Structural Insights and 3D Diffusion Pathways ... - ACS Publications

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Structural insights and 3D diffusion pathways within the lithium superionic conductor Li GePS 10

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Dominik A. Weber, Anatoliy Senyshyn, Kai S. Weldert, Sebastian Wenzel, Wenbo Zhang, Rene Kaiser, Stefan Berendts, Jürgen Janek, and Wolfgang G. Zeier Chem. Mater., Just Accepted Manuscript • DOI: 10.1021/acs.chemmater.6b02424 • Publication Date (Web): 02 Aug 2016 Downloaded from http://pubs.acs.org on August 4, 2016

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Chemistry of Materials

Structural insights and 3D diffusion pathways within the lithium superionic conductor Li10GeP2S12 Dominik A. Webera, Anatoliy Senyshynb, Kai S. Welderta, Sebastian Wenzela, Wenbo Zhanga, René Kaisera, Stefan Berendtsc, Jürgen Janeka* and Wolfgang G. Zeiera* a

Physikalisch-Chemisches Institut, Justus-Liebig-Universität Gießen, Heinrich-Buff-Ring 10-14, 35392 Gießen, Germany

b

Heinz Maier-Leibnitz Zentrum, Technische Universität München, 85748 Garching, Germany

c

Institut für Chemie, Technische Universität Berlin, Straße des 17. Juni 135, 10623 Berlin, Germany

Abstract Inspired by the ongoing debate about the ion dynamics in the lithium superionic conductor Li10GeP2S12 (LGPS), we present neutron powder diffraction data in combination with analyses of differential bond valence and nuclear density maps to elucidate the underlying diffusion pathways in Li10GeP2S12. LGPS exhibits quasiisotropic three-dimensional lithium diffusion pathways, which is a combination of one-dimensional diffusion channels crossing two diffusion planes. Furthermore, ultrasonic speeds of sound measurements are used to understand the lattice dynamics and obtain the Debye temperature of LGPS. Temperature dependent X-ray diffraction is performed in order to understand the local temperature-dependent behavior of the prevalent structural backbone, as well as the thermal stability of the material. At elevated temperatures, the superionic conducting Li10GeP2S12 phase partially decomposes into Li4P2S6, explaining the deterioration of the ionic conductivity upon heating.

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1. Introduction Since the first report on Li10GeP2S12 (LGPS) as lithium superionic conductor,1 the excitement for a future using all-solid-state batteries, replacing today’s liquid electrolyte batteries, has grown.2 Many subsequent reports on Li10GeP2S12 describe the modification of the material in order to increase the conductivity for its potential use as a solid electrolyte in solid-state batteries (SSB). This includes the isoelectronic substitution of Ge with Sn or Si,3–7 as well as the alteration of the lithium amount within the general sum formula Li10+xM1+xP2−xS12 (M = Si, Sn, Ge).6,8–11 While the class of thiophosphate superionic conductors has shown or is predicted to exhibit instabilities against a Li metal anode,7,12–17 a new electrolyte composition, Li9.54Si1.74P1.44S11.7Cl0.3, in the Li10GeP2S12 structure type has attracted interest and seems to lead to good performance in an all-solid-state battery.18 However, despite the advances in controlling the ionic conductivity and even finding new compositions within this structure type, there are still conflicting reports on the underlying ion dynamics. As shown for other electrolytes such as lithium garnets,19,20 it is necessary to understand the structural chemistry as well as the structural effects on the diffusion pathways in order to optimize the ionic transport. Figure 1 shows a structural representation of Li10GeP2S12. The structure consists of an immobile framework that includes (Ge/P)S4 and PS4 tetrahedra as well as a potentially immobile octahedral LiS6 complex. The Li-S bond has a reasonable average bond length of 2.65 Å. Therefore, the LiS6 octahedron (or Li2 site) has been considered an integral part of the framework. The LiS6 octahedra share edges with the (Ge/P)S4 tetrahedra forming chains in the direction that are bridged along by PS4 tetrahedra. On the one hand, the transport of lithium ions and the high conductivity of LGPS has been attributed to a one-dimensional diffusion channel employing two Li positions (Li1 and Li3, respectively) along the direction, in between the chains formed by (Ge/P)S4 tetrahedra (Fig. 1b).1,6,8–11 It is still widely believed that this purely one-dimensional channel transport is the only mechanism operating at room temperature, and that possible three-dimensional diffusion only occurs at higher temperatures.9 On the other hand, theoretical calculations and nuclear magnetic resonance investigations suggest an additional in-plane diffusion mechanism already at room temperature.7,17,21 This additional diffusion mechanism should effectively


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circumvent blocking of the lithium diffusion within the tunnels by zero-dimensional defects (point defects), which is an intrinsic problem in one-dimensional ion conductors.

a)

b)

PS4 Li(2)S6 M/PS4

b c a

b a

Li1

Li3

Li4

Figure 1: Crystal structure of Li10GeP2S12 in a polyhedral representation, obtained via Rietveld refinement against neutron diffraction data. a) Rigid structural framework of chains of (Ge/P)S4 tetrahedra and (Li2)S6 octahedra which are sharing common edges. The PS4 tetrahedra bridge these chains in the direction. b) Polyhedral representation of the structural framework along the direction. The one-dimensional diffusion pathway along the direction is indicated using the corresponding populated Wyckoff sites.

Inspired by the conflicting reports on the nature of the lithium diffusion in Li10GeP2S12, we investigated the diffusion pathways using neutron powder diffraction in combination with a differential bond valence analysis as well as an analysis of the nuclear density maps, reconstructed by the maximum entropy method. Furthermore, temperature-dependent X-ray diffraction was carried out to determine thermal stability and structural parameters providing more insight into the underlying structural chemistry and bonding situation in Li10GeP2S12. In addition, speed of sound measurements provide a Debye temperature of 181 K, showing that classical phonon behavior can be expected at room temperature.

2. Experimental Section Synthesis. All preparations and sample treatments were carried out under argon



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(Praxair, 5.0) atmosphere. Lithium sulfide (Li2S, Sigma Aldrich, 99.98 %), phosphorus pentasulfide (P2S5, Sigma Aldrich, 99 %), sulfur (S8, Sigma Aldrich, > 99.95 %) and germanium sulfide (GeS, Sigma Aldrich, 99.99 %) were mixed in the stoichiometric ratio. A 5 wt% sulfur excess was added to compensate the loss of sulfur at higher temperatures. The precursors were ball milled (Fritsch Pulverisette 7 premium line) dry in batches of 2 g for 22 h at 500 rpm using a ZrO2 milling set (45 mL bowl and 60 g of balls with a diameter of 3 mm) with intermediate cooling times (every 10 min for 15 min) to prevent excessive heating of the sample. The obtained mixture was then filled into a quartz ampoule (15 mm inner diameter and 10-12 cm in length) which was sealed under vacuum with an oxyhydrogen torch and subsequently heated in a chamber furnace (Nabertherm) to 500 °C within 18.5 h, annealed for 36 h, and then cooled to room temperature. The powder was then milled for 10 min and pressed into pellets (10 mm diameter) for the electrochemical measurements and kept as powder for the diffraction studies. Electrical conductivity measurements. Gold electrodes were vapor deposited onto the pellet, and the contacted pellet was afterwards sealed in a pouch cell. Conductivity measurements in the temperature range from −40 °C to 100 °C (climate chamber, Weiss Klimatechnik, one hour equilibration time) were carried out using a Biologic SP300 impedance analyzer in the range of 7 MHz to 100 mHz with a sinus amplitude of 20 mV. For highly conductive samples, the frequency range of the employed setup is insufficient to resolve the complete spectra and the high frequency region (>1 MHz region) is unreliable. Therefore, the data points in the high frequency region were removed from the fit of the impedance spectra. Furthermore, data measured above 100 °C were also deemed unreliable due to a decomposition process, as described below. Fitting procedure was conducted using the RelaxIS software package (rhd instruments, Version 3). X-ray powder diffraction. Temperature-dependent structural investigations were carried out by means of X-ray powder diffraction, using a STOE Stadi P powder diffractometer with Mo Kα1 radiation (λ = 0.709320 Å, Ge(111)-monochromator) in Debye-Scherrer geometry and an imaging plate detector (active detector area between 2θ = −31.983° and 113.008° with a step size of 0.03°). Li10GeP2S12 powder was enclosed in a SiO2 capillary (1.0 mm in diameter) and sealed by epoxy resin in the argon atmosphere of the glove box. The capillary rotated during the measurements to


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avoid any effects caused by preferred orientation of crystallites. Diffraction data were recorded for 30 minutes each in a temperature range from 25 °C to 600 °C with heating steps of 25 K and a rate of 10 K/min. At each temperature step, an equilibration period of 5 minutes was applied prior to the measurement. After the last temperature step, the sample was cooled to room temperature at a rate of 10 K/min and another diffraction pattern was recorded. For structural refinements, the program package FullProf Suite (version February 2016)22 was used in cyclic mode. Peak profiles in the range of 6° ≤ 2θ ≤ 40° were fitted with a pseudo Voigt function, the background was described by linear interpolation between a set of manual points with refinable heights. As a starting point for the refinements, the structural model published by S. Adams and R. Prasada Rao was used.23 During the refinements, atomic displacement parameters of all Li atoms were fixed to Biso = 2 Å2 and the atomic positions were not allowed to vary. For all other atoms, these parameters were refined, and only the strongly deviating thermal parameters of the sulfur atoms were constrained to each other. Neutron powder diffraction. High-resolution neutron powder diffraction data collection on the Li10GeP2S12 sample was performed in Debye-Scherrer geometry at Heinz Maier-Leibnitz Zentrum (research reactor FRM II, Garching b. München, Germany) on the high-resolution diffractometer SPODI.24 Monochromatic neutrons (λ = 1.54832 Å) were obtained from the thermal neutron beam at a 155° take-off angle using the 551 reflection of a vertically focused composite Ge monochromator of 200 mm height. The vertical position-sensitive multidetector (300 mm vertical sensitivity range at 1.117 m sample-to-detector distance) consisting of 80 3He tubes and covering an angular range of 160° 2θ was used for data collection. The Li10GeP2S12 sample (approx. 2 cm3 in volume) was filled into a thin-wall (0.15 mm) vanadium can of 10 mm in diameter under argon atmosphere and then metalsealed using indium wire. The vanadium container was then mounted on a capillary spinner enabling sample rotation and, thus, minimizing effects of preferred crystallite orientations. Two-dimensional powder diffraction data of the continuously rotated sample were collected and corrected for geometrical aberrations and detector nonlinearities.25 The Rietveld and crystal structure independent (Le Bail) refinements were carried out using the software package FullProf.22 The peak profile shape was described by a pseudo-Voigt function using the modified Thomson-Cox-Hastings



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setting. The instrumental resolution was determined using Na2Ca3Al2F14 as reference material. The background of the diffraction pattern was fitted using a linear interpolation between selected data points in non-overlapping regions. The scale factor, lattice parameter, fractional coordinates of atomic sites and their isotropic displacement parameters, zero angular shift, profile shape parameters and half-width (Caglioti) parameters were varied during the refinement. Due to the high neutron absorption coefficient of 6Li, the standard correction for cylindrical samples with µR = 1.436 as been performed. In recent years, the crystal structure of Li10GeP2S12 and related solid

solutions was studied intensively to establish a relationship between the structure and the Li-ion conductivity.1,6–11,17,23 Among various reports, most advanced information on the crystal structure of Li10GeP2S12 was gained by means of force-field simulations,23 X-ray single crystal diffraction26 and neutron scattering.9,26 The reported results have been found to be quite consistent, i.e. Li10GeP2S12 crystallizes in its own structure type with the space group P42/nmc. The crystal structure of Li10GeP2S12 consists of negatively charged isolated [PS4]3- and [GeS4]4- units with lithium located at interstitial positions. Sulfur atoms in Li10GeP2S12 are distributed over three Wyckoff 8g sites with .m. symmetry. Germanium has been found sharing the 4d site with phosphorus, whereas the 2b site is occupied by phosphorus only. The description of the lithium subsystem is slightly controversial: Initially Kamaya et al.1 reported three lithium sites on the Wyckoff positions 16h, 4d and 8f. However, theoretical calculations by Adams and Prasada Rao23 revealed an additional lithium position at the 4c site, which was further confirmed by X-ray single crystal and neutron powder diffraction.9,23 Adams and Prasada Rao23 reported their results on the Li10GeP2S12 crystal structure on the basis of force-field molecular dynamics simulations in a conventional setting for the P42/nmc space group, whereas other authors23 performing Rietveld refinement, have chosen origin choice 1 in their models. Since symmetry choice 2 is commonly accepted as standard in the International Tables for Crystallography, this setting has been used for the modeling of the crystal structure presented here. Bond valence and maximum entropy method analyses. The analysis of the lithium coordination environment is a crude approximation for understanding diffusion, but it is often performed as a first step for the determination of probable lithium diffusion


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pathways in a material. There are several other methods to predict the preferred ion diffusion pathways with higher reliability on the basis of the underlying crystal structure. For example, the differential bond valence model became of significant relevance for prediction of conduction pathways in recent years, primarily due to its simplicity, robustness and high predictive power.27,28The bond valence modeling is a simple tool, which gives a necessary first approximation for the mechanism of lithium diffusion.9,23 It is assumed that the ion transport from one Li equilibrium site to the other one follows a pathway, along which the valence mismatch ΔV = |V - Vnom| between the bond valence V and the nominal valence Vnom remains as small as possible.29 The total bond valence sum V of cation A can be expressed as: 𝑉 =

𝑆!!! ,

Eq. 1

!

in which the individual bond valences 𝑆!!! are calculated using the bond lengths 𝑅!!! between atom A and the adjacent anions X: 𝑆!!! = exp

𝑅 − 𝑅!!! 𝑏

Eq. 2

with the tabulated parameters R and b. Using the bond valence method, it is possible to identify energetically favorable transport pathways and visualize diffusion using a bond valence mismatch landscape.28 Due to the unique features of thermal neutrons, neutron diffraction provides a number of advantages for studies of complex lithium-containing systems compared to X-ray based techniques. Analysis of the periodic distribution of nuclear densities in the target material often yields experimental input in order to distinguish preferable and possible diffusion pathways. Single crystals of sufficient size and quality are usually not available; hence, neutron powder diffraction is applied. However, termination effects in Fourier maps, caused by powder averaging and limited data statistics, often seriously limit the direct analysis of diffusion pathways in disordered systems. Therefore, the determination of electron/nuclear density maps from the “limited” powder diffraction datasets by the maximum-entropy method (MEM) is often applied as an alternative.30 The method is based on the estimation of 3D scattering densities from a limited amount of information by maximizing information entropy under restraints, consistent with experimental observations. Termination effects often occur to be less pronounced in a MEM evaluation,31 leading to more



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precise scattering density maps. Lithium with its natural isotope composition possesses a negative scattering length (bLi = −1.9 fm). Therefore, negative nuclear density maps can be used to understand the diffusion pathway landscape. Reconstruction of the negative nuclear density maps using the maximum entropy method (MEM)30 was performed using the program PRIMA.32 Ultrasonic speed of sound measurements. Speed of sound measurements were performed on consolidated discs using an Epoch 600 (Olympus) and 5 MHz transducers for longitudinal and transverse speeds of sound, respectively. 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 (< 100 nm) of gold in order to prevent side reactions of the couplant and the sample. All measurements were performed under Argon. The measurement uncertainty from the speed of sound data mostly results from an uncertainty in the thickness. As the measurement uncertainty for the thickness of the pellet is much higher than that of the gold layer, the thin layer of Au is not expected to add significantly to the measurement uncertainty. In addition, no damping of the signal is expected due to the high crystallinity and high speed of sound in Au. While in typical sound measurements an uncertainty of 2 % can be achieved,33 here we assume an uncertainty of ~ 5 % due to the extreme mechanical softness of the pellets.

3. Results and Discussion Electrical conductivity measurements. The inset in Figure 2 shows the measured impedance response of the synthesized Li10GeP2S12 pellet at −40 °C. An equivalent circuit consisting of a constant phase element (CPE) in series with two parallel CPE/resistor components was used to fit the impedance data at lower temperatures. Two semi-circles (parallel CPE/resistor) were necessary in order to obtain a reasonable fit. However, the bulk and grain boundary contributions could not be fully and reliably de-convoluted and therefore the herein reported conductivities correspond to the overall sample conductivity. Due to the fast conducting nature of the material, no semi-circles result at high temperatures. Therefore, the impedance response is fitted using only one constant phase element to account for the gold electrodes. This leads to an uncertainty in the obtained ionic conductivity, and the data above 100 °C have to be treated with care. The partial electronic conductivity of


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Li10GeP2S12 at room temperature is 9 nS/cm (electronic transference number of 3.6·10−7), determined by using a DC polarization method with Au electrodes as

recently reported by our group.13 An activation energy of Ea = 0.35 ± 0.02 eV for the ionic conduction is determined from an Arrhenius plot. The measured total room temperature ionic conductivity corresponds to 5 mS/cm. All measured data are in good agreement with values from the literature,1,6,11 demonstrating the successful synthesis of Li10GeP2S12.

350

300

-Im(Z) / Ω

6 4

0

0

5·103

Re(Z) / Ω

0

Li10GeP2S12, EA = 0.35 eV

−4 2.5

3

10,000

T=- 40°C, 233 K data fit

5·103

2

−2

b)

250

0 −10

8,000

−20

6,000

−30 4,000

−40

2,000

3.5

4

1000T-1 / K-1

4.5

0 101

−50

102

103

104

105

106

Phase angle φ (Z) / °

Temperature T / K 400 8

|Z| / Ω

a) ln(σT / Scm-1K)

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−60

Frequency ω/ Hz

Figure 2: a) Arrhenius plot for the total lithium ion conductivity of Li10GeP2S12 from −40 °C to 100 °C, exhibiting an activation barrier of 0.35 eV. The inset shows a representative impedance response at −40 °C, using Au electrodes. The blue line shows the equivalent circuit fit consisting of a constant phase element in series with two parallel CPE/resistor components. Selected data points, corresponding to the frequencies of 4.3 · 104 Hz and 900 Hz, respectively, are shown in red. b) Bode plot of the impedance |Z| and phase angle φ against the frequency ω for the corresponding impedance response at −40 °C.

Neutron powder diffraction structure analysis. The obtained neutron diffraction intensities are found to be consistent with the space group P42/nmc and the resulting lattice parameters are in fair agreement with literature values. Selected graphical results of the Rietveld refinement are displayed in Figure 3. The structural model by Adams and Prasada Rao23 was used as the starting model: Lattice parameters, atomic coordinates, isotropic displacement parameters and lithium occupations were refined and the best fit was obtained with the parameters listed in Table 1. The observed large displacement parameters for all atomic sites in Li10GeP2S12 may point to an underlying atomic disorder. The total lithium content in LixGeP2S12 was constrained to x = 10, while lithium occupations over the four lithium sites were allowed to vary. The



lithium occupation at the Li2 site with 4d symmetry systematically showed occupation values g > 1.0, which may correspond to site disorder. A symmetry reduction from 4d to 8g at the Li2 site was performed and resulted in a minor improvement of the fit residuals, a reduction of the isotropic displacement parameters and a site splitting of ca. 1.45(3) Å. In principle, the Li sites (except the Li2 site) in Li10GeP2S12 exhibit a slight lithium disorder, e.g. Li1 shows a splitting of 0.25(3) Å, Li3 ~ 0.10(2) Å and Li4 ~ 0.05(4) Å.

95

100

105

λ = 1.54832 Å Observed Calculated Difference

Intensity / a.u.

Intensity / a.u.

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2Θ / °

20

40

60

80

100

120

140

Diffraction angle 2Θ / ° Figure 3: Neutron powder diffraction data and results of the Rietveld refinement for Li10GeP2S12 at ambient temperature. Experimental data are shown as points, the red line denotes the calculated pattern and the difference profile is shown in blue underneath. Calculated positions of Bragg reflections are shown by green vertical tick marks. Obtained fit profile residuals are given in Table 1. The inset shows the goodness-of-fit for small d-spacing. Table 1: Refined structural parameters (atomic coordinates, Biso and occupancy) for Li10GeP2S12 at room temperature, obtained from Rietveld refinements of neutron powder data. The space group is P42/nmc (No. 137, setting 2). Obtained lattice parameters a = 8.7088(6) Å, c = 12.605(1) Å. Values in parentheses represent statistical errors on the last significant digit. Atomic coordinates

Wyckoff Atom

Biso / Å2

g(occ.)

Site

x/a

y/a

z/c

Li1

16h

0.498(2)

0.013(2)

0.4469(13)

8.3(8)

0.50(1)

Li2

8g

0.25

0.226(7)

0.195(2)

9.1(9)

0.54(1)

Li3

8f

0.496(2)

−0.004(2)

0.25

9.0(9)

0.62(3)


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Li4

4c

0.25

0.75

0.499(2)

7.0(9)

0.68(3)

Ge1\P1

4d

0.25

0.25

0.9422(3)

3.4(8)

0.5/0.5

P2

2b

0.25

0.25

0.75

4.1(2)

1.0

S1

8g

0.25

0.9392(7)

0.6600(5)

3.6(3)

1.0

S2

8g

0.25

0.0452(7)

0.3493(6)

3.6(3)

1.0

S3

8g

0.25

0.4446(8)

0.0395(5)

3.9(4)

1.0

Fit residuals (Rp, Rwp, Rexp, χ2): 1.73, 2.19, 2.11, 1.08

Determination of Li diffusion pathways. Analysis of the first neighboring (below 3.4 Å) lithium-lithium distances identified possible lithium diffusion either along the direction or the direction of the tetragonal lattice. An illustration of the underlying lithium framework in Li10GeP2S12 is shown in Figure 4. Lithium diffusion in the direction involves Li3 and two disordered Li1 sites and can be written as: –Li3–[Li1–Li1]–Li3–. Neighboring chains along are interconnected by lithium bonds along in different ways involving blocks formed by –Li3–Li2– Li3–Li2– or –Li4–Li1–Li4–Li1–. Horizontal planes can be connected either by infinite chains along the c-direction or by a Li1–Li3 bridge. b)

a) z=1 z = 3/4 z = 1/2 z=

c a

1/ 4

b a

z=0 Li1

Li2

Li3

Li4

Figure 4: Lithium framework in Li10GeP2S12 as determined from neutron powder diffraction. Different lithium sites are marked by color, and interatomic distances are presented using a bicolor scheme. For clarity, the refined atomic split sites for Li3 and Li4 are not shown. The thin black lines denote the tetragonal lattice. Li1-Li3 and Li4-Li1 sites exhibit the smallest LiLi distances. The z-coordinates are shown for an easier comparison with Figures 5 and 6.



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Using R and b parameters tabulated by Adams,29 with the differential bond valence model, the 3D distribution of the valence mismatch for Li10GeP2S12 was calculated. Calculations were performed by summing up bond valence contributions to sulfur anions up to a cut-off distance of 5.5 Å for any point within 3D grid. The framework formed by germanium, phosphorus, and sulfur in Li10GeP2S12 was used as an input.

a)

x=1

x = 3/4

b)

z = 3/4

z = 2/3

z = 1/4

x = 1/2

x = 1/4

x=0

z=1

z = 5/6

z = 1/2

z = 1/3

z = 1/6

z=0

ΔV-1, (v.u.)-1 10 8 6 4 2 0

Figure 5: Differential bond valence iso-surfaces (green) for Li in Li10GeP2S12 (ΔV = 0.10 v.u., 0.1 Å resolution) in two different orientations. The valence mismatch ΔV is shown as the reciprocal of the valence units. Black spheres denote lithium location. Slices of obtained differential bond valence iso-surfaces at different locations, x and z, in the unit cell depict the probable lithium distribution along a) the direction and b) along . A lithium pathway along a) the direction and b) additional pathways along can be identified.

Slices of the obtained differential bond valence iso-surfaces, corresponding to the probable lithium distribution in Li10GeP2S12, are shown in Figure 5. The channels created by connecting the minimum-valence-mismatch indicate a complex threedimensional character of the lithium conductivity in Li10GeP2S12. Lithium diffusion along the direction (Fig. 1 and 4) is most obvious. However, components along


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and appear to also be present and cannot be excluded. In addition, this differential bond-valence analysis predicts lithium clustering (Fig. 5a, x/a = 1/4, 3/4) having an octahedral shape and being formed by Li3 in the apical positions and Li4 in the vertices. Similar to a Hirschfeld surface analysis, the differential bond valence method usually provides an overview of the possible diffusion pathways28 that are permitted from the geometrical point of view. However, the difference bond valence method cannot be used for the prediction of activation energies. In addition, one has to keep in mind that predicted diffusion pathways can be variously populated and, therefore, the most prominent diffusion pathways have to be identified. Employing the maximum entropy method to analyze the collected neutron diffraction data provides negative nuclear density maps that can be used to understand the diffusion pathway landscape. The corresponding files for the iso-surface models can be found in the Supporting Information. The negative nuclear density maps for Li10GeP2S12 reconstructed from experimental structural factors (Tab. 1) are shown in Figure 6 along with slices at various positions within the unit cell. x=1

a)

x = 3/4

z=0

b)

z = 1/2

z = 2/3

x = 1/2

z = 1/6

z = 3/4

x = 1/4

z = 1/4

z = 5/6

x=0

z = 1/3

z=1 b(x103), fm/Å3 0 -3 -6 -9 -12 -15

Figure 6: Maximum entropy method (MEM) reconstructed negative nuclear density maps in Li10GeP2S12 (surface threshold −0.015 fm/Å3, cell grid 256 x 256 x 512) and slices in a) (001) and b) (110) planes, x and z, respectively. Green spheres denote lithium locations. a) shows the diffusion tunnels within Li10GeP2S12 along the direction, whereas the Li distribution along can be seen in b).



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The MEM analysis of lithium nuclear densities confirms the three-dimensional character of lithium diffusion in Li10GeP2S12 at room temperature, in which lithium diffusion is observed along (i.e. at z = 0, 1/4, 1/2, 3/4; Fig. 6b) besides the already confirmed lithium diffusion along the direction. In contrast to the results obtained using the differential bond valence method (Fig. 5), no continuous chain of negative neutron scattering can be drawn at z = 1/6, 1/3, 2/3 and 5/6, respectively. The chosen threshold indicates an essentially lower probability for lithium diffusion along at these z coordinates. The negative nuclear density distribution

clearly

shows

diffusion

along

the

Li3−[Li1−Li1]−Li3,

Li3−[Li2−Li2]−Li3 and Li4−[Li1−Li1]−Li4 pathways. Hereby, crossing of these pathways suggests three-dimensional diffusion of lithium ions in Li10GeP2S12 already at room temperature. The (Li2)S6 octahedral as well as the Li4 interstitial position are involved in this three-dimensional diffusion. A nuclear magnetic resonance investigation21 has recently shown that the Li-ion migration mechanism indeed involves the Li4 site, but it was not possible to confirm diffusion involving the Li2 site. However, determination of lithium diffusion pathways in Li10GeP2S12 seems to be very challenging via nuclear magnetic resonance spectroscopy, as sometimes even the three-dimensional diffusion cannot be detected.34 The obtained neutron diffraction data in combination with a maximum entropy analysis unambiguously support the relevance of the Li3−[Li2−Li2]−Li3 pathway ( direction; z = 1/4, 3/4) for lithium diffusion as well. These results are well in accordance with theoretical predictions7,23 involving these two lithium sites and reflect the partial occupancies (Table 1) of the lithium positions.23 Determination of activation barriers for Li diffusion. A more accurate identification of lithium diffusion pathways in Li10GeP2S12 requires their reconsideration in the frame of probabilities of lithium location. Assuming the normalized nuclear density map to have the feature of a probability field, lithium ion motion can be analyzed in the framework of Boltzmann statistics.35 One-particle potentials (OPP) for the different lithium diffusion pathways were recalculated from negative nuclear densities and are shown in Figure 7. Three most prominent pathways for lithium diffusion, namely Li3−[Li1−Li1]−Li3, Li3−[Li2−Li2]−Li3 and Li4−[Li1−Li1]−Li4, are identified to occur in Li10GeP2S12. Assuming the activation energy to be proportional to the hopping rate,36 the most


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probable path for lithium diffusion is the one along the direction of the tetragonal lattice. This pathway is characterized by two different energy barriers of approximately 0.04 eV and 0.11 eV, which correspond to the [Li1−Li1] disorder and the diffusion from the Li1 to the Li3 site, respectively. Along , two pathways may be identified, i.e. Li1−Li4 in the direction and the Li3–Li2 jump resulting in a lithium transport along [110]. For a better visualization of the corresponding lithium framework, see Figure 4. Lithium transport along the [110] direction will occur through a disordered [Li1–Li1] site and then a Li1−Li4 chain characterized by an energy barrier of approximately 0.13 eV. In contrast to the [Li1−Li1] disorder, the energy barrier characterizing the [Li2−Li2] split site has been found marginally small, resulting in a 0.12 eV energy barrier at the Li2−Li3 section as a factor determining the probability of lithium hopping in the [110] direction. It has to be noted that a large magnitude of structural disorder in Li10GeP2S12 may enable a direct Li1−Li1 and Li3−Li3 hopping along between two adjacent lithium chains, but the overall probability for this process is certainly lower than along the three Li3−[Li1−Li1]−Li3, Li3−[Li2−Li2]−Li3 and Li4−[Li1−Li1]−Li4 pathways.

a)

b)

0.04 eV

b

0.11 eV

a

c

c)

d)

OPP, eV

0.13 eV

0.04 eV 0.12 eV

OPP, eV

OPP, eV

Figure 7: a) 3D lithium distribution nuclear density maps for different sections, (100), (010), (110) and (110) within the unit cell. b)−d) Selected slices with Miller indices (100), (110) and



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Page 16 of 28

(1𝟏𝟎) and the corresponding lithium one-particle-potential (OPP) recalculated from negative nuclear

densities

within

1D

slices

along

lines

connecting

Li3−[Li1−Li1]−Li3,

Li3−[Li2−Li2]−Li3 and Li4−[Li1−Li1]−Li4.

These different diffusion pathways have theoretically been proposed in the recent literature and activation energies for the processes have been determined using density functional theory methods and solid-state nuclear magnetic resonance spectroscopy. The one-dimensional conduction path has been predicted to have activation energies of 0.17 eV17 and 0.19 eV23. The in-plane diffusion is governed by an activation barrier of 0.28 eV17 and 0.30 eV23. From solid-state nuclear magnetic resonance spectroscopy, the activation barriers for the diffusion along the channels and in-plane were determined to be 0.16 eV and 0.26 eV, respectively.21 While the qualitative comparison of activation energies using the OPP method is valid, the obtained activation barriers are not quantitative. First, in the OPP approach the ion is treated individually as an Einstein oscillator within the classical limit to Boltzmann statistics. Second, while the maximum OPP corresponds to the migration activation energy, any additional defect formation energies are neglected.37 However, the obtained results indicate that lithium conductivity in Li10GeP2S12 at room temperature is three-dimensional and that the total diffusion pathway consists of three different segments. Small differences in the obtained activation energy barriers for different directions indicate a quasi-isotropic character of 3D lithium conductivity in Li10GeP2S12 at ambient temperature. Lattice dynamics of Li10GeP2S12. A prerequisite for using the one-particle potential approach is that the atomic vibrations and the ion hopping behavior must be governed by Boltzmann statistics, and that correlation is negligible. In other words, the lattice dynamics of an ion conductor are very important for the motion of mobile ions,38 and knowledge of material parameters such as the Debye temperature is paramount. Recently, ultrasonic speeds of sound measurements have been used to obtain the elastic properties of various thiophosphates39,40 and oxides, such as garnets.41–43 In these reports, the focus has been laid on obtaining Young’s modulus, proving the mechanical softness of the thiophosphates and the stiffness of the oxides, respectively. In addition to the mechanical properties, the Debye temperatures of solids can be obtained by speed of sound measurements:44–47


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!/!

𝑣! ℏ 6𝜋 ! 𝜃! = 𝑘! 𝑉

Eq. 3

,

with the Debye temperature 𝜃! , the Boltzmann constant kB, Planck´s constant ħ and the average volume per atom V. The mean speed of sound, vm, can be calculated via: ! 𝑣! =

𝑣!!!

3 , + 2𝑣!!!

Eq. 4

using the measured longitudinal and transverse speeds of sound, vl and vt, respectively. In addition to the elastic properties47 and the Debye temperature, the Grüneisen parameter γ can be calculated using the Poisson ratio vp:48 𝑣 1 − 2 𝑣! 3 1 + 𝑣! ! 𝛾= ∙ , 𝑣! = 𝑣 2 2 − 3𝑣! 2−2 ! 𝑣!

!

Eq. 5

! .

The Grüneisen parameter reflects the lattice anharmonicity and hence the phonons in the crystal lattice that deviate from harmonic oscillations.49 Measuring these properties directly gives information about the lattice dynamics and the temperature at which one can expect classical behavior of the lattice vibrations. The obtained parameters are given in Table 2. In addition, speed of sound measurements can be used to estimate the minimum thermal conductivity of the material, 𝜅!"# , which may be useful for understanding thermal conductivity and how fast heat in an all-solidstate battery may be removed:33,44 𝜅!"#

1 𝜋 = 2 6

! !

!

𝑘! 𝑉 !! 2𝑣! + 𝑣! .

Eq. 6

Table 2: Average parameters obtained via speed of sound measurements of two sample pellets and the corresponding standard deviation. Longitudinal speed of sound vl, transverse speed of sound vt, mean speed of sound vm, Debye temperature ΘD, Poisson ratio vp, Grüneisen parameter γ, Young’s modulus E, shear modulus G, bulk modulus K and minimum lattice thermal conductivity 𝜅!"# . 𝑣! /

𝑣! /

𝑣! /

𝜃! /

𝑚𝑠 !!

𝑚𝑠 !!

𝑚𝑠 !!

𝐾

2360±70

1480±80

1630±90

181±10

𝑣!

0.18

𝛾

1.20

𝐸 /

𝐺 /

𝐾 /

𝜅!"# /

𝐺𝑃𝑎

𝐺𝑃𝑎

𝐺𝑃𝑎

𝑊𝑚 !! 𝐾 !!

10.5±0.9

4.5±0.5

5.4±0.1

0.41

The obtained speeds of sound and corresponding Debye temperatures for Li10GeP2S12



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are very low compared to other crystalline materials.49 This may either be a consequence of the long and soft metal-sulfur bond,49 but also the fast ion motion of lithium may play a role.44 While the obtained elastic parameters are smaller than the ones calculated for Li10GeP2S12,50 they are within the range of experimentally obtained data for other Li-S-P compounds (15-25 GPa)40 and an order of magnitude smaller than for the mechanically stiff Li conducting garnets.41 Li10GeP2S12 exhibits an average Debye temperature below room temperature and any motion of the lattice can therefore well be described within the Boltzmann statistical approach. The Grüneisen parameter is slightly larger than what can be expected for a purely tetrahedrally coordinated compound,45,49 however, the values are not too large and the lattice vibrations at room temperature may be regarded as fairly harmonic. Thermal expansion behavior. The occurrence of a change in the ionic conductivity at temperatures above 200 °C has been recently reported.9 It appears to be correlated with a change of the ionic conductivity mechanism and, correspondingly, a different activation energy. According to Kwon et al., the observed phase transition may also lead to a change of the lithium diffusion pathway from the low temperature 1D to the high-temperature 3D diffusion.9 However, the results presented here using the maximum entropy analysis of neutron diffraction data, as well as former theoretical predictions and results from nuclear magnetic resonance spectroscopy,7,21,23 show that Li10GeP2S12 is already a three-dimensional lithium ion conductor at room temperature. Therefore, the question of the possibly different nature of ion dynamics at higher temperatures remains to be understood. In order to investigate the behavior of the lattice at higher temperatures, temperature dependent X-ray diffraction was performed up to 600 °C.


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Page 19 of 28

Figure 8: Temperature dependent X-ray diffraction patterns of Li10GeP2S12 in a small angular window, showing the thermal expansion of the unit cell (MoKα1 radiation).

Li4P2S6

25°C

Intensity / a.u.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


600°C

300°C

25°C

6

8

10

12

14

16

Diffraction angle 2Θ (MoKα1) / ° Figure 9: Selected angular window of the temperature dependent X-ray diffraction patterns of Li10GeP2S12, corresponding to the heating profile 25 °C – 300 °C – 600 °C – 25°C. Calculated Bragg reflections for Li10GeP2S12 are shown as green vertical tick marks. During the heating



1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

process, Li4P2S6 evolves as an impurity phase (grey area).

Figure 8 shows the obtained Bragg intensities with temperature, suggesting a seemingly regular linear thermal expansion of the lattice. However, two distinct features are found when analyzing the different diffraction patterns independently. Figure 9 shows representative diffraction patterns at four selected temperatures during the heating process and after cooling. While Li10GeP2S12 is phase pure at room temperature, the heating process up to 600°C results in formation of the secondary phase Li4P2S6. Analysis of the diffraction patterns suggest formation of this impurity phase between 300°C and 400°C, coinciding with the reported diffusive phase transition in Li10GeP2S12.9 Due to the thermodynamic instability of Li10GeP2S12,7 formation of Li4P2S6 is already possible at lower temperatures, however, a phase fraction in the order of a few percent needs to be present in order to detect it in the diffraction data. Seemingly, more Li4P2S6 is formed when Li10GeP2S12 decomposes with increasing temperature. Unfortunately, due to overlap of the reflections, a quantitative analysis via Rietveld refinement is not possible. At this stage, we cannot be sure of the decomposition reaction, as other side phases should be present that are not detected using X-ray diffraction. The impurity phase Li4P2S6 itself is a lithium ionic conductor as well, with reported conductivities of 2.3·10−6 S/cm at 100°C and similar activation energies compared to Li10GeP2S12 of 0.29 eV.51 Therefore, it is possible that the formation of the impurity phase with its much lower conductivity severely affects the overall measurable conductivity of the resulting composite sample. As decomposition reactions are heterogeneous in character, they will primarily occur at grain boundaries, and the poor conducting products may block ion transfer across the grain boundaries. A corresponding behavior has been observed in the chemically analogue compound Li7P3S11 in which Li4P2S6 forms above 300°C severely deteriorating the determined ionic conductivities.52,53 For a better understanding of the behavior of the crystal structure at higher temperatures, the temperature dependent lattice parameters are shown in Figure 10. The lattice parameter a increases linearly with temperature and a linear thermal expansion coefficient αL300K αL300K =

Δa

·

1

ΔT a300K

of 3.4·10−5 K−1 is found. However, the

lattice parameter c shows a highly anisotropic thermal expansion behavior above 300 °C that becomes more pronounced with increasing temperature. This anisotropic


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expansion leads to a much faster increase of the lattice in the direction compared to the direction. As recently shown in garnet-type solid electrolytes,20 expanding unit cells and the correspondingly expanding polyhedra affect the lithium ion conductivities in solids. Figure 11 shows the temperature dependent polyhedral volumes of the structural backbone obtained from Rietveld refinement against the X-ray diffraction patterns. Within the uncertainty of the refined atomic positions, the (Ge/P)S4 and PS4 tetrahedra seem to retain their volume with increasing temperature. In contrast, the LiS6 polyhedron volume (Li2 position) increases with increasing temperature. These trends can be easily understood in terms of bond strengths, which may be correlated to bond length in tetrahedral and octahedral compounds.49 In other words, the P-S and Ge-S bonds are much stronger than a Li-S bond since the Li-S bond is much longer. Therefore, the thermal expansion of the lattice drives the Li-S much faster apart than the more stiff Ge-S and P-S bonds. Considering the above elucidated three-dimensional diffusion pathways, a faster expansion of the lattice in the direction should broaden the diffusion pathways along , potentially increasing the Li-ion mobility in this plane faster than along .54 This seems reasonable, as the Li2 polyhedral volumes are indeed increasing, corresponding to a broader diffusion pathway. This behavior seems to be reversible, as the lattice parameters before heating are a = 8.792(2) Å and c = 12.722(4) Å and after the cool down a = 8.793(3) Å and c = 12.742(4) Å. A similar behavior of the anisotropic thermal expansion has recently been found in β-Li3PS4, influencing the diffusion pathways.54 However, more in-depth studies are necessary in order to confirm this hypothesis as the detrimental formation of the impurity phase Li4P2S6 overrules the potentially beneficial effects of an anisotropically expanding lattice in the ionic conductivity measurements at higher temperatures. In order to deconvolute these effects, elemental substitutions may be promising, if synthetically possible, which may alter the thermal expansion behavior and artificially increase the in-plane diffusion pathways.



a/Å

a)

b a

b)

PS4 Li(2)S6

c/Å

M/PS4

c b a

Temperature ϑ / °C

Figure 10: Thermal expansion of the lattice parameters a) a and b) c of the tetragonal unit cell. While the lattice parameter a exhibits a linear thermal expansion coefficient, the lattice parameter c shows strong anisotropic behavior above 400 °C. For better visualization of the occurring anisotropic thermal expansion, the unit cell and structural backbone of Li10GeP2S12 are shown in the inset. The broken lines serve as a guide to the eye with a constant linear thermal expansion coefficient αL300K of 3.4·10−5 K−1.

b)

a)

27

6

26

5

25

4 (Ge/P)S4 PS4

3 0

Volume Li(2)S6 / Å3

Volume (Ge/P)S4 / Å3

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 22 of 28

24 200

400

Temperature °C Temperature ϑϑ // °C

600

0

200

400

600

Temperature °C Temperature ϑ / °C

Figure 11: Temperature dependent volumes of a) the PS4, (Ge/P)S4 and b) Li(2)S6 polyhedra in the structural backbone of Li10GeP2S12 obtained from temperature dependent X-ray powder diffraction. While the tetrahedral volumes are almost constant with temperature, the thermal expansion leads to an increase of the Li2-S bond length and concurrently an increase in polyhedral volume of the Li2 site.


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4. Conclusions We have probed the lithium diffusion pathways in the fast ionic conductor Li10GeP2S12 using neutron powder diffraction bond valence and maximum entropy analysis. As recently suggested theoretically, a three-dimensional diffusion mechanism involving all lithium sites is already prevalent at room temperature. Based on the similarity of obtained activation energy barriers for different segments of 3D diffusion pathways, anisotropic bulk lithium conductivity in Li10GeP2S12 is suggested. Further, we used speed of sound measurements to gain more insights into the lattice dynamics of Li10GeP2S12 and obtained parameters such as the material’s Debye temperature, which is necessary to better understand ionic diffusion in solids. With a Debye temperature of ~180 K, the lattice dynamics of Li10GeP2S12 follow the classical Boltzmann statistics already at room temperature. In addition, we show that the thermodynamic instability of the material leads to formation of the impurity phase Li4P2S6 at higher temperatures, which lowers the obtainable ionic conductivity. The tetragonal lattice exhibits an anisotropic thermal expansion behavior, showing the high flexibility of the structure, which may ultimately affect the ionic conductivity in this material, if targeted substitution increases the in-plane diffusion pathways.

ASSOCIATED CONTENT

Supporting Information Files for the iso-surface model are included here. This material is available free of charge via the Internet at http://pubs.acs.org. AUTHOR INFORMATION

Corresponding Authors *[email protected]; *[email protected] Notes

The authors declare no competing financial interests. ACKNOWLEDGMENT

The authors acknowledge financial support by BASF SE within the International Network for Electrochemistry and Batteries. W.G.Z furthermore gratefully acknowledges the financial support through start-up funding provided by the Justus-



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Liebig-University Giessen.

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TOC graphic

Chain and in-plane Li+ diffusion in Li10GeP2S12


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