Diffuson-mediated thermal and ionic transport in superionic conductors

Institute of Inorganic and Analytical Chemistry, University of Münster, D-48149 Münster, Germany George W. Woodruff School of Mechanical Engineering, Georgia Institute of Technology, Atlanta, Georgia 30332, United States Institute of Physical Chemistry, University of Münster, D-48149 Münster, Germany International Graduate School for Battery Chemistry, Characterization, Analysis, Recycling and Application (BACCARA), University of Münster, D-48149 Münster, Germany Global Zero Emission Research Center, National Institute of Advanced Industrial Science and Technology (AIST), Tsukuba, Ibaraki 305-8569, Japan Interdisciplinary Materials Research Center, School of Materials Science and Engineering, Tongji University, 201804 Shanghai, China Institute of Materials Physics, University of Münster, D-48149 Münster, Germany Institute of Condensed Matter and Nanoscience, Université Catholique de Louvain, 1348 Ottignies-Louvain-la-Neuve, Belgium Dartmouth College, Thayer School of Engineering, New Hampshire, 03755 Hanover, United States Department of Materials Science and Engineering, Northwestern University, Illinois, 60208 Evanston, United States Federal Institute for Materials Research and Testing (BAM), D-12205 Berlin, Germany Institute of Condensed Matter, Theory and Solid-State Optics, Friedrich Schiller University, 07743 Jena, Germany Institut für Energieund Klimaforschung (IEK), IEK-12: Helmholtz-Institut Münster, Forschungszentrum Jülich, 48149 Münster, Germany


Introduction
Superionic conductors are sparking tremendous interest in multiple fields. While fast ionic conductors are searched for solid state batteries, 1 they also often possess low lattice thermal conductivities essential for high thermoelectric efficiencies. [2][3][4] In fact, many Ag + and Cu + superionic thermoelectric materials have thermal conductivities below the theoretical "minimum" thermal conductivity for solids, leading some to the suggestion that ion mobility makes these materials more liquid-like. 2,4 However, no physical connections between thermal and ionic transport have thus far been shown. Recently, a foundational misconception within the concept of "phonon liquid electron crystal" 2 was proven by showing that transverse phonon modes, which should not exist in a liquid, do persist above superionic phase transitions in solids. 5,6 Concurrently, a universal theory for heat conduction in solids has been developed that suggests low thermal conductivities can arise solely from (static) atomic disorder, strong anharmonicity and/or complex unit cells, all of which are inherent to fast ion conducting materials. 3,5,6 Lacking a direct comparison between ionic transport data and the thermal and vibrational properties in these systems, the interdependence of both transport processes remains elusive.
Fundamentally, thermal and ionic transport are related at the vibrational level as they both arise from fluctuations of the phonon occupation number , which is the instantaneous number of vibrations (phonon quanta) that are in a vibrational mode having frequency . [7][8][9] Thermal transport results from the thermodynamic drive to have a constant phonon occupation (thermal energy) throughout the entire material. Ion transport occurs when a thermal fluctuation is energetically capable of moving the ion between adjacent lattice sites.
The rate at which phonons move throughout a material depends on the character of the phonon modes. Historically, heat transport by phonons was thought to occur in a propagating manner on a length-scale much larger than interatomic distances (phonon-gas model). In disordered, anharmonic and structurally more complex solids, heat transport can alternatively be conducted by fundamentally different transport mechanisms, one of which is the diffuson. 10,11 In contrast to the phonon-gas model, diffusons correspond to heat transport via local (atomic) scale random walk. 12,13 This diffusive walk of heat 14 means that non-propagating vibrations in a solid transfer thermal energy between adjacent diffuson modes, 11,12 at a much smaller length scale compared to typical propagating modes (shown schematically in Figure 1 a, b).
Regardless of the character of the vibrational modes, fluctuations in vibrational energy remain the fundamental origin of thermal transport. 7 On the other hand, ionic transport is restricted by an activation barrier (nominally referred to as EA) that is largely determined by the potential energy landscape, i.e. the coordination environment of the mobile species (Figure 1 c). 15 With that, the magnitude of the fluctuation needed to move an ion is significantly increased, relative to thermal transport, since the kinetic energy of the ion has to surpass this activation barrier. Despite this conceptual difference there are strong underlying physical relationships suggesting a connection between thermal diffusons and ionic diffusion: 1) fast ionic transport is achieved in strongly disordered materials 16,17 and strong disorder promotes the prevalence of thermal transport via diffusons. 12 2) Both transport phenomena operate within an atomic-scale random walk, in which local vibrations carry and transfer heat energy or momentum of ions (Figure 1 b, d). 12,18 3) Anharmonic lattice vibrations enhance diffuson-mediated transport 14 and, although the characteristic vibration of a mobile ion is usually shown using a harmonic potential well, an ion jump is an intrinsically anharmonic process. 19 Motivated by these apparent similarities, this work aims to provide a stepping stone in unifying the concepts of thermal and ionic transport by experimentally accessing both processes and answering the following questions: 1) Are significant diffuson contributions to thermal conductivity present in superionic conductors? 2) Which vibrational modes characterize both transport processes, and 3) how does the magnitude of ion transport influence thermal transport?
Especially the latter is of significant relevance, with high ionic conductivities hindering the long-term stability of superionic conductors in thermoelectric devices 20

Results and Discussion
Structural features. To answer these questions, a successful isovalent substitution from Ag8SiSe6 to Ag8GeSe6 and Ag8SnSe6 is achieved as shown by the linear increase of the lattice volume (Supplemental Note 1, 2). 21 These materials are superionic conductors and have been intensely investigated for their thermoelectric transport properties due to the low thermal conductivity. 3,22,23 The different compositions have varying room-temperature structures, 24,25 but ultimately undergo a phase transition into the same cubic structure that is characterized by its strongly disordered Ag + sublattice (Figure 2 a and Supplemental Note 2). 3,22 Structurally, there are a large number of Ag + sites that are tetrahedrally coordinated and have face-sharing connectivity. These sites have an average occupation of only 25 %. While the low-temperature phase of Ag8SiSe6 is cubic with an ordered Ag + sublattice, 24 both Ag8GeSe6 and Ag8SnSe6 crystallize in an orthorhombic structure at room-temperature (Supplemental Note 2). In the orthorhombic phase, the Ag + sublattice consists of five distinct lattice sites, each either tetrahedrally or trigonally coordinated by Se 2arranged in a corner-or edge-sharing fashion, respectively. Near the phase transition, the lattice volume increases strongly in a small temperature window before it settles into a volumetric thermal expansion coefficient ranging from 7.7 to 8.7 × 10 -5 K -1 , with an average of 8.2 ± 0.5 × 10 -5 K -1 for all compositions (Supplemental Note 2, 3).
While the influence of local structural changes are usually employed as an important metric in the understanding of ionic conduction 26 , the underlying vibrational influences remain elusive.
Here, we focus on the vibrational characterization of these superionic conductors to give new insights relating vibrations, ionic jumps and thermal transport.

Vibrational frequencies.
The vibrational spectrum of a material has significant importance for the fields of ionic conduction and thermal conduction. For ionic transport, the attempt frequency of the mobile species is included in the Arrhenius pre-factor (Supplemental Note 4). 15,27 Often the Debye frequency D is used to approximate the attempt frequency despite not probing the vibrations of the mobile species directly. In fact, especially in disordered and soft materials, the Debye-frequency often fails its original purpose to estimate the maximum vibrationalfrequency, but rather falls into the center of the vibrational density of states. 12,28 With that, it provides a useful experimental insight to the average dynamics of the lattice, but it should not be expected to be an accurate descriptor for the vibrational properties of the mobile ion. 28 A better descriptor for the attempt frequency would be a vibrational frequency E specific for the mobile ion that is conceptually similar to an Einstein oscillator. 29,30 This concept is especially useful for materials with an underlying guest-host structural motif, like the mobile ion within a rigid sub-lattice in superionic conductors 2 or guest atoms and associated "rattler modes" in the skutterudites and clathrates, which have rather dispersion-less phonon branches (Supplemental Note 5). 31 Here, we find that the average Ag + vibrational frequency (1.2 ± 0.2 THz; range = 1.0 -1.5 THz) determined from X-ray diffraction measurements is consistent with the average Ag + frequency found computationally (2 THz), and perfectly aligns with the lowest frequency peak in the  Ag + ionic transport. The thermoelectric properties of these fast ionic conductors, such as ultralow thermal conductivity, are often inferred to be connected to ion mobility, but the corresponding magnitude of ionic conductivity is rarely (if ever) reported. Here, we evaluate the magnitude and temperature dependence of the ion conduction of Ag8(Si,Ge,Sn)Se6 solid solutions (see Supplemental Note 4).
The ionic conductivities vary strongly upon substitution with room-temperature values ranging from 0.088 to 5.0 S/m as determined by impedance spectroscopy. The respective Arrhenius behavior for Ag8SiSe6, Ag8GeSe6 and Ag8SnSe6 reveals significant changes to the activation barrier and, with that, the temperature dependence of ionic transport (Figure 3 a). Here, Ag8GeSe6 exhibits the lowest activation barrier of 0.05 eV while both Ag8SiSe6 and Ag8SnSe6 show larger barriers of ~0.3 eV at temperatures above 298 K. These results are confirmed by nuclear magnetic resonance spectroscopy (Figure 3 b), which indicates that the activation energies are predominantly a bulk property of the solid and not caused by microstructural differences. Furthermore, not only the disordered high-temperature cubic phase but also the ordered low-temperature phases of the Ag + argyrodites have significant ionic conduction.
For all investigated materials, the logarithm of the pre-exponential factor scales linearly with the activation barrier (Figure 3 c). This is a common occurrence in ionic materials, known as the Meyer-Neldel rule, [36][37][38] and is attributed to the interrelation of migration enthalpy (i.e. Δ ∝ ) and entropy (i.e. Δ ∝ ln ( )). This relation has been derived before using transition state theory and the multi-excitation entropy model. 36,37 Specifically, it is believed that the inverse slope of the Meyer-Neldel plot is related to the energy of the vibrational modes that participate in ionic conduction, as well as a characteristic number of vibrations. However, a consistent definition of these quantities has been lacking. 37 Nevertheless, an inverse slope of ~37 meV is in agreement with those reported for other superionic conductors. 39  This result provides confidence that it is, in fact, the Ag + vibrations that are responsible for ion transport within the multi-excitation model. Typically, changes in the activation barrier and magnitude of ionic conductivity are discussed from the static perspective, e.g., electronegativity, unit cell and pathway volumes. 15 The sitespecific and directional analysis of phonon modes proposed here allows for identification of prominent vibrational frequencies of the mobile ion from lattice dynamics calculations. The importance of tracking specific vibrational modes within molecular dynamics simulations was recently discussed by Gordiz et al. 41 They showed that the vibrational contributions to ionic conduction are heavily frequency-dependent. Herein, we also assess the relevance and vibrational character (phonon-gas-like or diffuson-like) of phonon modes for thermal transport.
With that, we aim to draw conclusions about the interdependence of both phenomena.
Diffuson-mediated thermal transport. The ultra-low thermal conductivities of superionic conductors are often attributed to (static and dynamic) atomic disorder, soft bonding, lowfrequency optical phonon modes, anharmonicity and complex crystal structure. 42 All of these parameters are associated with either lowering the phonon group-velocity or increasing the phonon scattering rate, in context of the phonon-gas model. 3,22,23 While these structural and vibrational features are likely to factor into the characteristics of thermal transport, the vicinity to the Ioffe-Regel limit and the associated concept of minimum lattice thermal conductivity suggests the phonon-gas model is likely an incomplete description of thermal transport in these systems. 12  (Figure 4 a). 12,43 To confirm the diffuson-like nature of thermal transport at high temperature, low-temperature thermal conductivities were measured. Here, the lattice thermal conductivity of Ag8GeSe6 shows that the flat-temperature dependence persists down to 100 K before a phonon-peak arises around 12 K (Figure 4 b). This exceptional temperature dependence was confirmed by corresponding low-temperature measurements of Ag8SiSe6 and Ag8SnSe6 (see Supplemental Note 8).
Two-channel modeling based on the calculated lattice dynamics for Ag8GeSe6 is utilized to explain these results. Here, the heat current operator matrix is analyzed regarding its diagonal (phonon-like) and off-diagonal (diffuson-like) elements such that the total lattice thermal conductivity of the two-channel model is = !" + $ %% . 13  The two-channel model additionally allows for a spectral analysis of the thermal conductivity of both channels (Figure 4 d, e). The contributions from the phonon gas channel arise from the lowest frequency modes below 1 THz, while the spectral thermal conductivity of the diffusonchannel is extended over the whole frequency range. Note that the spectral thermal conductivity is not expected to be zero at zero frequency but is an apparent artifact of uniform q-meshing. 45 Here, the minimal contributions of the high-frequency vibrations, dominated by M = Si, Ge and Sn (see Figure 2 c), explain why no significant changes to the thermal transport are observed in the solid-solutions. Nevertheless, at temperatures relevant for ion transport, the Ag + vibrations have predominantly diffuson-like character.
Evaluation of the mode-specific Grüneisen parameters (Supplemental Note 5) lends an explanation for the strong decline of the phonon gas channel (Figure 4 f). Here, large Grüneisen parameters are found in the same frequency range in which the phonon gas channel contributions are the strongest (Figure 4 d, e). Therefore, the extensive anharmonicity at low frequencies is believed to be the driving factor for the strong suppression of the phonon-channel and the subsequent transition to diffuson-dominated thermal transport. This was captured phenomenologically by our analytical scattering function. It is not surprising that such large Grüneisen parameters exist since anharmonic vibrations have been linked to ionic transport. 5,6,46 Having assessed the vibrational character of phonon modes from thermal transport behavior, it is also possible to characterize the vibrational character directly from the mode eigenvectors.
The spatial extension and "shape" of phonon modes has been used previously to distinguish vibrational characters. Here, we use the participation ratio for such analysis (Figure 4 g, details in Supplemental Note 10). 9,10 A participation ratio close to unity is indicative of a high spatial extension of a mode and typically found in simple crystalline materials like c-Si that have textbook phonon-gas transport, 10 and so a participation ratio of 1 can be called the phonon limit.
Participation ratios below ~0.1 are typical for localized modes that do not contribute to thermal transport, and the localization limit is, by definition, when only 1 atom (out of all the atoms in the unit cell) participates in the vibrational mode. 9 The calculated participation ratios are in the These results confirm diffuson-mediated thermal transport in the Ag + argyrodites at temperatures relevant for thermoelectrics and ionic conduction. Accordingly, the phonon-gas channel has only a minor contribution. It can be concluded that a complex crystal structure, resulting in many energetically close phonon modes, and relatively large anharmonicities are sufficient to achieve diffuson-dominated thermal transport in crystalline materials. This is to say that we do not need to explicitly invoke ion transport effects in order to capture the magnitude of thermal conductivity. Thus, significant diffuson-contributions can be expected in other structurally well-defined materials.
Connecting ionic and thermal diffusion. The results presented above have the surprising observation that the thermal conductivity of Ag + argyrodites is seemingly independent of ionic transport as the diffuson-based thermal conductivity does not change significantly, while the changes in ion transport (based on Nernst-Einstein diffusion coefficients, Supplemental Note 4) are large across the same temperature range (Figure 5 a). To comprehend the underlying differences of thermal and ionic transport one can examine the number of Ag + ions participating in the respective processes. While all Ag + vibrational modes (and with that all Ag + ions) contribute to thermal transport at high temperatures (e.g., Figure 4 e), the instantaneous fraction of mobile Ag + participating in ionic transport is given by the Boltzmann distribution, where F G) HI is the number density of mobile Ag + ions, F JGJKH is the total number density of Ag + ions, and L is the macroscopic activation barrier (Figure 5 b). Consequently, even with an exceptionally low activation barrier of 0.05 eV, comparable to other superionic conductors like Cu2Se, 48 only ~10% of the ions are thermally mobile (without an applied electric field) at any given time over the investigated temperature range. For higher activation barriers, this fraction stays well below 1%. This observation sheds light on the misconception of a "liquid-like" sublattice, given that temperatures of more than 1000 K would be necessary to even reach a 50% thermally mobile sublattice. While instantaneous mobile ion fractions on the order of several percent could be expected to significantly impact thermal transport within the phonon gas model, the nature of diffuson modes makes them less likely to be largely affected so long as the mode energies do not change drastically as more ions become mobile. This does not seem to be the case, nor does the magnitude of ionic conductivity suggest any significant thermal transport by ion mobility (see Supplemental Note 11).
At a deeper level, the difference between ionic and thermal transport is better understood by considering their similarities, that is the microscopic origin of both processes: phonon occupation fluctuations. Thermal transport in solids arises from any deviation ( − ) ≠ 0 from the equilibrium phonon occupation number , with being the instantaneous occupation of mode N (see Supplemental Note 6). Only the drift velocity of the phonon (e.g., phonon-gaslike or diffuson-like) determines the magnitude of its contribution to transport. In contrast, ionic motion is an activated process such that a critical fluctuation has to be reached for the ion to overcome the activation barrier. The magnitude of this critical fluctuation will depend on the vibrational mode(s) involved and the specific jump that is considered (for example see Figure   3 d-f). For a single vibrational mode that oscillates in the jump direction, the critical number of phonons O required for a jump to occur is where is the frequency of the phonon mode and corresponds to the activation barrier of Discussing ion jumps in the context of phonon fluctuations is a novel concept and may be a first step towards a lattice dynamical theory of ion transport. Generally, one could use the jumpdirection projected density of states for each Ag + site to elucidate jump probabilities for each frequency and each (compare Figure 3 d, e). From our frequency-dependent analysis, we reach an important conclusion that lower frequencies are more beneficial for ionic transport (Supplemental Note 6), independent of the material, temperature or activation barrier. This is a crucial insight for ion conduction research showing that the local jump dynamics need to be tailored rather than the global lattice dynamics. 27,28 The inherently low frequencies of Ag + vibrations in the Ag8MSe6 argyrodites may thus be a contributing factor to their superionic behavior.
The prominent contribution of Ag + vibrational modes to both thermal and ionic transport is made clear by comparing the vibrational frequencies of those found experimentally and from the calculated density of states ( Figure 5 d). First, the average diffuson frequency found from the spectral two-channel calculation ( $ %% = 2.3 THz) is in excellent agreement with the value estimated experimentally ( $ %% = 2.2 ± 0.2 THz, range 1.9 THz to 2.4 THz) using an analytical diffuson transport model (Supplemental Note 3). 12 Comparatively, an average ionic transport frequency is found by using the average of the partial Ag + density of states ( _`= 2.0 THz).
Meanwhile, the Einstein frequency determined from X-ray diffraction experiments ( = 1.2 THz) coincides with the maximum of the partial Ag + density of states, which consists of modes that contribute to large thermal displacements. The fact that these average frequencies all reside between 1 and 3 THz where Ag + makes up ~70% of the vibrational density of states, and that Ag + vibrations have been shown to be largely diffuson-like in nature means that diffusons are foundational to understanding both thermal and ionic transport in these materials.
Consequently, the vibrational character is likely important to understand transport in other ionic conductors, with or without ultralow thermal conductivity.  Outlook on engineering transport in ionic conductors. As we have shown, the magnitude of ionic conductivity can be changed independently without altering the thermal transport properties that are governed by diffusons. Therefore, strategies to lower the ionic conductivity, and consequently improve the long-term stability in thermoelectric devices, are not expected to have detrimental effects on the desirable low lattice thermal conductivities. This profound insight opens the possibility to tailor superionic conductors and expand their usage in commercial thermoelectric devices 20,49 by using the known design principles in superionic conductors. 15,40 In addition, our results lead to an important conclusion for the research of solid ionic conductors that are becoming increasingly relevant for the performance optimization of all solid-state batteries. 1,26 Here, the Meyer-Neldel behavior is a bottleneck in achieving higher ionic conductivities since a lowering of the activation barrier has detrimental effects on the conductivity pre-factor. 15,27,36,39 With that, reducing the slope of the Meyer-Neldel plot becomes a main goal. 39 By deriving the multi-excitation entropy using phonon-fluctuation considerations, we show that the Meyer-Neldel slope can be related to the prominent vibrational modes of ionic conduction and a characteristic number of phonons found to be in the range of equilibrium phonon occupations (Supplemental Note 6). 7 This leads to a novel concept and design principle in solid-electrolyte research: determining and manipulating phonon occupations (e.g. targeted phonon excitation 41 , through opto-electronic stimulation 50 ) in conjunction with tailoring the site-specific vibrational frequencies and activation energies.

Conclusion
This work demonstrates that diffuson-mediated thermal transport is dominant in Ag + argyrodite superionic conductors at temperatures relevant for thermoelectrics and ionic transport. An important consequence of this fact is that ionic conductivity can be varied by orders of magnitude without affecting the thermal conductivity. The similarities of the argyrodite structure with lithium and sodium superionic conductors means that diffusons are likely prevalent in battery materials as well. Thus, by understanding the origins of thermal and ionic transport, in particular via vibrational characteristics and phonon occupations, novel design concepts are possible to further improve functional ionic conducting materials.

Author contributions
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