Correlated Rattling of Sodium‐Chains Suppressing Thermal Conduction in Thermoelectric Stannides

Tin‐based intermetallics with tunnel frameworks containing zigzag Na chains that excite correlated rattling impinging on the framework phonons are attractive as thermoelectric materials owing to their low lattice thermal conductivity. The correlated rattling of Na atoms in the zigzag chains and the origin of the low thermal conductivity is uncovered via experimental and computational analyses. The Na atoms behave as oscillators along the tunnel, resulting in substantial interactions between Na atoms in the chain and between the chain and framework. In these intermetallic compounds, a shorter inter‐rattler distance results in lower thermal conductivity, suggesting that phonon scattering by the correlated rattling Na‐chains is enhanced. These results provide new insights into the behavior of thermoelectric materials with low thermal conductivity and suggest strategies for the development of such materials that utilize the correlated rattling.


Introduction
Thermoelectric materials generate electric power from heat sources with different temperatures, and are thus expected to find numerous applications as independent power sources; e.g., working in small and closed spaces for planetary probes. Such materials can be used as power sources for sensors and wearable devices connected to the Internet of Things (IoT) by exploiting the difference between the room temperature and waste-heat frameworks that encapsulate guest atoms. [8][9][10][11][12][13][14][15][16][17][18][19][20] The energy of rattling modes and κ lattice of caged compounds, including clathrates and skutterudites, can be reduced by increasing the free space available to guest atoms, [13] while the filling fraction, distribution, and fluctuation of rattlers significantly affect the scattering of low-and mid-frequency phonons, thereby reducing thermal conductivity. [9][10][11] Rattling Cu atoms with planar 3-coordinate bonding were recently observed in non-caged tetrahedrite compounds such as (Cu,Zn) 12 (Sb,As) 4 S 13 that exhibit intrinsically low κ lattice . [21,22] However, the precise mechanism by which rattling phonons reduce κ lattice remains under debate. [18][19][20][21][22] In addition to the cage frameworks found in clathrates and skutterudites, several Sn-based intermetallic compounds (Na 2+x X 2+x Sn 4−x (X = Al, Ga, Sn) and Na 2 ZnSn 5 ) exhibit unique "helical tunnel frameworks" comprising Sn atoms and X or Zn atoms that host Na atoms in quasi 1D open spaces (Figure 1; Figures S1 and S2, Supporting Information). [23][24][25][26][27][28][29][30] Polycrystalline samples of Na 2+x Ga 2+x Sn 4−x (x = 0.19) exhibit excellent thermoelectric properties, with a figure of merit (ZT = 0.98, 295 K) comparable to that of practical Bi 2 Te 3 -based thermoelectric materials near room temperature (ZT = 1). [27] Two polymorphs (hP-and tI-phases) of Na 2 ZnSn 5 with similar helical tunnel structures exhibit particularly low κ lattice values (1.1 and 0.61 W m −1 K −1 , respectively). [29] The previous studies of the helical-tunnel Na-stannides are summarized in the Supporting Information and the thermoelectric properties of the compounds developed herein are tabulated in Table 1. The data indicate helical-tunnel Na-stannides are a promising new class of PGEC materials, despite being unstable in air. [27][28][29] However, the precise role of the rattling Na atoms in 1D space and the mechanism by which they reduce κ lattice remains unclear. It is interesting to understand the role of the rattlers in 1D space and what differentiates these compounds from the compounds with isolated rattling atoms in the cage frameworks in terms of κ lattice through a comprehensive examination of several compounds with similar tunnel structures.
Adv. Mater. 2023, 35,2207646 Ga/Sn Ba ) ) ) ) ) ) ) ) Na (e) correlated rattler ) ) ) ) ) ) ) ) ) ) ) ) ) ) ) Na X/Sn  30 , having a cage framework structure. a) An overview of the crystal structure of Na 2 In 2 Sn 4 , b,c) the helical tunnel containing Na atoms (zigzag chains) as rattlers in Na 2 In 2 Sn 4 , and d) a schematic picture of the correlated rattling. e,f) The cages of β-Ba 8 Ga 16 Sn 30 that enclose Ba atoms as isolated rattlers. The Na atoms in Na-(X)-Sn compounds (a-d) exhibit large one-directional thermal motion along the helical tunnel and exhibit shorter inter-atom distances than rattlers isolated in the cages of clathrate compounds (e, f). The thermal ellipsoids in (b) are drawn at 75% probability. In this study, the mechanism by which PGEC compounds achieve a low κ lattice was elucidated, particularly in relation to the vibrational states of Na and host atoms of Na-stannides with helical tunnel frameworks. The specific heat, crystal structures, and phonon states of five such stannides were analyzed experimentally and by ab initio lattice dynamics. Na atoms are shown to oscillate anisotropically in the tunnel and coupled with each other as in a chain and with the framework lattice. The correlation between Na atoms plays an important role in lowering the κ lattice of each compound by triggering the anharmonicity of the phonons in the frameworks.

Thermoelectric Properties of Dense Bulk Samples
A densely packed pellet of a polycrystalline specimen is required to obtain a reliable estimate of κ lattice . The densities of previously prepared samples of the Na-stannides were insufficient with a relative density of ≈75%, [27,28] except for hP-and tI-Na 2 ZnSn 5 where ingots were obtained. [29] Thus, in this study, polycrystalline samples of Na and Na 2 In 2 Sn 4 with a relative density of 97% or greater were prepared using the press-sintering method ( Figure S3, Supporting Information). A monophasic sample can be synthesized reproducibly under the conditions described in Section 4; however, the electronic transport properties of the compounds still exhibit some sample dependence, [27] Table 1 and Figure 2.
The electrical resistivity (ρ) and thermal conductivity (κ total ) of the dense samples of Na  Figure 2) showed lower and higher values, respectively, than those of the samples with lower densities in preceding studies. [27,28] These measured values result in the ZT value of 0.34 at 295 K and 0.47 at 375 K for Na 1 [31][32][33] and Al 0.3 La 0.5 Co 4 Sb 12 skutterudite, [12] with caged frameworks, and tetrahedrites Cu 12−x Tr x Sb 4 S 13 (Tr = Mn, Fe, Co, Ni, Zn), [21] with non-caged structures. Accordingly, Na 2+x X 2 +x Sn 4−x (X = Al, Ga, In) are expected to be excellent thermoelectric materials owing to their low κ lattice values.
The κ lattice of filled and unfilled skutterudite compounds varies significantly depending on the occupancy of rattler atoms encapsulated in the cage-like framework. [9][10][11] The κ lattice of the tunnel compound having a solid-solution phase with the composition Na 2+x Ga 2+x Sn 4−x (0 ≤ x ≤ 0.25) may decrease as the number of rattlers increases with increasing x. However, the κ lattice of a sintered Na 2+x Ga 2+x Sn 4−x polycrystalline sample with x = 0 (0.60 W m −1 K −1 ) was essentially identical to the κ lattice (0.59 W m −1 K −1 ) of the sample with x = 0.19 (Table S1, Supporting Information). The effect of the non-stoichiometric composition of rattlers on thermal conductivity for tunnel compounds is expected to be small. Future systematic studies on the compounds with a wider solid solution range may be uncovered in more detail showing the effect of non-stoichiometry.

Specific Heat Capacities and Low-Temperature XRD Lattice-Dynamics
The specific heat capacities (C total ) and atomic displacement parameters (ADPs) obtained from X-ray diffraction (XRD) refinement have been analyzed to determine the vibrational states of the Na and host atoms in the Na-stannide system. Plots of the C total /T 3 of Na 1.82 Al 1.82 Sn 4.24 , Na 2.19 Ga 2.19 Sn 3.81 , Na 2 In 2 Sn 4 , and hP-and tI-Na 2 ZnSn 5 as a function of T show broad peaks at ≈10 K (Figure 3). Similar plots of clathrate compounds exhibit the same peak that was attributed to the rattling guest atoms. [14,15] The heat capacities of the Na-stannides were analyzed using Equations (1)-(4), assuming that the C total of the stannides could be expressed as a summation of the electronic (C c ), Debye (C D ), and Einstein (C E ) specific heat capacity components in the same way as that of clathrate compounds ( Table 2).
Here, γ is the electronic specific heat coefficient; N D and N E are the numbers of Debye and Einstein modes, respectively, θ D and θ E are the Debye and Einstein temperatures, respectively, and R is the gas constant.
The parameters of Equations (1)-(4) were optimized by fitting the measured C total /T 3 values under the constraint 3N = N D + N E ( Figure 3 and Table 2), where N is the number of atoms in each chemical formula. The N D and N E of the clathrate compounds are essentially identical to the number of host and guest atoms per chemical formula multiplied by three, respectively, implying each mode has three degrees of freedom, i.e., 3D motion. [14,15] Interestingly, the N E values obtained by the same fitting method for the Na-stannides synthesized in this study, ranged from 0.3 to 2.4, significantly lower than the theoretical N E values (5.46-6.57, assuming three times the number of guest Na atoms as per the chemical formulae). The low N E values strongly suggest that motion of Na atoms in the helical tunnels is restricted to a single degree of freedom, i.e., onedirectional Einstein vibration. Single-crystal X-ray analysis of all helical-tunnel Na-stannides reveals large, one-directional vibrations of Na atoms along the helical tunnels, as demonstrated by the elongation of thermal ellipsoids along the tunnel at the Na site ( Figure 1; Figure S2, Supporting Information). [27][28][29] The characteristic Debye (Θ D ) and Einstein (Θ E ) temperatures of the host and guest atoms were estimated from the temperature dependence of the equivalent isotropic ADPs (U eq ) and anisotropic ADPs (U ij ) by fitting with Equations (5) and (6) (Debye and Einstein models, respectively) together with the temperature-independent disorder parameters (d) ( Table 2, Figure 4; Tables S2-S10, Supporting Information). [13,23] 3 4 e xp 1 4 eq,aniso where h and k B are the Planck and Boltzmann constants, respectively, and m is the mean atomic mass.
Adv. Mater. 2023, 35, 2207646 Table 2. Characteristic parameters obtained from the analysis of heat capacities and single-crystal XRD patterns of the helical-tunnel Na stannides.
Heat capacity (polycrystalline) XRD (single crystal) Small negative γ values of −3.3 × 10 −4 J K −2 mol −1 (Na 2 In 2 Sn 4 ) and −1.6 × 10 −4 J K −2 mol −1 (hP-Na 2 ZnSn 5 ) were initially obtained, but they are theoretically meaningless and indicate that each compound is a degenerate semiconductor with a very small γ value. Therefore, the γ values of Na 2 In 2 Sn 4 and hP-Na 2 ZnSn 5 were set to 0 in the analysis of their heat capacities. The obtained N E , N D , and characteristic temperatures remained essentially unchanged (Table S11, Supporting Information); b) Ref. [29] Figure 4. a) Equivalent isotropic displacement parameters (U eq ) of the Na and (Al, Ga, or In)/Sn sites of Na 1.82 Al 1.82 Sn 4.24 (red squares), Na 2.19 Ga 2.19 Sn 3.81 (black diamonds), Na 2 In 2 Sn 4 (blue triangles), Na sites of hP-Na 2 ZnSn 5 (violet circles), and tI-Na 2 ZnSn 5 (green circles) [29] as a function of temperature. b) Principal axes of anisotropic atomic displacement ellipsoids are along the major axis (U anis_a ) of the Na site in each compound against T. The scale of the vertical axis in (b) is twice that in (a).
The Θ D values of the framework atoms (163-221 K) calculated from U eq are similar to those obtained by specific heat analysis (174-199 K). In contrast, the Θ E values of Na atoms obtained from the isotropic ADPs (U eq ) and from the major-axis component of the anisotropic ADPs, (U aniso_a ) were 90-118 K and 72-83 K, respectively, with the latter being closer to the θ E values obtained using the specific heat (38.5-57.7 K; Table 2). These analyses of the C total and ADP reveal that the Na atoms in the tunnel exhibit anisotropic rattling dynamics with largeamplitude oscillations along the tunnel.
The measured specific heat of compounds having lower κ lattice exhibits peak broadening and have a lower intensity. In particular, the specific heat of Na 1.82 Al 1.82 Sn 4.18 , which has the lowest κ lattice , is not adequately reproduced by a simple summation of the electronic, Debye, and Einstein specific heat capacities (Figure 3). This demonstrates that in addition to a reduction in the number of Einstein vibrational modes of Na atoms, the anisotropic vibrational modes deviate from the simple Einstein model (i.e., isolated harmonic modes where atoms do not affect each other). It suggests the existence of interactions among Na atoms, and we will discuss them further in the following sections.

Calculated Phonon Dispersions and Density of States
To investigate the low κ lattice of the helical-tunnel Na-stannides, the phonon dispersions, total and partial densities of states (DOSs), and temperature dependences of the specific heat and ADP of Na 2 Ga 2 Sn 4 were calculated by ab initio lattice dynamics using ordered structure models (Figure 5a; Table S12, Supporting Information). Einstein-like peaks are found ≈12.5 meV in the partial DOS of Na where those of Sn and Ga are modest. In addition, the phonon dispersions and partial DOSs (Figure 5b) show many avoided crossings in a wide energy region of 4-13 meV from the Γ point to the M, K, and A points, in which the heat-carrying acoustic-phonon branches are flattened by the interaction between the Na phonons and the acoustic phonons of the framework lattice. Avoided crossings have also been observed in some clathrates and are generally considered an important factor in the reduction of κ lattice (= ⅓C v ν 2 τ) owing to enhanced phonon scattering caused by a significant decrease in phonon lifetime (τ) and a reduction in the group velocity (ν) of the framework phonons near the resonant energies or across a wide energy range. [18][19][20]34] The partial DOS (PDOS) of Na of Na 2 Ga 2 Sn 4 exhibits two humps in the low-energy region at 3.9-6.1 meV and 7.1-9.5 meV, both roughly corresponding to those of metallic Na ( Figure S4, Supporting Information), overlapped with the partial DOSs of Sn and Ga (Figure 5b). Furthermore, the PDOS of Na is also observed to be overlapped with those of framework atoms up to 25 meV, indicating substantial interactions between the Na atoms and the framework. The temperature dependence of the calculated specific heat capacity (θ E = 59 K = 5.1 meV) obtained from the phonon structures is in good agreement with the measured specific heat capacity of Na 2.19 Ga 2.19 Sn 3.81 (θ E = 49 K = 4.2 meV), as shown in Figure 3b. The calculated atomic displacement ellipsoids of Na atoms are also elongated along the direction of the tunnel, in agreement with the ellipsoids refined by X-ray structure analysis (Figure 5a). The Θ E of Na atoms obtained from the calculated APDs is 74.1 K (=6.6 meV), which is in excellent agreement with that estimated from the measured ADPs of Na 2.19 Ga 2.19 Sn 3.81 (74 K; Figure S5 and Table S13, Supporting Information). These results indicate that Na atoms behave, at least in part, as 1D Einstein-like oscillators. Simultaneously, these atoms are coupled with framework atoms, which are described as swings along the tunnel. The dynamics of the Na atoms scatter the phonons dominated by the framework atoms over a wide energy range via the NaFW) interaction, thereby reducing κ lattice .

Correlated Rattling Examined through Ab Initio Lattice Dynamics
The κ lattice of the helical-tunnel Na-stannides varies by a factor of ≈2.5 (0.42-1.1 W m −1 K −1 ). Caged compounds with a larger amount of free space for the rattling of guest atoms typically exhibit smaller Θ E values, while the guest atom has a larger ADP, resulting in a lower κ lattice value; the κ lattice of a given  compound decreases with Θ D . [13,35] However, an analysis of the heat capacities and ADPs of each compound (Table 2; Figure S6, Supporting Information) reveals that the κ lattice of the helical-tunnel Na-stannides is independent of the Debye and Einstein temperatures (θ D , Θ D , θ Ε , Θ E ). The κ lattice of the tunnel compounds exhibits a strong positive dependence on the interatomic distance (d NaNa ) between Na atoms orderly arranged in the tunnel of Na 2 In 2 Sn 4 and the corresponding distances between Na sites considering their site occupancies in other compounds ( Figure S2, Supporting Information), as shown in Figure 6 and Figure S7 (Supporting Information). Na 1.82 Al 1.82 Sn 4.18 , which has the shortest d NaNa (3.694(8) Å), also shows the lowest κ lattice (0.42 W m −1 K −1 ), while hP-Na 2 ZnSn 5 , which has the longest d NaNa (3.872(11) Å), exhibits the highest κ lattice (1.1 W m −1 K −1 ). These d NaNa values are similar to that in metallic sodium (3.709 Å) [36] and are as short as 47-65% of the distances between rattling guest atoms encapsulated in the cage compounds (d BaBa in Ba 8 Ga 16 Sn 30 , 5.84 Å (α phase) and 6.08 Å (β phase); [37] d NaNa in Na 4 Si 23 , 5.70 Å; [38] d LaLa in La 0.5 Co 4 Sb 12 , 7.83 Å [12] ). This correlation between the κ lattice and d NaNa values of the helical-tunnel Na-stannides remains unchanged even if κ lattice values are estimated using other κ carrier values, e.g., even when Lorenz numbers described by Kim et al. [39] are used, for example, ( Figure S8, Supporting Information).
Despite the clear correlation between lattice thermal conductivity and d Na-Na , however, there is no clear link between these two factors of the correlation from the viewpoint of physics that is a mechanism behind thermal conduction in these compounds and thus the correlation remains a phenomenological explanation at best. To bridge the missing link between thermal conductivity and d Na−Na , thorough examinations of interatomic force constants have been made by means of ab initio lattice dynamics within the harmonic approximation, without using any of the experimentally observed values, except for thermal conductivity, for consistency of the analyses. Second order force constants determine all the phonon states and phonon-related properties within the harmonic approximation. Even though higher order force constants determine thermal conductivity, they are just corrections to the second order force constants for vibrating atoms instead of static atoms at their respective stable positions. Thus, we focus on the second order force constants as their magnitudes to explain: a) interatomic bonds that govern thermal conduction and b) the interatomic bonds that trigger phonon scattering that determines thermal conductivity. Thermal conductivity can be described according to the simplest theory as follows: where C V , v G , and λ are heat capacity, group velocity, and phonon mean free path, respectively. As can be seen from Figure S9 (Supporting Information), no clear correlation between the C V and v G and the d Na−Na is found, though v G may decrease with increasing d Na−Na . Here, we intentionally neglect the dependence of v G on d Na−Na in the following discussion for simplicity and clarity before recalling the dependence of v G on d Na−Na at the end of the discussion. This leaves phonon mean free path or phonon relaxation time that determines phonon scattering or thermal conductivity in these compounds. The scalarized force constants for NaNa bonds, , and for atoms in the framework, FW FW sum k − are tabulated in Table 3 together with their respective per-bond values, Na Na bond k − , Na FW bond k − and FW FW bond k − . Since the sum showed negligible dependence on cutoff distance if the nearest neighbor atoms are included, the sums were taken throughout the supercells that are longer than 10 Å along each axis. Although disparities in d Na−Na between experimental and computed values are seen, it can be simply attributed to the use of the ordered models while averaged values among many microscopic states or configurations of atoms by partial occupations were observed in experiments. Interactions among framework atoms measured by FW FW sum k − are much greater than other interatomic interactions given by Na FW sum k − and Na Na sum k − (Figure 7a), which manifests that the FW atoms dominate thermal conduction in these compounds.
In contrast to the caged compounds where rattling atoms are rather isolated in cages, Na Na bond k − are greater than that between Na atoms and framework atoms (FW), Na FW bond k − when per-bond values are compared (Table 3), which explains the substantial Debye contribution to specific heats of these compounds discussed above. On the other hand, when sum of these force constants is compared, Na FW sum k − , becomes greater than Na Na sum k − due to greater number of bonds between an Na atom and surrounding FW atoms. The Na FW sum k − , increased with increasing Na Na sum k − as shown in Figure 7b, which explains the influence of phonon states of Na atoms in a zigzag chain onto phonon states of framework atoms, or dynamics of framework atoms that dominate thermal conduction in these compounds. The Na Na sum k − decreases with increasing d Na−Na in the zigzag chain (Figure 7c) that indicates more pronounced Einstein-like behavior with increasing d Na−Na and vice versa.
These explain the reason why the correlation between κ lattice and d Na−Na was observed ( Figure 6) via dynamic interactions between Na atoms in the zigzag chain with FW atoms that determine κ lattice . As compared with caged compounds with Adv. Mater. 2023, 35, 2207646   Figure 6. Lattice thermal conductivity (κ lattice ) as a function of the Na-Na interatomic distance (d Na-Na ) in helical-tunnel compounds. rattlers, these compounds possess an extra degree of freedom to control κ lattice ; interaction within the zigzag chain of Na or separation of the Na atoms. With further increasing d Na−Na , it falls back to the rattling in the caged compounds when FW atoms dominate thermal conduction. Even if we recall the dependences of v G and FW FW sum k − on d Na−Na , this argument is not changed but rather enhanced and can be explained by phonon conduction in dynamic interaction of phonons in nanometer dimension. [40]

Overall Pictures of Correlated Rattling of Sodium Chains
Despite the differences among the helical-tunnel Na-stannides, including their crystal systems, the number of atoms in the unit cell, elements forming the framework, and ordering of elements, the observed correlation between κ lattice and d NaNa provides phenomenological evidence supported by underlying physics, i.e., dynamics of chained rattling atoms. Hybrid nature of Einstein-Debye oscillation of Na atoms in the tunnel framework is evidenced by analyses of heat capacity. Anisotropic vibrations of Na atoms are uncovered by low-temperature single-crystal X-ray analysis. Lattice dynamics calculations supported the experimentally observed results and revealed that Na atoms are coupled with each other like a chain and with the framework lattice. Thus, these Na chains are best described as "quasi-1D strongly correlated Na atoms" and explain the clear correlation between κ lattice and d NaNa in helical-tunnel Na-stannides. Unlike rattlers in the caged compounds, Na atoms in the helical tunnel are not spatially isolated and experience stronger forces among themselves as d NaNa decreases. This modifies phonon propagations through framework atoms, invoking anharmonicity of the phonons, when Na atoms vibrate in a coupled manner in the zigzag chain. The similar is also observed in thermoelectric layered cobaltites. [41] The conventional caged compounds consisting of closedcages provide a rattler atom with only one degree of freedom to disturb the acoustic modes dominated by the framework. In contrast, the quasi-1D chains of rattler atoms in the helicaltunnel Na-stannides provide one extra degree of freedom to manipulate framework phonons through interatomic interactions in the chain. The dependence of these interactions on the interatomic NaNa distance facilitates the precise control of the thermal conductivity of multiple frameworks. Further, the interaction can also be indirectly controlled even with a single choice of elements in a framework. The results indicate a correlated rattling with an extra degree of freedom owing to the interaction of Na atoms in the chain, thus paving the way for an alternative approach to tailor thermal conduction and develop superior thermoelectric materials. Further research into the tunnel compounds with close-packed rattling atoms in a chain and with a wider range of solid-solution compositions including rattling atoms would clarify the effects of the rattler-rattler interactions, rattler-framework interactions, and nonstoichiometry on the thermal transport properties and rattling phenomena.

Conclusion
The thermoelectric properties of the tin-based intermetallics with tunnel frameworks containing Na guest atoms Adv. Mater. 2023, 35,2207646   were investigated using dense bulk samples prepared by pressure-assisted sintering. The Ga-containing stannide (Na 2.19 Ga 2.19 Sn 3.81 ) showed the highest dimensionless figure of merit (ZT: 0.82 at 295 K and 1.20 at 387 K), while all compounds had low lattice thermal conductivities (κ lattice : 0.42-1.1 W m −1 K −1 ), which exhibit a strong positive correlation with interatomic distance of Na atoms in the tunnel. X-ray crystallographic and specific heat analyses revealed that Na atoms and framework behave as Einstein-like vibrating rattlers along the tunnel and as a Debye solid, respectively. While ab initio lattice dynamics suggest the presence of significant interactions both among Na atoms and between Na atoms and the framework, the chain-like Na rattlers with large anisotropic oscillations strongly scatter the phonons dominating the thermal conduction in the framework lattice via anharmonicity of the vibrational modes enhanced by the correlation between rattling atoms that become stronger with decreasing inter-rattler distance. These findings provide a deeper understanding of the mechanism by which κ lattice is reduced, and thus facilitate the development of the high-performance thermoelectric materials.

Experimental Section
Details of Experiments: All manipulations of air-sensitive materials were carried out in an Ar-filled glove box. Ternary stannides of Na 2+x X 2+x Sn 4−x (X = Al, Ga, In) and Na 2 ZnSn 5 are unstable in air and gradually decompose into Sn or Sn-In binary compounds, group 13 and 12 elements, and some sodium hydroxides. Sodium (lump, 99.95%, Nippon Soda), aluminum (powder, 99.99%, Rare Metallic Co. Ltd.), gallium (shot, 99.9999%, Dowa Electronics), indium (lump, 99.999%, Mitsubishi Metal Corporation), zinc (wire, 99.99%, Nilaco Co. Ltd.), tin (shot, ≥99.99%, Rare Metallic and Mitsuwa Chemicals Co., Ltd.), and NaSn powder (prepared in-house) were used as the starting materials. Na 2+x X 2+x Sn 4−x (X = Al, Ga, In) and two polytypes of Na 2 ZnSn 5 were synthesized according to previously reported procedures. [27][28][29][30] Because Na 2+x Al 2+x Sn 4−x could only be obtained as a solid solution with a less-than-stoichiometric Na content, [28] the most Na-rich composition (x = −0.18, Na 1.82 Al 1.82 Sn 4.18 ) was chosen because the rattling of Na is an important factor in the reduction of the lattice thermal conductivity. The Na-rich sample (x = 0.19 in Na 2.19 Ga 2.19 Sn 3.81 ), which gave the highest thermoelectric properties in the previous work on Na 2+x Ga 2+x Sn 4−x , was prepared for the same reason. [27] No solid solution with a nonstoichiometric range of compositions was found in Na 2 ZnSn 5 or Na 2 In 2 Sn 4 .
To synthesize polycrystalline samples of Na 1.82 Al 1.82 Sn 4.12 , a mixture of Sn, Al, and NaSn powders was pressed into disc-shaped compacts (ϕ10-11 × 3−4 mm 3 ) with a die at a pressure of ≈200 MPa. The compacts were loaded in boron nitride (BN) crucibles (99.5%, ϕ12 × 18 mm 3 , Syowa Denko K. K.), sealed in a stainless steel (SUS) container, and heated at 623 K for 10 h. The obtained samples were pulverized and mixed using an agate mortar and pestle and pressed into compacts again. The compacts were then heated at 623 K for 18-36 h in a crucible sealed in a SUS container, followed by furnace cooling to room temperature. The procedure involving pulverization, mixing, compaction, and heating was performed in triplicate to obtain homogeneous samples. The dense bulk samples (φ12 × 3-4 mm 3 ) were prepared at 573-593 K and 50 MPa for 20 min by pressure sintering.
To synthesize polycrystalline Na 2.19 Ga 2.19 Sn 3.91 samples, Na and Sn were first heated to 873 K for 4 h and then held at this temperature for 2 h in a BN crucible sealed in a SUS container. After heating, Ga was added to the sample in the crucible. The crucible was then sealed again in another SUS container and heated at 973 K for 2 h. Pulverization of the obtained samples, compaction, and heating at 693-713 K for 30-60 h was performed in duplicate. Sample densification was carried out at 673-693 K under 50 MPa for 20 min. The polycrystalline Na 2 In 2 Sn 4 samples were synthesized as follows: the starting materials were heated at 823 K for 10 h and then pulverized, compacted, and heated at 613 K for 18-36 h three times. The dense bulk samples were prepared at 578 K and 50 MPa for 20 min.
Ingots of hP-Na 2 ZnSn 5 were prepared by heating the starting material at 773 K for 10 h followed by furnace cooling to room temperature. Polycrystalline tI-Na 2 ZnSn 5 samples were obtained by heating the starting materials at 773 K for 10 h, followed by heating the powder mixture compacts of the obtained samples at 623 K for 36 h twice. The dense bulk samples were prepared at 573 K and 50 MPa for 20 min. The densities of the samples were determined from their weights and volumes. The crystalline phases of the samples were identified by XRD analysis using a Bragg-Brentano type powder diffractometer (Bruker AXS, D2 PHASER, CuKα).
Single crystals of Na 2+x X 2+x Sn 4−x (X = Al, Ga, In) were retrieved from the fragments of the samples prepared from the starting materials with the aforementioned compositions under the following conditions: X = Al, the sample was heated at 673 K and cooled to 653 K for 20 h; X = Ga, the sample was heated at 753 K and cooled to 673 K for 40 h; X = In, the sample was heated at 933 K and cooled to 833 K for 2 h, followed by cooling to 673 K for 400 h. The single crystals were sealed in glass capillaries (outer diameter of 0.5 mm) in a glove box prior to XRD analysis using Mo Kα radiation (λ = 0.71075 Å) on a single-crystal X-ray diffractometer (D8 QUEST, Bruker AXS Inc.) with a multilayered confocal mirror and a CMOS detector. Measurements were made at 300, 250, 200, 145, and 90 K. The XRD data collection, unit-cell refinement, and absorption correction were performed using the Bruker Instrument Service V4.2.0 and APEX3 (Bruker AXS Inc., 2016). [42,43] The crystal structures were refined using full-matrix least-squares on F 2 with the SHELXL-2014 software. [44] Visualization of the crystal structures was performed using the software package VESTA. [45] The electrical resistivity (ρ), Seebeck coefficient (S), and thermal conductivity (κ) of identical dense bulk samples were measured at 295−393 K by the direct current four-probe method (ρ) and the thermoelectric-power temperature-difference method (S) using a self-made cell, and by the hot disk method [46] (κ) using a commercial instrument (TPS-2500S, Hot Disk AB) with a single-side mode in an Ar-filled glove box. The specific heat capacities of fragments of the dense bulk samples were measured in the range of 2-300 K in a commercial calorimeter (PPMS, Quantum Design, Inc.) by the thermal relaxation method.
Details of Ab Initio Lattice Dynamics: To identify phonon states of Na-chains and frameworks, ab initio lattice dynamics calculations were performed using VASP [47,48] and phonopy [49] codes to obtain optimized atomic configurations and phonon states, respectively. An initial structure model of each compound was constructed such that Na atoms are equally separated and located at the second nearest neighbor site from X-substituted site that corresponds to the configuration showing the minimum total energy, according to preliminary calculations for possibly all the possible configurations for Na 2 Ga 2 Sn 4 . Electronic structure calculations were conducted using a plane-wave basis set with a plane wave cut-off energy, k-point mesh grid using Γ-point centered Monkhorst-Pack scheme, [50][51][52] and real-space mesh grid for fast Fourier transformation to ensure numerical accuracies of 0.1 meV per atom. The mesh grids in reciprocal and real spaces are common to a unitcell containing two Na atoms or the equivalent. The criteria for convergency are 1.0 × 10 −8 eV per unitcell and 1.0 × 10 −4 eV Å −1 per atom for total energy in electronic structure calculations and structure optimization of both lattice constants and atomic positions to ensure precise evaluations of force constants between atoms. Inner-core electrons are treated as pseudo-potentials using the Projector-augmented wave (PAW) method, [53] with an exception of the 2p electrons of Na, which are treated as valence electrons. The generalized gradient correction approximation (GGA) formulated by Perew, Berke, and Ernzerhof (GGA-PBE) [54] was used for the exchange-correlation functional, which is known to underestimate bond length and overestimate bond strength for a given atomic configuration. Phonon states were calculated within the harmonic approximation using the finite displacement method [55,56] wherein a displacement of 0.01 Å is given to an atom at a time to obtain the second order force constant matrices and then a dynamical matrix, the solution to which yields phonon eigenvalues and eigenvectors. Phonon partial density of states was calculated with enough fine mesh to ensure convergence with each atom's contribution decomposed without ambiguity. Anisotropic thermal ellipsoids of atomic displacements were calculated based on the cif definition from the phonon eigenvectors and eigenvalues following the scheme developed by Grosse-Kunstleve and Adams. [57] Values of C V and v G obtained by ab initio lattice dynamics are represented by the constant volume heat capacity of respective compounds at 300 K ( Figure S9a, Supporting Information) and group velocity around Γ point that corresponds to sound velocity in these compounds ( Figure S9b, Supporting Information), respectively, for simplicity and clarity.
In this study, the second order force constant matrices, Φ ij , were scalarized for ease of comparisons. The force exerted from atom i to atom j, F ij , is approximated as where r ij is a vector pointing from atom i to atom j, and Δr is an infinitesimal atomic displacement. From the force, the second order force constants between atom i and atom j, k ij , can be readily calculated as within the harmonic approximation. It should be noted that the sum can be readily converted to energies of harmonic oscillators using infinitesimal displacements of atoms, Δr.

Supporting Information
Supporting Information is available from the Wiley Online Library or from the author.