Anion charge-lattice volume dependent Li ion migration in compounds with the face-centered cubic anion frameworks

The proper design principles are essential for the efficient development of superionic conductors. However, the existing design principles are mainly proposed from the perspective of crystal structures. In this work, the face-centered cubic (fcc) anion frameworks were creatively constructed to study the effects of anion charge and lattice volume on the stability of lithium ion occupation and lithium ion migration. Both the large negative anion charges and large lattice volumes would increase the relative stabilities of lithium-anion tetrahedron, and make Li ions prefer to occupy the tetrahedral sites. For a tetrahedral Li ion migration to its adjacent tetrahedral site through an octahedral transition state, the smaller the negative anion charge is, the lower the lithium ion migration barrier will be. While for an octahedral Li ion migration to its adjacent octahedral site through a tetrahedral transition state, the larger negative anion charge is, the lower the lithium ion migration barrier will be. New design principles for developing superionic conductors with the fcc anion framework were proposed. Low Li ion migration barriers would be achieved by adjusting the non-lithium elements within the same crystal structure framework to obtain the desired electronegativity difference between the anion element and non-lithium cation element.


Introduction
Safety is the most important concern when using the commercial lithium ion batteries (LIBs) in the application scenarios of the large-scale energy storage, such as electric vehicles. Replacing the currently employed flammable liquid electrolytes in LIBs with the solid-state electrolyte (SSE) materials and collocating with the Li metal anodes to construct the all-solid-state lithium ion batteries (ASSLIBs) not only could solve the battery safety issues, but also remarkably enhance the energy density of battery systems [1] . Correspondingly, the construction of practical ASSLIBs needs SSE materials to achieve Li ion fast conduction with a low activation energy (E a < 300 meV) and a high Li ionic conductivity (10 -3 -10 -2 S cm -1 ) at room temperature. So far, some superionic conductors, such as Li 7 La 3 Zr 2 O 12 [2] , Li 1+x Al x Ti 2−x (PO 4 ) 3 [3] oxides and Li 10 GeP 2 S 12 [4] , Li 7 P 3 S 11 [5] sulfides have been widely studied as the SSE materials, and the state-of-the-art ionic conductivities of 12-17 mS cm -1 at room temperature are experimentally realized in Li 10 GeP 2 S 12 and Li 7 P 3 S 11 sulfides.
To efficiently develop more advanced superionic conductors for ASSLIBs, the better understanding of fast ion migration mechanism in the state-of-the-art superionic conductors and the development of proper design principles are quite essential. Ceder et. al. have proposed an important design principle for superionic conductors that the body-centred cubic (bcc) anion framework with face-sharing lithiumanion tetrahedra allows the low activation energy of Li ion migration [6] , which is successfully guiding the high-throughput screening of new superionic conductors [7] . G. Hautier et. al. found the distorted lithium-sulfur polyhedrons in LiTi 2 (PS 4 ) 3 provide the smooth energy landscape combining small activation barriers with numerous migration paths, and proposed the design concept of "frustrated energy landscape" for superionic conductors [8] . The mobile species with unfavored coordination environments are correlated with high ionic conductivities [9] , which is consistent with the concept of "frustrated energy landscape". Furthermore, the high-throughput screening of fast lithium ion conductors by Xiao et al. shows the activation energies of Li ion migration in the olivine-structures are lower than those of the layered-and even spinel-structures [10] .
However, the existing design principles of the face-sharing lithium-anion tetrahedron and frustrated energy landscape are mainly proposed from the perspective of crystal structure without the considerations of other factors, e.g. binding strength between the migrating Li ion and its adjacent anions, and the polarizability of anion [11] . It is generally accepted that the Columbic force dominates the interaction between Li cation and its adjacent anion framework in ionic materials [12][13][14] . In the spinel LiMn 2 O 4 , the different valence states of Mn ions and their arrangements surrounding Li ions have important effects on the activation barrier of Li diffusions [12] . Our previous study of the chalcopyritestructured LiMS 2 (M are transition metals, from Ti to Ni) materials with the same crystal structure demonstrates that the larger negative anion charges resulted from the larger electronegativity difference between M and S elements would increase the activation barrier for Li ion migration between the two stable tetrahedral sites through an octahedral transition site, namely Tet-Oct-Tet pathway [13] . In addition, Mo et al. constructed an artificial face-centered cubic (fcc) anion sublattice of the monovalent Sin comparison with the bivalent S 2with a constant anion volume, and found activation barrier for Li-ion migration along the Tet-Oct-Tet pathway in the monovalent Ssublattice is smaller than that of the bivalent S 2sublattice [14] . On the contrary, in Li 3 MI 6 (M=Sc, Y and La) compounds with stable octahedral Li occupations, the larger negative I anion charges would lower the activation barrier for Li ion migration along the Oct-Tet-Oct pathways [15] . The questions then become, why anion charge shows the reverse influence on Li migration barrier for the Tet-Oct-Tet and Oct-Tet-Oct pathway? Are there any connections among Li occupation pattern, anion charge and anion volume? Therefore, in this work, we made efforts to further understand the roles of anion charge as well as anion volume on the Li ion occupations and Li ion migrations, and proposed new design principles for developing superionic conductors.

Computational details
This work is based on the DFT calculations performed by using the Vienna ab-initio Simulation Package (VASP) software. The interaction between ion cores and valence electrons described by the projector augmented wave (PAW) method [16] . The generalized gradient approximation (GGA) [17] in the form of Perdew-Burke-Ernzerhof (PBE) exchange functional [18] was used to solve the quantum states of electron. The plane-wave energy cutoff is set to 500 eV. The Monkhorst-Pack method [19] with 1×1×2 k-point mesh is employed for the Brillouin zone sampling of the super lattice. The convergence criterions of energy and force are set to 10 −5 eV/atom and 0.01 eV/Å, respectively. The anion charges of lithium compounds were calculated by using the Atoms in Molecules method (Bader charge analysis) [20] . The energy variations and migration enthalpies of lithium ion migration in the fcc-type anion frameworks with 48 anions ( Figure S4) are calculated by the nudged elastic band (NEB) method [21] . The anion charges are changed by the uniform background charge of the framework system. Only the one migrating Li ion is allowed to relax, while the other anions are fixed in their initial positions, and this method can be also found in Ceder's work [6] .

Results and discussion
The topologies of the close-packed anions of the common lithium ionic conductor materials can be approximately classified into fcc, body-centered cubic (bcc) and hexagonal close-packed (hcp) frameworks [6] . The anion frameworks of LiCoO 2 , Li 2 MnO 3 , Li 4 Ti 5 O 12 , Li 2 S, LiTiS 2 , Li 3 YCl 6 and Li 3 YBr 6 [22] can be exactly matched to fcc frameworks. For Li 7 P 3 S 11 and Li 10 GeP 2 S 12 , the S anion sublattice can be roughly mapped to bcc lattices with some distortions. In both γ-Li 3 PS 4 and Li 4 GeS 4 , the S anion sublattices can be closely matched to hcp arrays [6] . In the aforementioned lithium compounds as well as more than half lithium compounds in the Materials Project (MP) database, Li ions mainly occupy the tetrahedral or octahedral sites, forming the stable tetrahedral or octahedral lithium-anion polyhedrons, as shown in the distribution of lithium coordination environments ( Figure   S1 in Supporting Information). We find that there are pairs of the adjacent anion tetrahedron and proposed. The anion charge and anion volume dependent Li occupation pattern and Li migration in the hcp anion arrangements will be further studied in another work.

Anion charge and anion volume dependent Li occupation and migration
First, we have calculated the Bader charges and occupation volumes of anion for some stable lithium oxides and sulfides from the MP database, to determine the reasonable value ranges of anion charge and volume, as listed in Table S1 and S2 in Supporting Information. Figure S3 shows that the scatter distributions of anion charge and volume of some common lithium oxides and sulfides around the fitted straight lines, approximately demonstrating a positive correlation between anion charge and anion volume. It is worth mentioning that the anion volumes are the volumes of unit cell averaged to every anion rather than the volumes of electron cloud of anion, so anion volumes are not only affected by the radii of anion but also by the radii of non-lithium cation. Then, an artificial fcc-type anion framework with 48 anions and one single Li ion ( Figure S4) was built to simulate the Li ion migration between two adjacent Oct and Tet central sites, as the local structure shown in Figure 1a. This computational strategy can make us directly capture the effect of anion charge and anion volume, which has been successfully used by Ceder et al. [6] Then, the nudged elastic band (NEB) calculations were performed to monitor the energy variations for Li ion migration from an Oct site to its adjacent Tet site with respect to different anion charges and anion volumes, as schematically shown in Figure   1b.    (Table S1 and Figure S3a). In addition, the larger  Figure 3d). This is because the relative energies of LiO 4 are higher than those of LiO 6 (E tet-oct > 0, While the increasing O anion volumes make E m first decrease and then increase when the negative O anion charges are more than -1.2e, which is consistent with the earlier study on the fcc S 2sublattice by Ceder et al [6] . In summary, the larger negative anion charges would deliver a high E m for the tetrahedral Li ion migration along the Tet-Oct-Tet pathways, but a lower E m for the octahedral Li ion migration along the Oct-Tet-Oct pathways.

Model validation
The energy barrier and jump distance for Li ion migration are determined by the total energy landscape of Li. The total energy landscape of Li ion in an ionic solid depends on the electrostatic interaction between Li ion and other ions, which can be further divided into a short-range Li-anion attractive interaction and a longer-range Li-cation repulsive interaction [8] . The Li-anion attractive interaction is modulated by the high-frequency alternations of the stable Li occupation sites separated by energy barriers that Li needs to overcome when squeezing through a small bottleneck to reach the adjacent stable site. While the Li-cation repulsive interactions show much longer modulations on the order of the distance between two cations. The resulting total energy landscapes are mainly set by the Li-anion interactions, so Li ion migration in an ionic compound can be approximatively reduced to Li ion migration in an anion framework model. It is also noted that the Li-cation repulsive interactions also contribute to the total energy landscape to some extents, and the weight of the Li-cation interaction in setting the total energy landscape is set by the arrangements and valance states of cation.
Combining the anion framework model (heat maps in Figure 2) with the calculated Bader charges and volumes of O/S anion (listed in Table S1 and S2) for some lithium oxides and sulfides, the corresponding E a were predicted, as shown in Figure 4 and Additionally, it can be seen from Figure 4 that E m from both model prediction and NEB calculations of Li four-coordinated compounds are relatively smaller than those of Li six-coordinated compounds, as least for the above-mentioned oxides and sulfides, which is consistent with the fact of most superionic conductors showing Li tetrahedral occupations, such as Li 3

PO 4 , Li 3 PS 4 , Li 7 P 3 S 11 and
Li 10 GeP 2 S 12 . Most importantly, beyond the compounds in Table S3, E m of Li migration in other lithium compounds with face-sharing tetrahedron and octahedron can be predicted by our anion framework model, associated with known anion charges and anion volumes. The AFLOW database contains many material compounds with structure and Bader charge information [24] , therefore, it is feasible to screen superionic conductors with low E m by combining our model with the AFLOW database without extra DFT calculations.  Table S3.
The above anion framework model analyses about the change trends of E m with respect to anion charge and anion volume are also confirmed by some reported materials. Our previous work on the chalcopyrite-structured LiMS 2 (M are transition metals, from Ti to Ni) materials with tetrahedral Li occupations shows that the smaller negative S anion charges resulted from the smaller electronegativity difference between transition metal and sulfur element would lead to lower E m for Li ion migration along the Tet-Oct-Tet pathways [13] . Mo et al. found E m for the tetrahedral Li ion migration along the Tet-Oct-Tet pathways in a fcc monovalent Sanion framework is much lower that of the bivalent S 2anion framework with the same anion volume [14] . For the spinel structured LiAlCl 4 , Similar effect can be also found in Li 10 MP 2 S 12 (M = Ge and Sn) materials with tetrahedral Li occupations. The higher electronegativity of Ge vs Sn (2.0 vs 1.7 [25] ) give rise to less electron densities on S anions in Li 10 GeP 2 S 12 , leading to the smaller negative anion charges, and thereby show relatively lower E m compared to Li 10 SnP 2 S 12 [26] , which are in good accordance with the AIMD simulations by S. P. Ong et al [27] . The above reported lithium compounds with tetrahedral Li occupations consistently obey the rule of the smaller negative anion charges leading to higher E m for the tetrahedral Li ion migration, proposed in the foregoing model analyses of Figure 3c. On the other hand, our previous research on the lithium iodides [28] , Li 3 MI 6 (M=Sc, Y and La) with octahedral Li occupations, shows that the largest I anion negative charges of Li 3 LaI 6 resulted from the most active La (Pauling electronegativity χ A , Sc (χ A = 1.36) > Y (χ A = 1.22) > La (χ A = 1.10) [29] ) lead to the lowest phonon DOS center of Li and smallest E m for Li ion migration along the Oct-Tet-Oct pathways, which are also in good agreement with the foregoing model analyses of Figure 3a. We also find E m change of Li ion migration along the Oct-Tet-Oct pathways in the gradually charged Li x CoO 2 [30] , P3m1-Li x TiS 2 [31] and P6 3 /mmc -Na x CoO 2 [32] cathodes match our charge-volume map, that is with more Li or Na extraction from these layered structures, the lattice parameter c as well as anion charge would decrease to some extents [31,33] , making the values of anion charge-volume locate at the more top left portion in the charge-volume heat maps ( Figure 2) and eventually increasing E m . In total, the chargevolume maps (change trends of E m with respect to anion charge and anion volume) of anion framework model are reasonable and creditable, although the predicted absolute values of E m from anion framework model may differ from the NEB data especially for those electrolytes with high Lication repulsive interactions or large anion framework distortions.

New principles for developing superionic conductors with fcc anion frameworks
In a ternary, quaternary and even more polynary alkali metal compounds, the anion charges are usually affected by the electronegativity of the non-alkali metal elements, as confirmed by some previous work [13,26] . The atomic radius and valence electron configuration of the non-alkali metal element determines its coordination environment and the crystal volume, eventually affecting the corresponding anion volumes. The above anion framework model analyses clearly show that anion charge and anion volume significantly affect alkali metal ion occupation and ion migration. It is expected to achieve low E m for alkali metal ion migration by adjusting the non-alkali metal element within the same crystal structure framework. Here, based on the above findings, general principles for developing new ternary ABC type lithium, sodium or even multivalent metal superionic conductors with fcc anion frameworks can be summarized: (i) for the superionic conductors with stable A ion octahedral occupation sites and small C anion volumes, the large electronegativity difference between the anion element C and six-coordinated non-mobile cation element B is essential for achieving excellently fast A ion migration, as shown in Figure 5a, and the corresponding non-mobile cation element B should give preference to the elements located at the left bottom of the periodic table with small electronegativity, as shown in Figure 5b. The chemical compositions of the recent reported two superionic conductors, Li 3 YCl 6 and Li 3 YBr 6 with Li octahedral occupations [22] , are completely in conformity with this octahedron principle; (ii) for the superionic conductor with stable A ion tetrahedral occupation sites and large C anion volumes, the small electronegativity difference between the anion element C and four-coordinated nonmobile cation element B is essential for achieving excellently fast A ion migration, as shown in Figure 5a, and the corresponding non-mobile cation element B should give preference to the elements located at the right top of the periodic table of elements with large electronegativity, which are close to but less than that of C element, as shown in Figure 5b. The chemical compositions of the most superionic conductors with Li tetrahedral occupations, such as Li 3 PS 4 , Li 7 P 3 S 11 and Li 10 GeP 2 S 12 , fit perfectly with this tetrahedron principle. We hope that these two guiding principles will contribute to the design and optimization of superionic conductors.

Conclusions
In summary, the fcc anion framework model show that anion charge and anion volume significantly affect lithium ion occupation and ion migration, which is confirmed by many reported materials. Both the larger negative anion charges and large anion volumes would enhance the relative stabilities of the tetrahedral Li occupation. For tetrahedral Li ion migration along the Tet-Oct-Tet pathways through an Oct transition state, the smaller negative anion charge is, the lower the lithium ion migration barrier is. While for octahedral Li ion migration along the Oct-Tet-Oct pathways through a Tet transition state, the larger negative anion charge is, the lower the lithium ion migration barrier is. Our anion framework model can be used for screening superionic conductors with low E m , along with the structure and Bader charge information in the AFLOW material database. Most importantly, new design principles for developing advanced superionic conductors with fcc anion frameworks were proposed. Adjusting the non-mobile cation element within the same crystal structure framework to obtain the desired electronegativity differences between the anion element and non-mobile cation element, eventually achieving low E m for ion migration.