Alloying effects on superionic conductivity in lithium indium halides for all-solid-state batteries

Alloying of anions is a promising engineering strategy for tuning ionic conductivity in halide-based inorganic solid electrolytes. We explain the alloying effects in Li3InBr6−xClx, in terms of strain, chemistry, and microstructure, using first-principles molecular dynamics simulations and electronic structure analysis. We find that strain and bond chemistry can be tuned through alloying and affect the activation energy and maximum diffusivity coefficient. The similar conductivities of the x = 3 and x = 6 compositions can be understood by assuming that the alloy separates into Br-rich and Cl-rich regions. Phase-separation increases diffusivity at the interface and in the expanded Cl-region, suggesting microstructure effects are critical. Similarities with other halide superionic conductors are highlighted.


047903-2
Zevgolis et al. APL Mater. 6, 047903 (2018) Superionic diffusivity is due to a small activation energy barrier, E a , and a large maximum diffusion coefficient, D 0 , through the Arrhenius expression, D = D 0 e −E a k B T , where k B is Boltzmann's constant and T is the temperature. Our study of Li 3 InBr 6x Cl x shows that it is necessary to computationally screen for new superionic electrolytes using both E a and D 0 . For example, Cl 6 has a smaller E a than Br 3 , but Br 3 has a larger diffusivity at 500 K, due to its larger D 0 .
Børn-Oppenheimer 14 MD were run on Li 3 InBr 6x Cl x using the Quantum ESPRESSO planewave density functional theory (DFT) code. 15 Using the Nernst-Einstein relationship, the diffusion coefficients, D, were calculated. Figure 1 shows example Br 3 supercells with the Li-ions removed and an ordered indium (purple) sublattice. A time step of 20 a.u. (0.97 fs) and wavefunction and charge density cutoffs of 30 Ry and 300 Ry were used after convergence tests were run. Ultrasoft pseudopotentials 16 for Li, In, Cl, and Br were employed with the Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional. 17 The pseudopotentials are Li.pbe-s-van ak.UPF, In.pbe-d-rrkjus.UPF, Cl.pbe-n-van.UPF, and Br.pbe-van mit.UPF.
We independently vary the composition, volume, and microstructure in the MD to isolate the effects of strain and chemistry. The four compositions were simulated as solid solutions with a random placement of Br and Cl in the supercell. For each alloy, the configurational complexity is too great to calculate the full ensemble. Thus, we optimize the geometry of 3-6 reference configurations and simulate the one with the lowest DFT ground state energy. The spread in energies, given in the supplementary material (Table SI. 6), is at most 36 meV/atom, within the magnitude of thermal fluctuations at our simulation temperatures. The volumes for Br 6 (4389 Å 3 ) and Cl 6 (3638 Å 3 ) were optimized, and the Br 3 (4112 Å 3 ) and Br 4 (4262 Å 3 ) computational volumes were scaled according to the experimental trend in volume versus composition (Fig. 6). The alloyed compositions were simulated at three volumes: 4389, 4262, and 4112 Å 3 . The computational lattice vectors for each supercell are given in Table SI.1 of the supplementary material. In addition, a nanophase (NP) separated supercell with a Cl-rich side and Br-rich side was simulated for the Br 3 composition (right cell in Fig. 1).
All compositions were simulated at multiple temperatures between 500 and 900 K to obtain E a and D 0 . MD simulations are equilibrated at temperature and then run for at least 25 ps. Across all compositions and volumes, except Cl 6 , we see the non-Arrhenius behavior, consistent with our previous study on Li 3 InBr 6 . 13 The non-Arrhenius behavior indicates a change in diffusion mechanism and a variable E a . The change in E a does not occur at the same temperature for all compositions but often occurs between 800 K and 900 K (see Fig. SI.1 of the supplementary material). Thus, we report E a between 500 and 800 K in Fig. 2.
Effect of volume and chemistry: The conventional paradigm of ionic conductivity suggests that diffusivity will increase as volume increases, but for the alloyed systems, the effect is much more profound. Diffusivity in Li 3 InBr 6x Cl x does not systematically increase with increasing volume because E a does not systematically decrease. Figure 2(b) shows that Br 3 at the central 4262 Å 3 volume (yellow) has the highest E a of that alloy computation, for example. The trend in D can only be understood by also considering the trends in D 0 . The largest D 0 for the Br 3 and Br 4 alloys occurs at the ∼3% expanded volumes [blue for Br 4 and yellow for Br 3 in Fig. 2(c)]. The far more subtle trend  in E a , which is a convolution of the size of the octahedral local minima and the tetrahedral transition state, is explored in the supplementary material. D 0 characterizes the shape of the local minimum energy wells (curvature), which we probe by considering the effects of volume and chemistry. We posit that large values of D 0 are due to an "ideal" bond length that causes maximum bond frustration, increasing the jump attempt frequency. If Li + only had purely ionic bonds, it would sit in the center of the halide octahedra and bonds would have no strict angular preferences. In contrast, we previously showed that in Li 3 InBr 6 , many Li-Br bonds have polar-covalent character, with a narrower-than expected distribution of angles and shortened bond distances. 13 Polar-covalent bonds were rigorously defined via distance cutoffs and angle cutoffs based on the centers of Maximally Localized Wannier Functions (MLWF). 18 In this work, we adopt the same definition and analysis to classify bonds as containing chiefly polar-covalent or chiefly ionic character. Further details on the selection of cutoff criteria can be found in the supplementary material (Figs. SI.1 and SI.2), as well as in Ref. 13.
Bonds in lithium indium halide exhibit frustration in the inability of Li to form simultaneous polar-covalent bonds with all the halides in the octahedral site, which we previously discovered contributes to fast diffusion in Br 6 . 13 The octahedral site is too large in Br 6 , so each Br only has 1-2 simultaneous polar-covalent bonds. Since we no longer consider Li or the halides to be purely ionic, we stop including the charge superscripts.
The ability of Li to form simultaneous polar-covalent bonds depends on the size of the octahedral site, which can be characterized by the distance r Xc from a halide (X) to the centroid of the site (c) in the relaxed supercell (Fig. 3). The distance from Li to the centroid r Lic depends on the cell volume and the Br/Cl composition; on average, r Lic is nonzero because of the oscillatory motion and a strong polar-covalent interaction, as described in our previous work. 13 We compute r Lic by extracting the Li-halide distance, r LiX , from the Li-X pair correlation function and subtracting r Xc . We report r Lic /r Xc in Fig. 3 and note the increasing trend with volume, contrary to what is expected from purely ionic bonds. At large volumes, Li interacts primarily with one halide, is far from the centroid, and cannot be easily captured in a polar-covalent bond by other halides in the octahedral site, which affects the jump attempt frequency and thus D 0 . We point out that similar behavior was observed in AgBr x I 1x alloys, for which silver-halide distances were maintained across large composition regions while distances to the site centroids increased with decreasing x. 19 The Goldilocks effect is observed. If the Li-centroid is too large, Li cannot form simultaneous polar-covalent bonds. If the Li-centroid is too small, Li can form many simultaneous polar-covalent bonds easily. In these extreme simulations, frustration is reduced. The "just right" octahedral size leads to the largest D 0 values for each alloy: Br 4 with a volume of 4389 Å 3 and Br 3 with a volume of 4262 Å 3 [ Fig. 2(c)]. While Br 3 with a volume of 4389 Å 3 also seems to have an intermediate Li-centroid distance, it is reasonable to expect that the octahedral site is too large using the Br 6 volume with a Br 3 composition; this D 0 has an intermediate value of 6.6 × 10 4 cm 2 /s. The D 0 value of the Br 4 alloy at the small 4112 Å 3 volume could not be easily calculated because the data were severely non-Arrhenius over the range 500-800 K (see the supplementary material), so it is not provided in Fig. 2.
We count the number of polar-covalent bonds that an anion has, on average, for each volume and report the polar-covalent bonds as a percentage of total neighbors in Fig. 4. The general trends in polar-covalent bond character in the alloys are in accord with Fajan's rules, which state that Br-Li bonds are more polar-covalent than Cl-Li bonds. 20 We get essential information about the polar-covalent bond strength by examining the effect of the temperature on polar-covalent bonds for Br 3   bonds are tenuous and increasing the temperature can break the bonds. Figure 4 shows a significant change in the number of polar-covalent between 700 and 900 K for the 4262 Å 3 volume, which has the most frustrated Li-Br bonds. Increasing the temperature between 700 and 800 K breaks many of the polar-covalent bonds and helps explain the high D 0 . In contrast, the small Br 3 cell has a larger percentage of polar-covalent bonds, which decreases significantly only at 900 K. The large Br 3 cell has a small and constant percentage of polar-covalent bonds. A similar steep trend (not shown) occurs in Br 4 with the 4389 Å 3 volume, which has the highest D 0 for that composition. The trends shown in Fig. 4 are robust with respect to changing the cutoffs defining ionic versus polar-covalent bonds, as described in the supplementary material. Highly frustrated bonds affect D 0 and E a , contributing to the non-Arrhenius behavior. While frustration in the octahedral site affects E a , variations in E a are more tied directly to the volume of the tetrahedral site, as discussed in the supplementary material. We find polar-covalent bonds forming at 2.9 Å and 2.65 Å for Li-Br and Li-Cl, respectively, regardless of composition (Figs. SI.1 and SI.2 of the supplementary material). Thus, Li has to be much closer to Cl than Br to form a bond with polar-covalent character. It follows that more Br neighbors lead to increased competition between polar-covalent bonds, which we previously showed to increase the jump rate in Li 3 InBr 6 . 13 At the x = 3 composition, most Li ions have 3 Br neighbors. If we define Π(Br > 3) as the percentage of Li with an excess (>3) of Br neighbors, then the average value of Π ave (Br > 3) = 1.8%. However, if we consider only jumping Li, then Π jump (Br > 3) > Π ave (Br > 3) indicates that excess Br neighbors lead to more jumps. This is seen in Fig. 5 as a function of temperature and volume. For instance, at 500 K and with a 4112 Å 3 volume, the likelihood for having excess Br neighbors grows for jumping Li by 11.5% in absolute terms. As the temperature or volume increases, the effect of excess Br neighbors decreases because the frustration (competition) between these Li-Br bonds diminishes. Indeed, there is building consensus that frustration, the inability for a system with mobile ions to settle in a significantly deep energy minimum, affects superionic conductivity. 9,13,21-23 Frustrated bonds can set up a chain of correlated diffusion events, further increasing conductivity.
Effect of phase separation: Slightly expanded octahedral sites lead to a higher D 0 . However, strain can also be induced by a phase separation of Br-rich and Cl-rich regions, either globally or at the nanoscale. We find that the trend in conductivity with alloying 12 is understood by assuming a nanophase (NP) microstructure for the Br 3 composition, which can be directly simulated (Fig. 1).  Li 3 InBr 6x Cl x does not follow Vegard's Law, 24 which states that the lattice parameter of an alloy is the linear weighted average of the lattice parameters of the parent compounds. Instead there are two trends of volume versus alloying: a Br-rich trend and a Cl-rich trend (Fig. 6). The slope of the Cl-rich region is steep because substituting larger Br anions into the Cl sub-lattice causes the volume to increase significantly. In contrast, substituting a smaller Cl into the Br sub-lattice does not cause as much strain. The abrupt change in slope between the two regimes indicates that the system likely exhibits some phase separating tendency.
The hysteresis in volume upon Cl substitution (red arrows in Fig. 6) after the x = 3 composition 3,12,25 further suggests the presence of some phase separating tendency. Some of the Cl-rich compositions follow the Br-rich volume-trend, while other samples at the same composition follow the Cl-rich trend. Samples with two different volumes indicate different structures. X-ray diffraction further indicates a phase-separation at the x = 3 composition. 12 Accordingly, we simulated Li diffusion within a NP supercell of Br 3 (structure in Fig. 1). Figure 2 shows that at 500 K, the NP at 4262 Å 3 has a higher D than the solid solutions of Br 4 and Br 3 at almost any volume (except for Br 3 at 4389 Å 3 , which is an unphysical ∼6% volume expansion). At 500 K, the NP has an order of magnitude higher diffusion than the solid solution and is 3× larger than Li 3 InBr 4 Cl 2 . Thus, Br 3 with a nanophase microstructure reproduces the experimental trend in conductivity with alloying: Br 6 , Br 3 (NP), Br 4 , and Cl 6 (from largest to smallest). 12 Note that we simulated the NP with a 4112 Å 3 volume at 700 K and it has higher diffusivity than the solid solution with the same volume (Table SI.1 of the supplementary material). Thus, we assume that the NP will also have a higher diffusivity than the solid solution with the small volume at 500 K.
We chose to simulate the NP at the medium volume at multiple temperatures because the volume is closer to the relaxed (zero pressure) Br 3 volume (see the supplementary material). More importantly, we want to identify the effect of the microstructure on frustration, and the medium volume solid solution Br 3 has the highest D 0 .
By calculating the energy of relaxed supercells, we find that the NP has essentially equal energy as the solid solution (Table SI.1 of the supplementary material). Furthermore, the higher diffusivity of the NP microstructure results in a higher cation configurational entropy, which may further stabilize the system. These energy differences support phase separation in the experimental conductivity measurements. As discussed below, the NP material has higher diffusivity at low temperature for two reasons: the expanded volume of the Cl region leads to a higher Li concentration and even higher Li diffusion. Additionally, more Li-ions jump in the interface between the Br and Cl regions.
The expanded NP Cl-region (compared to the Cl 6 volume) enriches the Li concentration compared to the Br region. We calculate that the percentage of Li in the Cl-rich side, Π ave (Cl-rich), is 8%-9% percentage points larger than the Br-rich side for the simulations at different volumes and temperatures (Table SI. theoretical formalism has also been suggested to describe the enrichment of the mobile ion species in terms of space-charge layer formation at a solid interface. 10,11,[26][27][28] Our analyses at varying temperatures show that the expansion and compression of the Cl and Br regions change E a and increase or decrease jump events, respectively. Beyond Li enrichment, Li ions are also more likely to jump in the Cl-rich side; Π jump (Cl-rich) is greater than Π ave (Cl-rich). Figure 5(a) shows Π jump (Cl-rich) Π ave (Cl-rich) > 0 for all simulations (see also Table SI.5 of the supplementary material). More Li ions jump on the Cl-rich side at small volumes (green > yellow) and lower temperatures. The increase in diffusion in the Br region as the temperature increases supports the hypothesis that E a is larger in the Br region.
The interface between the Cl-rich and Br-rich regions (the channel in Fig. 1) is also strained. The effects of variations in volume and bond chemistry can lead to frustration of the Li-Br bonds. In the center of the interface, Li has 3 Br and 3 Cl bonds, which is used to distinguish Li in the interface versus the bulk. The percentage of jumping Li in the interface Π jump (interface) is always higher than the average percentage of Li in the interface, Π ave (interface), as shown in Fig. 7(b). E a in the interface/channel is lower than the bulk, so more jumping occurs at lower temperatures; as temperature increases, the likelihood of jumping in the interface decreases, as expected.
Note that the intermediate volume has the largest Π jump (interface) Π ave (interface), probably due to the ideal volume for frustrated polar-covalent bonds, similar to the solid solution shown in Fig. 5.
In conclusion, we show how bond frustration and phase separation affect diffusivity in halide alloys. Calculation of E a and D 0 across varying Cl/Br compositions and volumes highlights the importance of a low E a and a large D 0 for superionic conductivity. We find that a large D 0 can be understood through bond frustration; the effect of strain and chemistry can be controlled through alloying. The "ideal" Li-Br bond length to maximize frustration and thus the jump attempt frequency occurs in Li 3 InBr 3 Cl 3 at a slightly expanded (3%) volume due to competition between polar-covalent bonds with the Br in the octahedral site. Finally, we show that the experimental trend in the conductivity with alloying can be understood if Li 3 InBr 3 Cl 3 has a phase-separated microstructure rather than a solid solution. Phase separation has the effect of enriching the Li concentration in the Cl-rich region, creating a space charge layer similar to that proposed for other halide superionic conductors. 10,11,[26][27][28] These insights into halide alloying will provide an important key to finding better and faster solid electrolytes.
See supplementary material for additional information on supercell volumes and energies, diffusion coefficients, octahedral and tetrahedral site sizes, polar-covalent character of bonds, nano-phase Li and jump locations, and geometry optimization.