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On the Thermal Conductivity and Local Lattice Dynamical Properties of NASICON Solid Electrolytes
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  • Thorben Böger
    Thorben Böger
    Institute of Inorganic and Analytical Chemistry, University of Münster, Münster D-48149, Germany
    International Graduate School for Battery Chemistry, Characterization, Analysis, Recycling and Application (BACCARA), University of Münster, Münster D-48149, Germany
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  • Tim Bernges
    Tim Bernges
    Institute of Inorganic and Analytical Chemistry, University of Münster, Münster D-48149, Germany
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  • Matthias T. Agne
    Matthias T. Agne
    Department of Chemistry and Biochemistry, University of Oregon, Eugene, Oregon 97403, United States of America
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  • Pieremanuele Canepa
    Pieremanuele Canepa
    Department of Materials Science and Engineering, National University of Singapore, 117575Singapore
    Department of Chemical and Biomolecular Engineering, National University of Singapore, 117585Singapore
    Department of Electrical & Computer Engineering, University of Houston, Houston, Texas 77204, United States of America
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    Orcidhttps://orcid.org/0000-0002-5168-9253
  • Frank Tietz
    Frank Tietz
    Institute of Energy Materials and Devices (IMD-2), Forschungszentrum Jülich, Jülich D-52425, Germany
    Institute of Energy Materials and Devices (IMD), IMD-4: Helmholtz-Institut Münster, Forschungszentrum Jülich, Münster 48149, Fed. Rep. Germany
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  • Wolfgang G. Zeier*
    Wolfgang G. Zeier
    Institute of Inorganic and Analytical Chemistry, University of Münster, Münster D-48149, Germany
    International Graduate School for Battery Chemistry, Characterization, Analysis, Recycling and Application (BACCARA), University of Münster, Münster D-48149, Germany
    Institute of Energy Materials and Devices (IMD), IMD-4: Helmholtz-Institut Münster, Forschungszentrum Jülich, Münster 48149, Fed. Rep. Germany
    *Email: [email protected]
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    Orcidhttps://orcid.org/0000-0001-7749-5089
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Journal of the American Chemical Society

Cite this: J. Am. Chem. Soc. 2024, 146, 47, 32678–32688
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https://doi.org/10.1021/jacs.4c12034
Published November 13, 2024

Copyright © 2024 The Authors. Published by American Chemical Society. This publication is licensed under

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Abstract

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The recent development of solid-state batteries brings them closer to commercialization and raises the need for heat management. The NASICON material class (Na1+xZr2PxSi3–xO12 with 0 ≤ x ≤ 3) is one of the most promising families of solid electrolytes for sodium solid-state batteries. While extensive research has been conducted to improve the ionic conductivity of this material class, knowledge of thermal conductivity is scarce. At the same time, the material’s ability to dissipate heat is expected to play a pivotal role in determining efficiency and safety, both on a battery pack and local component level. Dissipation of heat, which was, for instance, generated during battery operation, is important to keep the battery at its optimal operating temperature and avoid accelerated degradation of battery materials at interfaces. In this study, the thermal conductivity of NaZr2P3O12 and Na4Zr2Si3O12 is investigated in a wide temperature range from 2 to 773 K accompanied by in-depth lattice dynamical characterizations to understand underlying mechanisms and the striking difference in their low-temperature thermal conductivity. Consistently low thermal conductivities are observed, which can be explained by the strong suppression of propagating phonon transport through the structural complexity and the intrinsic anharmonicity of NASICONs. The associated low-frequency sodium ion vibrations lead to the emergence of local random-walk heat transport contributions via so-called diffusons. In addition, the importance of lattice dynamics in the discussion of ionic transport as well as the relevance of bonding characteristics typical for mobile ions on thermal transport, is highlighted.

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Introduction

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Solid-state batteries use ion-conducting solids instead of liquid electrolytes and have been extensively studied in recent years. (1,2) Especially limited lithium resources and a priority on cost-effectiveness make solid-state sodium batteries a promising option. (1,2) Various categories of solid-state sodium-ion conductors are under examination, including the Na3PS4 family, (3) borate-hydrides, (4) the newly found oxyhalides, (5) Na-β/β″-alumina, (6) and the Na1+xZr2PxSi3–xO12 (0 ≤ x ≤ 3) substitution series. (7) The latter demonstrates superior thermal, chemical, and electrochemical stability, reducing the risk of thermal runaways, e.g., at the electrode/electrolyte interfaces. (8) First reported by Hong (9) and Goodenough et al., (10) these compounds are best known under the name Na+ superionic conductors (NASICONs) given their high ionic conductivity. The ionic conductivity of the series is the highest for x ≈ 2.4, with the endmembers, NaZr2P3O12 and Na4Zr2Si3O12, themselves being rather poor ionic conductors. (11) By substitution of zirconium with other transition metals and even phosphorus and silicon with heavier homologues, a large variety of derived compounds can be obtained and the ionic conductivity can be increased to higher values. (7,12)
Typically, the focus in these materials is on maximizing sodium-ion conduction and its relation to structural features. (13−15) To date, only a few studies aim to understand thermodynamic properties, (16−19) such as heat capacity, or enthalpy and entropy of formation, or even the lattice dynamics (20,21) in these materials. Recently, Morgan et al. (20) and Zhen et al. (21) employed ab initio phonon calculations for NaZr2P3O12 to improve the accuracy of chemical shift calculations for nuclear magnetic resonance and understand thermal expansion in NaZr2P3O12. Morgan et al. (20) further used lattice dynamics calculations within the harmonic approximation to report anisotropic thermal displacement parameters and highlighted the importance of anharmonicity in this system. Zhen et al. (21) reported phonon band structures and phonon density of states (DOS) and utilized the quasi-harmonic approximation to derive Grüneisen parameters. Although anharmonicity and phonon band structures are the basis for thermal transport in materials, not much is known about the thermal conductivity of these ion conductors.
When moving to large-scale applications, the thermal conductivity, which is dominated by the material’s lattice dynamics in electronic insulators, is of great importance. (22) Hence, thermal conductivity characterizes the ability of solid electrolytes to dissipate heat generated during cycling. (23) Good heat dissipation allows the heat to be removed faster, resulting in a lower equilibrium temperature of the battery (pack). High temperatures due to slow heat dissipation can drive the system out of its stable or at least optimal operating temperature. Therefore, ensuring all battery components are inside an optimal temperature window is indispensable to optimize battery performance and lifetime and to avoid device failure by thermal runaway. These considerations can be safeguarded by simulations predicting the temperature distribution inside a solid-state battery cell during cycling. (24,25) As simulations rely on accurate experimental input parameters, comprehensive experimental knowledge about the thermal properties of all materials and a fundamental understanding of how heat distributes within a cell are required to evaluate solid-state battery safety.
Temperature gradients drive macroscopic thermal transport in the materials. At a microscopic scale, temperature differences translate to an inhomogeneous occupation of vibrational states, which thermodynamics seeks to balance. (26) This balancing occurs by a net diffusion of heat-carrying phonons from the warmer to the colder region, which can macroscopically be observed as thermal conduction. The so-called phonon-gas model has traditionally been used in crystalline materials to describe this balancing rate. (27) In this model, lattice vibrations, quantified as quasiparticles called phonons, carry thermal energy proportional to their heat capacity, mean free path length, and velocity. These phonons propagate through the crystal like atoms in an ideal gas until a scattering event occurs. (27,28) In amorphous materials, the lack of long-range atomic order results in very small mean free path lengths that are in the order of interatomic distances and concurrently, very low thermal conductivities are observed. (29,30) Thermal conductivities can often fall below the lower limit explained by the phonon-gas model, suggesting that the mean free path lengths of the phonons are below interatomic distances. (30,31) Therefore, transport can no longer be treated solely by gas-like phonons. (30) Instead, a second mechanism of phonon heat transport has been proposed, conducting heat via coupling of energetically degenerate vibrations. (32,33) These nonpropagating phonon modes are called diffusons, (30) which can be thought of as an atomic scale random-walk diffusion of thermal energy caused by a temperature gradient. (32) Diffuson transport has also been reported in several crystalline materials with complex unit cells, (32) leading to a large number of phonon modes, a sufficient degree of atomic disorder, (34,35) or high levels of intrinsic anharmonicity. (34,36) Computational methods have supported the hypothesis of diffuson-mediated thermal transport. (37,38)
Both the phonon-gas and the diffuson-type transport can occur in a material simultaneously, leading to theories accounting for thermal transport via both mechanisms in a parallel phonon-gas channel and a diffuson channel. (32,33,39) As illustrated in Figure 1 (left), the thermal transport mechanisms can be differentiated empirically by the frequency range of their main contributions and their temperature dependence. (30,33) Low energy (frequency) lattice vibrations, especially acoustic modes, typically contribute to the phonon-gas channel, whereas higher energy modes are rather diffuson-like. (30,35) The Ioffe–Regel limit can be used as a rough guide to separate the phonon-gas and diffuson regime. It defines phonons whose lifetime τ is smaller than the inverse of their angular frequency ω (τ < 1/ω) as diffusons and is used in analytical models to divide the vibrational spectrum into phonon gas- and diffuson-like phonons. (30,35) In more sophisticated models, like the two-channel model based on the Wigner transport equation developed by Simoncelli et al., (33) the thermal transport is calculated mode- and wave vector-dependent, resulting in a more gradual transition between phonon-gas-dominated and diffuson-dominated frequencies.

Figure 1

Figure 1. Schematic of the density of states (middle), indicating frequency ranges commonly associated with phonon-gas and diffuson character. Representation of typical thermal conductivity trends from phonon-gas and diffuson contributions (left). Low frequencies can be associated with softer potentials and larger vibrational amplitudes displacing the (mobile) ion further from its equilibrium position. In contrast, higher frequencies cause the ion to perform more periods of vibrations with smaller amplitudes (right).

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Commonly, thermal transport via propagating phonons is macroscopically manifested by a rapid increase in thermal conductivity at the lowest temperatures because of a rising heat capacity. A decrease proportional to the inverse of temperature occurs at higher temperatures due to enhanced phonon–phonon Umklapp scattering (Figure 1, upper left). (29) The transition between both domains leads to a “phonon peak” in the thermal conductivity at low temperatures (usually below 50 K). Anharmonic interactions, necessary to describe phonon–phonon interactions and thermal conductivity, require a more sophisticated model than well-defined energy states for phonon modes. Anharmonicity gives rise to a distribution of phonon energies characterized by a Lorentzian profile centered around their harmonic frequencies. The breadth of this distribution is referred to as the phonon linewidth. (26) Consequently, a certain phonon linewidth overlap of neighboring phonon modes emerges. Unlike the phonon-gas model, diffuson contributions increase with temperature because of increasing phonon occupation and overlap. At elevated temperatures, saturation of thermal conductivity occurs due to the inherent limit of fully overlapping modes, and diffusons are expected to conduct thermal energy at a constant rate (Figure 1, lower left). (32,33)
Motivated by the need to explore the thermal conductivity of solid electrolytes, this study examines the lattice dynamics of the endmembers of the NASICON series, namely, NaZr2P3O12 and Na4Zr2Si3O12, and elucidates the relevance of lattice dynamics for ionic and thermal transport in these materials. Although their ionic conductivities are low, understanding the dynamics in the NASICON endmembers can help comprehend the influence of dynamics on the entire series. The choice of the endmembers allows us to deconvolute potential outcomes of Na+ content, avoiding the complexities of sodium/vacancy and P/Si disorder. (11) The thermal transport of NaZr2P3O12 and Na4Zr2Si3O12 is experimentally investigated in a wide temperature range from 2 to 773 K, revealing diffuson contributions to thermal transport in these solid electrolytes. The enhanced anharmonicity of the sodium sublattice in Na4Zr2Si3O12 compared to NaZr2P3O12 is assessed by mode-dependent Grüneisen parameters, rationalizing the differences in thermal conductivity observed in both compositions. Combining a spatial analysis of the average vibrational frequencies of sodium ions with the spectral breakdown of thermal conductivity underscores the importance of lattice dynamics in the discussion of the ionic transport properties (Figure 1, right). Additionally, this analysis shows that vibrations that are typically deemed important for ionic transport exhibit low frequencies and distinct anharmonicity. This work provides a conceptually different perspective on the interplay between thermal and ionic transport in solid electrolytes. A unique perspective on how energy landscapes shape characteristic frequencies in ionic conductors is provided. It demonstrates that the sublattice and site occupation of the mobile species, pivotal for ionic conductivity, also affects phonon linewidths and thermal transport in solid ionic conductors.

Results and Discussion

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Crystal Structure and Thermal Expansion

NaZr2P3O12 and Na4Zr2Si3O12 both crystallize in a trigonal lattice within the R3̅c space group, independent of temperature. (14) In this structure, the Zr4+ ions are coordinated by oxygen in octahedrons that are corner-sharing with PO43– and SiO44– polyanions, respectively. Vice versa, each tetrahedron is connected to one octahedron at each corner. (7,40) In NaZr2P3O12, sodium ions only occupy the 6-fold, antiprismatic coordinated sodium site Na(1) (orange, Figure 2a). In addition to the occupation of the Na(1) position, an irregularly 8-fold coordinated sodium site Na(2) is occupied in Na4Zr2Si3O12 (yellow, Figure 2a). The irregular shape of the Na(2) polyhedron results in four distinct bond lengths. In both NaZr2P3O12 and Na4Zr2Si3O12, the respective sodium positions are fully occupied.

Figure 2

Figure 2. (a) Hexagonal representation of the unit cell of Na4Zr2Si3O12 with coordination environments and the connectivity of the Na+ sites. The only differences in the NaZr2P3O12 structure are the vacant Na(2) sites and PO43– instead of SiO44– tetrahedra. (b) Temperature dependence of the Na–O bond distances. For the Na(2) site in Na4Zr2Si3O12 the average bond distance is given, and the temperature dependence of all four individual bond lengths is shown in Figure S7. (c) Connectivity of the sodium polyhedra, visualizing that the diffusion path from the Na(1) to Na(2) site is bent. Only 2 out of the 6 adjacent Na(3) polyhedra around the central Na(1) antiprism are shown for visual clarity. The connectivity to all adjacent Na(2) and Na(3) sites can be best seen in Figure 8a–c. (d) Diverging bond angles within the Na(1)O6 polyhedra due to the rotation of ZrO68– and (P/Si)O43/4– polyhedra building up the rigid framework of the unit cell, leading to a distortion of the sodium ion cavity.

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The diffusion pathway between the two sodium sites Na(1) and Na(2) involves an intermediate, 5-fold coordinated position (typically referred to as Na(3)) located between Na(1) and Na(2). The Na(1) antiprisms are elongated along the c-axis and linked to six adjacent Na(2) and Na(3) sites, whereas each Na(2) site is only neighbored by two Na(1) sites. (41,42) While ion motion is expected to involve the intermediate Na(3) position, resulting in a bent diffusion pathway from the Na(1) to the Na(2) site (Figure 2c), this site is not permanently occupied in any of the endmembers. This fully ordered sodium sublattice causes both compounds to have low ionic conductivities (see Section S2). (14) As the Na(2) site serves as an intermediate position in NaZr2P3O12, the path between two permanently occupied sites involves twice as many individual jumps than in Na4Zr2Si3O12, leading to an even lower ionic conductivity.
To ensure the successful synthesis of NaZr2P3O12 and Na4Zr2Si3O12 and to acquire detailed structural information, temperature-dependent X-ray diffraction and Rietveld refinements were performed (details in Section S3). Diffraction confirms the high phase purity of the materials (less than 1 wt % ZrO2) and agrees with the structural parameters of previous reports. (20,43) The refined diffractograms and tabulated structural parameters at 150, 298, and 623 K are given in the Supporting Information (Section S3). Besides the structural evolution, these measurements allow for validation of the lattice dynamics calculations and provide information about the vibrational frequencies of the atoms.
With increasing temperature, anisotropic expansion of the unit cell can be observed for both compositions. The c-lattice parameter expands, while the a- and b-lattice parameters decrease (Figures S4 and S5), which seems caused by subtle correlated rotations of the ZrO68– and (P/Si)O43/4– polyhedra (Figure S6). (44) The ZrO68– and (P/Si)O43/4– polyhedra are assumed to be rigid units. Despite the decrease in the crystallographic a, b-direction, the unit cell volumes increase linearly (Figure S4 and S5). Volumetric thermal expansion coefficients can be determined from the linear increase in the temperature. The volumetric thermal expansion coefficients for NaZr2P3O12 and Na4Zr2Si3O12 are 1.2(5)·10–5 K–1 and 2.2(2)·10–5 K–1, respectively, in good agreement with the literature. (45,46)
Analyzing the temperature dependence of the bond lengths reveals differences in their local expansion behavior. The bonds within the rigid framework composed of the (P/Si)O43/4– tetrahedra and ZrO68– octahedra do not change significantly across the temperature range investigated, confirming the assumption of rigid unit modes (Figures S4 and S5). In contrast, the Na–O bond length of the octahedrally coordinated Na(1) site exhibits significant thermal expansion of 5.04(11)·10–5 K–1 and 4.3(2)·10–5 K–1 in NaZr2P3O12 and Na4Zr2Si3O12, respectively (Figure 2b), corresponding to an increase of 2.2% and 1.9% between 150 and 600 K. Using an expansion coefficient of the bond length allows for direct comparisons with the dilatation of other distances in the unit cell. In agreement with the anisotropic thermal expansion of the unit cell, the expansion of the Na(1)-polyhedra is more pronounced in the crystallographic c-direction, leading to increased distortion of the coordination octahedra instead of a general volume expansion (Figure 2d). The additional 8-fold coordinated Na(2) position occupied in Na4Zr2Si3O12 is characterized by four distinct bond lengths, two decreasing and two expanding upon heating (Figure S7). The absolute values of their expansion coefficients are comparable to or even surpass those found for the Na(1)–O bonds. However, the average bond length of Na(2)–O remains constant within the experimental uncertainty and over the entire temperature range (Figure 2b). Imagining the ZrO68– and (P/Si)O43/4– polyhedra as rigid unit modes, any changes in their arrangements will deform the softer sodium polyhedra. Here, a “softer” polyhedron refers to lower bond strength, indicated by lower charges, longer bonds, and lower average vibrational frequencies (Figure 4a).
Alongside the volumetric expansion of the lattice and coordination polyhedra, a linear increase of the isotropic thermal displacement parameter of sodium on the Na(1) and Na(2) sites can be observed (Figure 3a). Former investigations and lattice dynamics calculations quantitatively confirm those values. (20,43) From a lattice dynamics perspective, these high amplitudes of thermal vibration hint at loosely bound atoms with low force constants, as expected from the ion-conducting nature of the material. Low force constants of bonding lead to more deformable polyhedra, corroborating the significant thermal expansion coefficients of sodium–oxygen bonds discussed above. As the vibrational frequency of a bond scales with the square root of the force constants, the vibrational frequencies of sodium ions are expected to be comparably low. Vibrational frequencies can be approximated from the change of isotropic thermal displacements with temperature via Einstein frequencies (details in Section S4). The Einstein model treats each ion as vibrating as a harmonic oscillator independently of the others at a single frequency. With that, the vibrational frequency can be directly calculated from the experimental isotropic displacement parameter. The displacement parameters of the individual sodium sites differ, so a different frequency can be calculated for each distinct Na+ site. The steeper slope observed for the Na(1) site in NaZr2P3O12 yields a lower Einstein frequency of 2.49(4) THz compared to the sodium sites in Na4Zr2Si3O12 (3.77(7) THz and 3.25(5) THz for the Na(1) and Na(2) site, Figure 3b), suggesting lower force constants and weaker bonding interactions for NaZr2P3O12, caused by the longer Na(1)-O bond in NaZr2P3O12. This observation is supported by bond valence sum (BVS) analyses, which reveal a bond valence sum of 0.79 at room temperature for the Na(1) site in NaZr2P3O12, opposing to 0.91 in Na4Zr2Si3O12. The BVS can be seen as the effective number of electrons that an ion uses for bonding. Hence, sodium ions should approach a BVS of 1. Lower BVS generally indicates loosely bound species and, therefore, more mobile and weaker interacting ions. Considering the longer Na(1)–O bond distances and the more elongated coordination polyhedron in NaZr2P3O12 compared to Na4Zr2Si3O12 these findings align with the structural analysis. To analyze the bonding strength even further, the Crystal Orbital Hamilton Populations (47) were calculated for both compounds (for computational details, see Section S1.4). The results corroborate the previous findings: Although there are no antibonding states below the Fermi level, the bonding energy of the sodium ions is much lower than those of zirconium, silicon, or phosphorus ions (see Figure S8).

Figure 3

Figure 3. (a) Isotropic thermal displacement parameters from lattice dynamics calculations (dashed lines), values obtained from Rietveld refinement (symbols), and their respective linear fit (solid line). Diamond shapes correspond to values reported by Morgan et al. (20) and Qui et al., (43) respectively. (b) Einstein frequencies of sodium ions on the Na(1) and Na(2) site in NaZr2P3O12 (NZP) and Na4Zr2Si3O12 (NZS), respectively, deduced from the temperature dependence of the isotropic thermal displacement parameters.

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Figure 4

Figure 4. (a) Average frequencies of the total, atom-, site- and jump-direction-projected phonon DOS in NaZr2P3O12 in comparison to those found for Na4Zr2Si3O12. (b) Atom-, site- jump-direction-projected phonon DOS of sodium in NaZr2P3O12 and Na4Zr2Si3O12. In NaZr2P3O12, atom- and site-projected phonon DOS are identical as there is only one sodium site. (c) Connectivity of the Na+ polyhedra, visualizing the direction of projection for the jump-direction-projected phonon DOS (onto the Na(3) site). (d) Distribution of omnidirectional eigenvectors eq,s and (e) distribution of eigenvectors projected onto the Na(3) site obtained by scaling the eigenvector by the dot product of the eigenvector and projection vector p. The hat notes that the respective vector is scaled to unity. The eigenvectors are a function of wave vector q and phonon branch index s.

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Lattice Dynamics

Given the inherent importance of lattice vibrations for thermal and ionic conductivities, ab initio lattice dynamics calculations were conducted for both NASICON compositions. From these calculations, e.g., the frequency distribution of phonon modes (i.e., the phonon density of states, phonon DOS) and the mode-resolved Grüneisen parameter as a measure for anharmonicity can be assessed to guide our understanding of transport in this material class. The dispersion relations in the form of a phonon band structure (Figure S9) reveal that most bands are almost dispersionless. The phonon DOS, integrated over the first Brillouin zone, contains information about the distribution of all of the phonon modes in the frequency space. The phonon DOS can be projected onto atomic species or distinct lattice sites to yield insights into their vibrational characteristics. For simplicity, the general vibrational characteristics of both compositions are not discussed in terms of the actual phonon DOS, but mainly in the form of average frequencies. However, the total phonon DOS and its projections are listed in the Supporting Information (Figure S10).
The overall shape of the phonon DOS does not change when phosphorus is replaced with silicon and sodium is added for charge compensation (Figure S10). Consequently, both compositions have similar average frequencies of the total phonon DOS, i.e., 13.6 and 11.4 THz for NaZr2P3O12 and Na4Zr2Si3O12, respectively. The consistent average frequency of the total phonon DOS can be explained by the similarities of the ZrO6–PO4 framework compared to the ZrO6–SiO4 framework. This structural similarity is reflected in the vibrational frequencies (Figure S10). As the ions in the framework are responsible for approximately 90% of all vibrations, the average vibrational frequencies between both compositions are also not altered significantly.
For further analyses of the vibrational spectrum, the phonon DOS can be projected onto each lattice site. Summing up contributions from all lattice sites of an atom yields the respective atom-projected phonon DOS. The similarity of the atom-projected phonon DOS of P and Si is also evident in their average vibrational frequencies with values of 18.8 and 17.1 THz, respectively (Figure 4a). We attribute these high-frequency vibrations to the rigidity of the PO43– and SiO44– frameworks, resulting in high force constants. All vibration modes involving Na+ ions are located at significantly lower frequencies, with average frequencies of 3.6 and 4.2 THz for both compositions (Figure 4a). Analyzing the two Na positions in Na4Zr2Si3O12 separately reveals a lower average frequency for the Na(2) site than the Na(1) site. Given the direct correlation of vibrational frequency and force constant, these low frequencies are a consequence of the weak bonding of sodium ions, resulting in high amplitudes of thermal vibration. Although still weakly bound, the bonding on the Na(1) position in Na4Zr2Si3O12 is expected to be stronger (higher average frequency) than that on the Na(2) site, which is confirmed by the structural analyses revealing shorter bond lengths and slightly lower absolute thermal expansion coefficients.
As Einstein frequencies measure only occupied phonon modes following the Bose–Einstein distribution, their frequencies are lower than the average frequencies obtained from lattice dynamics calculations. Weighing the phonon DOS by the phonon occupations of their frequencies, the average frequencies from lattice dynamics calculations at 300 K are reduced to 3.7 THz and 2.5 THz for the Na(1) and Na(2) site, respectively, which is in close agreement with the Einstein frequencies (Figure 3b and Figure S11). (48)
While the site-projected phonon DOS results in a complete spectrum of vibrations, taking displacements of the ions in all directions into account, it is also possible to project the phonon DOS into a specific spatial direction to probe the frequency of vibrations in that direction (Figure 4b–e). Projecting the phonon DOS in a specific direction can be regarded as only considering part of the phonon eigenvectors in that direction (Figure 4d,e). Again, the vibrational frequency scales directly with the force constant and is a proxy for the resistance ions experience when displaced in specific directions. Consequently, projecting the phonon DOS along the migration vector of the Na+ ion examines how different sodium vibrations contribute to the Na+ motion in the direction of the intermediate Na site (Na(3)), which in turn can quantify the restoring forces the ion faces during its diffusional motion (Figure 4b).
Considering Na4Zr2Si3O12 as both sodium sites, Na(1) and Na(2) are occupied, showing a significant reduction in average vibrational frequency from the site average to the jump-direction-projected average (4.8 to 4.2 THz and 4.0 to 3.3 THz for the Na(1) and Na(2) site, respectively) and suggesting that vibrations displacing the sodium ions into the jump direction possess frequencies even lower than the site average (Figure 4a,b). Previously, it was suggested that ions with lower average phonon frequencies are more likely to overcome the activation barrier for a jump to an adjacent site. (49) At first, this seems counterintuitive, as the lower energy of low-frequency vibrations leads to a higher phonon occupation number that accumulates the energy of the activation barrier. However, the Bose–Einstein statistic indicates a significantly higher occupation number for low-frequency vibrations, which in turn lead to a higher probability of low-frequency vibrations to provide sufficient energy for the ion to overcome the activation barrier compared to vibrations of higher frequency. The low frequencies along the migration vector may, hence, be an essential factor in understanding ionic transport from a phonon perspective. This focus on low frequency contradicts conventional arguments derived from transition state theory, which posits that high attempt frequencies enhance ionic transport. (50) However, this reasoning neglects that high-frequency vibrations are significantly less occupied. (49) While this Perspective examines the influence of phonons on ionic transport, phonons and their phonon DOS are considered more often when discussing thermal transport than when discussing ionic transport.

Thermal Conductivity in the Framework of Two-Channel Transport

Although the thermal conductivity is relatively insensitive to the overall shape of the vibrational spectrum, changes, especially at low frequencies, can alter the thermal conductivity’s features. (51) Therefore, the thermal conductivity of NaZr2P3O12 and Na4Zr2Si3O12 is assessed experimentally in a broad temperature range from 2 to 773 K, which is necessary to identify the underlying transport mechanisms from the characteristic temperature dependencies. As NASICONs are electronic insulators, the contribution of the mobile electrons to the thermal conductivity is several orders of magnitude lower than their lattice thermal conductivity (for details, see Section S7). Both compositions exhibit similar thermal conductivities and temperature dependencies (Figure 5a) above approximately 350 K, but significant differences can be observed at low temperatures.

Figure 5

Figure 5. Ab initio modeled (lines) and experimental (symbols) thermal conductivities of NaZr2P3O12 and Na4Zr2Si3O12. (a) Side-to-side comparison of the materials with a crystallite size of 0.5 μm used to calculate the ab initio thermal conductivities. The anisotropic structure of both compounds results in anisotropic thermal conductivity. The lines shown for the total thermal conductivity represent the directions in space with the highest and lowest thermal conductivity. For visual clarity, phonon-gas and diffuson conductivities are averaged in all spatial directions. (b) Extension of panel (a) by different crystallite sizes to emphasize their influence (b) on the height of the phonon peak in both materials. Note that both panels are scaled differently due to a much more distinct phonon peak in NaZr2P3O12. The impact on the diffuson channel is negligible and therefore not shown.

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A phonon peak, characteristic of phonon-gas transport and the onset of dominant phonon–phonon scattering, is observed for NaZr2P3O12 at 45 K (Figure 5a left). With increasing temperature, the thermal conductivity shows a T–1-type decline, suggesting phonon scattering before saturating to constant values around 400 K. Generally, the high-temperature magnitudes are in agreement with previous reports by Rohde et al. (52) for the solid solutions Na2.7Zr2P1.3Si1.7O12 and Na3Zr2PSi2O12. In contrast to NaZr2P3O12, no pronounced phonon peak is observed in Na4Zr2Si3O12 (Figure 5a, right). Instead, the initial increase in thermal conductivity via the phonon-gas channel causes a substantial increase in the total thermal conductivity up to 30 K followed by a less pronounced increase in thermal conductivity via diffusons.
A unified approach must be used to understand the temperature dependencies and the differences between both materials, considering phonon-gas and diffuson transport. Phonon-gas- and diffuson-type transport contributions are evaluated using an ab initio two-channel model proposed by Simoncelli et al. (33) The model is based on calculations of third-order force constants that capture anharmonic interactions and can thereby predict phonon scattering (rates) and phonon linewidth broadening. Subsequently, third-order force constants are used to calculate the scattering of propagating phonons (phonon-gas channel) and the overlap and coupling of diffuson modes. This allows the calculation of the thermal conductivity contribution of every pair of two phonon branches, s and s’, at each temperature and q-point, instead of using a simple cutoff to distinguish between phonon gas- and diffuson-type conductivity. For s = s’, thermal transport is phonon gas-like; otherwise, for s ≠ s’, thermal transport occurs via the diffuson channel. Computational details are given in the Supporting Information (Section S1.4).
The total thermal conductivity predicted by the ab initio model describes the experimental results well, not requiring additional fourth-order force constants, which are necessary to describe the thermal conductivity in other materials. (53,54) As the NASICON structure is anisotropic, the thermal conductivity is slightly different along the c-axis compared to the ab-plane. The range of thermal conductivity due to anisotropy is highlighted by the shaded area in Figure 5a. The total magnitude of the phonon peak is strongly related to the rate of defect scattering, e.g., grain boundaries (Figure 5b), making precise knowledge of the microstructure necessary. Scanning electron microscopy reveals similar microstructures in both compounds (see Section S8), with secondary particle sizes in the low micrometer range and below. However, these particle sizes only represent upper bounds for the crystallite sizes, but their actual size remains unknown. Additionally, the crystallite size within a sample is not monodisperse, as assumed by the phonon lifetime model used here, but will have some distribution.
Within these approximations, a crystallite size of 0.5 μm was estimated that fits the data of Na4Zr2Si3O12 best. The same analytical grain size was used for both compounds to avoid obfuscating the comparison between the two compounds and their analysis. However, in NaZr2P3O12 (Figure 5a left), the phonon peak at low temperatures is overestimated by the ab initio modeling using this crystallite size, and smaller crystallite sizes of ≈0.1 μm describe the phonon peak in NaZr2P3O12 more accurately. Figure 5b highlights the influence of varying crystallite sizes on the magnitude of the phonon peak magnitude.
Comparing the results for both NASICON compositions and distinguishing the contributions by phonon-gas-like and diffuson-like transport reveals that the significant difference is a stronger suppression of the phonon-gas contributions. This leads to differences in the temperature dependencies and an earlier (in temperature) increase and saturation of the diffuson contributions in Na4Zr2Si3O12. Given the similarity in heat capacity, speed of sound, phonon band structure, and sample treatment of both NASICON compositions, the stronger suppression of the phonon-gas channel seems to be driven by stronger phonon scattering in Na4Zr2Si3O12 caused by the additional sodium ions. There are no indications that other factors, such as strain, dislocations, or increased mass contrast, reduce the phonon peak in Na4Zr2Si3O12.
To understand this difference from a fundamental lattice dynamics perspective, the spectral distributions of the thermal conductivities of both channels (Figure 6), Grüneisen parameters, and phonon linewidths for both compositions have been evaluated (Figure 7). Phonon-gas contributions are found predominantly at low frequencies, mostly below 5 THz. In contrast, dominant diffuson contributions spread the entire frequency range (Figure 6), aligning with the two-channel thermal conductivity theory. (30) At elevated temperatures, contributions to the thermal conductivity at higher frequencies arise (lower panels in Figure 6). This can be reasoned by the enhanced occupation of high-frequency phonon modes at higher temperatures. With increasing temperature, the phonon-gas contributions are diminished.

Figure 6

Figure 6. Spectral thermal conductivity of phonon-gas and diffuson channel, denoted as κpg and κdiff, respectively, at 100 (top panels) and 800 K (lower panels). Due to the high phonon peak at low temperatures in NaZr2P3O12, an inset is added to show the entire spectrum in the upper left panel. The percentages give the contribution of each channel to the total thermal conductivity.

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Figure 7

Figure 7. (a) Grüneisen parameter of NaZr2P3O12 (left panels) and Na4Zr2Si3O12 (right panels). The ratio of sodium partial phonon DOS to total phonon DOS, which is the contribution of sodium ions to vibrations at a given frequency, is color-coded. (b) Log-scaled phonon linewidths averaged over the Brillouin zone depicted at different temperatures. Both quantities exhibit highly anharmonic modes at low frequencies, especially for Na4Zr2Si3O12 at high temperatures. (c,d) Outer panels: Portion of phonon band structure at 100 and 800 K with color-coded phonon linewidths of (c) NaZr2P3O12 and (d) Na4Zr2Si3O12, respectively. Inner panels: Phonon spectral energy density of phonon bands in a selected frequency range, showing the effect of the phonon linewidth (broadening) on the phonon overlap, highlighted with green shading.

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In NaZr2P3O12, substantial contributions from the phonon-gas channel are evident across all temperatures, whereas the phonon-gas channel is largely suppressed at high temperatures in Na4Zr2Si3O12 (compare lower and upper panels in Figure 6). In contrast to the phonon-gas channel, diffuson-like thermal transport is more pronounced at higher temperatures. In line with the earlier increase of the diffuson channel in Na4Zr2Si3O12, higher diffuson contributions are found compared to NaZr2P3O12 (compared to the left and right panels in Figure 6). The comparison of Grüneisen parameters γ (Figure 7a) highlights distinct differences between the compositions. Na4Zr2Si3O12 exhibits a significantly stronger peak of the Grüneisen parameter at lower frequencies, i.e., γmax = 36.0, at 1.1 THz for Na4Zr2Si3O12 as compared to γmax = 10.3, at 2.4 THz for NaZr2P3O12. Especially this peak at low frequencies, where phonon-gas contributions are the strongest, can be considered as the underlying reason the phonon-gas contributions in this frequency range are completely suppressed at 800 K in Na4Zr2Si3O12 leading to the observed temperature dependence of the phonon-gas channel (Figure 5a). The significantly smaller and more evenly spread Grüneisen parameter in NaZr2P3O12 allows for a higher phonon peak and relevant phonon-gas contributions across all temperatures.
The average Grüneisen parameters, which can be seen as a measure of the anharmonicity in the entire structure, are not extraordinarily high (γ = 1.51 for Na4Zr2Si3O12 and γ = 0.55 for NaZr2P3O12 at 300 K). Thus, the compounds are not particularly anharmonic, but the Na+ vibrations that dominate at frequencies with high Grüneisen parameters (color-coded in Figure 7a). Examining the sodium ion contribution to the phonon DOS per site reveals that the very high Grüneisen parameters at low frequency in Na4Zr2Si3O12 are associated with high contributions from ions on the Na(2) site, i.e., the sodium introduced by substitution of phosphorus with silicon (Figure S12). The distinct anharmonicity of the Na(2) site might be caused by its loose bonding (cf. Figure S7 and Figure 4b) compared to the Na(1) site and especially ions of the rigid framework, as indicated by its large thermal displacements (Figure 3). The Na(2) site is not occupied in NaZr2P3O12, which contributes to the lower anharmonicity observed in the phosphorus endmember. Considering that the high Grüneisen parameter is caused by the introduction of additional sodium ions to the Na(2) site, the strong reduction of the phonon-gas channel in Na4Zr2Si3O12 can be ultimately attributed to the introduction of additional sodium ions into the lattice too. The high Grüneisen parameters for sodium-ion vibrations are consistent with the low average frequencies and large amplitudes of the vibrations. Large amplitudes of vibration entailing large displacements from the equilibrium positions promote anharmonicity, as they cause more significant differences between harmonic and anharmonic bonding models (cf., Figure 8d).

Figure 8

Figure 8. Local environment and vibration of the sodium sites: Spatial distributions of average site frequencies in NaZr2P3O12 (a) and Na4Zr2Si3O12 (b) when projected in a specific direction. The neighboring Na(3) and oxygen sites are shown for orientation. (c) Calculated anisotropic thermal displacement ellipsoids of the sodium ions in Na4Zr2Si3O12 at 300 K with 95% probability. The thermal displacement in the direction of diffusion is higher than average. Especially the displacement parameters of the Na(2) site are highly anisotropic. The Na(3) sites displayed in (a) to (c) are unoccupied and only added as a visual reference for the direction of the diffusion path. Their displacement parameters are arbitrarily chosen. (d) High average frequencies correspond to a more rigid potential well, smaller displacements (labeled dharm. and danh.), and less deviation of harmonic and anharmonic potential than lower frequencies. The eigenvalues of the harmonic oscillator, marked by the gray lines, have a larger spacing the more rigid the potential becomes.

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While an increase in anharmonicity at low frequencies can explain the strong suppression of the phonon-gas channel, evaluating the phonon linewidth gives an explanation of why the diffuson contributions arise and govern the total thermal conductivity at lower temperatures in Na4Zr2Si3O12 than in NaZr2P3O12 (Figure 5a). In the harmonic approximation, the phonon bands are lines of zero width. However, considering anharmonicity leads to a finite width of the bands, known as phonon linewidth. Consequently, the frequency of each band becomes a probability distribution. The phonon linewidth directly influences this probability distribution of frequencies (called the phonon spectral energy density Φ), as it is the half-width at half-maximum of the distribution, which takes the form of a Lorentzian distribution:
Φ(ν)=1π(Γ(ν−ν0)2+Γ2)
(1)
where Γ is the phonon linewidth, ν the frequency, and ν0 the center frequency. Φ, Γ, and ν0 are all functions of wave vector q, and phonon branch index s. As the phonon linewidths rise along with anharmonicity, just as the Grüneisen parameter, higher linewidth broadening is expected for Na4Zr2Si3O12. The linewidths in Na4Zr2Si3O12 are almost an order of magnitude higher, and with that, average linewidths of ≈0.1 THz are reached in NaZr2P3O12 only at 500 K, while these values are already reached at 200 K in Na4Zr2Si3O12 (Figure 7b). Despite these large linewidths, the scattering rate, which is directly related to the phonon linewidth, is approximately 1 order of magnitude lower than the phonon frequency. This in turn means that the phonon lifetime is still larger than the inverse of the angular frequency (τ > 1/ω), not passing the Ioffe–Regel limit. Increased linewidths directly facilitate the overlap of adjacent (in frequency) phonon modes (Figure 7c,d) and thereby directly increase the possibility of diffuson-type thermal transport. (33) The broader the linewidths become, the greater the modes overlap, ultimately approaching the inherent limit of completely overlapping, marking an upper limit for the diffuson thermal conductivity and resulting in a saturating behavior. Given the lower temperatures for the same linewidth in Na4Zr2Si3O12, rise and saturation of the diffuson channel occur at lower temperatures too. Additionally, sizable primitive unit cells facilitate the overlap of phonon modes since more atoms in the primitive unit cell lead to more and, therefore, (in frequency) more closely spaced bands. While both endmembers already have comparably large unit cells, thus not requiring large linewidths for the phonon modes to overlap, Na4Zr2Si3O12 has slightly more bands (124) than NaZr2P3O12 (108), reducing the phonon linewidth needed even further.
Not only thermal but also ionic transport is an inherently anharmonic phenomenon, as the harmonic oscillator model fails to capture the saddle point between two lattice sites. (55) Consequently, large thermal displacements, ion conduction, and anharmonicity are interrelated phenomena. This exact behavior can be found in Na4Zr2Si3O12. The additional Na(2) sites not only possess higher isotropic thermal displacement parameters than the Na(1) site but also their thermal displacement is highly anisotropic (Figure 8c), thereby amplifying the overall anharmonicity within the compound. The distinct anisotropic shape of the thermal displacement ellipsoids of the Na(2) site can be related to a distorted bonding environment with four different Na(2)–O bond lengths varying between 2.51 and 3.05 Å, resulting in spatially varying restoring forces, seen by the wider spread of average frequencies with the projection direction than on the Na(1) site (Figure 8b).
This anisotropy results from the vibrational spectrum being a function of the direction in which the spectrum is examined (cf. Figure 4 and 4e). From a phonon perspective, low frequencies are especially important to overcome the activation barrier for ion migration. (49) Low phonon frequencies enhance ionic transport via two mechanisms. First, lower frequencies indicate lower force constants and therefore a softer potential, lowering the activation barrier. Here, Na4Zr2Si3O12 exhibits not only lower frequencies but also a lower activation barrier than NaZr2P3O12 as seen from impedance spectroscopy (Figure S2). Second, for a given activation energy, lower frequencies of the mobile ion enhance its probability to overcome the activation barrier. (49) The spatial distribution of average vibrational frequencies (Figure 8b) allows us to visually evaluate ionic transport and diffusion path via lattice dynamics. Projecting the phonon DOS through the face of a coordination polyhedron will result in lower frequencies than a projection directly onto one of the coordinating ions, especially if the site behind the face is unoccupied. Therefore, the multitude of six adjacent Na(3) vacancies laying roughly in the ab-plane around the Na(1) site results in a band of low frequencies. As the Na(2) site is only neighbored by two Na(3) sites, two isolated areas of low frequency are observed.
These spatially variant projections of the phonon DOS are symmetric with respect to the sign of the projection vector, i.e., the projected phonon DOS, and thus, the average frequencies on opposing ends of the sphere are the same. Therefore, the area with the lowest average frequency does not align with both adjacent Na(3) sites, which form an angle of ≈140° with the Na(2) site. In contrast to the atom- and site-projected average frequencies, Grüneisen parameters, and phonon linewidths, these projections hold directional information. The average frequencies in the direction of ionic transport toward the adjacent intermediate Na(3) sites are extraordinarily low (e.g., 48% reduction for the Na(2) site in Na4Zr2Si3O12). Consequently, the dominant direction of the anisotropic thermal displacement parameters (Figure 8c) aligns well with the vector for ion migration. Low force constants correspond to a low average frequency and a high displacement from the equilibrium position (Figure 8d). This observation suggests that vibrations governing ionic transport exhibit predominantly low frequencies (low energies), and thus, low-frequency phonons should be mainly considered in the discussion of ionic transport, which corroborates the argument that low frequencies are most important to overcome activation barriers made from a phonon occupation perspective. Our data suggest that the number of low-energy attempt frequencies is very high because even at relatively low temperatures the low-energy modes contributed significantly to the Boltzmann partition function. The resulting rationale is therefore to prioritize the consideration of low-frequency phonons in the context of ionic transport dynamics. In this context, materials design strategies should aim for hierarchical bonding, promoting high frequencies perpendicular to and low frequencies in the direction of ionic transport due to their different bond strengths.
Additionally, pronounced anharmonicity linked with weakly bound and mobile ions is not limited to these NASICON compounds but should be applied to all solid electrolytes. As this anharmonicity is identified as the leading cause of low thermal conductivity, this might explain why generally low thermal conductivities are found in solid electrolytes. (56) The difference in the temperature dependence of thermal conductivity between NaZr2P3O12 and Na4Zr2Si3O12 and the results of the lattice dynamics analysis can be summarized as follows:
1)

Phonon-gas transport is suppressed significantly by large anharmonicities evoked by Na+ vibrations at low frequencies in Na4Zr2Si3O12.

2)

At the same time, the increased anharmonicities facilitate mode coupling and diffuson transport.

3)

Average frequencies in the direction of ionic transport are extraordinarily low. These vibrational modes are, therefore, of particular importance for ionic transport, and local structural modifications focusing on these modes may help in designing faster ionic conductors.

Conclusions

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In this study, the thermal conductivity and lattice dynamics of two NASICON solid electrolytes, NaZr2P3O12 and Na4Zr2Si3O12, were investigated through a combination of experiments and ab initio calculations. This investigation reveals that thermal transport at high temperatures is dominated by diffusons. Temperature-dependent X-ray diffraction experiments and lattice dynamics calculations reveal low average vibrational frequencies of sodium ions, corresponding to low force constants and weakly bound ions. By analyzing the spatial distribution of average phonon frequencies, particularly low frequencies in the direction of ionic transport become evident. These below-average frequencies demonstrate that vibrational modes with high anharmonicity and low frequencies, i.e., high phonon occupations, should be beneficial for ionic transport and not those with high frequencies as typically considered. The relevance of low-frequency vibrations marks an important point for the future design of ionic conductors, as one part of the design strategies often employed is the increase of attempt frequencies. The distinct anharmonicity of sodium vibrations causes a clear reduction of thermal transport via the phonon-gas channel and suppression of the phonon peak in Na4Zr2Si3O12. Examination of the phonon linewidths simultaneously revealed an increased overlap of adjacent modes promoting diffuson-type thermal transport at lower temperatures. With respect to the inherent anharmonicity of ion transport present in all solid electrolytes, this anharmonicity of the mobile ion sublattice appears to be a dominant factor in the low thermal conductivity observed for many solid electrolytes.

Supporting Information

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The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/jacs.4c12034.

  • Experimental procedures, computational details, ionic conductivities, details on Rietveld refinements and structural analysis, theory of Einstein frequencies, Crystal Orbital Hamilton Populations, band structures, phonon density of states, electronic contribution to the thermal conductivity, scanning electron microscopy (PDF)

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Most electronic Supporting Information files are available without a subscription to ACS Web Editions. Such files may be downloaded by article for research use (if there is a public use license linked to the relevant article, that license may permit other uses). Permission may be obtained from ACS for other uses through requests via the RightsLink permission system: http://pubs.acs.org/page/copyright/permissions.html.

Author Information

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  • Corresponding Author
    • Wolfgang G. Zeier - Institute of Inorganic and Analytical Chemistry, University of Münster, Münster D-48149, Germany;  International Graduate School for Battery Chemistry, Characterization, Analysis, Recycling and Application (BACCARA), University of Münster, Münster D-48149, Germany;  Institute of Energy Materials and Devices (IMD), IMD-4: Helmholtz-Institut Münster, Forschungszentrum Jülich, Münster 48149, Fed. Rep. Germany;  Orcidhttps://orcid.org/0000-0001-7749-5089;  Email: [email protected]
  • Authors
    • Thorben Böger - Institute of Inorganic and Analytical Chemistry, University of Münster, Münster D-48149, Germany;  International Graduate School for Battery Chemistry, Characterization, Analysis, Recycling and Application (BACCARA), University of Münster, Münster D-48149, Germany
    • Tim Bernges - Institute of Inorganic and Analytical Chemistry, University of Münster, Münster D-48149, Germany
    • Matthias T. Agne - Department of Chemistry and Biochemistry, University of Oregon, Eugene, Oregon 97403, United States of America
    • Pieremanuele Canepa - Department of Materials Science and Engineering, National University of Singapore, 117575Singapore;  Department of Chemical and Biomolecular Engineering, National University of Singapore, 117585Singapore;  Department of Electrical & Computer Engineering, University of Houston, Houston, Texas 77204, United States of America;  Orcidhttps://orcid.org/0000-0002-5168-9253
    • Frank Tietz - Institute of Energy Materials and Devices (IMD-2), Forschungszentrum Jülich, Jülich D-52425, Germany;  Institute of Energy Materials and Devices (IMD), IMD-4: Helmholtz-Institut Münster, Forschungszentrum Jülich, Münster 48149, Fed. Rep. Germany
  • Notes
    The authors declare no competing financial interest.

Acknowledgments

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This study is funded by the European Union (ERC, DIONISOS, 101123802). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. Th.B. is a member of the International Graduate School for Battery Chemistry, Characterization, Analysis, Recycling and Application (BACCARA), which is funded by the Ministry of Culture and Science of the State of North Rhine Westphalia, Germany. The simulations for this work were performed on the computer cluster PALMA II of the University of Münster. Lukas Ketter is acknowledged for his help in taking the SEM pictures. Lara M. Gronych, Oliver M. Maus, and Matthias Hartmann are acknowledged for their help in acquiring X-ray diffractograms. We further acknowledge support by the Deutsche Forschungsgemeinschaft (DFG) under project number 459785385. The Welch Foundation is acknowledged for providing P.C. a Robert A. Welch professorship at the Texas Center for Superconductivity. All data of this study are available in datasafe. by Universität Münster under the DOI 10.17879/96968603446.

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    Chemical Reviews (Washington, DC, United States) (2020), 120 (14), 6820-6877CODEN: CHREAY; ISSN:0009-2665. (American Chemical Society)
    A review. Solid-state batteries were attracting wide attention for next generation energy storage devices due to the probability to realize higher energy d. and superior safety performance compared with the state-of-the-art lithium ion batteries. However, there are still intimidating challenges for developing low cost and industrially scalable solid-state batteries with high energy d. and stable cycling life for large-scale energy storage and elec. vehicle applications. This review presents an overview on the scientific challenges, fundamental mechanisms, and design strategies for solid-state batteries, specifically focusing on the stability issues of solid-state electrolytes and the assocd. interfaces with both cathode and anode electrodes. First, the authors give a brief overview on the history of solid-state battery technologies, followed by introduction and discussion on different types of solid-state electrolytes. Then, the assocd. stability issues, from phenomena to fundamental understandings, are intensively discussed, including chem., electrochem., mech., and thermal stability issues; effective optimization strategies are also summarized. State-of-the-art characterization techniques and in situ and operando measurement methods deployed and developed to study the aforementioned issues are summarized as well. Following the obtained insights, perspectives are given in the end on how to design practically accessible solid-state batteries in the future.
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  9. 9
    Hong, H.-P. Crystal structures and crystal chemistry in the system Na1+xZr2SixP3–xO12. Mater. Res. Bull. 1976, 11 (2), 173– 182,  DOI: 10.1016/0025-5408(76)90073-8
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    9
    Crystal structures and crystal chemistry in the system Na1+xZr2SixP3-xO12
    Hong, H. Y. P.
    Materials Research Bulletin (1976), 11 (2), 173-82CODEN: MRBUAC; ISSN:0025-5408.
    As part of a search for skeleton structures for fast alkali-ion transport, the system Na1+xZr2SixP3-xO12 was prepd., analyzed structurally and ion exchanged reversibly with Li+, Ag+, and K+ ions. Single-crystal x-ray anal. was used to identify the compn. NaZr2P3O12 and to refine its structure, which has rhombohedral space group R3c with cell parameters ar 8.815(1) and cr 22.746(7)Å. A small distortion to monoclinic symmetry occurs in the interval 1.8 ≤ x ≤ 2.2. The structure for Na3Zr2Si2PO12, proposed from powder data, has space group C2/c with am 15.586(9), bm 9.029(4), cm 9.205(5)Å, and β 123.70(5)°. Both structures contain a rigid, 3-dimensional network of PO4 or (SiO4) tetrahedra sharing corners with ZrO6 octahedra and a 3-dimensionally linked interstitial space. Of the 2 distinguishable alkali-ion sites in the rhombohedral structure, one is completely occupied in both end members, the occupancy of the other varies across the system from 0 to 100%. Several properties are compared with the fast Na+-ion conductor β-alumina.
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  10. 10
    Goodenough, J. B.; Hong, H.-P.; Kafalas, J. A. Fast Na+-ion transport in skeleton structures. Mater. Res. Bull. 1976, 11 (2), 203– 220,  DOI: 10.1016/0025-5408(76)90077-5
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    10
    Fast sodium(1+) ion transport in skeleton structures
    Goodenough, J. B.; Hong, H. Y. P.; Kafalas, J. A.
    Materials Research Bulletin (1976), 11 (2), 203-20CODEN: MRBUAC; ISSN:0025-5408.
    Skeleton structures were explored exptl. for fast Na+-ion transport. A skeleton structure consists of a rigid skeletal array of atoms stabilized by electrons donated by alkali ions partially occupying sites in a 3-dimensionally linked interstitial space. Fast Na+-ion transport was demonstrated in several structures, and the system Na1+xZr2P3-xSixO12 has Na+-ion resistivity at 300° of ρ300 .ltorsim.5Ω-cm for x≈2, which is competitive with the best β''-alumina. An activation energy εa≈0.29 eV is about 0.1 eV larger that that of β''-alumina.
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  11. 11
    Deng, Z.; Mishra, T. P.; Mahayoni, E.; Ma, Q.; Tieu, A. J. K.; Guillon, O.; Chotard, J.-N.; Seznec, V.; Cheetham, A. K.; Masquelier, C.; Gautam, G. S.; Canepa, P. Fundamental investigations on the sodium-ion transport properties of mixed polyanion solid-state battery electrolytes. Nat. Commun. 2022, 13 (1), 4470,  DOI: 10.1038/s41467-022-32190-7
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    11
    Fundamental investigations on the sodium-ion transport properties of mixed polyanion solid-state battery electrolytes
    Deng, Zeyu; Mishra, Tara P.; Mahayoni, Eunike; Ma, Qianli; Tieu, Aaron Jue Kang; Guillon, Olivier; Chotard, Jean-Noel; Seznec, Vincent; Cheetham, Anthony K.; Masquelier, Christian; Gautam, Gopalakrishnan Sai; Canepa, Pieremanuele
    Nature Communications (2022), 13 (1), 4470CODEN: NCAOBW; ISSN:2041-1723. (Nature Portfolio)
    Lithium and sodium (Na) mixed polyanion solid electrolytes for all-solid-state batteries display some of the highest ionic conductivities reported to date. However, the effect of polyanion mixing on the ion-transport properties is still not fully understood. Here, we focus on Na1+xZr2SixP3-xO12 (0 ≥ x ≥ 3) NASICON electrolyte to elucidate the role of polyanion mixing on the Na-ion transport properties. Although NASICON is a widely investigated system, transport properties derived from expts. or theory vary by orders of magnitude. We use more than 2000 distinct ab initio-based kinetic Monte Carlo simulations to map the compositional space of NASICON over various time ranges, spatial resolns. and temps. Via electrochem. impedance spectroscopy measurements on samples with different sodium content, we find that the highest ionic cond. (i.e., about 0.165 S cm-1 at 473 K) is exptl. achieved in Na3.4Zr2Si2.4P0.6O12, in line with simulations (i.e., about 0.170 S cm-1 at 473 K). The theor. studies indicate that doped NASICON compds. (esp. those with a silicon content x ≥ 2.4) can improve the Na-ion mobility compared to undoped NASICON compns.
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  12. 12
    Ma, Q.; Tsai, C.-L.; Wei, X.-K.; Heggen, M.; Tietz, F.; Irvine, J. T. S. Room temperature demonstration of a sodium superionic conductor with grain conductivity in excess of 0.01 S cm–1 and its primary applications in symmetric battery cells. J. Mater. Chem. A 2019, 7 (13), 7766– 7776,  DOI: 10.1039/C9TA00048H
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    12
    Room temperature demonstration of a sodium superionic conductor with grain conductivity in excess of 0.01 S cm-1 and its primary applications in symmetric battery cells
    Ma, Qianli; Tsai, Chih-Long; Wei, Xian-Kui; Heggen, Marc; Tietz, Frank; Irvine, John T. S.
    Journal of Materials Chemistry A: Materials for Energy and Sustainability (2019), 7 (13), 7766-7776CODEN: JMCAET; ISSN:2050-7496. (Royal Society of Chemistry)
    The lack of suitable candidate electrolyte materials for practical application limits the development of all-solid-state Na-ion batteries. Na3+xZr2Si2+xP1-xO12 was the very first series of NASICONs discovered some 40 years ago; however, sepn. of bulk cond. from total cond. at room temp. is still problematic. It has been suggested that the effective Na-ion cond. is ∼10-4 S cm-1 at room temp. for Na3+xZr2Si2+xP1-xO12 ceramics; however using a soln.-assisted solid-state reaction for prepn. of Na3+xZr2Si2+xP1-xO12, a total cond. of 5 × 10-3 S cm-1 was achieved for Na3.4Zr2Si2.4P0.6O12 at 25 °C, higher than the values previously reported for polycryst. Na-ion conductors. A bulk cond. of 1.5 × 10-2 S cm-1 was revealed by high frequency impedance spectroscopy (up to 3 GHz) and verified by low temp. impedance spectroscopy (down to -100 °C) for Na3.4Zr2Si2.4P0.6O12 at 25 °C, indicating further the potential of increasing the related total cond. A Na/Na3.4Zr2Si2.4P0.6O12/Na sym. cell showed low interface resistance and high cycling stability at room temp. A full-ceramic cell was fabricated and tested at 28 °C with good cycling performance.
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  13. 13
    Rajagopalan, R.; Zhang, Z.; Tang, Y.; Jia, C.; Ji, X.; Wang, H. Understanding crystal structures, ion diffusion mechanisms and sodium storage behaviors of NASICON materials. Energy Storage Mater. 2021, 34, 171– 193,  DOI: 10.1016/j.ensm.2020.09.007
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    Boilot, J. P.; Collin, G.; Colomban, P. Relation structure-fast ion conduction in the NASICON solid solution. J. Solid State Chem. 1988, 73 (1), 160– 171,  DOI: 10.1016/0022-4596(88)90065-5
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    Relation structure - fast ion conduction in the Nasicon solid solution
    Boilot, J. P.; Collin, G.; Colomban, P.
    Journal of Solid State Chemistry (1988), 73 (1), 160-71CODEN: JSSCBI; ISSN:0022-4596.
    Crystal detns. of the rhombohedral phase (space group R‾3c), for different compns. (2 < x < 2.4) in the true Nasicon solid soln. Na1+xZr2SixP3-xO12, were performed at different temps. by x-ray diffraction. As a consequence of interionic repulsions, the partial occupation of a mid-Na interstitial site within the conduction path was obsd. The compn. dependence of the mid-Na occupation factor, max. at x = 2, explains the max. of the c hexagonal parameter and of the Na(I)-O av. distance obsd. at about x = 2. Moreover, structural results clearly suggest that the enhanced cond. at x = 2 arises from Na interactions instead of geometry changes of the framework.
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  15. 15
    Kumar, P. P.; Yashonath, S. Structure, Conductivity, and Ionic Motion in Na1+xZr2SixP3-xO12: A Simulation Study. J. Phys. Chem. B 2002, 106 (28), 7081– 7089,  DOI: 10.1021/jp020287h
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    15
    Structure, Conductivity, and Ionic Motion in Na1+xZr2SixP3-xO12: A Simulation Study
    Kumar, P. Padma; Yashonath, S.
    Journal of Physical Chemistry B (2002), 106 (28), 7081-7089CODEN: JPCBFK; ISSN:1089-5647. (American Chemical Society)
    Const.-pressure, const.-temp. variable-shape simulation cell Monte Carlo and microcanonical ensemble mol. dynamics simulation of superionic conducting rhombohedral phase of Nasicon, Na1+xZr2SixP3-xO12, 0 ≤ x ≤ 3, at a temp. of 600 K is reported. Changes in structure, cond., hop path, site occupancies, bond lengths of framework atoms with compn. are discussed. Av. Na(1)-O distance shows a peak at x = 2, while Na(2)-O distance shows a monotonic increase. Sum of the sodium occupancies at Na(1) and mid-Na sites adds up to a const. value of one which supports the conclusion of J. P. Boilot et al. (1988) based on X-ray diffraction. Occupancy of Na(1) site attains a min. at x = 2. The predominant conduction channel (which carries more than 90% of the sodium ions) is found to be the one connecting Na(1)-mid-Na-Na(2). D. contours for sodium, depicting this conduction channel, are reported. Free energy profile along the conduction channel suggests that entropy contribution cannot be neglected. The mid-Na site is not assocd. with a free energy min.
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  16. 16
    Abello, L.; Chhor, K.; Barj, M.; Pommier, C.; Delmas, C. Heat capacity and Na+ ion disorder in Nasicon-type solid electrolytes Na3M2P3O12 (M2= Fe2, Cr2, ZrMg) in the temperature range 10 to 300 K. J. Mater. Sci. 1989, 24 (9), 3380– 3386,  DOI: 10.1007/BF01139069
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    Pet’kov, V. I.; Asabina, E. A.; Markin, A. V.; Smirnova, N. N.; Kitaev, D. B. Thermodynamic data of the NZP compounds family. J. Therm. Anal. Calorim. 2005, 80 (3), 695– 700,  DOI: 10.1007/s10973-005-0716-4
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    Pet’kov, V. I.; Asabina, E. A.; Markin, A. V.; Smirnova, N. N. Synthesis, characterization and thermodynamic data of compounds with NZP structure. J. Therm. Anal. Calorim. 2008, 91 (1), 155– 161,  DOI: 10.1007/s10973-007-8370-7
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    Maier, J.; Warhus, U.; Gmelin, E. Thermodynamic and electrochemical investigations of the Nasicon solid solution system. Solid State Ionics 1986, 18–19, 969– 973,  DOI: 10.1016/0167-2738(86)90294-8
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    Morgan, E. E.; Evans, H. A.; Pilar, K.; Brown, C. M.; Clément, R. J.; Maezono, R.; Seshadri, R.; Monserrat, B.; Cheetham, A. K. Lattice Dynamics in the NASICON NaZr2(PO4)3 Solid Electrolyte from Temperature-Dependent Neutron Diffraction, NMR, and Ab Initio Computational Studies. Appl. Phys. Lett. 2022, 34 (9), 4029– 4038,  DOI: 10.1021/acs.chemmater.2c00212
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    Zhen, X.; Sanson, A.; Sun, Q.; Liang, E.; Gao, Q. Role of alkali ions in the near-zero thermal expansion of NaSICON-type AZr2(PO4)3(A = Na,K,Rb,Cs) and Zr2(PO4)3 compounds. Phys. Rev. B 2023, 108 (14), 144102  DOI: 10.1103/PhysRevB.108.144102
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    Agne, M. T.; Böger, T.; Bernges, T.; Zeier, W. G. Importance of Thermal Transport for the Design of Solid-State Battery Materials. PRX Energy 2022, 1 (3), 31002,  DOI: 10.1103/PRXEnergy.1.031002
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    Kantharaj, R.; Marconnet, A. M. Heat Generation and Thermal Transport in Lithium-Ion Batteries: A Scale-Bridging Perspective. Nanoscale Microscale Thermophys. Eng. 2019, 23 (2), 128– 156,  DOI: 10.1080/15567265.2019.1572679
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    Gu, J.; Xu, R.; Chen, B.; Zhou, J. NMC811-Li6PS5Cl-Li/In All-Solid-State Battery Capacity Attenuation Based on Temperature-Pressure-Electrochemical Coupling Model. J. Electrochem. Soc. 2023, 170 (4), 040504  DOI: 10.1149/1945-7111/accaac
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    Naik, K. G.; Vishnugopi, B. S.; Mukherjee, P. P. Heterogeneities affect solid-state battery cathode dynamics. Energy Storage Mater. 2023, 55, 312– 321,  DOI: 10.1016/j.ensm.2022.11.055
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    Fultz, B. Vibrational thermodynamics of materials. Prog. Mater. Sci. 2010, 55 (4), 247– 352,  DOI: 10.1016/j.pmatsci.2009.05.002
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    Vibrational thermodynamics of materials
    Fultz, Brent
    Progress in Materials Science (2010), 55 (4), 247-352CODEN: PRMSAQ; ISSN:0079-6425. (Elsevier Ltd.)
    A review. The literature on vibrational thermodn. of materials is reviewed. The emphasis is on metals and alloys, esp. on the progress over the last decade in understanding differences in the vibrational entropy of different alloy phases and phase transformations. Some results on carbides, nitrides, oxides, hydrides and lithium-storage materials are also covered. Principles of harmonic phonons in alloys are organized into thermodn. models for unmixing and ordering transformations on an Ising lattice, and extended for non-harmonic potentials. Owing to the high accuracy required for the phonon frequencies, quant. predictions of vibrational entropy with anal. models prove elusive. Accurate tools for such calcns. or measurements were challenging for many years, but are more accessible today. Ab initio methods for calcg. phonons in solids are summarized. The exptl. techniques of calorimetry, inelastic neutron scattering, and inelastic X-ray scattering are explained with enough detail to show the issues of using these methods for investigations of vibrational thermodn. The explanations extend to methods of data anal. that affect the accuracy of thermodn. information. It is sometimes possible to identify the structural and chem. origins of the differences in vibrational entropy of materials, and the no. of these assessments is growing. There has been considerable progress in our understanding of the vibrational entropy of mixing in solid solns., compd. formation from pure elements, chem. unmixing of alloys, order-disorder transformations, and martensitic transformations. Systematic trends are available for some of these phase transformations, although more examples are needed, and many results are less reliable at high temps. Nanostructures in materials can alter sufficiently the vibrational dynamics to affect thermodn. stability. Internal stresses in polycrystals of anisotropic materials also contribute to the heat capacity. Lanthanides and actinides show a complex interplay of vibrational, electronic, and magnetic entropy, even at low temps. A "quasiharmonic model" is often used to extend the systematics of harmonic phonons to high temps. by accounting for the effects of thermal expansion against a bulk modulus. Non-harmonic effects beyond the quasiharmonic approxn. originate from the interactions of thermally-excited phonons with other phonons, or with the interactions of phonons with electronic excitations. In the classical high temp. limit, the adiabatic electron-phonon coupling can have a surprisingly large effect in metals when temp. causes significant changes in the electron d. near the Fermi level. There are useful similarities in how temp., pressure, and compn. alter the conduction electron screening and the interat. force consts. Phonon-phonon "anharmonic" interactions arise from those non-harmonic parts of the interat. potential that cannot be accounted for by the quasiharmonic model. Anharmonic shifts in phonon frequency with temp. can be substantial, but trends are not well understood. Anharmonic phonon damping does show systematic trends, however, at least for fcc metals. Trends of vibrational entropy are often justified with at. properties such as at. size, electronegativity, electron-to-atom ratio, and mass. Since vibrational entropy originates at the level of electrons in solids, such rules of thumb prove no better than similar rules devised for trends in bonding and structure, and tend to be worse. Fortunately, the required tools for accurate exptl. investigations of vibrational entropy have improved dramatically over the past few years, and the required ab initio methods have become more accessible. Steady progress is expected for understanding the phenomena reviewed here, as investigations are performed with the new tools of expt. and theory, sometimes in integrated ways.
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    Toberer, E. S.; Zevalkink, A.; Snyder, G. J. Phonon engineering through crystal chemistry. J. Mater. Chem. 2011, 21 (40), 15843,  DOI: 10.1039/c1jm11754h
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    Phonon engineering through crystal chemistry
    Toberer, Eric S.; Zevalkink, Alex; Snyder, G. Jeffrey
    Journal of Materials Chemistry (2011), 21 (40), 15843-15852CODEN: JMACEP; ISSN:0959-9428. (Royal Society of Chemistry)
    Mitigation of the global energy crisis requires tailoring the thermal cond. of materials. Low thermal cond. is crit. in a broad range of energy conversion technologies, including thermoelecs. and thermal barrier coatings. Here, we review the chem. trends and explore the origins of low thermal cond. in cryst. materials. A unifying feature in the latest materials is the incorporation of structural complexity to decrease the phonon velocity and increase scattering. With this understanding, strategies for combining these mechanisms can be formulated for designing new materials with exceptionally low thermal cond.
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    Hanus, R. C.; Gurunathan, R.; Lindsay, L.; Agne, M. T.; Shi, J.; Graham, S.; Synder, J. G. Thermal transport in defective and disordered materials. Appl. Phys. Rev. 2021, 8 (3), 31311,  DOI: 10.1063/5.0055593
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    Hunklinger, S.; Enss, C., Eds. Solid State Physics; Walter de Gruyter GmbH & Co KG, 2022. DOI:  DOI: 10.1515/9783110666502 .
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    Allen, P. B.; Feldman, J. L.; Fabian, J.; Wooten, F. Diffusons, locons and propagons: Character of atomic vibrations in amorphous Si. Philos. Mag. B 1999, 79 (11–12), 1715– 1731,  DOI: 10.1080/13642819908223054
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    Diffusions, locons, and propagons: character of atomic vibrations in amorphous Si
    Allen, Philip B.; Feldman, Joseph L.; Fabian, Jaroslav; Wooten, Frederick
    Philosophical Magazine B: Physics of Condensed Matter: Statistical Mechanics, Electronic, Optical and Magnetic Properties (1999), 79 (11/12), 1715-1731CODEN: PMABDJ; ISSN:0958-6644. (Taylor & Francis Ltd.)
    Numerical studies of amorphous Si show that the lowest 4% of vibrational modes are plane wave-like ("propagons") and the highest 3% of modes are localized ("locons"). The rest are neither plane wave-like nor localized. We call them "diffusons". Since diffusons are by far the most numerous, we try to characterize them by calcg. such properties as the wave-vector and polarization (which do not seem to be useful), "phase quotient" (a measure of the change of vibrational phase between first-neighbor atoms), spatial polarization memory and diffusivity. Localized states are characterized by finding decay lengths, inverse participation ratios and coordination nos. of the participating atoms.
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    Lv, W.; Henry, A. Examining the Validity of the Phonon Gas Model in Amorphous Materials. Sci. Rep. 2016, 6, 37675,  DOI: 10.1038/srep37675
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    Examining the Validity of the Phonon Gas Model in Amorphous Materials
    Lv, Wei; Henry, Asegun
    Scientific Reports (2016), 6 (), 37675CODEN: SRCEC3; ISSN:2045-2322. (Nature Publishing Group)
    The idea of treating phonon transport as equiv. to transport through a gas of particles is termed the phonon gas model (PGM), and it has been used almost ubiquitously to try and understand heat conduction in all solids. However, most of the modes in disordered materials do not propagate and thus may contribute to heat conduction in a fundamentally different way than is described by the PGM. From a practical perspective, the problem with trying to apply the PGM to amorphous materials is the fact that one cannot rigorously define the phonon velocities for non-propagating modes, since there is no periodicity. Here, we tested the validity of the PGM for amorphous materials by assuming the PGM is applicable, and then, using a combination of lattice dynamics, mol. dynamics (MD) and exptl. thermal cond. data, we back-calcd. the phonon velocities for the vibrational modes. The results of this approach show that if the PGM was valid, a large no. of the mid and high frequency modes would have to have either imaginary or extremely high velocities to reproduce the exptl. thermal cond. data. Furthermore, the results of MD based relaxation time calcns. suggest that in amorphous materials there is little, if any, connection between relaxation times and thermal cond. This then strongly suggests that the PGM is inapplicable to amorphous solids.
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    Hanus, R. C.; George, J.; Wood, M.; Bonkowski, A.; Cheng, Y.; Abernathy, D. L.; Manley, M. E.; Hautier, G.; Snyder, G. J.; Hermann, R. P. Uncovering design principles for amorphous-like heat conduction using two-channel lattice dynamics. Mater. Today Phys. 2021, 18, 100344  DOI: 10.1016/j.mtphys.2021.100344
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    Uncovering design principles for amorphous-like heat conduction using two-channel lattice dynamics
    Hanus, Riley; George, Janine; Wood, Max; Bonkowski, Alexander; Cheng, Yongqiang; Abernathy, Douglas L.; Manley, Michael E.; Hautier, Geoffroy; Snyder, G. Jeffrey; Hermann, Raphael P.
    Materials Today Physics (2021), 18 (), 100344CODEN: MTPAD5; ISSN:2542-5293. (Elsevier Ltd.)
    The physics of heat conduction puts practical limits on many technol. fields such as energy prodn., storage, and conversion. It is now widely appreciated that the phonon-gas model does not describe the full vibrational spectrum in amorphous materials, since this picture likely breaks down at higher frequencies. A two-channel heat conduction model, which uses harmonic vibrational states and lattice dynamics as a basis, has recently been shown to capture both crystal-like (phonon-gas channel) and amorphous-like (diffuson channel) heat conduction. While materials design principles for the phonon-gas channel are well established, similar understanding and control of the diffuson channel is lacking. In this work, in order to uncover design principles for the diffuson channel, we study structurally-complex cryst. Yb14 (Mn,Mg)Sb11, a champion thermoelec. material above 800 K, exptl. using inelastic neutron scattering and computationally using the two-channel lattice dynamical approach. Our results show that the diffuson channel indeed dominates in Yb14MnSb11 above 300 K. More importantly, we demonstrate a method for the rational design of amorphous-like heat conduction by considering the energetic proximity phonon modes and modifying them through chem. means. We show that increasing (decreasing) the mass on the Sb-site decreases (increases) the energy of these modes such that there is greater (smaller) overlap with Yb-dominated modes resulting in a higher (lower) thermal cond. This design strategy is exactly opposite of what is expected when the phonon-gas channel and/or common anal. models for the diffuson channel are considered, since in both cases an increase in at. mass commonly leads to a decrease in thermal cond. This work demonstrates how two-channel lattice dynamics can not only quant. predict the relative importance of the phonon-gas and diffuson channels, but also lead to rational design strategies in materials where the diffuson channel is important.
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  33. 33
    Simoncelli, M.; Marzari, N.; Mauri, F. Unified theory of thermal transport in crystals and glasses. Nat. Phys. 2019, 15 (8), 809– 813,  DOI: 10.1038/s41567-019-0520-x
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    33
    Unified theory of thermal transport in crystals and glasses
    Simoncelli, Michele; Marzari, Nicola; Mauri, Francesco
    Nature Physics (2019), 15 (8), 809-813CODEN: NPAHAX; ISSN:1745-2473. (Nature Research)
    Crystals and glasses exhibit fundamentally different heat conduction mechanisms: the periodicity of crystals allows for the excitation of propagating vibrational waves that carry heat, as first discussed by Peierls, while in glasses the lack of periodicity breaks Peierls's picture and heat is mainly carried by the coupling of vibrational modes, often described by a harmonic theory introduced by Allen and Feldman. Anharmonicity or disorder are thus the limiting factors for thermal cond. in crystals or glasses. Hitherto, no transport equation has been able to account for both. Here, we derive such an equation, resulting in a thermal cond. that reduces to the Peierls and Allen-Feldman limits, resp., in anharmonic crystals or harmonic glasses, while also covering the intermediate regimes where both effects are relevant. This approach also solves the long-standing problem of accurately predicting the thermal properties of crystals with ultralow or glass-like thermal cond., as we show with an application to a thermoelec. material representative of this class.
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  34. 34
    Niedziela, J. L.; Bansal, D.; May, A. F.; Ding, J.; Lanigan-Atkins, T.; Ehlers, G.; Abernathy, D. L.; Said, A.; Delaire, O. Selective breakdown of phonon quasiparticles across superionic transition in CuCrSe2. Nat. Phys. 2019, 15 (1), 73– 78,  DOI: 10.1038/s41567-018-0298-2
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    34
    Selective breakdown of phonon quasiparticles across superionic transition in CuCrSe2
    Niedziela, Jennifer L.; Bansal, Dipanshu; May, Andrew F.; Ding, Jingxuan; Lanigan-Atkins, Tyson; Ehlers, Georg; Abernathy, Douglas L.; Said, Ayman; Delaire, Olivier
    Nature Physics (2019), 15 (1), 73-78CODEN: NPAHAX; ISSN:1745-2473. (Nature Research)
    Superionic crystals exhibit ionic mobilities comparable to liqs. while maintaining a periodic cryst. lattice. The at. dynamics leading to large ionic mobility have long been debated. A central question is whether phonon quasiparticles-which conduct heat in regular solids-survive in the superionic state, where a large fraction of the system exhibits liq.-like behavior. Here we present the results of energy- and momentum-resolved scattering studies combined with first-principles calcns. and show that in the superionic phase of CuCrSe2, long-wavelength acoustic phonons capable of heat conduction remain largely intact, whereas specific phonon quasiparticles dominated by the Cu ions break down as a result of anharmonicity and disorder. The weak bonding and large anharmonicity of the Cu sublattice are present already in the normal ordered state, resulting in low thermal cond. even below the superionic transition. These results demonstrate that anharmonic phonon dynamics are at the origin of low thermal cond. and superionicity in this class of materials.
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  35. 35
    Bernges, T.; Peterlechner, M.; Wilde, G.; Agne, M. T.; Zeier, W. G. Analytical model for two-channel phonon transport engineering. Mater. Today Phys. 2023, 35, 101107  DOI: 10.1016/j.mtphys.2023.101107
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    Analytical model for two-channel phonon transport engineering
    Bernges, Tim; Peterlechner, Martin; Wilde, Gerhard; Agne, Matthias T.; Zeier, Wolfgang G.
    Materials Today Physics (2023), 35 (), 101107CODEN: MTPAD5; ISSN:2542-5293. (Elsevier Ltd.)
    The redn. of vibrational contributions to thermal transport and the search for material classes with intrinsically low lattice thermal conductivities are at the heart of thermoelec. research. Both engineering the heat transport of known thermoelecs. and searching for new material candidates is guided by understanding the physics of low thermal conduction. Spectral anal. models (e.g., the Callaway model) for propagating phonon transport have proved to be a powerful tool for interpreting exptl. results and providing metrics for materials design. Now, however, it is known that another mechanism of phonon heat transport can occur in complex cryst. materials. Called diffusons, they describe the non-propagating at. scale random-walk of thermal energy between energetically proximal phonon modes. While anal. models exist to describe both transport behaviors independently, an anal. model accounting for both transport channels simultaneously is necessary to interpret and design so-called 2-channel thermal transport. In this work, we propose an anal. 2-channel transport model that partitions the vibrational d. of states into two transport regimes and subsequently accounts for both transport mechanisms. The model is then used to explain the exptl. thermal conductivities of the solid soln. series Ag9-xGa1-xGexSe6. In this series, substitution leads to the stabilization of a highly vacant Ag+ substructure, which is expected to induce strong point-defect phonon scattering. While the propagating phonons are strongly scattered at low temps., the diffuson channel is apparently unaffected. By establishing materials design metrics for 2-channel thermal transport from anal. theory, exptl. investigations of materials with astonishingly low lattice thermal conductivities can now be better guided and informed.
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  36. 36
    Acharyya, P.; Ghosh, T.; Pal, K.; Rana, K. S.; Dutta, M.; Swain, D.; Etter, M.; Soni, A.; Waghmare, U. V.; Biswas, K. Glassy thermal conductivity in Cs3Bi2I6Cl3 single crystal. Nat. Commun. 2022, 13 (1), 5053,  DOI: 10.1038/s41467-022-32773-4
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    Glassy thermal conductivity in Cs3Bi2I6Cl3 single crystal
    Acharyya, Paribesh; Ghosh, Tanmoy; Pal, Koushik; Rana, Kewal Singh; Dutta, Moinak; Swain, Diptikanta; Etter, Martin; Soni, Ajay; Waghmare, Umesh V.; Biswas, Kanishka
    Nature Communications (2022), 13 (1), 5053CODEN: NCAOBW; ISSN:2041-1723. (Nature Portfolio)
    As the periodic at. arrangement of a crystal is made to a disorder or glassy-amorphous system by destroying the long-range order, lattice thermal cond., κL, decreases, and its fundamental characteristics changes. The realization of ultralow and unusual glass-like κL in a cryst. material is challenging but crucial to many applications like thermoelecs. and thermal barrier coatings. Herein, we demonstrate an ultralow (∼0.20 W/m·K at room temp.) and glass-like temp. dependence (2-400 K) of κL in a single crystal of layered halide perovskite, Cs3Bi2I6Cl3. Acoustic phonons with low cut-off frequency (20 cm-1) are responsible for the low sound velocity in Cs3Bi2I6Cl3 and make the structure elastically soft. While a strong anharmonicity originates from the low energy and localized rattling-like vibration of Cs atoms, synchrotron X-ray pair-distribution function evidence a local structural distortion in the Bi-halide octahedra and Cl vacancy. The hierarchical chem. bonding and soft vibrations from selective sublattice leading to low κL is intriguing from lattice dynamical perspective as well as have potential applications.
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    Xia, Y.; Gaines, D.; He, J.; Pal, K.; Li, Z.; Kanatzidis, M. G.; Ozoliṇš, V.; Wolverton, C. A unified understanding of minimum lattice thermal conductivity. Proc. Natl. Acad. Sci. U.S.A. 2023, 120 (26), e2302541120  DOI: 10.1073/pnas.2302541120
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    Zhou, H.; Tiwari, J.; Feng, T. Understanding the flat thermal conductivity of La2Zr2O7 at ultrahigh temperatures. Phys. Rev. Mater. 2024, 8 (4), 043804  DOI: 10.1103/PhysRevMaterials.8.043804
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    Isaeva, L.; Barbalinardo, G.; Donadio, D.; Baroni, S. Modeling heat transport in crystals and glasses from a unified lattice-dynamical approach. Nat. Commun. 2019, 10 (1), 3853,  DOI: 10.1038/s41467-019-11572-4
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    Modeling heat transport in crystals and glasses from a unified lattice-dynamical approach
    Isaeva Leyla; Baroni Stefano; Barbalinardo Giuseppe; Donadio Davide; Baroni Stefano
    Nature communications (2019), 10 (1), 3853 ISSN:.
    We introduce a novel approach to model heat transport in solids, based on the Green-Kubo theory of linear response. It naturally bridges the Boltzmann kinetic approach in crystals and the Allen-Feldman model in glasses, leveraging interatomic force constants and normal-mode linewidths computed at mechanical equilibrium. At variance with molecular dynamics, our approach naturally and easily accounts for quantum mechanical effects in energy transport. Our methodology is carefully validated against results for crystalline and amorphous silicon from equilibrium molecular dynamics and, in the former case, from the Boltzmann transport equation.
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  40. 40
    Deng, Z.; Sai Gautam, G.; Kolli, S. K.; Chotard, J.-N.; Cheetham, A. K.; Masquelier, C.; Canepa, P. Phase Behavior in Rhombohedral NaSiCON Electrolytes and Electrodes. Chem. Mater. 2020, 32 (18), 7908– 7920,  DOI: 10.1021/acs.chemmater.0c02695
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    Phase Behavior in Rhombohedral NaSiCON Electrolytes and Electrodes
    Deng, Zeyu; Sai Gautam, Gopalakrishnan; Kolli, Sanjeev Krishna; Chotard, Jean-Noel; Cheetham, Anthony K.; Masquelier, Christian; Canepa, Pieremanuele
    Chemistry of Materials (2020), 32 (18), 7908-7920CODEN: CMATEX; ISSN:0897-4756. (American Chemical Society)
    The replacement of the presently used liq. electrolytes by a nonflammable solid electrolytes is an important avenue to create safer batteries. The natrium superionic CONductor (NaSiCON) Na1+xZr2SixP3-xO12 (0 ≤ x ≤ 3) that displays high bulk ionic cond. and good stability toward other NaSiCON-based electrodes is a good solid electrolyte in NaSiCON-based batteries. Despite the sizeable share of research on Na1+xZr2SixP3-xO12, the structural and thermodn. properties of NaSiCON require better understanding for more efficient synthesis and optimization as a solid electrolyte, which often follows chem. intuition. Here, we analyze the thermodn. properties of the rhombohedral NaSiCON electrolyte by constructing the Na1+xZr2SixP3-xO12 phase diagram, based on d. functional theory calcns., a cluster expansion framework, and Monte Carlo simulations. Specifically, we build the phase diagram as a function of temp. and compn. (0 ≤ x ≤ 3) for the high-temp. rhombohedral structure, which has been also obsd. in several pos. electrode materials, such as Na3Ti2(PO4)3, Na3V2(PO4)3, Na3Cr2(PO4)3, and Na3Fe2(PO4)3. Through the phase diagram, we identify the concn. domains providing the highest Na+-ion cond. and previously unreported phase-sepn. behavior across three different single-phase regions. Furthermore, we note the similarities in the phase behavior between Na1+xZr2SixP3-xO12 and other NaSiCON-based monotransition metal electrodes and discuss the potential competition between thermodn. and kinetics in exptl. obsd. phase sepn. Our work is an important addn. in understanding the thermodn. of NaSiCON-based materials and in the development of inexpensive Na-ion batteries. From our results, we propose that the addn. of SiO44- moieties to single-transition metal NaSiCON-phosphate-based electrodes will significantly slow the kinetics toward phase sepn.
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    Boilot, J. P.; Collin, G.; Colomban, P. Crystal structure of the true nasicon: Na3Zr2Si2PO12. Mater. Res. Bull. 1987, 22 (5), 669– 676,  DOI: 10.1016/0025-5408(87)90116-4
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    Crystal structure of the true Nasicon: Na3Zr2Si2PO12
    Boilot, J. P.; Collin, G.; Colomban, P.
    Materials Research Bulletin (1987), 22 (5), 669-76CODEN: MRBUAC; ISSN:0025-5408.
    Room-temp. Nasicon (Na3.12(10)Zr2.00(1)Si2.12P0.88O12) is monoclinic, space group B2/b, with a 15.669(7), b 9.246(6), c 9.055(7) Å, and γ 124°12(4); Z = 4; final R = 7.51% for 1860 reflections. Nasicon at 623 K (Na3.09(8)Zr2.01(1)Si2.09P0.91O12) is rhombohedral, space group R‾3c, with a 9.074(2) and c 23.057(4) Å; Z = 6; final R = 2.93% for 704 reflections. Evidence of the total occupancy of the Zr octahedron was found, showing that only the Si/P nonstoichiometry mechanism is present in the Nasicon crystal. For the 2 temps. which were investigated, the structures are very close to that of the Nasicon analog Na3Sc2P3O12. However the Si/P substitution prevents the Na long-range ordering even in the monoclinic low-temp. phase and therefore the cross over to the rhombohedral symmetry only involves very small at. displacements. For both structures, a new Na position (mid-Na) is displayed in the conduction channel, intermediate between the usual Na(1) and Na(2) sites.
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  42. 42
    Zou, Z.; Ma, N.; Wang, A.; Ran, Y.; Song, T.; Jiao, Y.; Liu, J.; Zhou, H.; Shi, W.; He, B.; Wang, Da; Li, Y.; Avdeev, M.; Shi, S. Relationships Between Na+ Distribution, Concerted Migration, and Diffusion Properties in Rhombohedral NASICON. Adv. Energy Mater. 2020, 10 (30), 2001486,  DOI: 10.1002/aenm.202001486
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    Relationships Between Na+ Distribution, Concerted Migration, and Diffusion Properties in Rhombohedral NASICON
    Zou, Zheyi; Ma, Nan; Wang, Aiping; Ran, Yunbing; Song, Tao; Jiao, Yao; Liu, Jinping; Zhou, Hang; Shi, Wei; He, Bing; Wang, Da; Li, Yajie; Avdeev, Maxim; Shi, Siqi
    Advanced Energy Materials (2020), 10 (30), 2001486CODEN: ADEMBC; ISSN:1614-6840. (Wiley-Blackwell)
    Rhombohedral NaZr2(PO4)3 is the prototype of all the NASICON-type materials. The ionic diffusion in these rhombohedral NASICON materials is highly influenced by the ionic migration channels and the bottlenecks in the channels which have been extensively studied. However, no consensus is reached as to which one is the preferential ionic migration channel. Moreover, the relationships between the Na+ distribution over the multiple available sites, concerted migration, and diffusion properties remain elusive. Using ab initio mol. dynamics simulations, here it is shown that the Na+ ions tend to migrate through the Na1-Na3-Na2-Na3-Na1 channels rather than through the Na2-Na3-Na3-Na2 channels. There are two types of concerted migration mechanisms: two Na+ ions located at the adjacent Na1 and Na2 sites can migrate either in the same direction or at an angle. Both mechanisms exhibit relatively low migration barriers owing to the potential energy conversion during the Na+ ions migration process. Redistribution of Na+ ions from the most stable Na1 sites to Na2 on increasing Na+ total content further facilitates the concerted migration and promotes the Na+ ion mobility. The work establishes a connection between the Na+ concn. in rhombohedral NASICON materials and their diffusion properties.
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    Qui, D. T.; Capponi, J. J.; Joubert, J. C.; Shannon, R. D. Crystal structure and ionic conductivity in Na4Zr2Si3O12. J. Solid State Chem. 1981, 39 (2), 219– 229,  DOI: 10.1016/0022-4596(81)90335-2
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    Crystal structure and ionic conductivity in sodium zirconium silicate (Na4Zr2Si3O12)
    Qui, D. Tran; Capponi, J. J.; Joubert, J. C.; Shannon, R. D.
    Journal of Solid State Chemistry (1981), 39 (2), 219-29CODEN: JSSCBI; ISSN:0022-4596.
    Na ion conductivities of Na4-xZr2Si3-xPxO12 range from 3.5 × 10-4 (ohm-cm)-1 for x = 0 to 1.9 × 10-1 (ohm-cm)-1 for x = 1.0 at 300°. Structure refinements of single-crystal Na4Zr2Si3O12 were carried out at 25, 300, and 620°. Little change occurs in bond distances and angles in the (Zr2Si3O12)-4 framework whereas the Na(1)-O6 and Na(2)-O8 polyhedra enlarge dramatically with increase of temp. The large thermal motion of Na(1) and Na(2) is probably related to the Na mobility in this structure. Of the four possible Na ion pathways, only two have openings large enough to allow reasonable mobility. The first, connecting a Na(1) site to a Na(2) site, is somewhat smaller (1.86 Å) than a Na ion (2.30 Å) at room temp. but increases substantially to 2.22 Å at 620°. The second, connecting a Na(2) site to a Na(2) site, is larger (2.37 Å) and increases to 2.66 Å at 620°. Difference Fourier maps show significant electron d. along Na(2)-Na(2) paths and Na(2) thermal ellipsoids have major axes close to these paths.
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    Lenain, G. E.; McKinstry, H. A.; Alamo, J.; Agrawal, D. K. Structural model for thermal expansion in MZr2P3O12 (M = Li, Na, K, Rb, Cs). J. Mater. Sci. 1987, 22 (1), 17– 22,  DOI: 10.1007/BF01160546
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    Structural model for thermal expansion in alkali metal zirconium phosphates MZr2P3O12 (M = Li, Na, K, Rb, Cs)
    Lenain, G. E.; McKinstry, H. A.; Alamo, J.; Agrawal, D. K.
    Journal of Materials Science (1987), 22 (1), 17-22CODEN: JMTSAS; ISSN:0022-2461.
    A structural model is proposed to describe the highly anisotropic thermal expansion in the NaZr2P3O12 structure as a result of the thermal motion of the polyhedra in the structure. In the proposed model the rotations of the PO4 tetrahedra are coupled to the rotation of the Zr octahedra. Of the 2 versions considered, the 1st one allows angular distortions to occur only in the ZrO6 octahedra; the 2nd one permits all polyhedra to be distorted.
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    Oota, T.; Yamai, I. Thermal Expansion Behavior of NaZr2(PO4)3Type Compounds. J. Am. Ceram. Soc. 1986, 69 (1), 1– 6,  DOI: 10.1111/j.1151-2916.1986.tb04682.x
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    Pet’kov, V. I.; Orlova, A. I.; Kazantsev, G. N.; Samoilov, S. G.; Spiridonova, M. L. Thermal Expansion in the Zr and 1-, 2-Valent Complex Phosphates of NaZr2(PO4)3 (NZP) Structure. J. Therm. Anal. Calorim. 2001, 66 (2), 623– 632,  DOI: 10.1023/A:1013145807987
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    Thermal expansion in the Zr and 1-, 2-valent complex phosphates of NaZr2(PO4)3 (NZP) structure
    Pet'kov, V. I.; Orlova, A. I.; Kazantsev, G. N.; Samoilov, S. G.; Spiridonova, M. L.
    Journal of Thermal Analysis and Calorimetry (2001), 66 (2), 623-632CODEN: JTACF7; ISSN:1418-2874. (Kluwer Academic Publishers)
    A5-4xZrxZr(PO4)3 (A = Na, K; 0 ≤ x ≤ 1.25), Na1-xCd0.5xZr2(PO4)3 (0 ≤ χ ≤ 1), Na5-xCd0.5xZr(PO4)3 (O ≤ χ ≤ 4) compns. which belong to the NZP structural family were synthesized using the sol-gel method. The lattice thermal expansions of members of these rows were detd. up to 600°C by high-temp. X-ray diffractometry. The axial thermal expansion coeffs. change from -5.8·10-6 to 7.5·10-6 °C-1 (αa) and from 2.6·10-6 to 22·10-6 °C-1 (αc). These results, in addn. to those for other NZP compds., allow us to explain their low thermal expansion. The mechanism can be attributed to strongly bonded three-dimensional network structure, the existence of structural holes capable of damping some of the thermal vibrations and anisotropy in the thermal expansion of the lattice.
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    Dronskowski, R.; Bloechl, P. E. Crystal orbital Hamilton populations (COHP): energy-resolved visualization of chemical bonding in solids based on density-functional calculations. J. Phys. Chem. 1993, 97 (33), 8617– 8624,  DOI: 10.1021/j100135a014
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    Crystal orbital Hamilton populations (COHP): energy-resolved visualization of chemical bonding in solids based on density-functional calculations
    Dronskowski, Richard; Bloechl, Peter E.
    Journal of Physical Chemistry (1993), 97 (33), 8617-24CODEN: JPCHAX; ISSN:0022-3654.
    After giving a concise overview of the current knowledge in the field of quantum mech. bonding indicators for mols. and solids, the authors show how to obtain energy-resolved visualization of chem. bonding in solids by d.-functional electronic structure calcns. From a band structure energy partitioning scheme, i.e., rewriting the band structure energy as a sum of orbital pair contributions, the authors derive what is to be defined as crystal orbital Hamilton populations (COHP). In particular, a COHP(ε) diagram indicates bonding, nonbonding, and antibonding energy regions within a specified energy range while an energy integral of a COHP gives access to the contribution of an atom or a chem. bond to the distribution of 1-particle energies. A further decompn. into specific AOs or symmetry-adapted linear combinations of AOs (hybrids) can easily be performed by using a projector technique involving unitary transformations. Because of its structural simplicity and the availability of reliable thermodn. data, the authors study the bonding within cryst. silicon (diamond phase) 1st. As a basis set, both bcc. screened and diamond screened at.-centered tight-binding linear muffin-tin orbitals (TB-LMTOs) are used throughout. The shape of COHP vs. energy diagrams and the significance of COHP subcontributions (s-s, sp3-sp3) are analyzed. Specifically, the difference between the COHPs energy integral (1-particle bond energy) and the exptl. bond energy is critically examd. While abs. values for the 1-particle bond energy show a high basis set dependence due to changing on-site (crystal field) COHP terms, the shape of off-site (bonding) COHP functions, elucidating the local bonding principle within an extended structure, remains nearly unaffected.
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    Krenzer, G.; Kim, C.-E.; Tolborg, K.; Morgan, B. J.; Walsh, A. Anharmonic lattice dynamics of superionic lithium nitride. J. Mater. Chem. A 2022, 10 (5), 2295– 2304,  DOI: 10.1039/D1TA07631K
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    Bernges, T.; Hanus, R.; Wankmiller, B.; Imasato, K.; Lin, S.; Ghidiu, M.; Gerlitz, M.; Peterlechner, M.; Graham, S.; Hautier, G.; Pei, Y.; Hansen, M. R.; Wilde, G.; Snyder, G. J.; George, J.; Agne, M. T.; Zeier, W. G. Considering the Role of Ion Transport in Diffuson-Dominated Thermal Conductivity. Adv. Energy Mater. 2022, 12 (22), 2200717,  DOI: 10.1002/aenm.202200717
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    Considering the Role of Ion Transport in Diffuson-Dominated Thermal Conductivity
    Bernges, Tim; Hanus, Riley; Wankmiller, Bjoern; Imasato, Kazuki; Lin, Siqi; Ghidiu, Michael; Gerlitz, Marius; Peterlechner, Martin; Graham, Samuel; Hautier, Geoffroy; Pei, Yanzhong; Hansen, Michael Ryan; Wilde, Gerhard; Snyder, G. Jeffrey; George, Janine; Agne, Matthias T.; Zeier, Wolfgang G.
    Advanced Energy Materials (2022), 12 (22), 2200717CODEN: ADEMBC; ISSN:1614-6840. (Wiley-Blackwell)
    Next-generation thermal management requires the development of low lattice thermal cond. materials, as obsd. in ionic conductors. For example, thermoelec. efficiency is increased when thermal cond. is decreased. Detrimentally, high ionic cond. leads to thermoelec. device degrdn. Battery safety and design also require an understanding of thermal transport in ionic conductors. Ion mobility, structural complexity, and anharmonicity have been used to explain the thermal transport properties of ionic conductors. However, thermal and ionic transport are rarely discussed in direct comparison. Herein, the ionic cond. of Ag+ argyrodites is found to change by orders of magnitude without altering the thermal cond. Thermal cond. measurements and two-channel lattice dynamics modeling reveal that the majority of Ag+ vibrations have a non-propagating diffuson-like character, similar to amorphous materials. It is found that high ionic mobility is not a requirement for diffuson-mediated transport. Instead, the same bonding and structural traits that can lead to fast ionic conduction also lead to diffuson-mediated transport. Bridging the fields of solid-state ionics and thermal transport, it is proposed that a vibrational perspective can lead to new design strategies for functional ionic conducting materials. As a first step, the authors relate the so-called Meyer-Neldel behavior in ionic conductors to phonon occupations.
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    Famprikis, T.; Canepa, P.; Dawson, J. A.; Islam, M. S.; Masquelier, C. Fundamentals of inorganic solid-state electrolytes for batteries. Nat. Mater. 2019, 18 (12), 1278– 1291,  DOI: 10.1038/s41563-019-0431-3
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    Fundamentals of inorganic solid-state electrolytes for batteries
    Famprikis, Theodosios; Canepa, Pieremanuele; Dawson, James A.; Islam, M. Saiful; Masquelier, Christian
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  • Abstract

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    Figure 1

    Figure 1. Schematic of the density of states (middle), indicating frequency ranges commonly associated with phonon-gas and diffuson character. Representation of typical thermal conductivity trends from phonon-gas and diffuson contributions (left). Low frequencies can be associated with softer potentials and larger vibrational amplitudes displacing the (mobile) ion further from its equilibrium position. In contrast, higher frequencies cause the ion to perform more periods of vibrations with smaller amplitudes (right).

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    Figure 2

    Figure 2. (a) Hexagonal representation of the unit cell of Na4Zr2Si3O12 with coordination environments and the connectivity of the Na+ sites. The only differences in the NaZr2P3O12 structure are the vacant Na(2) sites and PO43– instead of SiO44– tetrahedra. (b) Temperature dependence of the Na–O bond distances. For the Na(2) site in Na4Zr2Si3O12 the average bond distance is given, and the temperature dependence of all four individual bond lengths is shown in Figure S7. (c) Connectivity of the sodium polyhedra, visualizing that the diffusion path from the Na(1) to Na(2) site is bent. Only 2 out of the 6 adjacent Na(3) polyhedra around the central Na(1) antiprism are shown for visual clarity. The connectivity to all adjacent Na(2) and Na(3) sites can be best seen in Figure 8a–c. (d) Diverging bond angles within the Na(1)O6 polyhedra due to the rotation of ZrO68– and (P/Si)O43/4– polyhedra building up the rigid framework of the unit cell, leading to a distortion of the sodium ion cavity.

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    Figure 3

    Figure 3. (a) Isotropic thermal displacement parameters from lattice dynamics calculations (dashed lines), values obtained from Rietveld refinement (symbols), and their respective linear fit (solid line). Diamond shapes correspond to values reported by Morgan et al. (20) and Qui et al., (43) respectively. (b) Einstein frequencies of sodium ions on the Na(1) and Na(2) site in NaZr2P3O12 (NZP) and Na4Zr2Si3O12 (NZS), respectively, deduced from the temperature dependence of the isotropic thermal displacement parameters.

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    Figure 4

    Figure 4. (a) Average frequencies of the total, atom-, site- and jump-direction-projected phonon DOS in NaZr2P3O12 in comparison to those found for Na4Zr2Si3O12. (b) Atom-, site- jump-direction-projected phonon DOS of sodium in NaZr2P3O12 and Na4Zr2Si3O12. In NaZr2P3O12, atom- and site-projected phonon DOS are identical as there is only one sodium site. (c) Connectivity of the Na+ polyhedra, visualizing the direction of projection for the jump-direction-projected phonon DOS (onto the Na(3) site). (d) Distribution of omnidirectional eigenvectors eq,s and (e) distribution of eigenvectors projected onto the Na(3) site obtained by scaling the eigenvector by the dot product of the eigenvector and projection vector p. The hat notes that the respective vector is scaled to unity. The eigenvectors are a function of wave vector q and phonon branch index s.

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    Figure 5

    Figure 5. Ab initio modeled (lines) and experimental (symbols) thermal conductivities of NaZr2P3O12 and Na4Zr2Si3O12. (a) Side-to-side comparison of the materials with a crystallite size of 0.5 μm used to calculate the ab initio thermal conductivities. The anisotropic structure of both compounds results in anisotropic thermal conductivity. The lines shown for the total thermal conductivity represent the directions in space with the highest and lowest thermal conductivity. For visual clarity, phonon-gas and diffuson conductivities are averaged in all spatial directions. (b) Extension of panel (a) by different crystallite sizes to emphasize their influence (b) on the height of the phonon peak in both materials. Note that both panels are scaled differently due to a much more distinct phonon peak in NaZr2P3O12. The impact on the diffuson channel is negligible and therefore not shown.

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    Figure 6

    Figure 6. Spectral thermal conductivity of phonon-gas and diffuson channel, denoted as κpg and κdiff, respectively, at 100 (top panels) and 800 K (lower panels). Due to the high phonon peak at low temperatures in NaZr2P3O12, an inset is added to show the entire spectrum in the upper left panel. The percentages give the contribution of each channel to the total thermal conductivity.

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    Figure 7

    Figure 7. (a) Grüneisen parameter of NaZr2P3O12 (left panels) and Na4Zr2Si3O12 (right panels). The ratio of sodium partial phonon DOS to total phonon DOS, which is the contribution of sodium ions to vibrations at a given frequency, is color-coded. (b) Log-scaled phonon linewidths averaged over the Brillouin zone depicted at different temperatures. Both quantities exhibit highly anharmonic modes at low frequencies, especially for Na4Zr2Si3O12 at high temperatures. (c,d) Outer panels: Portion of phonon band structure at 100 and 800 K with color-coded phonon linewidths of (c) NaZr2P3O12 and (d) Na4Zr2Si3O12, respectively. Inner panels: Phonon spectral energy density of phonon bands in a selected frequency range, showing the effect of the phonon linewidth (broadening) on the phonon overlap, highlighted with green shading.

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    Figure 8

    Figure 8. Local environment and vibration of the sodium sites: Spatial distributions of average site frequencies in NaZr2P3O12 (a) and Na4Zr2Si3O12 (b) when projected in a specific direction. The neighboring Na(3) and oxygen sites are shown for orientation. (c) Calculated anisotropic thermal displacement ellipsoids of the sodium ions in Na4Zr2Si3O12 at 300 K with 95% probability. The thermal displacement in the direction of diffusion is higher than average. Especially the displacement parameters of the Na(2) site are highly anisotropic. The Na(3) sites displayed in (a) to (c) are unoccupied and only added as a visual reference for the direction of the diffusion path. Their displacement parameters are arbitrarily chosen. (d) High average frequencies correspond to a more rigid potential well, smaller displacements (labeled dharm. and danh.), and less deviation of harmonic and anharmonic potential than lower frequencies. The eigenvalues of the harmonic oscillator, marked by the gray lines, have a larger spacing the more rigid the potential becomes.

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  • References

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      Deng, Zeyu; Mishra, Tara P.; Mahayoni, Eunike; Ma, Qianli; Tieu, Aaron Jue Kang; Guillon, Olivier; Chotard, Jean-Noel; Seznec, Vincent; Cheetham, Anthony K.; Masquelier, Christian; Gautam, Gopalakrishnan Sai; Canepa, Pieremanuele
      Nature Communications (2022), 13 (1), 4470CODEN: NCAOBW; ISSN:2041-1723. (Nature Portfolio)
      Lithium and sodium (Na) mixed polyanion solid electrolytes for all-solid-state batteries display some of the highest ionic conductivities reported to date. However, the effect of polyanion mixing on the ion-transport properties is still not fully understood. Here, we focus on Na1+xZr2SixP3-xO12 (0 ≥ x ≥ 3) NASICON electrolyte to elucidate the role of polyanion mixing on the Na-ion transport properties. Although NASICON is a widely investigated system, transport properties derived from expts. or theory vary by orders of magnitude. We use more than 2000 distinct ab initio-based kinetic Monte Carlo simulations to map the compositional space of NASICON over various time ranges, spatial resolns. and temps. Via electrochem. impedance spectroscopy measurements on samples with different sodium content, we find that the highest ionic cond. (i.e., about 0.165 S cm-1 at 473 K) is exptl. achieved in Na3.4Zr2Si2.4P0.6O12, in line with simulations (i.e., about 0.170 S cm-1 at 473 K). The theor. studies indicate that doped NASICON compds. (esp. those with a silicon content x ≥ 2.4) can improve the Na-ion mobility compared to undoped NASICON compns.
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    12. 12
      Ma, Q.; Tsai, C.-L.; Wei, X.-K.; Heggen, M.; Tietz, F.; Irvine, J. T. S. Room temperature demonstration of a sodium superionic conductor with grain conductivity in excess of 0.01 S cm–1 and its primary applications in symmetric battery cells. J. Mater. Chem. A 2019, 7 (13), 7766– 7776,  DOI: 10.1039/C9TA00048H
      12
      Room temperature demonstration of a sodium superionic conductor with grain conductivity in excess of 0.01 S cm-1 and its primary applications in symmetric battery cells
      Ma, Qianli; Tsai, Chih-Long; Wei, Xian-Kui; Heggen, Marc; Tietz, Frank; Irvine, John T. S.
      Journal of Materials Chemistry A: Materials for Energy and Sustainability (2019), 7 (13), 7766-7776CODEN: JMCAET; ISSN:2050-7496. (Royal Society of Chemistry)
      The lack of suitable candidate electrolyte materials for practical application limits the development of all-solid-state Na-ion batteries. Na3+xZr2Si2+xP1-xO12 was the very first series of NASICONs discovered some 40 years ago; however, sepn. of bulk cond. from total cond. at room temp. is still problematic. It has been suggested that the effective Na-ion cond. is ∼10-4 S cm-1 at room temp. for Na3+xZr2Si2+xP1-xO12 ceramics; however using a soln.-assisted solid-state reaction for prepn. of Na3+xZr2Si2+xP1-xO12, a total cond. of 5 × 10-3 S cm-1 was achieved for Na3.4Zr2Si2.4P0.6O12 at 25 °C, higher than the values previously reported for polycryst. Na-ion conductors. A bulk cond. of 1.5 × 10-2 S cm-1 was revealed by high frequency impedance spectroscopy (up to 3 GHz) and verified by low temp. impedance spectroscopy (down to -100 °C) for Na3.4Zr2Si2.4P0.6O12 at 25 °C, indicating further the potential of increasing the related total cond. A Na/Na3.4Zr2Si2.4P0.6O12/Na sym. cell showed low interface resistance and high cycling stability at room temp. A full-ceramic cell was fabricated and tested at 28 °C with good cycling performance.
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    13. 13
      Rajagopalan, R.; Zhang, Z.; Tang, Y.; Jia, C.; Ji, X.; Wang, H. Understanding crystal structures, ion diffusion mechanisms and sodium storage behaviors of NASICON materials. Energy Storage Mater. 2021, 34, 171– 193,  DOI: 10.1016/j.ensm.2020.09.007
      There is no corresponding record for this reference.
    14. 14
      Boilot, J. P.; Collin, G.; Colomban, P. Relation structure-fast ion conduction in the NASICON solid solution. J. Solid State Chem. 1988, 73 (1), 160– 171,  DOI: 10.1016/0022-4596(88)90065-5
      14
      Relation structure - fast ion conduction in the Nasicon solid solution
      Boilot, J. P.; Collin, G.; Colomban, P.
      Journal of Solid State Chemistry (1988), 73 (1), 160-71CODEN: JSSCBI; ISSN:0022-4596.
      Crystal detns. of the rhombohedral phase (space group R‾3c), for different compns. (2 < x < 2.4) in the true Nasicon solid soln. Na1+xZr2SixP3-xO12, were performed at different temps. by x-ray diffraction. As a consequence of interionic repulsions, the partial occupation of a mid-Na interstitial site within the conduction path was obsd. The compn. dependence of the mid-Na occupation factor, max. at x = 2, explains the max. of the c hexagonal parameter and of the Na(I)-O av. distance obsd. at about x = 2. Moreover, structural results clearly suggest that the enhanced cond. at x = 2 arises from Na interactions instead of geometry changes of the framework.
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    15. 15
      Kumar, P. P.; Yashonath, S. Structure, Conductivity, and Ionic Motion in Na1+xZr2SixP3-xO12: A Simulation Study. J. Phys. Chem. B 2002, 106 (28), 7081– 7089,  DOI: 10.1021/jp020287h
      15
      Structure, Conductivity, and Ionic Motion in Na1+xZr2SixP3-xO12: A Simulation Study
      Kumar, P. Padma; Yashonath, S.
      Journal of Physical Chemistry B (2002), 106 (28), 7081-7089CODEN: JPCBFK; ISSN:1089-5647. (American Chemical Society)
      Const.-pressure, const.-temp. variable-shape simulation cell Monte Carlo and microcanonical ensemble mol. dynamics simulation of superionic conducting rhombohedral phase of Nasicon, Na1+xZr2SixP3-xO12, 0 ≤ x ≤ 3, at a temp. of 600 K is reported. Changes in structure, cond., hop path, site occupancies, bond lengths of framework atoms with compn. are discussed. Av. Na(1)-O distance shows a peak at x = 2, while Na(2)-O distance shows a monotonic increase. Sum of the sodium occupancies at Na(1) and mid-Na sites adds up to a const. value of one which supports the conclusion of J. P. Boilot et al. (1988) based on X-ray diffraction. Occupancy of Na(1) site attains a min. at x = 2. The predominant conduction channel (which carries more than 90% of the sodium ions) is found to be the one connecting Na(1)-mid-Na-Na(2). D. contours for sodium, depicting this conduction channel, are reported. Free energy profile along the conduction channel suggests that entropy contribution cannot be neglected. The mid-Na site is not assocd. with a free energy min.
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    16. 16
      Abello, L.; Chhor, K.; Barj, M.; Pommier, C.; Delmas, C. Heat capacity and Na+ ion disorder in Nasicon-type solid electrolytes Na3M2P3O12 (M2= Fe2, Cr2, ZrMg) in the temperature range 10 to 300 K. J. Mater. Sci. 1989, 24 (9), 3380– 3386,  DOI: 10.1007/BF01139069
      There is no corresponding record for this reference.
    17. 17
      Pet’kov, V. I.; Asabina, E. A.; Markin, A. V.; Smirnova, N. N.; Kitaev, D. B. Thermodynamic data of the NZP compounds family. J. Therm. Anal. Calorim. 2005, 80 (3), 695– 700,  DOI: 10.1007/s10973-005-0716-4
      There is no corresponding record for this reference.
    18. 18
      Pet’kov, V. I.; Asabina, E. A.; Markin, A. V.; Smirnova, N. N. Synthesis, characterization and thermodynamic data of compounds with NZP structure. J. Therm. Anal. Calorim. 2008, 91 (1), 155– 161,  DOI: 10.1007/s10973-007-8370-7
      There is no corresponding record for this reference.
    19. 19
      Maier, J.; Warhus, U.; Gmelin, E. Thermodynamic and electrochemical investigations of the Nasicon solid solution system. Solid State Ionics 1986, 18–19, 969– 973,  DOI: 10.1016/0167-2738(86)90294-8
      There is no corresponding record for this reference.
    20. 20
      Morgan, E. E.; Evans, H. A.; Pilar, K.; Brown, C. M.; Clément, R. J.; Maezono, R.; Seshadri, R.; Monserrat, B.; Cheetham, A. K. Lattice Dynamics in the NASICON NaZr2(PO4)3 Solid Electrolyte from Temperature-Dependent Neutron Diffraction, NMR, and Ab Initio Computational Studies. Appl. Phys. Lett. 2022, 34 (9), 4029– 4038,  DOI: 10.1021/acs.chemmater.2c00212
      There is no corresponding record for this reference.
    21. 21
      Zhen, X.; Sanson, A.; Sun, Q.; Liang, E.; Gao, Q. Role of alkali ions in the near-zero thermal expansion of NaSICON-type AZr2(PO4)3(A = Na,K,Rb,Cs) and Zr2(PO4)3 compounds. Phys. Rev. B 2023, 108 (14), 144102  DOI: 10.1103/PhysRevB.108.144102
      There is no corresponding record for this reference.
    22. 22
      Agne, M. T.; Böger, T.; Bernges, T.; Zeier, W. G. Importance of Thermal Transport for the Design of Solid-State Battery Materials. PRX Energy 2022, 1 (3), 31002,  DOI: 10.1103/PRXEnergy.1.031002
      There is no corresponding record for this reference.
    23. 23
      Kantharaj, R.; Marconnet, A. M. Heat Generation and Thermal Transport in Lithium-Ion Batteries: A Scale-Bridging Perspective. Nanoscale Microscale Thermophys. Eng. 2019, 23 (2), 128– 156,  DOI: 10.1080/15567265.2019.1572679
      There is no corresponding record for this reference.
    24. 24
      Gu, J.; Xu, R.; Chen, B.; Zhou, J. NMC811-Li6PS5Cl-Li/In All-Solid-State Battery Capacity Attenuation Based on Temperature-Pressure-Electrochemical Coupling Model. J. Electrochem. Soc. 2023, 170 (4), 040504  DOI: 10.1149/1945-7111/accaac
      There is no corresponding record for this reference.
    25. 25
      Naik, K. G.; Vishnugopi, B. S.; Mukherjee, P. P. Heterogeneities affect solid-state battery cathode dynamics. Energy Storage Mater. 2023, 55, 312– 321,  DOI: 10.1016/j.ensm.2022.11.055
      There is no corresponding record for this reference.
    26. 26
      Fultz, B. Vibrational thermodynamics of materials. Prog. Mater. Sci. 2010, 55 (4), 247– 352,  DOI: 10.1016/j.pmatsci.2009.05.002
      26
      Vibrational thermodynamics of materials
      Fultz, Brent
      Progress in Materials Science (2010), 55 (4), 247-352CODEN: PRMSAQ; ISSN:0079-6425. (Elsevier Ltd.)
      A review. The literature on vibrational thermodn. of materials is reviewed. The emphasis is on metals and alloys, esp. on the progress over the last decade in understanding differences in the vibrational entropy of different alloy phases and phase transformations. Some results on carbides, nitrides, oxides, hydrides and lithium-storage materials are also covered. Principles of harmonic phonons in alloys are organized into thermodn. models for unmixing and ordering transformations on an Ising lattice, and extended for non-harmonic potentials. Owing to the high accuracy required for the phonon frequencies, quant. predictions of vibrational entropy with anal. models prove elusive. Accurate tools for such calcns. or measurements were challenging for many years, but are more accessible today. Ab initio methods for calcg. phonons in solids are summarized. The exptl. techniques of calorimetry, inelastic neutron scattering, and inelastic X-ray scattering are explained with enough detail to show the issues of using these methods for investigations of vibrational thermodn. The explanations extend to methods of data anal. that affect the accuracy of thermodn. information. It is sometimes possible to identify the structural and chem. origins of the differences in vibrational entropy of materials, and the no. of these assessments is growing. There has been considerable progress in our understanding of the vibrational entropy of mixing in solid solns., compd. formation from pure elements, chem. unmixing of alloys, order-disorder transformations, and martensitic transformations. Systematic trends are available for some of these phase transformations, although more examples are needed, and many results are less reliable at high temps. Nanostructures in materials can alter sufficiently the vibrational dynamics to affect thermodn. stability. Internal stresses in polycrystals of anisotropic materials also contribute to the heat capacity. Lanthanides and actinides show a complex interplay of vibrational, electronic, and magnetic entropy, even at low temps. A "quasiharmonic model" is often used to extend the systematics of harmonic phonons to high temps. by accounting for the effects of thermal expansion against a bulk modulus. Non-harmonic effects beyond the quasiharmonic approxn. originate from the interactions of thermally-excited phonons with other phonons, or with the interactions of phonons with electronic excitations. In the classical high temp. limit, the adiabatic electron-phonon coupling can have a surprisingly large effect in metals when temp. causes significant changes in the electron d. near the Fermi level. There are useful similarities in how temp., pressure, and compn. alter the conduction electron screening and the interat. force consts. Phonon-phonon "anharmonic" interactions arise from those non-harmonic parts of the interat. potential that cannot be accounted for by the quasiharmonic model. Anharmonic shifts in phonon frequency with temp. can be substantial, but trends are not well understood. Anharmonic phonon damping does show systematic trends, however, at least for fcc metals. Trends of vibrational entropy are often justified with at. properties such as at. size, electronegativity, electron-to-atom ratio, and mass. Since vibrational entropy originates at the level of electrons in solids, such rules of thumb prove no better than similar rules devised for trends in bonding and structure, and tend to be worse. Fortunately, the required tools for accurate exptl. investigations of vibrational entropy have improved dramatically over the past few years, and the required ab initio methods have become more accessible. Steady progress is expected for understanding the phenomena reviewed here, as investigations are performed with the new tools of expt. and theory, sometimes in integrated ways.
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    27. 27
      Toberer, E. S.; Zevalkink, A.; Snyder, G. J. Phonon engineering through crystal chemistry. J. Mater. Chem. 2011, 21 (40), 15843,  DOI: 10.1039/c1jm11754h
      27
      Phonon engineering through crystal chemistry
      Toberer, Eric S.; Zevalkink, Alex; Snyder, G. Jeffrey
      Journal of Materials Chemistry (2011), 21 (40), 15843-15852CODEN: JMACEP; ISSN:0959-9428. (Royal Society of Chemistry)
      Mitigation of the global energy crisis requires tailoring the thermal cond. of materials. Low thermal cond. is crit. in a broad range of energy conversion technologies, including thermoelecs. and thermal barrier coatings. Here, we review the chem. trends and explore the origins of low thermal cond. in cryst. materials. A unifying feature in the latest materials is the incorporation of structural complexity to decrease the phonon velocity and increase scattering. With this understanding, strategies for combining these mechanisms can be formulated for designing new materials with exceptionally low thermal cond.
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    28. 28
      Hanus, R. C.; Gurunathan, R.; Lindsay, L.; Agne, M. T.; Shi, J.; Graham, S.; Synder, J. G. Thermal transport in defective and disordered materials. Appl. Phys. Rev. 2021, 8 (3), 31311,  DOI: 10.1063/5.0055593
      There is no corresponding record for this reference.
    29. 29
      Hunklinger, S.; Enss, C., Eds. Solid State Physics; Walter de Gruyter GmbH & Co KG, 2022. DOI:  DOI: 10.1515/9783110666502 .
      There is no corresponding record for this reference.
    30. 30
      Allen, P. B.; Feldman, J. L.; Fabian, J.; Wooten, F. Diffusons, locons and propagons: Character of atomic vibrations in amorphous Si. Philos. Mag. B 1999, 79 (11–12), 1715– 1731,  DOI: 10.1080/13642819908223054
      30
      Diffusions, locons, and propagons: character of atomic vibrations in amorphous Si
      Allen, Philip B.; Feldman, Joseph L.; Fabian, Jaroslav; Wooten, Frederick
      Philosophical Magazine B: Physics of Condensed Matter: Statistical Mechanics, Electronic, Optical and Magnetic Properties (1999), 79 (11/12), 1715-1731CODEN: PMABDJ; ISSN:0958-6644. (Taylor & Francis Ltd.)
      Numerical studies of amorphous Si show that the lowest 4% of vibrational modes are plane wave-like ("propagons") and the highest 3% of modes are localized ("locons"). The rest are neither plane wave-like nor localized. We call them "diffusons". Since diffusons are by far the most numerous, we try to characterize them by calcg. such properties as the wave-vector and polarization (which do not seem to be useful), "phase quotient" (a measure of the change of vibrational phase between first-neighbor atoms), spatial polarization memory and diffusivity. Localized states are characterized by finding decay lengths, inverse participation ratios and coordination nos. of the participating atoms.
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    31. 31
      Lv, W.; Henry, A. Examining the Validity of the Phonon Gas Model in Amorphous Materials. Sci. Rep. 2016, 6, 37675,  DOI: 10.1038/srep37675
      31
      Examining the Validity of the Phonon Gas Model in Amorphous Materials
      Lv, Wei; Henry, Asegun
      Scientific Reports (2016), 6 (), 37675CODEN: SRCEC3; ISSN:2045-2322. (Nature Publishing Group)
      The idea of treating phonon transport as equiv. to transport through a gas of particles is termed the phonon gas model (PGM), and it has been used almost ubiquitously to try and understand heat conduction in all solids. However, most of the modes in disordered materials do not propagate and thus may contribute to heat conduction in a fundamentally different way than is described by the PGM. From a practical perspective, the problem with trying to apply the PGM to amorphous materials is the fact that one cannot rigorously define the phonon velocities for non-propagating modes, since there is no periodicity. Here, we tested the validity of the PGM for amorphous materials by assuming the PGM is applicable, and then, using a combination of lattice dynamics, mol. dynamics (MD) and exptl. thermal cond. data, we back-calcd. the phonon velocities for the vibrational modes. The results of this approach show that if the PGM was valid, a large no. of the mid and high frequency modes would have to have either imaginary or extremely high velocities to reproduce the exptl. thermal cond. data. Furthermore, the results of MD based relaxation time calcns. suggest that in amorphous materials there is little, if any, connection between relaxation times and thermal cond. This then strongly suggests that the PGM is inapplicable to amorphous solids.
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    32. 32
      Hanus, R. C.; George, J.; Wood, M.; Bonkowski, A.; Cheng, Y.; Abernathy, D. L.; Manley, M. E.; Hautier, G.; Snyder, G. J.; Hermann, R. P. Uncovering design principles for amorphous-like heat conduction using two-channel lattice dynamics. Mater. Today Phys. 2021, 18, 100344  DOI: 10.1016/j.mtphys.2021.100344
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      Uncovering design principles for amorphous-like heat conduction using two-channel lattice dynamics
      Hanus, Riley; George, Janine; Wood, Max; Bonkowski, Alexander; Cheng, Yongqiang; Abernathy, Douglas L.; Manley, Michael E.; Hautier, Geoffroy; Snyder, G. Jeffrey; Hermann, Raphael P.
      Materials Today Physics (2021), 18 (), 100344CODEN: MTPAD5; ISSN:2542-5293. (Elsevier Ltd.)
      The physics of heat conduction puts practical limits on many technol. fields such as energy prodn., storage, and conversion. It is now widely appreciated that the phonon-gas model does not describe the full vibrational spectrum in amorphous materials, since this picture likely breaks down at higher frequencies. A two-channel heat conduction model, which uses harmonic vibrational states and lattice dynamics as a basis, has recently been shown to capture both crystal-like (phonon-gas channel) and amorphous-like (diffuson channel) heat conduction. While materials design principles for the phonon-gas channel are well established, similar understanding and control of the diffuson channel is lacking. In this work, in order to uncover design principles for the diffuson channel, we study structurally-complex cryst. Yb14 (Mn,Mg)Sb11, a champion thermoelec. material above 800 K, exptl. using inelastic neutron scattering and computationally using the two-channel lattice dynamical approach. Our results show that the diffuson channel indeed dominates in Yb14MnSb11 above 300 K. More importantly, we demonstrate a method for the rational design of amorphous-like heat conduction by considering the energetic proximity phonon modes and modifying them through chem. means. We show that increasing (decreasing) the mass on the Sb-site decreases (increases) the energy of these modes such that there is greater (smaller) overlap with Yb-dominated modes resulting in a higher (lower) thermal cond. This design strategy is exactly opposite of what is expected when the phonon-gas channel and/or common anal. models for the diffuson channel are considered, since in both cases an increase in at. mass commonly leads to a decrease in thermal cond. This work demonstrates how two-channel lattice dynamics can not only quant. predict the relative importance of the phonon-gas and diffuson channels, but also lead to rational design strategies in materials where the diffuson channel is important.
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    33. 33
      Simoncelli, M.; Marzari, N.; Mauri, F. Unified theory of thermal transport in crystals and glasses. Nat. Phys. 2019, 15 (8), 809– 813,  DOI: 10.1038/s41567-019-0520-x
      33
      Unified theory of thermal transport in crystals and glasses
      Simoncelli, Michele; Marzari, Nicola; Mauri, Francesco
      Nature Physics (2019), 15 (8), 809-813CODEN: NPAHAX; ISSN:1745-2473. (Nature Research)
      Crystals and glasses exhibit fundamentally different heat conduction mechanisms: the periodicity of crystals allows for the excitation of propagating vibrational waves that carry heat, as first discussed by Peierls, while in glasses the lack of periodicity breaks Peierls's picture and heat is mainly carried by the coupling of vibrational modes, often described by a harmonic theory introduced by Allen and Feldman. Anharmonicity or disorder are thus the limiting factors for thermal cond. in crystals or glasses. Hitherto, no transport equation has been able to account for both. Here, we derive such an equation, resulting in a thermal cond. that reduces to the Peierls and Allen-Feldman limits, resp., in anharmonic crystals or harmonic glasses, while also covering the intermediate regimes where both effects are relevant. This approach also solves the long-standing problem of accurately predicting the thermal properties of crystals with ultralow or glass-like thermal cond., as we show with an application to a thermoelec. material representative of this class.
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    34. 34
      Niedziela, J. L.; Bansal, D.; May, A. F.; Ding, J.; Lanigan-Atkins, T.; Ehlers, G.; Abernathy, D. L.; Said, A.; Delaire, O. Selective breakdown of phonon quasiparticles across superionic transition in CuCrSe2. Nat. Phys. 2019, 15 (1), 73– 78,  DOI: 10.1038/s41567-018-0298-2
      34
      Selective breakdown of phonon quasiparticles across superionic transition in CuCrSe2
      Niedziela, Jennifer L.; Bansal, Dipanshu; May, Andrew F.; Ding, Jingxuan; Lanigan-Atkins, Tyson; Ehlers, Georg; Abernathy, Douglas L.; Said, Ayman; Delaire, Olivier
      Nature Physics (2019), 15 (1), 73-78CODEN: NPAHAX; ISSN:1745-2473. (Nature Research)
      Superionic crystals exhibit ionic mobilities comparable to liqs. while maintaining a periodic cryst. lattice. The at. dynamics leading to large ionic mobility have long been debated. A central question is whether phonon quasiparticles-which conduct heat in regular solids-survive in the superionic state, where a large fraction of the system exhibits liq.-like behavior. Here we present the results of energy- and momentum-resolved scattering studies combined with first-principles calcns. and show that in the superionic phase of CuCrSe2, long-wavelength acoustic phonons capable of heat conduction remain largely intact, whereas specific phonon quasiparticles dominated by the Cu ions break down as a result of anharmonicity and disorder. The weak bonding and large anharmonicity of the Cu sublattice are present already in the normal ordered state, resulting in low thermal cond. even below the superionic transition. These results demonstrate that anharmonic phonon dynamics are at the origin of low thermal cond. and superionicity in this class of materials.
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    35. 35
      Bernges, T.; Peterlechner, M.; Wilde, G.; Agne, M. T.; Zeier, W. G. Analytical model for two-channel phonon transport engineering. Mater. Today Phys. 2023, 35, 101107  DOI: 10.1016/j.mtphys.2023.101107
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      Analytical model for two-channel phonon transport engineering
      Bernges, Tim; Peterlechner, Martin; Wilde, Gerhard; Agne, Matthias T.; Zeier, Wolfgang G.
      Materials Today Physics (2023), 35 (), 101107CODEN: MTPAD5; ISSN:2542-5293. (Elsevier Ltd.)
      The redn. of vibrational contributions to thermal transport and the search for material classes with intrinsically low lattice thermal conductivities are at the heart of thermoelec. research. Both engineering the heat transport of known thermoelecs. and searching for new material candidates is guided by understanding the physics of low thermal conduction. Spectral anal. models (e.g., the Callaway model) for propagating phonon transport have proved to be a powerful tool for interpreting exptl. results and providing metrics for materials design. Now, however, it is known that another mechanism of phonon heat transport can occur in complex cryst. materials. Called diffusons, they describe the non-propagating at. scale random-walk of thermal energy between energetically proximal phonon modes. While anal. models exist to describe both transport behaviors independently, an anal. model accounting for both transport channels simultaneously is necessary to interpret and design so-called 2-channel thermal transport. In this work, we propose an anal. 2-channel transport model that partitions the vibrational d. of states into two transport regimes and subsequently accounts for both transport mechanisms. The model is then used to explain the exptl. thermal conductivities of the solid soln. series Ag9-xGa1-xGexSe6. In this series, substitution leads to the stabilization of a highly vacant Ag+ substructure, which is expected to induce strong point-defect phonon scattering. While the propagating phonons are strongly scattered at low temps., the diffuson channel is apparently unaffected. By establishing materials design metrics for 2-channel thermal transport from anal. theory, exptl. investigations of materials with astonishingly low lattice thermal conductivities can now be better guided and informed.
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    36. 36
      Acharyya, P.; Ghosh, T.; Pal, K.; Rana, K. S.; Dutta, M.; Swain, D.; Etter, M.; Soni, A.; Waghmare, U. V.; Biswas, K. Glassy thermal conductivity in Cs3Bi2I6Cl3 single crystal. Nat. Commun. 2022, 13 (1), 5053,  DOI: 10.1038/s41467-022-32773-4
      36
      Glassy thermal conductivity in Cs3Bi2I6Cl3 single crystal
      Acharyya, Paribesh; Ghosh, Tanmoy; Pal, Koushik; Rana, Kewal Singh; Dutta, Moinak; Swain, Diptikanta; Etter, Martin; Soni, Ajay; Waghmare, Umesh V.; Biswas, Kanishka
      Nature Communications (2022), 13 (1), 5053CODEN: NCAOBW; ISSN:2041-1723. (Nature Portfolio)
      As the periodic at. arrangement of a crystal is made to a disorder or glassy-amorphous system by destroying the long-range order, lattice thermal cond., κL, decreases, and its fundamental characteristics changes. The realization of ultralow and unusual glass-like κL in a cryst. material is challenging but crucial to many applications like thermoelecs. and thermal barrier coatings. Herein, we demonstrate an ultralow (∼0.20 W/m·K at room temp.) and glass-like temp. dependence (2-400 K) of κL in a single crystal of layered halide perovskite, Cs3Bi2I6Cl3. Acoustic phonons with low cut-off frequency (20 cm-1) are responsible for the low sound velocity in Cs3Bi2I6Cl3 and make the structure elastically soft. While a strong anharmonicity originates from the low energy and localized rattling-like vibration of Cs atoms, synchrotron X-ray pair-distribution function evidence a local structural distortion in the Bi-halide octahedra and Cl vacancy. The hierarchical chem. bonding and soft vibrations from selective sublattice leading to low κL is intriguing from lattice dynamical perspective as well as have potential applications.
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    37. 37
      Xia, Y.; Gaines, D.; He, J.; Pal, K.; Li, Z.; Kanatzidis, M. G.; Ozoliṇš, V.; Wolverton, C. A unified understanding of minimum lattice thermal conductivity. Proc. Natl. Acad. Sci. U.S.A. 2023, 120 (26), e2302541120  DOI: 10.1073/pnas.2302541120
      There is no corresponding record for this reference.
    38. 38
      Zhou, H.; Tiwari, J.; Feng, T. Understanding the flat thermal conductivity of La2Zr2O7 at ultrahigh temperatures. Phys. Rev. Mater. 2024, 8 (4), 043804  DOI: 10.1103/PhysRevMaterials.8.043804
      There is no corresponding record for this reference.
    39. 39
      Isaeva, L.; Barbalinardo, G.; Donadio, D.; Baroni, S. Modeling heat transport in crystals and glasses from a unified lattice-dynamical approach. Nat. Commun. 2019, 10 (1), 3853,  DOI: 10.1038/s41467-019-11572-4
      39
      Modeling heat transport in crystals and glasses from a unified lattice-dynamical approach
      Isaeva Leyla; Baroni Stefano; Barbalinardo Giuseppe; Donadio Davide; Baroni Stefano
      Nature communications (2019), 10 (1), 3853 ISSN:.
      We introduce a novel approach to model heat transport in solids, based on the Green-Kubo theory of linear response. It naturally bridges the Boltzmann kinetic approach in crystals and the Allen-Feldman model in glasses, leveraging interatomic force constants and normal-mode linewidths computed at mechanical equilibrium. At variance with molecular dynamics, our approach naturally and easily accounts for quantum mechanical effects in energy transport. Our methodology is carefully validated against results for crystalline and amorphous silicon from equilibrium molecular dynamics and, in the former case, from the Boltzmann transport equation.
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    40. 40
      Deng, Z.; Sai Gautam, G.; Kolli, S. K.; Chotard, J.-N.; Cheetham, A. K.; Masquelier, C.; Canepa, P. Phase Behavior in Rhombohedral NaSiCON Electrolytes and Electrodes. Chem. Mater. 2020, 32 (18), 7908– 7920,  DOI: 10.1021/acs.chemmater.0c02695
      40
      Phase Behavior in Rhombohedral NaSiCON Electrolytes and Electrodes
      Deng, Zeyu; Sai Gautam, Gopalakrishnan; Kolli, Sanjeev Krishna; Chotard, Jean-Noel; Cheetham, Anthony K.; Masquelier, Christian; Canepa, Pieremanuele
      Chemistry of Materials (2020), 32 (18), 7908-7920CODEN: CMATEX; ISSN:0897-4756. (American Chemical Society)
      The replacement of the presently used liq. electrolytes by a nonflammable solid electrolytes is an important avenue to create safer batteries. The natrium superionic CONductor (NaSiCON) Na1+xZr2SixP3-xO12 (0 ≤ x ≤ 3) that displays high bulk ionic cond. and good stability toward other NaSiCON-based electrodes is a good solid electrolyte in NaSiCON-based batteries. Despite the sizeable share of research on Na1+xZr2SixP3-xO12, the structural and thermodn. properties of NaSiCON require better understanding for more efficient synthesis and optimization as a solid electrolyte, which often follows chem. intuition. Here, we analyze the thermodn. properties of the rhombohedral NaSiCON electrolyte by constructing the Na1+xZr2SixP3-xO12 phase diagram, based on d. functional theory calcns., a cluster expansion framework, and Monte Carlo simulations. Specifically, we build the phase diagram as a function of temp. and compn. (0 ≤ x ≤ 3) for the high-temp. rhombohedral structure, which has been also obsd. in several pos. electrode materials, such as Na3Ti2(PO4)3, Na3V2(PO4)3, Na3Cr2(PO4)3, and Na3Fe2(PO4)3. Through the phase diagram, we identify the concn. domains providing the highest Na+-ion cond. and previously unreported phase-sepn. behavior across three different single-phase regions. Furthermore, we note the similarities in the phase behavior between Na1+xZr2SixP3-xO12 and other NaSiCON-based monotransition metal electrodes and discuss the potential competition between thermodn. and kinetics in exptl. obsd. phase sepn. Our work is an important addn. in understanding the thermodn. of NaSiCON-based materials and in the development of inexpensive Na-ion batteries. From our results, we propose that the addn. of SiO44- moieties to single-transition metal NaSiCON-phosphate-based electrodes will significantly slow the kinetics toward phase sepn.
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    41. 41
      Boilot, J. P.; Collin, G.; Colomban, P. Crystal structure of the true nasicon: Na3Zr2Si2PO12. Mater. Res. Bull. 1987, 22 (5), 669– 676,  DOI: 10.1016/0025-5408(87)90116-4
      41
      Crystal structure of the true Nasicon: Na3Zr2Si2PO12
      Boilot, J. P.; Collin, G.; Colomban, P.
      Materials Research Bulletin (1987), 22 (5), 669-76CODEN: MRBUAC; ISSN:0025-5408.
      Room-temp. Nasicon (Na3.12(10)Zr2.00(1)Si2.12P0.88O12) is monoclinic, space group B2/b, with a 15.669(7), b 9.246(6), c 9.055(7) Å, and γ 124°12(4); Z = 4; final R = 7.51% for 1860 reflections. Nasicon at 623 K (Na3.09(8)Zr2.01(1)Si2.09P0.91O12) is rhombohedral, space group R‾3c, with a 9.074(2) and c 23.057(4) Å; Z = 6; final R = 2.93% for 704 reflections. Evidence of the total occupancy of the Zr octahedron was found, showing that only the Si/P nonstoichiometry mechanism is present in the Nasicon crystal. For the 2 temps. which were investigated, the structures are very close to that of the Nasicon analog Na3Sc2P3O12. However the Si/P substitution prevents the Na long-range ordering even in the monoclinic low-temp. phase and therefore the cross over to the rhombohedral symmetry only involves very small at. displacements. For both structures, a new Na position (mid-Na) is displayed in the conduction channel, intermediate between the usual Na(1) and Na(2) sites.
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    42. 42
      Zou, Z.; Ma, N.; Wang, A.; Ran, Y.; Song, T.; Jiao, Y.; Liu, J.; Zhou, H.; Shi, W.; He, B.; Wang, Da; Li, Y.; Avdeev, M.; Shi, S. Relationships Between Na+ Distribution, Concerted Migration, and Diffusion Properties in Rhombohedral NASICON. Adv. Energy Mater. 2020, 10 (30), 2001486,  DOI: 10.1002/aenm.202001486
      42
      Relationships Between Na+ Distribution, Concerted Migration, and Diffusion Properties in Rhombohedral NASICON
      Zou, Zheyi; Ma, Nan; Wang, Aiping; Ran, Yunbing; Song, Tao; Jiao, Yao; Liu, Jinping; Zhou, Hang; Shi, Wei; He, Bing; Wang, Da; Li, Yajie; Avdeev, Maxim; Shi, Siqi
      Advanced Energy Materials (2020), 10 (30), 2001486CODEN: ADEMBC; ISSN:1614-6840. (Wiley-Blackwell)
      Rhombohedral NaZr2(PO4)3 is the prototype of all the NASICON-type materials. The ionic diffusion in these rhombohedral NASICON materials is highly influenced by the ionic migration channels and the bottlenecks in the channels which have been extensively studied. However, no consensus is reached as to which one is the preferential ionic migration channel. Moreover, the relationships between the Na+ distribution over the multiple available sites, concerted migration, and diffusion properties remain elusive. Using ab initio mol. dynamics simulations, here it is shown that the Na+ ions tend to migrate through the Na1-Na3-Na2-Na3-Na1 channels rather than through the Na2-Na3-Na3-Na2 channels. There are two types of concerted migration mechanisms: two Na+ ions located at the adjacent Na1 and Na2 sites can migrate either in the same direction or at an angle. Both mechanisms exhibit relatively low migration barriers owing to the potential energy conversion during the Na+ ions migration process. Redistribution of Na+ ions from the most stable Na1 sites to Na2 on increasing Na+ total content further facilitates the concerted migration and promotes the Na+ ion mobility. The work establishes a connection between the Na+ concn. in rhombohedral NASICON materials and their diffusion properties.
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    43. 43
      Qui, D. T.; Capponi, J. J.; Joubert, J. C.; Shannon, R. D. Crystal structure and ionic conductivity in Na4Zr2Si3O12. J. Solid State Chem. 1981, 39 (2), 219– 229,  DOI: 10.1016/0022-4596(81)90335-2
      43
      Crystal structure and ionic conductivity in sodium zirconium silicate (Na4Zr2Si3O12)
      Qui, D. Tran; Capponi, J. J.; Joubert, J. C.; Shannon, R. D.
      Journal of Solid State Chemistry (1981), 39 (2), 219-29CODEN: JSSCBI; ISSN:0022-4596.
      Na ion conductivities of Na4-xZr2Si3-xPxO12 range from 3.5 × 10-4 (ohm-cm)-1 for x = 0 to 1.9 × 10-1 (ohm-cm)-1 for x = 1.0 at 300°. Structure refinements of single-crystal Na4Zr2Si3O12 were carried out at 25, 300, and 620°. Little change occurs in bond distances and angles in the (Zr2Si3O12)-4 framework whereas the Na(1)-O6 and Na(2)-O8 polyhedra enlarge dramatically with increase of temp. The large thermal motion of Na(1) and Na(2) is probably related to the Na mobility in this structure. Of the four possible Na ion pathways, only two have openings large enough to allow reasonable mobility. The first, connecting a Na(1) site to a Na(2) site, is somewhat smaller (1.86 Å) than a Na ion (2.30 Å) at room temp. but increases substantially to 2.22 Å at 620°. The second, connecting a Na(2) site to a Na(2) site, is larger (2.37 Å) and increases to 2.66 Å at 620°. Difference Fourier maps show significant electron d. along Na(2)-Na(2) paths and Na(2) thermal ellipsoids have major axes close to these paths.
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    44. 44
      Lenain, G. E.; McKinstry, H. A.; Alamo, J.; Agrawal, D. K. Structural model for thermal expansion in MZr2P3O12 (M = Li, Na, K, Rb, Cs). J. Mater. Sci. 1987, 22 (1), 17– 22,  DOI: 10.1007/BF01160546
      44
      Structural model for thermal expansion in alkali metal zirconium phosphates MZr2P3O12 (M = Li, Na, K, Rb, Cs)
      Lenain, G. E.; McKinstry, H. A.; Alamo, J.; Agrawal, D. K.
      Journal of Materials Science (1987), 22 (1), 17-22CODEN: JMTSAS; ISSN:0022-2461.
      A structural model is proposed to describe the highly anisotropic thermal expansion in the NaZr2P3O12 structure as a result of the thermal motion of the polyhedra in the structure. In the proposed model the rotations of the PO4 tetrahedra are coupled to the rotation of the Zr octahedra. Of the 2 versions considered, the 1st one allows angular distortions to occur only in the ZrO6 octahedra; the 2nd one permits all polyhedra to be distorted.
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    45. 45
      Oota, T.; Yamai, I. Thermal Expansion Behavior of NaZr2(PO4)3Type Compounds. J. Am. Ceram. Soc. 1986, 69 (1), 1– 6,  DOI: 10.1111/j.1151-2916.1986.tb04682.x
      There is no corresponding record for this reference.
    46. 46
      Pet’kov, V. I.; Orlova, A. I.; Kazantsev, G. N.; Samoilov, S. G.; Spiridonova, M. L. Thermal Expansion in the Zr and 1-, 2-Valent Complex Phosphates of NaZr2(PO4)3 (NZP) Structure. J. Therm. Anal. Calorim. 2001, 66 (2), 623– 632,  DOI: 10.1023/A:1013145807987
      46
      Thermal expansion in the Zr and 1-, 2-valent complex phosphates of NaZr2(PO4)3 (NZP) structure
      Pet'kov, V. I.; Orlova, A. I.; Kazantsev, G. N.; Samoilov, S. G.; Spiridonova, M. L.
      Journal of Thermal Analysis and Calorimetry (2001), 66 (2), 623-632CODEN: JTACF7; ISSN:1418-2874. (Kluwer Academic Publishers)
      A5-4xZrxZr(PO4)3 (A = Na, K; 0 ≤ x ≤ 1.25), Na1-xCd0.5xZr2(PO4)3 (0 ≤ χ ≤ 1), Na5-xCd0.5xZr(PO4)3 (O ≤ χ ≤ 4) compns. which belong to the NZP structural family were synthesized using the sol-gel method. The lattice thermal expansions of members of these rows were detd. up to 600°C by high-temp. X-ray diffractometry. The axial thermal expansion coeffs. change from -5.8·10-6 to 7.5·10-6 °C-1 (αa) and from 2.6·10-6 to 22·10-6 °C-1 (αc). These results, in addn. to those for other NZP compds., allow us to explain their low thermal expansion. The mechanism can be attributed to strongly bonded three-dimensional network structure, the existence of structural holes capable of damping some of the thermal vibrations and anisotropy in the thermal expansion of the lattice.
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    47. 47
      Dronskowski, R.; Bloechl, P. E. Crystal orbital Hamilton populations (COHP): energy-resolved visualization of chemical bonding in solids based on density-functional calculations. J. Phys. Chem. 1993, 97 (33), 8617– 8624,  DOI: 10.1021/j100135a014
      47
      Crystal orbital Hamilton populations (COHP): energy-resolved visualization of chemical bonding in solids based on density-functional calculations
      Dronskowski, Richard; Bloechl, Peter E.
      Journal of Physical Chemistry (1993), 97 (33), 8617-24CODEN: JPCHAX; ISSN:0022-3654.
      After giving a concise overview of the current knowledge in the field of quantum mech. bonding indicators for mols. and solids, the authors show how to obtain energy-resolved visualization of chem. bonding in solids by d.-functional electronic structure calcns. From a band structure energy partitioning scheme, i.e., rewriting the band structure energy as a sum of orbital pair contributions, the authors derive what is to be defined as crystal orbital Hamilton populations (COHP). In particular, a COHP(ε) diagram indicates bonding, nonbonding, and antibonding energy regions within a specified energy range while an energy integral of a COHP gives access to the contribution of an atom or a chem. bond to the distribution of 1-particle energies. A further decompn. into specific AOs or symmetry-adapted linear combinations of AOs (hybrids) can easily be performed by using a projector technique involving unitary transformations. Because of its structural simplicity and the availability of reliable thermodn. data, the authors study the bonding within cryst. silicon (diamond phase) 1st. As a basis set, both bcc. screened and diamond screened at.-centered tight-binding linear muffin-tin orbitals (TB-LMTOs) are used throughout. The shape of COHP vs. energy diagrams and the significance of COHP subcontributions (s-s, sp3-sp3) are analyzed. Specifically, the difference between the COHPs energy integral (1-particle bond energy) and the exptl. bond energy is critically examd. While abs. values for the 1-particle bond energy show a high basis set dependence due to changing on-site (crystal field) COHP terms, the shape of off-site (bonding) COHP functions, elucidating the local bonding principle within an extended structure, remains nearly unaffected.
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    48. 48
      Krenzer, G.; Kim, C.-E.; Tolborg, K.; Morgan, B. J.; Walsh, A. Anharmonic lattice dynamics of superionic lithium nitride. J. Mater. Chem. A 2022, 10 (5), 2295– 2304,  DOI: 10.1039/D1TA07631K
      There is no corresponding record for this reference.
    49. 49
      Bernges, T.; Hanus, R.; Wankmiller, B.; Imasato, K.; Lin, S.; Ghidiu, M.; Gerlitz, M.; Peterlechner, M.; Graham, S.; Hautier, G.; Pei, Y.; Hansen, M. R.; Wilde, G.; Snyder, G. J.; George, J.; Agne, M. T.; Zeier, W. G. Considering the Role of Ion Transport in Diffuson-Dominated Thermal Conductivity. Adv. Energy Mater. 2022, 12 (22), 2200717,  DOI: 10.1002/aenm.202200717
      49
      Considering the Role of Ion Transport in Diffuson-Dominated Thermal Conductivity
      Bernges, Tim; Hanus, Riley; Wankmiller, Bjoern; Imasato, Kazuki; Lin, Siqi; Ghidiu, Michael; Gerlitz, Marius; Peterlechner, Martin; Graham, Samuel; Hautier, Geoffroy; Pei, Yanzhong; Hansen, Michael Ryan; Wilde, Gerhard; Snyder, G. Jeffrey; George, Janine; Agne, Matthias T.; Zeier, Wolfgang G.
      Advanced Energy Materials (2022), 12 (22), 2200717CODEN: ADEMBC; ISSN:1614-6840. (Wiley-Blackwell)
      Next-generation thermal management requires the development of low lattice thermal cond. materials, as obsd. in ionic conductors. For example, thermoelec. efficiency is increased when thermal cond. is decreased. Detrimentally, high ionic cond. leads to thermoelec. device degrdn. Battery safety and design also require an understanding of thermal transport in ionic conductors. Ion mobility, structural complexity, and anharmonicity have been used to explain the thermal transport properties of ionic conductors. However, thermal and ionic transport are rarely discussed in direct comparison. Herein, the ionic cond. of Ag+ argyrodites is found to change by orders of magnitude without altering the thermal cond. Thermal cond. measurements and two-channel lattice dynamics modeling reveal that the majority of Ag+ vibrations have a non-propagating diffuson-like character, similar to amorphous materials. It is found that high ionic mobility is not a requirement for diffuson-mediated transport. Instead, the same bonding and structural traits that can lead to fast ionic conduction also lead to diffuson-mediated transport. Bridging the fields of solid-state ionics and thermal transport, it is proposed that a vibrational perspective can lead to new design strategies for functional ionic conducting materials. As a first step, the authors relate the so-called Meyer-Neldel behavior in ionic conductors to phonon occupations.
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    50. 50
      Famprikis, T.; Canepa, P.; Dawson, J. A.; Islam, M. S.; Masquelier, C. Fundamentals of inorganic solid-state electrolytes for batteries. Nat. Mater. 2019, 18 (12), 1278– 1291,  DOI: 10.1038/s41563-019-0431-3
      50
      Fundamentals of inorganic solid-state electrolytes for batteries
      Famprikis, Theodosios; Canepa, Pieremanuele; Dawson, James A.; Islam, M. Saiful; Masquelier, Christian
      Nature Materials (2019), 18 (12), 1278-1291CODEN: NMAACR; ISSN:1476-1122. (Nature Research)
      A review. In the crit. area of sustainable energy storage, solid-state batteries have attracted considerable attention due to their potential safety, energy-d. and cycle-life benefits. This Review describes recent progress in the fundamental understanding of inorg. solid electrolytes, which lie at the heart of the solid-state battery concept, by addressing key issues in the areas of multiscale ion transport, electrochem. and mech. properties, and current processing routes. The main electrolyte-related challenges for practical solid-state devices include utilization of metal anodes, stabilization of interfaces and the maintenance of phys. contact, the solns. to which hinge on gaining greater knowledge of the underlying properties of solid electrolyte materials.
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    51. 51
      Qian, X.; Zhou, J.; Chen, G. Phonon-engineered extreme thermal conductivity materials. Nat. Mater. 2021, 20 (9), 1188– 1202,  DOI: 10.1038/s41563-021-00918-3
      51
      Phonon-engineered extreme thermal conductivity materials
      Qian, Xin; Zhou, Jiawei; Chen, Gang
      Nature Materials (2021), 20 (9), 1188-1202CODEN: NMAACR; ISSN:1476-1122. (Nature Portfolio)
      A review. Materials with ultrahigh or low thermal cond. are desirable for many technol. applications, such as thermal management of electronic and photonic devices, heat exchangers, energy converters and thermal insulation. Recent advances in simulation tools (first principles, the atomistic Green's function and mol. dynamics) and exptl. techniques (pump-probe techniques and microfabricated platforms) have led to new insights on phonon transport and scattering in materials and the discovery of new thermal materials, and are enabling the engineering of phonons towards desired thermal properties. We review recent discoveries of both inorg. and org. materials with ultrahigh and low thermal cond., highlighting heat-conduction physics, strategies used to change thermal cond., and future directions to achieve extreme thermal conductivities in solid-state materials.
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    52. 52
      Rohde, M.; Mohsin, I. U. I.; Ziebert, C.; Seifert, H. J. Ionic and Thermal Transport in Na-Ion-Conducting Ceramic Electrolytes. Int. J. Thermophys. 2021, 42 (10), 136,  DOI: 10.1007/s10765-021-02886-x
      52
      Ionic and Thermal Transport in Na-Ion-Conducting Ceramic Electrolytes
      Rohde, Magnus; Mohsin, Ijaz U. I.; Ziebert, Carlos; Seifert, Hans Juergen
      International Journal of Thermophysics (2021), 42 (10), 136CODEN: IJTHDY; ISSN:0195-928X. (Springer)
      We have studied the ionic and thermal transport properties along with the thermodn. key properties of a Na-ion-conducting phosphate ceramic. The system Na1+xAlxTi2-x(PO4)3 (NATP) with x = 0.3 was taken as a NASICON-structured model system which is a candidate material for solid electrolytes in post-Li energy storage. The com. available powder (NEI Coorp., USA) was consolidated using cold isostatic pressing before sintering. In order to compare NATP with the"classical" NASICON system, Na1+xZr2(SiO4)x(PO4)3-x (NaZSiP) was synthesized with compns. of x = 1.7 and x = 2, resp., and characterized with regard to their ionic and thermal transport behavior. While ionic cond. of the NaZSiP compns. was about more than two orders of magnitude higher than in NATP, the thermal cond. of the NASICON compd. showed an opposite behavior. The room temp. value was about a factor two higher in NATP compared to NaZSiP. While the thermal cond. decreases with increasing temp. in NATP, it increases with increasing temp. in NaZSiP. However, the overall change of this thermal transport parameter over the measured temp. range from room temp. up to 800 °C appeared to be relatively small.
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    53. 53
      Kuang, H.-L.; Wu, C.-W.; Zeng, Y.-J.; Chen, X.-K.; Zhou, W.-X. The amplification effect of four-phonon scattering in CdX (X = Se, Te): The role of mid-frequency phonons. Int. J. Therm. Sci. 2024, 205, 109254  DOI: 10.1016/j.ijthermalsci.2024.109254
      There is no corresponding record for this reference.
    54. 54
      Chen, X.-K.; Zhang, E.-M.; Wu, D.; Chen, K.-Q. Strain-Induced Medium-Temperature Thermoelectric Performance of Cu4TiSe4: The Role of Four-Phonon Scattering. Phys. Rev. Appl. 2023, 19 (4), 044052  DOI: 10.1103/PhysRevApplied.19.044052
      There is no corresponding record for this reference.
    55. 55
      Muy, S.; Schlem, R.; Shao-Horn, Y.; Zeier, W. G. Phonon–Ion Interactions: Designing Ion Mobility Based on Lattice Dynamics. Adv. Energy Mater. 2021, 11 (15), 2002787,  DOI: 10.1002/aenm.202002787
      55
      Phonon-Ion Interactions: Designing Ion Mobility Based on Lattice Dynamics
      Muy, Sokseiha; Schlem, Roman; Shao-Horn, Yang; Zeier, Wolfgang G.
      Advanced Energy Materials (2021), 11 (15), 2002787CODEN: ADEMBC; ISSN:1614-6840. (Wiley-Blackwell)
      A review. This review is focused on the influence of lattice dynamics on the ionic mobility in superionic conductors in particular solid-state Li-ion conductors. After a succinct review of the static view of ionic conduction, the role of polarizability as the underlying cause of lattice softness is discussed in connection with the anharmonicity and the roles of lattice dynamics on ionic cond. as proposed in early theories in the 70's and 80's by Mahan, Zeller, Rice and Roth are reviewed with the emphasis on various proposed correlations between Debye and Einstein frequency as well as other specific vibrational modes with the activation energy. The role of lattice dynamics on the correlation between the pre-exponential factor and activation energy, i.e. the Meyer-Neldel rule is also presented with emphasis on the entropy of migration and its dependence on the vibrational spectrum of the lattice. Moreover, a recent computational high-throughput screening based on the av. vibrational frequency is also discussed to illustrate the application of lattice dynamics descriptors to design new lithium conductors. Finally, several open questions regarding the fundamental understanding of the role of lattice dynamics and new strategies to tune ionic cond. based on these concepts are presented.
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    56. 56
      Cheng, Z.; Zahiri, B.; Ji, X.; Chen, C.; Chalise, D.; Braun, P. V.; Cahill, D. G. Good Solid-State Electrolytes Have Low, Glass-Like Thermal Conductivity. Small 2021, 17 (28), e2101693  DOI: 10.1002/smll.202101693
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