SUPPLEMENTARY INFORMATION

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1 Design principles for solid-state lithium superionic conductors Yan Wang 1, William Davidson Richards 1, Shyue Ping Ong 1,2, Lincoln J. Miara 3, Jae Chul Kim 1, Yifei Mo 1,4 and Gerbrand Ceder 1,5,6 * 1 Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA 2 Department of NanoEngineering, University of California, San Diego, La Jolla, California 92093, USA 3 Samsung Advanced Institute of Technology-USA, 1 Cambridge Center, Suite 702, Cambridge, Massachusetts 02142, USA 4 Department of Materials Science and Engineering, University of Maryland, College Park, Maryland 20742, USA 5 Department of Materials Science and Engineering, University of California, Berkeley, California 94720, USA 6 Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA * gceder@berkeley.edu NATURE MATERIALS 1

2 Figure S1. A flowchart of the structural matching algorithm for Li 10 GeP 2 S 12 with its anion sublattice closely matches to a bcc lattice. The algorithm finds the supercell and affine transformation M mapping the input exact bcc lattice onto the lattice of the Li 10 GeP 2 S 12 that minimizes the root-mean-square (rms) distance from the S atoms in the transformed (and slightly distorted) bcc-like supercell to the corresponding S atoms in the Li 10 GeP 2 S 12. Only affine transformations preserving bcc supercell lattice angles to within 3 degrees, and supercell lattice vector lengths to within 5% are considered in the matching, and the maximum allowed rms is set to be 0.3(V/n) 1/3 1.0 (Å) for the mapping, where V is the volume of Li 10 GeP 2 S 12 and n is the number of S atoms in Li 10 GeP 2 S 12. The conventional unit-cell parameters (a, b, c, α, β, and γ) of the transformed lattice are the results for the structural matching. 2 NATURE MATERIALS

3 SUPPLEMENTARY INFORMATION Table S1. The structural matching results for Li 10 GeP 2 S 12, Li 7 P 3 S 11, Li 2 S, Li 4 GeS 4 and γ- Li 3 PS 4 (low temperature phase, space group Pmn2 1 ). The structures are obtained from the Inorganic Crystal Structure Database (ICSD) 1. a, b, c, α, β, and γ are the conventional unit-cell parameters of the transformed lattice. R is the rms distance between the sulfur sublattice of each structure and the transformed lattice. Materials Anion lattice type a (Å) b (Å) c (Å) α ( ) β ( ) γ ( ) R (Å) Li 10 GeP 2 S 12 bcc Li 7 P 3 S 11 bcc Li 2 S fcc Li 4 GeS 4 hcp γ-li 3 PS 4 hcp NATURE MATERIALS 3

4 Figure S2. Histogram of volume per sulfur atom for materials in the ICSD containing lithium and sulfur but no other anion species (N, O, Se, F, Cl, Br, I) or hydrogen. The data set from ICSD we use is cleaned by removing duplicate structures using an affine mapping algorithm 2. 4 NATURE MATERIALS

5 SUPPLEMENTARY INFORMATION Figure S3. An octahedrally coordinated Li (green atom) in the bcc sulfur lattice (yellow atoms). This site is found to be unstable over the entire range of considered volumes. NATURE MATERIALS 5

6 Table S2. The calculated energy difference between Li occupancies in an octahedral site (E oct ) and a tetrahedral site (E tet ) in the bcc sulfur lattice. The octahedral site is found to be unstable over the entire range of considered volumes. Volume per S (Å 3 ) E oct -E tet (ev) NATURE MATERIALS

7 SUPPLEMENTARY INFORMATION Figure S4. Calculated minimal energy paths for Li-ion migration in the bcc sulfur lattice at different volumes. There is no activation barrier when V 62.1 Å 3. NATURE MATERIALS 7

8 Figure S5. Calculated minimal energy paths for Li-ion migration in the fcc sulfur lattice at different volumes. We note that at the volume of 34.0 Å, the octahedral and tetrahedral site energies are very close, but the activation energy is still large as compared to bcc at the same volume due to a smaller three-coordinated channel size (the face-sharing triangle between tetrahedral and octahedral sites) than that of bcc (the distorted triangle between two tetrahedral sites). 8 NATURE MATERIALS

9 SUPPLEMENTARY INFORMATION a b c Figure S6. Calculated minimal energy paths with types T-O-T (a), T-T (b) and O-O (c) for Li-ion migration in the hcp sulfur lattice at different volumes. NATURE MATERIALS 9

10 Figure S7. Arrhenius plot of simulated Li-ion diffusivities for Li 7 P 3 S 11 from ab initio molecular dynamics. The activation energy is estimated to be about 0.19 ev. 10 NATURE MATERIALS

SUPPLEMENTARY INFORMATION a b Figure S8. a, Histogram of volume per O atom for materials in the ICSD containing lithium and oxygen but no other anion species (N, S, Se, F, Cl, Br, I) or hydrogen.

11 SUPPLEMENTARY INFORMATION a b Figure S8. a, Histogram of volume per O atom for materials in the ICSD containing lithium and oxygen but no other anion species (N, S, Se, F, Cl, Br, I) or hydrogen. The data set from ICSD we use is cleaned by removing duplicate structures using an affine mapping algorithm 2. b, Activation barrier obtained for the Li-ion migration paths in the bcc and fcc O 2- sublattices as a function of volume. Solid lines are guides to the eye. NATURE MATERIALS 11

12 a b Figure S9. a, Histogram of volume per Br atom for materials in the ICSD containing lithium and bromine but no other anion species (N, O, S, Se, F, Cl, I) or hydrogen. The data set from ICSD we use is cleaned by removing duplicate structures using an affine mapping algorithm 2. b, Activation barrier obtained for the Li-ion migration paths in the bcc and fcc Br - sublattices as a function of volume. Solid lines are guides to the eye. 12 NATURE MATERIALS

13 SUPPLEMENTARY INFORMATION a b Figure S10. a, Histogram of volume per S atom for materials in the ICSD containing Na and S but no other anion species (N, O, Se, F, Cl, Br, I) or hydrogen. The data set from ICSD we use is cleaned by removing duplicate structures using an affine mapping algorithm 2. b, Activation barrier obtained for the Na-ion migration paths in the bcc and fcc S 2- sublattices as a function of volume. Solid lines are guides to the eye. NATURE MATERIALS 13

14 a b Figure S11. a, Histogram of volume per S atom for materials in the ICSD containing Mg and S but no other anion species (N, O, Se, F, Cl, Br, I) or hydrogen. The data set from ICSD we use is cleaned by removing duplicate structures using an affine mapping algorithm 2. b, Activation barrier obtained for the Mg-ion migration paths in the bcc and fcc S 2- sublattices as a function of volume. Solid lines are guides to the eye. 14 NATURE MATERIALS

15 SUPPLEMENTARY INFORMATION a b Figure S12. Li sites energies in Li 10 MP 2 S 10 (M = Si, Ge, Sn). a, Ground-state structure of Li 10 MP 2 S 10. The arrangement of Li, M and P atoms are determined using the ground state structure obtained from density functional theory calculations. The Li sites are classified according to the newly determined crystal structure of Li 10 GeP 2 S 12 (ref. 3). Li2-a and Li2-b (Li4-a and Li4-b) sites are different due to the ordering of M/P in the structure. b, Li site energies in Li 10 MP 2 S 10. The site energy is defined as EE!"#$ = EE!"!#$ EE!"#"$#%, where EE!"!#$ is the total energy of the pristine unit-cell and EE!"#"$#% is the total energy of the unit-cell containing a Li vacancy at the site. The reference energy (0 ev) is set to be the average site energy of all Li sites. It is clear that for Li 10 GeP 2 S 10 the site energy differences between Li1 and Li3 (determine the c-axis Li-ion transport), Li1 and Li2- a/li4-b (determine the ab-plane Li-ion transport) are smaller than those of Li 10 SnP 2 S 10 but larger than those of Li 10 SiP 2 S 10. NATURE MATERIALS 15

16 Figure S13. Calculated DFT charge densities (upper panels) and 2D sections (lower panels) for an fcc S 2- lattice with a single Li + ion (left panels), and a Mg-doped Li 2 S compound with explicit cations (right panels). Li, S and Mg atoms are colored as green, yellow and orange, respectively. Both calculations use a volume of 46.6 Å 3 per S atom, same as the Li 2 S structure in the ICSD. In the calculation of the S 2- lattice, the charge is compensated by a uniform positive charge background to correct the image interaction. In the upper panels, same charge isosurfaces (0.025 e /aa!!, aa! is Bohr radius) are plotted. Very similar charge distributions are obtained for Li 2 S with explicit cations and the S 2- lattice without cations. 16 NATURE MATERIALS

17 SUPPLEMENTARY INFORMATION Figure S14. Li-ion migration minimal energy pathways in (a) a fcc S 2- lattice with a single Li + ion; (b) a Mg-doped Li 2 S compound with explicit cations. Li, S and Mg atoms are colored as green, yellow and orange, respectively. Both calculations use a volume of 46.6 Å 3 per S atom, same as the Li 2 S structure in the ICSD. The Li-ion migration path is marked as connected green balls in each structure. (c) The calculated Li-ion activation energy along the pathways. Only the migrating Li atom is allowed to relax while the other atoms are fixed. Similar barriers are obtained in the Li 2 S with explicit cations and in the S 2- lattice without other cations. The Li-ion migration barrier in the Li 2 S with explicit cations is slightly larger because of the electrostatic repulsion between the migrating Li + and the other cations. NATURE MATERIALS 17

18 Figure S15. 2D sections of calculated DFT charge densities (units: e /aa!!, aa! is Bohr radius) along Li-ion diffusion channels for Li 10 GeP 2 S 10 with explicit cations (left panels) and its S 2- lattice without cations. In the S 2- lattice, the charge is compensated by a uniform positive charge background to correct the image interaction. Li, S, Ge and P atoms are colored as green, yellow, blue and purple, respectively. Very similar charge distributions are obtained near the Li-ion diffusion channels for Li 10 GeP 2 S 10 with explicit cations and for the S 2- lattice without cations. 18 NATURE MATERIALS

19 SUPPLEMENTARY INFORMATION References 1. Inorganic Crystal Structure Database Ong, S. P. et al. Python Materials Genomics (pymatgen): A robust, open-source python library for materials analysis. Comput. Mater. Sci. 68, (2013). 3. Kuhn, A., Köhler, J. & Lotsch, B. V. Single-crystal X-ray structure analysis of the superionic conductor Li 10 GeP 2 S 12. Phys. Chem. Chem. Phys. 15, (2013). NATURE MATERIALS 19

SUPPORTING INFORMATION. Phase stability, electrochemical stability and ionic. P, and X = O, S or Se) family of superionic.

SUPPORTING INFORMATION. Phase stability, electrochemical stability and ionic. P, and X = O, S or Se) family of superionic. SUPPORTING INFORMATION Phase stability, electrochemical stability and ionic conductivity in the Li 10±1 MP 2 X 12 (M = Ge, Si, Sn, Al or P, and X = O, S or Se) family of superionic conductors Shyue Ping