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SUPPLEMENTARY INFORMATIONDOI: 10.1038/NMAT4369

NATURE MATERIALS | www.nature.com/naturematerials 1 S-‐1

Supplementary Information

Design Principles for Solid-state Lithium Superionic

Conductors

Yan Wang1, William Davidson Richards1, Shyue Ping Ong1,2,

Lincoln J. Miara3, Jae Chul Kim1, Yifei Mo1,4 and Gerbrand Ceder1,5,6*

1Department of Materials Science and Engineering, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

2Department of NanoEngineering, University of California, San Diego, La Jolla, California 92093, USA

3Samsung Advanced Institute of Technology-USA, 1 Cambridge Center, Suite 702, Cambridge, Massachusetts 02142, USA

4Department of Materials Science and Engineering, University of Maryland, College Park, Maryland 20742, USA

5Department of Materials Science and Engineering, University of California, Berkeley, California 94720, USA

6Materials Sciences Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA

*E-mail: [email protected]

Design principles for solid-state lithium superionic conductors

© 2015 Macmillan Publishers Limited. All rights reserved

http://dx.doi.org/10.1038/nmat4369

2 NATURE MATERIALS | www.nature.com/naturematerials

SUPPLEMENTARY INFORMATION DOI: 10.1038/NMAT4369

S-‐2

Figure S1. A flowchart of the structural matching algorithm for Li10GeP2S12 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

Li10GeP2S12 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

Li10GeP2S12. 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 Li10GeP2S12 and n is the number of S atoms in

Li10GeP2S12. The conventional unit-cell parameters (a, b, c, α, β, and γ) of the

transformed lattice are the results for the structural matching.

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Table S1. The structural matching results for Li10GeP2S12, Li7P3S11, Li2S, Li4GeS4 and γ-

Li3PS4 (low temperature phase, space group Pmn21). 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)

Li10GeP2S12 bcc 4.35 4.35 4.20 90.0 90.0 90.0 0.58

Li7P3S11 bcc 4.19 4.29 4.20 90.2 87.4 90.7 0.82

Li2S fcc 5.76 5.76 5.76 90.0 90.0 90.0 0.0

Li4GeS4 hcp 4.01 4.01 6.15 90.0 90.0 122.15 0.27

γ-Li3PS4 hcp 3.80 3.80 6.14 90.0 90.0 118.9 0.32



NATURE MATERIALS | www.nature.com/naturematerials 3


S-‐2

Figure S1. A flowchart of the structural matching algorithm for Li10GeP2S12 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

Li10GeP2S12 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

Li10GeP2S12. 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 Li10GeP2S12 and n is the number of S atoms in

Li10GeP2S12. The conventional unit-cell parameters (a, b, c, α, β, and γ) of the

transformed lattice are the results for the structural matching.

S-‐3

Table S1. The structural matching results for Li10GeP2S12, Li7P3S11, Li2S, Li4GeS4 and γ-

Li3PS4 (low temperature phase, space group Pmn21). 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)

Li10GeP2S12 bcc 4.35 4.35 4.20 90.0 90.0 90.0 0.58

Li7P3S11 bcc 4.19 4.29 4.20 90.2 87.4 90.7 0.82

Li2S fcc 5.76 5.76 5.76 90.0 90.0 90.0 0.0

Li4GeS4 hcp 4.01 4.01 6.15 90.0 90.0 122.15 0.27

γ-Li3PS4 hcp 3.80 3.80 6.14 90.0 90.0 118.9 0.32





S-‐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 algorithm2.

S-‐5

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.





S-‐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


mapping algorithm2.

S-‐5

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.





S-‐6

Table S2. The calculated energy difference between Li occupancies in an octahedral site

(Eoct) and a tetrahedral site (Etet) in the bcc sulfur lattice. The octahedral site is found to

be unstable over the entire range of considered volumes.

Volume per S (Å3) 28.5 34 40.0 46.6 54.0 62.1 70.8

Eoct-Etet (eV) 0.74 0.55 0.40 0.29 0.20 0.14 0.09

S-‐7

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 Å3.





S-‐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).

S-‐9

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.





S-‐10

Figure S7. Arrhenius plot of simulated Li-ion diffusivities for Li7P3S11 from ab initio

molecular dynamics. The activation energy is estimated to be about 0.19 eV.

S-‐11

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


mapping algorithm2. b, Activation barrier obtained for the Li-ion migration paths in the

bcc and fcc O2- sublattices as a function of volume. Solid lines are guides to the eye.





S-‐10

Figure S7. Arrhenius plot of simulated Li-ion diffusivities for Li7P3S11 from ab initio

molecular dynamics. The activation energy is estimated to be about 0.19 eV.

S-‐11

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



bcc and fcc O2- sublattices as a function of volume. Solid lines are guides to the eye.





S-‐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



bcc and fcc Br- sublattices as a function of volume. Solid lines are guides to the eye.

S-‐13

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

algorithm2. b, Activation barrier obtained for the Na-ion migration paths in the bcc and

fcc S2- sublattices as a function of volume. Solid lines are guides to the eye.





S-‐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



bcc and fcc Br- sublattices as a function of volume. Solid lines are guides to the eye.

S-‐13

a

b

Figure S10. a, Histogram of volume per S atom for materials in the ICSD containing Na



algorithm2. b, Activation barrier obtained for the Na-ion migration paths in the bcc and






S-‐14

a

b

Figure S11. a, Histogram of volume per S atom for materials in the ICSD containing Mg



algorithm2. b, Activation barrier obtained for the Mg-ion migration paths in the bcc and


S-‐15

a

b

Figure S12. Li sites energies in Li10MP2S10 (M = Si, Ge, Sn). a, Ground-state structure of

Li10MP2S10. 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 Li10GeP2S12 (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 Li10MP2S10. 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 Li10GeP2S10 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 Li10SnP2S10

but larger than those of Li10SiP2S10.





S-‐16

Figure S13. Calculated DFT charge densities (upper panels) and 2D sections (lower

panels) for an fcc S2- lattice with a single Li+ ion (left panels), and a Mg-doped Li2S

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 Å3 per S atom,

same as the Li2S structure in the ICSD. In the calculation of the S2- 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 Li2S with explicit cations and the S2-

lattice without cations.

S-‐17

Figure S14. Li-ion migration minimal energy pathways in (a) a fcc S2- lattice with a

single Li+ ion; (b) a Mg-doped Li2S 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 Å3 per S atom, same as the Li2S 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 Li2S with explicit cations and in

the S2- lattice without other cations. The Li-ion migration barrier in the Li2S with explicit

cations is slightly larger because of the electrostatic repulsion between the migrating Li+

and the other cations.





S-‐18

Figure S15. 2D sections of calculated DFT charge densities (units: |e|/𝑎𝑎!!, 𝑎𝑎! is Bohr

radius) along Li-ion diffusion channels for Li10GeP2S10 with explicit cations (left panels)

and its S2- lattice without cations. In the S2- 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 Li10GeP2S10 with explicit

cations and for the S2- lattice without cations.

S-‐19

References

1. Inorganic Crystal Structure Database. http://icsd.fiz-karlsruhe.de/icsd/

2. Ong, S. P. et al. Python Materials Genomics (pymatgen): A robust, open-source

python library for materials analysis. Comput. Mater. Sci. 68, 314–319 (2013).

3. Kuhn, A., Köhler, J. & Lotsch, B. V. Single-crystal X-ray structure analysis of the

superionic conductor Li10GeP2S12. Phys. Chem. Chem. Phys. 15, 11620–16622

(2013).



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