Influence of Lattice Dynamics on Na+ Transport in the Solid Electrolyte

Article pubs.acs.org/cm

Cite This: Chem. Mater. 2017, 29, 8859-8869

Influence of Lattice Dynamics on Na+ Transport in the Solid Electrolyte Na3PS4−xSex Thorben Krauskopf,† Constantin Pompe,† Marvin A. Kraft,† and Wolfgang G. Zeier*,†,‡ †

Institute of Physical Chemistry, Justus-Liebig-University Giessen, Heinrich-Buff-Ring 17, D-35392 Giessen, Germany Center for Materials Research (LaMa), Justus-Liebig-University Giessen, Heinrich-Buff-Ring 16, D-35392 Giessen, Germany

‡

S Supporting Information *

ABSTRACT: Li+- and Na+-conducting thiophosphates have attracted much interest because of their intrinsically high ionic conductivities and the possibility to be employed in solid-state batteries. Inspired by the recent finding of the influence of changing lattice vibrations and induced lattice softening on the ionic transport of Li+-conducting electrolytes, here we explore this effect in the Na+ conductor Na3PS4−xSex. Ultrasonic speed of sound measurements are used to monitor a changing lattice stiffness and Debye frequencies. The changes in the lattice dynamics are complemented by X-ray diffraction and electrochemical impedance spectroscopy. With systematic alteration of the polarizability of the anion framework, a softening of the lattice can be observed that leads to a reduction of the activation barrier for migration as well as a decreased Arrhenius prefactor. This work shows that, similar to Li+ transport, the softening of the average vibrational frequencies of the lattice has a tremendous effect on Na+-ionic transport and that ion−phonon interactions need to be considered in solid electrolytes. conductors.17−29 Further, Jansen30 put forward the idea of a “paddle-wheel” or “revolving-door” mechanism in which concerted polyhedral motion assists the ionic diffusion process, an idea that was just recently corroborated by Benedek and coworkers in oxygen-ion-conducting materials.31 Furthermore, just recently, Shao-Horn and co-workers theoretically explored the influence of the phonon frequencies and energy of the phonon band center on the ionic conduction process.12,13 However, it has proven to be a challenge to investigate the influence of a soft lattice on the ionic transport experimentally by changing the lattice polarizability. On one hand, changes in the anion sublattice often induce structural changes, for instance, in Na3PS4−xOx,32−38 or may induce changes in the carrier density leading to strong convolution effects. Recently, Kraft et al. were able to experimentally show the effect of a changing lattice polarizability on the ionic conductivity in the Li+-conducting argyrodite Li6PS5X.11 With changes in the anion content, the lattice softness was altered, leading to the expected trend in decreasing activation barrier with increasing lattice softness. However, the decreasing softness further led to a changing Arrhenius prefactor because the vibrational frequencies of the jumping ion as well as the migrational entropy are also affected. Inspired by the findings in Li+ conductors that a softening of the lattice induces a decrease not only in the activation barrier,

1. INTRODUCTION Solid ion conductors are currently being investigated for possible applications in solid-state batteries.1 Especially because of their intrinsically high ionic conductivities, multiple material classes such as the NASICON family,2,3 the Li+-conducting garnets,4,5 and the Li+- and Na+-conducting thiophosphates have attracted much interest.6−9 In general, the ionic conductivity σ of solids is governed by an activated hopping process of ions with the activation barrier EA that determines the mobility, as well as the Arrhenius prefactor σ0: σ σ = 0 e−EA / kBT (1) T The overall conductivity then depends on the Arrhenius prefactor, which, for instance, includes parameters such as the charge carrier density, and the activation barrier. In particular, the enthalpy of migration, i.e., the activation barrier, is usually strongly related to the underlying crystal structure, in which, for instance, the polyhedral connectivity and even the basic crystallographic anion packing exert a strong influence.10 In addition to these structural requirements, a softer anion lattice with more polarizable anions has been corroborated to lower the activation barrier,11−13 explaining the high ionic conductivity in many of the Li+- and Na+-conducting thiophosphates.6,14−16 The importance of the polarizability of the anion sublattice, i.e., the lattice softness, had first been realized in the 1970s/1980s.17−21 For an understanding of the influence of the oscillator strength on ionic motion, the focus was on understanding the phonon spectra of ionic © 2017 American Chemical Society

Received: August 16, 2017 Revised: September 11, 2017 Published: September 25, 2017 8859

DOI: 10.1021/acs.chemmater.7b03474 Chem. Mater. 2017, 29, 8859−8869

Article

Chemistry of Materials

radiation [λ1 = 1.540 598 0 Å; λ2 = 1.544 426 0 Å; I(λ2)/I(λ1) = 0.5]. Measurements were carried out in the 2θ range between 10° and 100° with a step size of 0.0334°. The counting time per step was 200 s. For the structural characterization of (HT)-Na3PS4−xSex, a PANalytical Empyrean powder diffractometer in Bragg−Brentano θ−θ geometry with the same Cu Kα radiation and a PIXcel detector was used. Measurements were carried out in the 2θ range between 10° and 90° with a step size of 0.026°. Counting time per step was 300 s. All powders were sealed in an airtight sample holder and covered with Kapton polyimide film (7.5 μm). Rietveld Analysis. Rietveld refinements were carried out using the Fullprof software.40 Profiles were fitted with a pseudo Voigt function, and the background was described using linear interpolation between a set of manual points with refinable heights. For the cubic modifications, the structure of c-Na3PSe4 from Zhang et al. with the space group I4̅ 3m and, for the tetragonal modifications, the structural data of t-Na3PS4 from Jansen with the space group P4̅21c were used as the initial starting model.38,41 Because of the high background of the polyimide film, the range between 10° and 22° was excluded for the refinement. The following parameters were initially refined: (1) peak shape, background coefficients, and lattice constants using the Le Bail method; and afterward in a Rietveld refinement, the (2) scale factor, (3) fractional atomic coordinates, and (4) isotropic atomic displacements parameters. The occupancies were initially restricted to the nominal stoichiometry of the compounds. At the end, all refined parameters were fixed, and only the occupancies of S and Se were allowed to refine, because a simultaneous refinement of all atomic parameters did not converge to reasonable and stable values. The standard deviations of all parameters except S and Se occupancies were adopted from the second-to-last refinement. Rwp, Rexp, and the goodness of fit S were used to assess the quality of the refined structural models.42 Uncertainties of the bond lengths and lattice parameters are obtained from the refinements, and the polyhedral volumes using the structural program VESTA. Scanning Electron Microscopy. The microstructure of the electrolyte pellets was investigated using the Carl Zeiss Ultra field emission SEM instrument, Merlin. The air-sensitive samples were transferred from the glovebox to the vacuum chamber of the SEM with the Leica transfer module system, EM VCT50. Electrochemical Impedance Spectroscopy. Electrical conductivities were measured by AC impedance spectroscopy, using a custom-built setup. Powder samples were placed between two stainless steel rods with 10 mm diameter and pressed uniaxial at 3 t (≈380 MPa) for 5 min. Electrochemical impedance spectroscopy (EIS) was conducted in the temperature range 253−333 K using an SP300 impedance analyzer (Biologic) at frequencies from 7 MHz to 100 mHz with an amplitude of 10 mV. During the measurement, proper contact was ensured by external pressure. (BM)-Na3PS4−xSex was measured twice because a large scattering of the transport variables was observed. Therefore, the measurement uncertainties mainly stem from uncertainty in pellet thickness and are obtained by propagation of the standard deviations of both measured values. Ultrasonic Speed-of-Sound Measurements. Pulse−echo speed-of-sound measurements were performed on consolidated discs using an Epoch 600 instrument (Olympus) with 5 MHz transducers for longitudinal and transverse speeds of sound. Since the samples are highly air-sensitive and enclosing the pellets in pouches would prevent penetration of transverse signals, the samples were coated with a thin layer (873 K and seems to also crystallize in a tetragonal phase with lattice parameters seemingly close to the cubic polymorph.39 Figure 1 shows the crystal structure of both tetragonal and cubic modifications of the sodium tetra-thiophosphate Na3PS4. The cubic structure crystallizes with all PS43− in a body-centered cubic arrangement and one crystallographic Na position (Wyckoff 6b, as shown in Figure 1a), in which Na+ occupies the octahedral sites. A minor rotation of the PS43− tetrahedra around the [111] axis leads to the tetragonal structure, and the Na position splits into two crystallographically independent positions (Wyckoff 4d and 2a). Because of the rotation of the PS43− tetrahedra and displacement of Na+, the lattice parameter c is slightly elongated leading to the transition into the tetragonal space group. Using anion substitution, the selenophosphate Na3PSe4 can be synthesized, which crystallizes in the cubic phase at room temperature and transforms into the tetragonal phase below ∼270 K.41,50,51 In 2012, Hayashi et al. reported on the stabilization of the cubic Na3PS4 phase at room temperature using a synthetic ballmilling approach.7 Reaching an ionic conductivity of up to 4.6 × 10−4 S cm−1 at room temperature has led to a resurgence of interest in this phase.52 Consequently, for an increase in the diffusion pathways and possibly the lattice polarizability, substitutions of S with Se were performed that form full solid solutions and affect the ionic conductivity.39,41 In addition, substitution of P5+ with As5+ significantly increases the conductivity53,54 due to a widening of diffusion pathways and possibly altered Na−S interactions.

2% can be achieved, the obtained values are systematically underestimated because only ≈78% (BM) and ≈90% (HT) dense pellets were obtained. The uncertainties were calculated using a linear error propagation of the larger uncertainties (i.e., thickness of the pellet and standard deviation of the transit time of the sound waves). As the uncertainties do not reflect the influence of a scattering microstructure, the real uncertainties are expected to be slightly larger as seen in the Li+-argyrodites.11 Ultrasonic measurements are a widely used and extremely accurate approach to determine the elastic properties and speed of sound in materials. Such methods are a direct probe of strength of the bonds themselves.43−46 Further, the obtained longitudinal and transverse speeds of sound νlong and νtrans can be used to calculate the mean speed of sound νmean, as well as the Debye temperature ΘD and Debye frequency υD via eqs 2−4.47−49

vmean 3 =

3 v long −3 + 2vtrans−3

ℏ ⎛ 6π 2 ⎞ ⎜ ⎟ kB ⎝ V ⎠

(2)

1/3

ΘD =

υD =

vmean

⎛ 3 ⎞1/3 ⎜ ⎟ v ⎝ 4πV ⎠ mean

(3)

(4)

These equations show that a decreasing speed of sound directly relates to a decreasing Debye temperature and Debye frequency, both of which are parameters that reflect the phonon vibrational frequencies of a material.

3. RESULTS Crystal Structure of Na3PS4−xSex. Initially prepared by Jansen and co-workers38 in the tetragonal modification using classical high-temperature solid-state synthesis, a cubic hightemperature phase was found to crystallize above 534 K. The low-temperature α phase crystallizes in the tetragonal space 8861


Article


Figure 2. X-ray diffraction data and result of the Rietveld refinement of c-Na3PS2Se2 using the (a) ball-milling (BM) and (b) high temperature (HT) approach. Experimental data are shown as points; the red line denotes the calculated pattern, and the difference profile is shown in blue. Calculated positions of Bragg reflections are shown by green vertical tick marks. The obtained profile residuals are given in the figure, showing a high goodnessof-fit and phase purity of the samples. (c, d) All the diffraction patterns with changing Se composition of both the (BM)- and (HT)-routes, respectively. In the (HT)-route, the cubic phase is only present for x ≥ 1.5; below x = 1.5, the pure tetragonal phase is formed. The (BM)-synthesis leads to the cubic structure over the whole range of solid solution. However, in the compositions of (BM)-c-Na3PS4 and (BM)-c-Na3PS3.5Se0.5, small reflections of the tetragonal phase can be observed, indicating the presence of a small mass fraction of the tetragonal phase.

routes. All other diffraction patterns and corresponding results of the Rietveld refinements can be found in the Supporting Information. As expected, the (BM)-synthesis leads to the cubic structure in all compositions, while the (HT)-synthesis leads to the tetragonal phase for x ≤ 1.5 as shown in Figure 2c,d, respectively. With increasing Se fraction, the cubic structure becomes more stable in the high-temperature synthesis as the phase transition moves below room temperature. Figure 2c,d shows the Bragg reflections moving toward lower angles, corresponding to an increasing lattice, as expected for a solid solution in which the smaller S2− (1.84 Å)58 is replaced by the larger Se2− (1.98 Å)58. Ball-milling-assisted synthesis has recently shown that it can lead to an underlying amorphous thiophosphate phase, which can influence the ionic transport.59 Therefore, the pattern of (BM)-c-Na3PS4 was additionally collected and refined with an internal LaB6 standard (SRM 660b)60 (data not shown), and no amorphous thiophosphatecontaining phase can be detected within the resolution limit of the technique. At lower selenium contents, i.e, (BM)-c-Na3PS4 and (BM)-Na3PS3.5Se0.5, small amounts of the tetragonal phase were observed in the (BM)-synthesized products. Some other minor impurities can be found in the compositions of (BM)-cNa3PSSe3, (BM)-c-Na3PS0.5Se3.5, and (BM)-c-Na3PSe4. These minor phases cannot be indexed unequivocally and, therefore, are not expected to influence the transport significantly because of the small weight fractions. A comparison between the reflections of the two synthetic routes shows broader reflections for the (BM)-synthesis. A Williamson−Hall plot (see the Supporting Information, Figure S1) shows smaller crystallite size for (BM)-Na3PS4 and some possible strain, compared to the high-temperature synthesis.

The local jump processes in both phases, tetragonal and cubic, are shown in Figure 1d,e, respectively. In the cubic phase, the jump process occurs from the Na1 site to Na1 (i.e., 6b−6b) via a very small and strongly distorted tetrahedral site (Wyckoff 12d), providing an ideal three-dimensional diffusion pathway. The 12d site has recently experimentally been found to be populated;55 however, density functional theory (DFT) with molecular dynamics (MD) simulations suggest that an occupancy of the 12d site leads to a destabilization of the structure at 0 K and therefore suggests that it is unstable.56 In the tetragonal structure, the different Na positions lead to multiple possible jump processes, in which a jump from the Na1 (4d) to the Na2 (2a) site through a triangle site seems to be the most favorable. While 4d−4d jumps are generally possible, they would include jumps via edge-sharing tetrahedra and are therefore unlikely.57 Both the cubic and tetragonal structures exhibit diffusion pathways directly bypassing transition states that are coordinated by the chalcogenide anions S and Se, respectively. As full solid solutions are possible that do not change the carrier density, replacing the sulfur with selenium in Na3PS4−xSex seems like the ideal model system to study the effect of the changing lattice polarizability on the ionic transport. Structural Characterization. Solid solutions Na3PS4−xSex were synthesized via a traditional high-temperature synthesis (HT) and a ball-milling route (BM), which has been shown to stabilize the cubic phase at room temperature.7 Both routes, (HT) and (BM), are investigated in this study to see if there are differences in the structures and transport between the two synthetic routes. Figure 2 shows exemplary Rietveld refinements of the diffraction data of Na3PS2Se2 using both synthesis 8862


Article


Figure 3. Structural data of Na3PS4−xSex obtained from the Rietveld refinements of X-ray powder Bragg data. (a) Lattice parameters and (b) Se occupancy. (c, d) Changing Na−S/Se bond lengths as well as the volume (area) of the “transition-state”-like structure of the (HT) and (BM) synthesis, respectively. (e) P−S/Se bond lengths with x, and (f) changing dihedral angles, moving toward a more cubic arrangement. The shown uncertainties correspond to the 3σ of the statistical uncertainties (σ), obtained from the Rietveld refinements. The vertical dashed lines in parts c, e, and f mark the transition between the tetragonal and cubic structure in the (HT)-route.

linearly and fully concurs with the synthetic molar fractions, within the uncertainty of the data. Refinements of Na on the 12d position, i.e., the transition state, in the cubic phase only leads to a minor improvement of the profile residuals as well as a negligible occupancy on the transition state. Rietveld refinements of synchrotron data also show that in the related compound c-Na3PSe4 no residual electron density lies on the 12d position,50 and any occupancy may even be related to an underlying tetragonal phase that results in a pseudo-occupancy refinement. As Klerk et al.56 assume that the finding of electron density of Na on the 12d site may be related to the high number of jumps from 6b to 6b, rather than a real, stabilized occupancy, occupancy refinements of the 12d position are omitted from the refinements performed here. With increasing unit cell size, as well as the transition from the tetragonal to cubic cell, changes in the local bond lengths and polyhedra are expected. Figure 3c,d shows the changing Na−S/Se bond lengths as well as the volume in the “transitionstate”-like structure, through which the diffusion occurs. As

Figure 3 shows the structural data as obtained from Rietveld refinements against the laboratory diffraction data of the (HT) and (BM) synthetic routes. The tables with the obtained structural information on the different compositions can be found in the Supporting Information, Tables S1−S19. The lattice parameter a increases linearly with increasing selenium fraction in the cubic phase synthesized via the ball-milling route, because of the increasing average anion radius, following Vegard’s law. The lattice parameters observed here of all samples correspond well to the literature values for the (BM)syntheses55,61 and the (HT)-syntheses.38,39,41,50,62 Using the high-temperature synthesis, the tetragonal phase crystallizes at low Se fractions. With increasing Se content, the lattice parameters increase and converge into the cubic structure at x = 1.5. This transition was recently observed by Bo et al.;39 however, in this work, a smaller step size in selenium fraction is used. Figure 3b shows the Se occupancies, as obtained from the Rietveld refinements. As expected, the Se occupancy increases 8863


Article


Figure 4. (a) Representative Arrhenius plots of (HT)-Na3PS4−xSex obtained via temperature-dependent impedance spectroscopy. The vertical line represents room temperature. As c-Na3PSe4 transforms to t-Na3PSe4 below 270 K, both are denoted in the legend. (b) Selected Nyquist plots of (HT)-Na3PS4 showing the quality of the impedance data and the fit with the shown equivalent circuit.

Figure 5. Activation barriers for ionic motion of the (a) (HT)-Na3PS4−xSex and (b) (BM)-Na3PS4−xSex, showing a decreasing activation barrier with increasing Se fraction. In addition, the Arrhenius prefactor σ0 decreases as well. The vertical dashed lines in parts a and c mark the transition between the tetragonal and cubic structure in the (HT)-route.

These structural data show the influence of the synthetic parameters on the structure. The isoelectric anionic substitution induces the same changes in the lattice (bond length, polyhedral volumes, and occupancies), regardless of the synthetic approach. As the diffusion pathways bypass a window coordinated by the S and Se anion, the question arises how the ionic transport and dynamical lattice properties are affected. Ionic Transport. Figure 4a shows representative Arrhenius plots of different selected compositions of the high-temperature phases. The temperature-dependent ionic conductivity was measured using impedance spectroscopy, and a representative temperature-dependent impedance response can be found in Figure 4b. The impedance data were fit to an equivalent circuit consisting of one parallel constant phase element (CPE)/ resistor in series with a CPE, representing the blocking

mentioned above, in the case of the tetragonal unit cell, the diffusion pathway occurs via a triangle site and, in the cubic cell, a distorted tetrahedron. With increasing unit cell size, the triangle and the tetrahedron expand providing a broader diffusion pathway for Na+ in the unit cell. The increasing bottleneck size coincides with an increase of the Na−S/Se bond lengths as well as the P−S/Se bond length (Figure 3e). In addition to the effect of an increasing anion radius, the substitution of S with Se in the tetragonal cell leads to a reduction of the two dihedral angles that describe the rotation of the PS43− groups. With increasing Se fraction, the dihedral angles decrease, leading to less rotation along the [111] axis and to fully converged PS43− tetrahedra at x = 1.5, which then results in the cubic crystal structure. 8864


Article

Chemistry of Materials electrodes. With increasing temperature, the semicircle decreases because of faster ionic transport. The apex-frequency of the room-temperature data is 1.0 × 106 Hz and corresponds to a geometrical capacitance of 21.6 pF cm−2, and an ideality factor of α = 0.85. Scanning electron micrographs (see the Supporting Information, Figure S2) show how the cold-pressed samples of both synthesis routes exhibit grains that are wellfused together. While bulk and grain boundary contributions cannot be deconvoluted, and the conductivities correspond to the overall sample conductivity, the obtained capacitances correspond well to bulk transport.63 The room-temperature conductivity values of (HT)Na3PS4−xSex and (BM)-Na3PS4−xSex can be found in the Supporting Information, Figure S3. Whereas, in (HT)Na3PS4−xSex, the room-temperature ionic conductivity seems to increase with increasing selenium content as shown by Bo et al.39 and Zhang et al.,41 there is no clear trend for the (BM)synthesized compounds. (BM)-Na3PS4 shows, with 1.5(2) × 10−4 S cm−1, the maximum value of the ionic conductivity in all samples. In contrast, (HT)-Na3PS4 exhibits the minimum value of the room-temperature conductivity with 1.8(1) × 10−5 S cm−1. To decide whether the low ionic conductivity of (HT)Na3PS4 results from structural or microstructural features, the tetragonal polymorph was also prepared by a mechanochemical approach (see the Experimental section). Indeed, (BM)-tNa3PS4 shows a similar ionic conductivity [1.6(1) × 10−4 S cm−1] as its cubic polymorph. This shows that structureindependent factors, such as defect concentration (as suggested theoretically)39,56,62,64 and microstructure, are likely responsible for the discrepancy in ionic conductivity in both tetragonal and cubic polymorphs, and not the crystal structure itself. Here, a more crystalline (larger crystallite size) sample exhibits lower ionic conductivities. The similar conductivity values of both structures (BM)-t-Na3PS4 and (BM)-c-Na3PS4 support the theoretical predictions by Zhu et al.,64 Yu et al.,65 Klerk et al.,56 and Chu et al.,62 which suggested only minor differences in the transport between both polymorphs. In addition, in the (HT)-c-Na3PSe4 phase, cooling during the measurement to temperatures below 285 K leads to a changing slope, suggesting the formation of the tetragonal Na3PSe4 phase (for more detailed data, see the Supporting Information, Figure S4). This phase transition has recently been found by Pompe51 and cannot be found in the other phases in the temperature regime measured here, which is in good agreement with the theoretical calculated phase stabilities.39 This phase transition does not occur in (BM)-c-Na3PSe4 and may be explained by a concurrent stabilization of the cubic modification resulting from the mechanochemical synthetic approach. Using the linear Arrhenius behavior (see Figure 4) of the conductivity, the activation barrier EA and the prefactor σ0 can be extracted and are shown in Figure 5. With increasing Se content, both the activation barrier as well as the Arrhenius prefactor decrease, corresponding well with the Meyer−Neldel rule66−70 (see Figure 6). The lower decrease in activation energy and higher prefactors in (BM)-Na3PS4−xSex may be attributed to the smaller crystallite size, which has been shown to affect the ionic conductivity (see above). In addition, the harsh conditions of the ball-milling synthesis may lead to intrinsic defects and higher overall conductivity values. As these materials are of superionic nature, the influence of defect formation enthalpies (not the defect concentration) to the measured EA is neglected in the following discussion. A similar behavior of changing the lattice polarizability has

Figure 6. log of pre-exponential factor σ0 versus activation energy EA for (BM)- and (HT)-Na3PS4−xSex. The linear relationship shows that the Meyer−Neldel rule is satisfied in all synthesized samples. Small deviations at higher activation barriers for the tetragonal samples seem to be due to the structural transition.

recently been shown by Kraft et al., in which the increasing lattice softness not only affected the activation energy but also decreased the prefactor in the ionic conductivity.11 Lattice Softness. For determination of the influence of the anion polarizability on the lattice dynamics, ultrasonic pulse− echo measurements were performed. Representative speed-ofsound responses can be found in the Supporting Information, Figure S5. The extracted transverse and longitudinal speeds of sound are shown in Figure 7. As recently employed by Kraft et al.11 to measure the lattice softness of ionic conductors, speedof-sound measurements exhibit a high degree of accuracy for measuring elastic constants.14,43,71−73 The speed of sound in a material directly relates to the slope of the acoustic phonon branches and is therefore a good measure of the phononic properties and lattice stiffness. For instance, a high speed of sound relates to a high average bond strength, a high Debye frequency, and high Debye temperatures, all of which represent a stiffer, more rigid lattice.43 Figure 7 shows that, with increasing Se fraction, the speed of sound decreases because of a softening of the lattice. The softening of the lattice corresponds to the introduction of the more polarizable Se anion. Interestingly, the speed of sound in the tetragonal regime is higher, which can be explained by the lattice constant. The larger lattice of the tetragonal structure (larger c-lattice parameter) leads to a smaller Brillouin zone and an increase in the slope of the acoustic branches, directly showing the value of speed-of-sound measurements. Using eq 4, the Debye frequencies of the lattice can be calculated. With increasing Se content, the decreasing speed of sound and increasing lattice softness lead to a decrease in the Debye frequencies (Figure 7c,d). As recently shown by Kraft et al.,11 the changing anion framework induces a softening of the lattice, and at the same time, a decrease in the prefactor and activation energy of the ionic motion can be observed.

4. DISCUSSION The observed changes in the structure, the ionic transport, as well as the softening of the lattice provide evidence of a correlation between the phononic nature of a material and the ionic conductivity. On one hand, the increasing bottleneck volume and unit cell volume are expected to influence the activation barrier. On the other hand, a decrease in the activation barrier as well as the prefactor of the conductivity can 8865


Article


Figure 7. Longitudinal, transverse, and mean speeds of sound of the (a) (HT)-Na3PS4−xSex and (b) (BM)-Na3PS4−xSex solid solutions. (c, d) With increasing anion polarizability, the lattice softens leading to a decreasing speed of sound and a decreasing Debye frequency. The vertical dashed lines in parts a and c mark the transition between the tetragonal and cubic structure in the (HT)-route.

changing oscillator frequency as well as a changing entropy of migration. Using the formalism of transition-state theory of ionic motion, the jump frequency can be estimated from the activation barrier:26

be found, which seem to coincide with the softening of the lattice. Using conventional hopping theory, the prefactor σ0 is given by74 σ0 =

zn(Ze)2 ΔSm / kB 2 e a0 ν 0 kB

(5)

ν0 =

governed by a geometrical factor z (to take into account different diffusion geometries and correlation factors), the carrier density of mobile ions n, the entropy of migration ΔSm, the jump distance a0, the charge of the ions Ze, and the jump frequency ν0. The jump frequency itself corresponds to the oscillator frequency of the moving cation on its lattice site, i.e., the attempt frequency, as well as the jump probability. In terms of transition-state theory, the jump frequency can be described as the inverse of the lifetime of the excited state.26 In the current study, the constant carrier density leads to only three changing variables, i.e., the jump distance, the oscillator frequency, and the entropy of migration. The entropy of migration depends on the phononic properties as well and can, for a small vibration approximation, be expressed as75,76 ⎛ ∏ 3N ν I ⎞ i ΔSm = kB ln⎜⎜ 3Ni =−11 S ⎟⎟ ⎝ ∏i = 1 νi ⎠

1 a0

2EA M Na+

(7)

with the jump distance, a0, and the mass of Na+, MNa. Figure 8 shows the obtained jump frequencies against the Debye frequencies as obtained by speed-of-sound measurements. A direct correlation can be seen that the activation barriers are

(6)

with the normal frequencies ν1, ..., νN for vibrations about the initial and saddle point, I and S, respectively. Whereas nearly all vibrational frequencies affect the entropy of migration, the migration enthalpy is related to the one vibrational mode that carries the ion across the saddle point.75 This attempt frequency is typically approximated by the Debye frequency.26 Therefore, a softening of the lattice leads to lower Debye frequencies of the lattice (see Figure 7) and is expected to affect not only the activation barrier, but also the prefactor through a

Figure 8. Oscillator frequencies as obtained from the measured activation barriers and crystallographic jump distances against the obtained Debye frequencies. A correlation (dashed line as a guide to the eye) exists linking the lattice softness and Debye frequency to the jump frequency of a moving ion. The data point of (HT)-Na3PS4 is excluded, because of the large structural discrepancy between the cubic and tetragonal phase and the different diffusion pathways in the tetragonal phase. 8866


Article


that not only Li+ but also Na+ transport can be severely affected by the phononic properties of a material. These data suggest that, for an improvement of the ionic conductivity in solid electrolytes, it may be necessary to decouple the dynamic motions of the excited from the initial state, instead of aiming at an overall softer lattice.

influenced by a softening of the lattice. It needs to be noted that the extracted Debye frequencies are slightly lower than the calculated jump frequencies ν0. First, the samples are not fully dense, which leads to slightly lower speeds of sound, and second, the data provided here measure the full unit cell with the different covalent P−S/Se bonds and the more ionic Na− S/Se interactions. In addition, the jump frequencies in the ballmilled samples are slightly higher, possibly because of microstructural differences as discussed above. These data show that the softening of the lattice leads to lower oscillatory frequencies of the lattice and the moving cation. The changing vibronic frequencies affect the entropy of migration and the changing jump frequency, and with it the enthalpy of migration (i.e., the activation barrier). In addition to the changing oscillator frequency and migration enthalpy, the prefactor decreases over multiple orders of magnitude when compared to ν0. It can be assumed that the changing vibrational frequencies of the lattice affect the entropy of migration more strongly through the ratio of the vibrational partition functions, expressed as the product of normal frequency modes in eq 6. The influence of the lattice vibrations on the prefactor and the enthalpy of migration may be the reason for the origin of the Meyer−Neldel rule (see Figure 6).66 However, for corroboration of this statement, vibrational entropies and free energies need to be computed, which has posed to be challenging. Recent progress in computational chemistry could enable more precise predictions in the future.77 The observed correlation between a softer, more polarizable lattice and the ionic transport shows that not only the activation barrier, but also the prefactor of the conductivity is affected by a softer lattice. The work presented here shows that the influence of a dynamic lattice is also apparent in Na+-ion conductors, after recently being shown for Li+-conducting argyrodites by Kraft et al.11 Indeed, this ambiguous influence has been largely overlooked in the past and shows that a better understanding of the influence of a dynamic lattice on ionic transport is necessary. With an understanding of this influence of the softer vibrational modes on the transport, it may be necessary to find structures with soft vibrational modes in the excited state and stiffer phonon modes in the initial ground state, to obtain high prefactors and concurrent low enthalpies of migration.

■

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.chemmater.7b03474. Structural data as obtained from Rietveld refinements against X-ray powder Bragg data, a Williamson−Hall plot, SEM images, room-temperature conductivity of all compounds, an exemplary Arrhenius plot, an exemplary longitudinal and transversal sound response, and diffraction patterns and corresponding results of the Rietveld refinements (PDF)

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Wolfgang G. Zeier: 0000-0001-7749-5089 Notes

The authors declare no competing financial interest.

■

ACKNOWLEDGMENTS We gratefully acknowledge the financial support through startup funding by the Justus-Liebig-University Giessen. We thank Jan Peilstöcker for the capillary measurements.

■

REFERENCES

(1) Janek, J.; Zeier, W. G. A solid future for battery development. Nat. Energy 2016, 1, 16141. (2) Guin, M.; Tietz, F. Survey of the transport properties of sodium superionic conductor materials for use in sodium batteries. J. Power Sources 2015, 273, 1056−1064. (3) Vogel, E. M.; Cava, R. J.; Rietman, E. Na+ ion conductivity and crystallographic cell characterization in the Hf-nasicon system Na1+xHf2SixP3−xO12. Solid State Ionics 1984, 14, 1−6. (4) Thangadurai, V.; Kaack, H.; Weppner, W. J. F. Novel Fast Lithium Ion Conduction in Garnet-Type Li5La3M2O12 (M = Nb, Ta). J. Am. Ceram. Soc. 2003, 86, 437−440. (5) Cussen, E. J. The structure of lithium garnets: cation disorder and clustering in a new family of fast Li+ conductors. Chem. Commun. 2006, 412−413. (6) Kamaya, N.; Homma, K.; Yamakawa, Y.; Hirayama, M.; Kanno, R.; Yonemura, M.; Kamiyama, T.; Kato, Y.; Hama, S.; Kawamoto, K. A lithium superionic conductor. Nat. Mater. 2011, 10, 682−686. (7) Hayashi, A.; Noi, K.; Sakuda, A.; Tatsumisago, M. Superionic glass-ceramic electrolytes for room-temperature rechargeable sodium batteries. Nat. Commun. 2012, 3, 856. (8) Richards, W. D.; Tsujimura, T.; Miara, L. J.; Wang, Y.; Kim, J. C.; Ong, S. P.; Uechi, I.; Suzuki, N.; Ceder, G. Design and synthesis of the superionic conductor Na10SnP2S12. Nat. Commun. 2016, 7, 11009. (9) Rao, R. P.; Adams, S. Studies of lithium argyrodite solid electrolytes for all-solid-state batteries. Phys. Status Solidi A 2011, 208, 1804−1807. (10) Wang, Y.; Richards, W. D.; Ong, S. P.; Miara, L. J.; Kim, J. C.; Mo, Y.; Ceder, G. Design principles for solid-state lithium superionic conductors. Nat. Mater. 2015, 14, 1026.

5. CONCLUSION In this work, the effect of the lattice polarizability on the Na+ conduction in Na3PS4−xSex has been studied, providing further evidence of the influence of lattice vibrations on ionic transport. Using different synthetic methods, the stability of the different polymorphs has been explored, and the changes in the structure have been monitored using X-ray diffraction. Speed-of-sound measurements were used to obtain information about the lattice dynamics, i.e., the softness of the lattice and Debye frequencies, when substituting S with Se. Temperaturedependent impedance spectroscopy has been used to monitor changes in the Na+-ionic transport. First, this work shows that the ionic conductivities of the tetragonal and cubic Na3PS4 are comparable when prepared similarly, which had been suggested theoretically. Second, speed-of-sound measurements show that the increasing polarizability of the anion framework induces a softening of the lattice, leading to lower Debye frequencies. This softening of the vibrational frequencies affects both the Arrhenius prefactor as well as the activation barrier for ionic migration, showing 8867


Article

Chemistry of Materials (11) Kraft, M. A.; Culver, S. P.; Calderon, M.; Böcher, F.; Krauskopf, T.; Senyshyn, A.; Dietrich, C.; Zevalkink, A.; Janek, J.; Zeier, W. G. Influence of Lattice Polarizability on the Ionic Conductivity in the Lithium Superionic Argyrodites Li6PS5X (X = Cl, Br, I). J. Am. Chem. Soc. 2017, 139, 10909−10918. (12) Wakamura, K. Roles of phonon amplitude and low-energy optical phonons on superionic conduction. Phys. Rev. B: Condens. Matter Mater. Phys. 1997, 56, 11593−11599. (13) Bachman, J. C.; Muy, S.; Grimaud, A.; Chang, H.-H.; Pour, N.; Lux, S. F.; Paschos, O.; Maglia, F.; Lupart, S.; Lamp, P. Inorganic solidstate electrolytes for lithium batteries: mechanisms and properties governing ion conduction. Chem. Rev. 2016, 116, 140−162. (14) Weber, D. A.; Senyshyn, A.; Weldert, K. S.; Wenzel, S.; Zhang, W.; Kaiser, R.; Berendts, S.; Janek, J.; Zeier, W. G. Structural insights and 3D diffusion pathways within the lithium superionic conductor Li10GeP2S12. Chem. Mater. 2016, 28, 5905−5915. (15) Bron, P.; Johansson, S.; Zick, K.; Schmedt auf der Günne, J.; Dehnen, S.; Roling, B. Li10SnP2S12: an affordable lithium superionic conductor. J. Am. Chem. Soc. 2013, 135, 15694−15697. (16) Kanno, R.; Hata, T.; Kawamoto, Y.; Irie, M. Synthesis of a new lithium ionic conductor, thio-LISICON−lithium germanium sulfide system. Solid State Ionics 2000, 130, 97−104. (17) Zeller, H. R.; Brüesch, P.; Pietronero, L.; Strässler, S. Lattice Dynamics and Ionic Motion in Superionic Conductors. In Superionic Conductors; Mahan, G., Ed.; Springer, 1976; pp 201−215. (18) Sen, P. N.; Huberman, B. A. Low-frequency response of superionic conductors. Phys. Rev. Lett. 1975, 34, 1059. (19) Huberman, B. A.; Sen, P. N. Dielectric response of a superionic conductor. Phys. Rev. Lett. 1974, 33, 1379. (20) Allen, S. J., Jr; Remeika, J. P. Direct measurement of the attempt frequency for ion diffusion in Ag and Na β-Alumina. Phys. Rev. Lett. 1974, 33, 1478. (21) Pardee, W. J.; Mahan, G. D. Disorder and ionic polarons in solid electrolytes. J. Solid State Chem. 1975, 15, 310−324. (22) Knauth, P. Inorganic solid Li ion conductors: An overview. Solid State Ionics 2009, 180, 911−916. (23) Bührer, W.; Nicklow, R. M.; Brüesch, P. Lattice dynamics of β(silver iodide) by neutron scattering. Phys. Rev. B: Condens. Matter Mater. Phys. 1978, 17, 3362. (24) Fischer, K.; Bilz, H.; Haberkorn, R.; Weber, W. Covalency and Deformability of Ag+‑Ions in the Lattice Dynamics of Silver Halides. Phys. Status Solidi B 1972, 54, 285−294. (25) Brüesch, P.; Pietronero, L.; Strässler, S.; Zeller, H. R. Brownian motion in a polarizable lattice: Application to superionic conductors. Phys. Rev. B 1977, 15, 4631. (26) Rice, M. J.; Roth, W. L. Ionic transport in super ionic conductors: a theoretical model. J. Solid State Chem. 1972, 4, 294−310. (27) Ebbsjo, I.; Vashishta, P.; Dejus, R.; Skold, K. Inelastic neutron scattering and dynamics of ions in the super-ionic conductor Ag2S. J. Phys. C: Solid State Phys. 1987, 20, L441. (28) Schmalzl, K.; Strauch, D.; Schober, H. Lattice-dynamical and ground-state properties of CaF2 studied by inelastic neutron scattering and density-functional methods. Phys. Rev. B: Condens. Matter Mater. Phys. 2003, 68, 144301. (29) Alben, R.; Burns, G. Lattice dynamics of the superionic conductor AgI. Phys. Rev. B 1977, 16, 3746. (30) Jansen, M. Volume Effect or Paddle-Wheel MechanismFast Alkali-Metal Ionic Conduction in Solids with Rotationally Disordered Complex Anions. Angew. Chem., Int. Ed. Engl. 1991, 30, 1547−1558. (31) Li, X.; Benedek, N. A. Enhancement of Ionic Transport in Complex Oxides through Soft Lattice Modes and Epitaxial Strain. Chem. Mater. 2015, 27, 2647−2652. (32) Pompetzki, M.; Jansen, M. Natriummonothiophosphat (V): Kristallstruktur und Natriumionenleitfähigkeit. Z. Anorg. Allg. Chem. 2002, 628, 641−646. (33) Pompetzki, M.; Jansen, M. Natriumtrithiophosphat (V): Kristallstruktur und Natriumionenleitfähigkeit. Z. Anorg. Allg. Chem. 2003, 629, 1929−1933.

(34) Pompetzki, M.; Dinnebier, R. E.; Jansen, M. Sodium dithiophosphate(V): Crystal structure, sodium ionic conductivity and dismutation. Solid State Sci. 2003, 5, 1439−1444. (35) Wiench, V. D. M.; Jansen, M. Ü ber Na3PO4: Versuche zur Reindarstellung, Kristallstruktur der Hochtemperaturform [1]. Z. Anorg. Allg. Chem. 1980, 461, 101−108. (36) Lissel, E.; Jansen, M.; Jansen, E.; Will, G. Bestimmung der Kristallstruktur von t-Na3PO4 mit Röntgen-und Neutronenpulvertechniken. Z. Krist-Cryst. Mater. 1990, 192, 233−244. (37) Hruschka, H.; Lissel, E.; Jansen, M. Na-Ion conduction in the solid solutions of Na3PO4/Na2SO4 and Na3AlF6/Na2SO4. Solid State Ionics 1988, 28, 159−162. (38) Jansen, M.; Henseler, U. Synthesis, structure determination, and ionic conductivity of sodium tetrathiophosphate. J. Solid State Chem. 1992, 99, 110−119. (39) Bo, S.-H.; Wang, Y.; Ceder, G. Structural and Na-ion conduction characteristics of Na3PSxSe4−x. J. Mater. Chem. A 2016, 4, 9044−9053. (40) Rodríguez-Carvajal, J. Recent advances in magnetic structure determination by neutron powder diffraction. Phys. B 1993, 192, 55− 69. (41) Zhang, L.; Yang, K.; Mi, J.; Lu, L.; Zhao, L.; Wang, L.; Li, Y.; Zeng, H. Na3PSe4: A novel chalcogenide solid electrolyte with high ionic conductivity. Adv. Energy Mater. 2015, 5, 1501294. (42) Young, R. A. Introduction to the Rietveld Method; Oxford University Press: New York, 1993. (43) Zeier, W. G.; Zevalkink, A.; Gibbs, Z. M.; Hautier, G.; Kanatzidis, M. G.; Snyder, G. J. Thinking like a chemist: intuition in thermoelectric materials. Angew. Chem., Int. Ed. 2016, 55, 6826−6841. (44) Heinrich, C. P.; Day, T. W.; Zeier, W. G.; Snyder, G. J.; Tremel, W. Effect of Isovalent Substitution on the Thermoelectric Properties of the Cu2ZnGeSe4−xSx Series of Solid Solutions. J. Am. Chem. Soc. 2014, 136, 442−448. (45) Weldert, K. S.; Zeier, W. G.; Day, T. W.; Panthöfer, M.; Snyder, G. J.; Tremel, W. Thermoelectric transport in Cu7PSe6 with high copper ionic mobility. J. Am. Chem. Soc. 2014, 136, 12035−12040. (46) Toberer, E. S.; Zevalkink, A.; Snyder, G. J. Phonon engineering through crystal chemistry. J. Mater. Chem. 2011, 21, 15843−15852. (47) Anderson, O. L. A simplified method for calculating the Debye temperature from elastic constants. J. Phys. Chem. Solids 1963, 24, 909−917. (48) Zevalkink, A.; Toberer, E. S.; Zeier, W. G.; Flage-Larsen, E.; Snyder, G. J. Ca3AlSb3: an inexpensive, non-toxic thermoelectric material for waste heat recovery. Energy Environ. Sci. 2011, 4, 510−518. (49) May, A. F.; Snyder, G. J. Introduction to Modeling Thermoelectric Transport at High Temperatures. In Thermoelectrics Handbook: Thermoelectrics and Its Energy Harvesting; Rowe, D. M., Ed.; CRC Press: Boca Raton, FL, 2012. (50) Bo, S.-H.; Wang, Y.; Kim, J. C.; Richards, W. D.; Ceder, G. Computational and Experimental Investigations of Na-Ion Conduction in Cubic Na3PSe4. Chem. Mater. 2016, 28, 252−258. (51) Pompe, C. Strukturchemie und elektrische Leitfähigkeiten von Natriumchalkogenometallaten. Dissertation, Universität Regensburg, Regensburg, 2016. (52) Hayashi, A.; Noi, K.; Tanibata, N.; Nagao, M.; Tatsumisago, M. High sodium ion conductivity of glass−ceramic electrolytes with cubic Na3PS4. J. Power Sources 2014, 258, 420−423. (53) Yu, Z.; Shang, S.; Seo, J.; Wang, D.; Luo, X.; Huang, Q.; Chen, S.; Lu, J.; Li, X.; Liu, Z. Exceptionally High Ionic Conductivity in Na3P0.62As0.38S4 with Improved Moisture Stability for Solid-State Sodium-Ion Batteries. Adv. Mater. 2017, 29, 1605561. (54) Shang, S.-L.; Yu, Z.; Wang, Y.; Wang, D.; Liu, Z.-K. Origin of Outstanding Phase and Moisture Stability in a Na 3 P 1−x As x S 4 Superionic Conductor. ACS Appl. Mater. Interfaces 2017, 9, 16261− 16269. (55) Tanibata, N.; Noi, K.; Hayashi, A.; Kitamura, N.; Idemoto, Y.; Tatsumisago, M. X-ray Crystal Structure Analysis of Sodium-Ion Conductivity in 94 Na3PS4· 6 Na4SiS4 Glass-Ceramic Electrolytes. ChemElectroChem 2014, 1, 1130−1132. 8868


Article


graphite intercalation compound. Phys. Rev. B: Condens. Matter Mater. Phys. 2008, 78, 214303. (77) Legrain, F.; Carrete, J.; van Roekeghem, A.; Curtarolo, S.; Mingo, N. How the Chemical Composition Alone Can Predict Vibrational Free Energies and Entropies of Solids. Chem. Mater. 2017, 29, 6220−6227.

(56) de Klerk, N. J. J.; Wagemaker, M. Diffusion mechanism of the sodium-ion solid electrolyte Na3PS4 and potential improvements of halogen doping. Chem. Mater. 2016, 28, 3122−3130. (57) Rao, R. P.; Chen, H.; Wong, L. L.; Adams, S. Na3+xMxP1‑xS4 (M= Ge4+, Ti4+, Sn4+) Enables High Rate All-Solid-State Na-ion Batteries Na2+2δFe2‑δ(SO4)3| Na3+xMxP1‑xS4| Na2Ti3O7. J. Mater. Chem. A 2017, 5, 3377−3388. (58) Shannon, R. D. Revised effective ionic radii and systematic studies of interatomic distances in halides and chalcogenides. Acta Crystallogr., Sect. A: Cryst. Phys., Diffr., Theor. Gen. Crystallogr. 1976, 32, 751−767. (59) Dietrich, C.; Sadowski, M.; Sicolo, S.; Weber, D. A.; Sedlmaier, S. J.; Weldert, K. S.; Indris, S.; Albe, K.; Janek, J.; Zeier, W. G. Local Structural Investigations, Defect Formation, and Ionic Conductivity of the Lithium Ionic Conductor Li4P2S6. Chem. Mater. 2016, 28, 8764− 8773. (60) Black, D. R.; Windover, D.; Henins, A.; Filliben, J.; Cline, J. P. Certification of standard reference material 660B. Powder Diffr. 2011, 26, 155−158. (61) Wenzel, S.; Leichtweiss, T.; Weber, D. A.; Sann, J.; Zeier, W. G.; Janek, J. Interfacial reactivity benchmarking of the sodium ion conductors Na3PS4 and sodium β-alumina for protected sodium metal anodes and sodium all-solid-state batteries. ACS Appl. Mater. Interfaces 2016, 8, 28216−28224. (62) Chu, I.-H.; Kompella, C. S.; Nguyen, H.; Zhu, Z.; Hy, S.; Deng, Z.; Meng, Y. S.; Ong, S. P. Room-Temperature All-solid-state Rechargeable Sodium-ion Batteries with a Cl − doped Na 3 PS 4 Superionic Conductor. Sci. Rep. 2016, 6, 33733. (63) Irvine, J. T. S.; Sinclair, D. C.; West, A. R. Electroceramics: characterization by impedance spectroscopy. Adv. Mater. 1990, 2, 132−138. (64) Zhu, Z.; Chu, I.-H.; Deng, Z.; Ong, S. P. Role of Na+ interstitials and dopants in enhancing the Na+ conductivity of the cubic Na3PS4 superionic conductor. Chem. Mater. 2015, 27, 8318−8325. (65) Yu, C.; Ganapathy, S.; de Klerk, N. J. J.; van Eck, E. R. H.; Wagemaker, M. Na-ion dynamics in tetragonal and cubic Na3PS4, a Na-ion conductor for solid state Na-ion batteries. J. Mater. Chem. A 2016, 4, 15095−15105. (66) Meyer, W. v.; Neldel, H. Ü ber die Beziehungen zwischen der Energiekonstanten und der Mengenkonstanten a in der Leitwerts Temperaturformel bei oxydischen Halbleitern. Z. Tech. Phys. 1937, 18, 588. (67) Ngai, K. L. Meyer−Neldel rule and anti Meyer−Neldel rule of ionic conductivity: Conclusions from the coupling model. Solid State Ionics 1998, 105, 231−235. (68) Nowick, A. S.; Lee, W. K.; Jain, H. Survey and interpretation of pre-exponentials of conductivity. Solid State Ionics 1988, 28, 89−94. (69) Knödler, D.; Pendzig, P.; Dieterich, W. Ion dynamics in structurally disordered materials: effects of random Coulombic traps. Solid State Ionics 1996, 86, 29−39. (70) Kreuer, K.-D.; Kohler, H.; Maier, J. In High Conductivity Ionic Conductors: Recent Trends and Applications; Takahashi, T., Ed.; World Scientific, 1989; pp 242−279. (71) Baranowski, L. L.; Heveran, C. M.; Ferguson, V. L.; Stoldt, C. R. Multi-Scale Mechanical Behavior of the Li3PS4 Solid-Phase Electrolyte. ACS Appl. Mater. Interfaces 2016, 8, 29573−29579. (72) Sakuda, A.; Hayashi, A.; Takigawa, Y.; Higashi, K.; Tatsumisago, M. Evaluation of elastic modulus of Li2S−P2S5 glassy solid electrolyte by ultrasonic sound velocity measurement and compression test. J. Ceram. Soc. Jpn. 2013, 121, 946−949. (73) Sakuda, A.; Hayashi, A.; Tatsumisago, M. Sulfide solid electrolyte with favorable mechanical property for all-solid-state lithium battery. Sci. Rep. 2013, 3, 2261. (74) Tilley, R. J. D. Defects in Solids; John Wiley & Sons, 2008. (75) Vineyard, G. H. Frequency factors and isotope effects in solid state rate processes. J. Phys. Chem. Solids 1957, 3, 121−127. (76) Toyoura, K.; Koyama, Y.; Kuwabara, A.; Oba, F.; Tanaka, I. First-principles approach to chemical diffusion of lithium atoms in a 8869


Influence of Lattice Dynamics on Na+ Transport in the Solid Electrolyte

Recommend Documents