Data for Analysis

Migration data. The energy barriers of elementary paths in 36 model APs (X3AB, where

X = Li, Na, or K, A = O, S, or Se, and B = F, Cl, Br, or I) were adopted from DFT calculations in Chapter 4. The AP lattice has a cation sublattice by vertex-sharing octahedra, thus cation defects can migrate via octahedron edges. APs with a high-symmetry cubic system have one distinct elementary path, whereas those with a lower-symmetry orthorhombic system have 12 paths (i.e., the number of edges). The number of samples is 322 and 281 for vacancy and interstitial dumbbell, respectively. The data is large enough to satisfy the rule of thumb that the number of samples needs to exceed five times of the number of descriptors to avoid the ‘curse of dimensionality’ (i.e., overfitting) and achieve reliable models.263,264 _{Also, this large data contains not only energetically}

preferred (low barrier) but also undesirable (high barrier) paths, which can improve the reliability of our ML model.

Descriptors. We collected 45 descriptors that may describe the ion mobility as listed in Table 5.1. Descriptors can be categorized by ‘lattice’, ‘mechanical’, ‘electronic’, ‘chemical’, or ‘compound species’ properties. The descriptors were collected based on DFT calculations in Chapter 4; available properties were adopted directly from the previous study, otherwise predicted low-energy structures of APs were used to obtain descriptors with methods discussed in the following paragraphs.

The properties of species (i.e., cation, octahedral anion, or framework anion) include ionic radius,187 _{atomic mass,}265 _{Pauling electronegativity,}266,267 _{polarizability, Bader charge, and Born}

effective charge. The Bader charge, which identifies individual atoms and estimates its actual net charge, was predicted using the Bader Charge Analysis code with previous DFT electron density Table 5.1 List of descriptors (abbreviations in parentheses) and its category. ‘Filter’ reduces the redundant descriptors denoted in the table (e.g., Vocta was reduced into Va).

Descriptor Category Filter Descriptor Category Filter

Volume per atom (Va) Lattice Ionic radius (rC) Cation ENC

Octahedron volume (Vocta) Lattice Va Atomic mass (mC) Cation ENC

Mass density (𝜌m) Lattice Electronegativity (ENC) Cation

Packing fraction (PF) Lattice Ion polarizability (𝛼!) Cation ENC

Packing fraction without cation (PFC) Lattice Bader charge (BCC) Cation

Tolerance factor (t) Lattice Born effective charge (BECC) Cation

Channel size (Df) Lattice Ionic radius (rO) Octahedral anion

Coordination number of cation (CNC) Lattice Atomic mass (mO) Octahedral anion

Path distance (CCD) Lattice Electronegativity (ENO) Octahedral anion rO

Cation – octahedral anion distance (COD) Lattice Ion polarizability (𝛼") Octahedral anion rO

Cation – framework anion distance (CFD) Lattice Bader charge (BCO) Octahedral anion

Distance between anion (AAD) Lattice Born effective charge (BECO) Octahedral anion

Path width by the first closest ion (PWc) Lattice Ionic radius (rF) Framework anion

Path width by the second closest ion (PW2c) Lattice Atomic mass (mF) Framework anion

Total path width (PW) Lattice Electronegativity (ENF) Framework anion

Phonon frequency (𝜔LEO) Lattice Ion polarizability (𝛼#) Framework anion

Bulk modulus (B) Mechanical Bader charge (BCF) Framework anion

Band gap (Eg) Electronic Born effective charge (BECF) Framework anion

Electronic dielectric constant (𝜀$) Electronic Ionicity between cation and _{chalcogen anion (IC-Ch)} Chemical Ionic dielectric constant (𝜀%) Electronic Ionicity between cation and _{halogen anion (IC-H)} Chemical Total dielectric constant (𝜀) Electronic 𝜀% Ionicity between anions (ICh-H) Chemical

Polarizability (𝛼.) Electronic Decomposition energy (Ed) Chemical

calculations.211–214 _{Since two cation sites involve in a path (i.e., end points of a path), the Bader}

charge and the Born effective charge of a sample (i.e., each elementary path) were estimated by averaging the values of two cations involved. The Bader and Born effective charges of octahedral and framework anions were evaluated by averaging the values of the same anions in a unit cell.

The packing fraction (PF), which represents the ratio of an occupied volume to the total volume, was obtained by the iterative method.39 _{The packing fraction without cation (PF}

C)

measures the occupied volume only by anions. The tolerance factor, which measures the degree of lattice distortions of (anti)perovskite structure,160 _{is calculated by 𝑡 = (𝑅}

X+ 𝑅B)/‘√2(𝑅X+ 𝑅A)“, where RX, RB, and RA are the atomic radii of cation (X site), anion at the octahedron center (A site),

and anion at the cubic framework (B site). The lattice structure tends to be cubic when t ~ 1, but it becomes orthorhombic as the degree of lattice distortion increases with smaller t (usually when t < 0.83). The channel size (i.e., the largest diameter of sphere that can pass freely through the anion sublattice) was claimed to be an important factor for ion mobility within a series of LGPS family.46

It was calculated by the Zeo++ code.268 _{The coordination number of cation was calculated by}

counting the number of neighbor ions which have the distance to the cation shorter than 1.2 times of the sum of their ionic radii. The coordination numbers of two cations were averaged since two cation sites involve in a path (the same as for the Bader charge).

Distance-based descriptors were discussed in a previous ML study, but their importance may vary.39 _{We included the distances between cations (CCD, i.e., linear path distance), between}

cation and octahedral/framework anions (COD/CFD, respectively), and between anions (AAD). COD and CFD were evaluated by finding the closest anion for each cation and the distance between them, and then averaging the distances of two cations involved in the path (the same as for the Bader charge). AAD was obtained by finding the closest neighbor anion for each anion and the distance between them, and then averaging the distances over all anions. The path width was evaluated by identifying ions close to the path and measuring the perpendicular distance from the path to the surface of ion (see Ref.39 _{for the detailed explanation). We listed three path widths,}

including the path width by the closest ion (PWc), the path width by the second-closest ion (PW2c),

and the total path width (PW) by the sum of PWc and PW2c. The lattice dynamics (i.e., phonon

the frequency of lowest-energy optical phonon (𝜔_LEO) to account for the effect of lattice dynamics.42

The total dielectric constant (𝜀) was obtained by the sum of electronic (𝜀_‚) and ionic (𝜀₍) contributions. A compound with a high polarizability would have a high ion mobility due to the lattice softening effect.44 _{The polarizability of compound was calculated by the Clausius-Mossotti}

relation as207

𝛼µ = 3 4𝜋‡

𝜀‚− 1

𝜀_‚+ 2ˆ 𝑣atom, (5.3)

where 𝑣_atom is the lattice volume per atom. The bond ionicities between cation and chalcogen anion (IC-Ch), between cation and halogen anion (IC-H), and between chalcogen and halogen anions (ICh- H) were calculated by the difference in electronegativities between species.39 The energy above the

convex hull (i.e., decomposition energy) was added in the list to include an aspect that most of superionic conductors are not stable at zero Kelvin.48 _{The formation energies of neutral vacancy}

and interstitial dumbbell are calculated by269

𝐸f = 𝐸AP_defect− Ÿ𝐸AP_pristine+ B 𝑛(𝜇( (

, (5.4)

where 𝐸_{AP_defect} and 𝐸_{AP_pristine} are the total energies of the AP supercell with and without a cation defect, i is the defect species (Li, Na, or K), and 𝑛( and 𝜇( are the number of the defect cation (-1 for vacancy and +1 for interstitial dumbbell) and its chemical potential. The defect formation energy for a certain elementary path is assigned by averaging the formation energies of two defects involved in the path (i.e., defects at end points of the path).