fact, the smaller ${\Theta }_{\text{E}}$ value for Li_0.5WO₃ (versus 151 K for NaWO₃) could be related to a larger Δ value of ~1 Å.

In order to see the effect of rattling on physical properties, we measured κ for NaWO₃, the result of which is plotted in Figure 5, together with the data for KWO₃. The κ for NaWO₃ is clearly larger than that for KWO₃ in the whole temperature range measured. The Wiedemann–Franz law was used to extract the electronic contribution of thermal conductivity κ_e. Using the 200 K data of ρ, we obtained κ_e = 1.6 W/m·IK for NaWO₃ and κ_e = 2.2 W/m·K for KWO₃. Hence, the difference in κ is ascribed to that of lattice thermal conductivity, κ_L, but is different from what can be expected from the weight difference of K⁺ and Na⁺ ions.^45,46 We thus conclude that rattling vibration of Na⁺ ions largely suppresses κ_L for NaWO₃. The suppression of κ_L is discussed for cage compounds in the context of the rattling effect.^13–15,46 More recently, the rattling and its relation with the suppression of κ_L are discussed in compounds without cage structure^47,48 such as LaOBiS_2–xSe_x, where rattling-like local vibration is found in a cation at the planar coordination geometry with lone pair electrons. These findings give us hope to explore compounds beyond well-known cage systems toward finding low κ_L.

Figure 5.

Temperature dependence of thermal conductivity κ of NaWO₃ and KWO₃.

CONCLUSION

We have extended the solubility limit of Na_xWO₃ to the full stoichiometry (x = 1) by exploiting a high-pressure route. NaWO₃ crystallizes in a perovskite structure with all in-phase octahedral rotations (a⁺a⁺a⁺). Despite the pristine perovskite phase, the specific heat measurement on NaWO₃ exhibits anharmonic phonon modes associated with low-energy rattling vibrations, which is absent in the ideal cubic perovskite KWO₃. The rattling behavior in NaWO₃ is caused by the presence of distinct crystallographic Na sites. The structural analysis revealed that the Na2 (6b) site is a rattler, as opposed to the isostructural quadruple AA′₃B₄O₁₂ perovskite, where the rattling site is the A (2a) cation. The present finding suggests that rattling phenomenon will be available for other ABO₃ perovskites with nonequivalent A sites. Conversely, a cation order may be possible by introducing different alkali metals (e.g, A = Na, A′ = K) in this structural type. Together with our recent study on stoichiometric phases of K_0.6WO₃ and KWO₃,²⁹ the high-pressure method turned out to be highly efficient: Li_xWO₃ (0.5 ≤ x ≤ 1) in the LiNbO₃ structure is also isolated.