doi.org/10.26434/chemrxiv.9942380.v1

Defect Chemistry of Disordered Solid-State Electrolyte Li10GeP2S12Prashun Gorai, Hai Long, Eric Jones, Shriram Santhanagopalan, Vladan Stevanovic

Submitted date: 05/10/2019 • Posted date: 08/10/2019Licence: CC BY-NC-ND 4.0Citation information: Gorai, Prashun; Long, Hai; Jones, Eric; Santhanagopalan, Shriram; Stevanovic, Vladan(2019): Defect Chemistry of Disordered Solid-State Electrolyte Li10GeP2S12. ChemRxiv. Preprint.

Several classes of materials, including thiophosphates, garnets, argyrodites, and anti-perovskites, have beenconsidered as electrolytes for all-solid-state batteries. Native point defects and dopants play a critical role inimpeding or facilitating fast ion conduction in these solid electrolytes. Despite its significance, comprehensivestudies of the native defect chemistry of well-known solid electrolytes is currently lacking, in part due theircompositional and structural complexity. Most of these solid-state electrolytes exhibit significant structuraldisorder, which requires careful consideration when modeling the point defect energetics. In this work, wemodel the native defect chemistry of a disordered solid electrolyte, Li10GeP2S12 (LGPS), by uniquelycombining ensemble statistics, accurate electronic structure, and modern first-principles defect calculations.We find that VLi, Lii, and PGe are the dominant defects. From these calculations, we determine the statistics ofdefect energetics; formation energies of the dominant defects vary over ~140 meV. Combined with ab initiomolecular dynamics simulations, we find that anti-sites PGe promote Li ion conductivity, suggesting LGPSgrowth under P-rich/Ge-poor conditions will enhance ion conductivity. To this end, we offer practicalexperimental guides to enhance ion conductivity.

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Li

Li

site disorder:

Ge

P

c

S

Li10GeP2S12

P/Ge

Defect Chemistry of Disordered Solid-State Electrolyte

Li10GeP2S12

Prashun Gorai,⇤a,b Hai Long,b Eric Jones,a Shriram Santhanagopalan,b Vladan Stevanovic⇤a,b

Several classes of materials, including thiophosphates, garnets, argyrodites, and anti-perovskites, havebeen considered as electrolytes for all-solid-state batteries. Native point defects and dopants play acritical role in impeding or facilitating fast ion conduction in these solid electrolytes. Despite its signifi-cance, comprehensive studies of the native defect chemistry of well-known solid electrolytes is currentlylacking, in part due their compositional and structural complexity. Most of these solid-state electrolytesexhibit significant structural disorder, which requires careful consideration when modeling the point de-fect energetics. In this work, we model the native defect chemistry of a disordered solid electrolyte,Li10GeP2S12 (LGPS), by uniquely combining ensemble statistics, accurate electronic structure, and mod-ern first-principles defect calculations. We find that VLi, Lii, and PGe are the dominant defects. From thesecalculations, we determine the statistics of defect energetics; formation energies of the dominant defectsvary over ⇠140 meV. Combined with ab initio molecular dynamics simulations, we find that anti-sites PGepromote Li ion conductivity, suggesting LGPS growth under P-rich/Ge-poor conditions will enhance ionconductivity. To this end, we offer practical experimental guides to enhance ion conductivity.

1 Introduction

All-solid-state batteries offer greater safety and higher energy andpower densities compared to the currently employed Li- and Na-ion batteries that utilize flammable liquid electrolytes.1,2 Severalclasses of materials have been considered for solid-state elec-trolytes, including thiophosphates (e.g. Li10GeP2S12),3,4 garnets(e.g. Li7La3Zr2O12 [c]),5,6 argyrodites (e.g. Li6PS5Cl),7,8 LISI-CONs (e.g. insert [c] ),9 NASICONs (e.g. insert [c]),10 anti-perovskites (e.g. Li3OCl)11,12 etc. Most of these solid-stateelectrolytes exhibit significant structural disorder, which enablesfast ion conduction.13,14 For instance, Li10GeP2S12 (LGPS) andderived compounds display complex site disorder, as shown inFigure 1. In LGPS, the available Li sites are partially occupiedand P/Ge exhibit substitutional disorder while the anion (S) sub-lattice is ordered.

Intrisic and extrinsic point defects play a critical role in deter-mining the properties of materials, particularly semiconductorsand insulators. First-principles defect calculations have provenimmensely useful in the development of materials for thermo-electrics,15,16 photovoltaics,17,18 power electronics19,20 etc. Insolid-state battery electrolytes, native point defects and dopantscould impede ion conduction by acting as diffusion path blockers,or enhance conduction by flattening the energy landscape for dif-fusion. Similarly, defects could influence ionic and electronic con-duction in cathode materials.21,22 While direct observation andmeasurement of point defects is quite challenging, first-principlesdefect calculations can provide insights about the dominant de-fects, and their concentrations as well as electronic carrier con-centrations. When combined with ab initio molecular dynamics

aColorado School of Mines, Golden, CO 80401. bNational Renewable Energy Labora-tory, Golden, CO 80401. ⇤E-mail: pgorai@mines.edu, vstevano@mines.edu

Li

Lisite disorder:

Ge P

Ge0.5P10.5

tetrahedra

P2tetrahedra

c

S

Li10GeP2S12

P/Ge

Fig. 1 Tetragonal crystal structure of Li10GeP2S12 contains P2S4tetrahedra and (Ge0.5P10.5)S4 tetrahedra with partially occupied Li sitesin the c-axis channels and in the a-b plane bridging sites. The Ssub-lattice and P2 sites are fully occupied.

simulations, the effect of point defects on the ion conductivitycan also be quantitatively probed. The native defect energetics ofordered compounds for cathodes such as olivines (LiFePO4),21

cobaltites (e.g. LiCoO2),23 silicates (e.g. Li2MnSiO4)22 have

1–9 | 1

been previously reported.

Despite its significance, comprehensive studies of the native de-fect chemistry of well-known solid electrolytes is currently lack-ing, in part due their compositional and structural complexity.In particular, the complex site disorder needs careful consid-eration when modeling the point defect chemistry.24,25 Recentattempts at modeling the point defect energetics in disorderedsolid-state electrolytes have utilized either a single ordered rep-resentation of the disordered structure26 or a low-temperatureordered phase.27 In this study, we model the native defect chem-istry of a disordered solid electrolyte, Li10GeP2S12, by adopting aunique methodology that combines ensemble statistics and first-principles defect calculations to account for the disorder.

Li10GeP2S12 and derived compounds are among the solid-stateelectrolytes exhibiting the highest ionic conductivities.3,28 It isbelieved that the soft anion lattice and the structural disorder ofLGPS enables high Li ion conductivity.29 The room-temperaturetetragonal (P42/nmc) crystal structure of LGPS is characterized by(Ge0.5P10.5)S4 and P2S4 tetrahedra, interspersed with partiallyoccupied Li sites, as shown in Figure 1. The one-dimensionalnetwork of Li ions that form along the c-axis channels are pri-marily responsible for Li ion diffusion in LGPS, although the im-portance of Li ion diffusion in the a-b plane has also been rec-ognized.30 Ever since its introduction in 2011, LGPS and relatedcompounds have been extensively studied both experimentally29

as well as theoretically.30–32 The theoretical studies employingfirst-principles calculations have focused primarily on the phasestability,30,31 and ion diffusion mechanism.30–32 In contrast, thenative defect chemistry of LGPS is largely unexplored, partly dueto the associated computational challenges. Recently, Oh et al.used first-principles defect calculations to map the defect chem-istry of LGPS.26 However, in this study, the disordered phase ofLGPS was represented by a single ordered structure. In a disor-dered material, a multitude of different local bonding environ-ments are possible, which could lead to variations in the forma-tion energies of the same defect. Therefore, it is fundamentallyimportant to account for the disorder and to determine the statis-tical variation in the defect formation energies.

In this study, we model the native defect chemistry of disor-dered LGPS by uniquely combining ensemble statistics, accurateelectronic structure, and modern first-principles defect calcula-tions. We use a Madelung energy minimization criteria in con-junction with ensemble statistics to select representative struc-tures that account for structural disorder. For four represen-tative structures, we perform state-of-the-art defect calculationsand find that VLi, Lii, and PGe are the dominant defects. We alsofind that the formation energies of the dominant defects can varyover ⇠140 meV across the representative structures. Combinedwith ab initio molecular dynamics simulations, we determine thatPGe defects promote Li ion conductivity, suggesting LGPS grownunder P-rich/Ge-poor conditions would enhance conductivity.

2 Computational Methods

2.1 Selecting Structures for Defect Calculations

The tetragonal (P42/nmc) crystal structure of LGPS (Figure 1)contains 4 unique Wyckoff sites of Li – Li1 (16h), Li2 (4d), Li3(8 f ), Li4 (4c), one of Ge – Ge1 (4d), two of P – P1 (4d), P2 (2b),and three of S – S1 (8g), S2 (8g), S3 (8g), where the Wyckoff sitesymbols are shown in parantheses.33 The Li1 and Li3 sites formthe one-dimensional network in the c-axis channels and Li2 andLi4 are the bridging sites lying in the a-b plane. While the anionsub-lattice (S1, S2, S3) and P2 site are fully occupied, the cationsub-lattices exhibit disorder, namely: (a) P1 and Ge1 sites havefractional occupation of 0.5, and (b) the 32 Li sites distributedover Li1-Li4 are occupied by 20 Li ions.

Given the complexity of the disorder in LGPS, the number ofpossible atomic configurations in the LGPS structure is extremelylarge. As a first step, we utilize a computationally effective elec-trostatic (Madelung) energy minimization criterion26,30 to select

Cou

nts

0

50

1000 structures(a)

(b)

100

E-Emin (meV/atom)

0 20

T = 823 Ks2

s3

s4

s1

prob

abilit

y (%

)

0.1

1

volume per atom ( 3)

19.4 19.6 19.8 20.0 20.2 20.4

Fig. 2 (a) Energy distribution of the 1000 DFT-relaxed, LGPSstructures selected using the electrostatic minimization criterion. Energyis expressed per atom, relative to the lowest energy structure. (b)Probability distribution of the 1000 relaxed structures calculated usingensemble statistics at T = 823 K. Labels s1, s2, s3, and s4 are the fourrepresentative structures chosen for defect calculations.

2 | 1–9

1000 configurations with lowest energies among all possible con-figurations in the 50-atom supercell. For this, we assume idealionic charges on Li (+1), Ge (+4), P (+5), and S (-2). Subse-quently, these 1000 structures are fully relaxed with density func-tional theory (DFT) using the standard GGA-PBE functional.34

More details of the computational setup are provided in the nextsection (Section 2.2). The energy distribution of these 1000 struc-tures (after DFT relaxation), also called thermodynamic densityof states, is shown in Figure 2(a).

Next, we adopt an ensemble statistical procedure to select acollection of ordered structures (among the 1000 relaxed struc-tures) to represent the disordered phase of LGPS. It has been re-cently shown that a disordered macrostate can be expressed asa thermodynamic average of structurally ordered microstates.35

This approach is predicated upon the statistical treatment of anensemble (distribution) of local minima and has been shown toreproduce well the structural features of amorphous and glassystates.35 In this work we apply ensemble statistics to the set of1000 ordered structures obtained in the previous step from theMadelung energies criterion. The thermodynamic contribution ofeach microstate to the disordered phase of LGPS is proportional tog(E)exp [�(E �Emin)/kBT ], where E is the energy of a microstate,g(E) is its degeneracy, and kB is the Boltzmann constant. Since thesymmetry of all “ordered” configurations turns out to be P1 afterDFT relaxations, the degeneracy of states (i.e. the multiplicity) isdecided on the basis of equality of DFT energies and relaxed vol-umes. The thermodynamic contribution of each microstate canbe expressed as a probability 1

Z g(E)exp [�(E �Emin)/kBT ] with Zbeing the normalization factor (partition function); Figure 2(b)shows the probability distribution of the 1000 structures as afunction of volume per atom assuming the typical synthesis tem-perature 823 K.3 An appropriate condition for selecting represen-tative structures is to choose those with high probability accord-ing to ensemble statistics. For performing defect calculations, wechose two highly probable structures with slightly different cellvolumes (19.74, 19.84 Å3/atom), which are, henceforth, referredto as structure 1 (s1 in Figure 2b), and structure 2 (s2 in Fig-ure 2b), respectively. To add diversity to the set of representativestructures, we also chose two more structures with different vol-umes (19.94, 20.02 Å3/atom) and lower probabilities, labelleds3 and s4 in Figure 2(b). We refrained from choosing structuresfrom the largest “cloud” of data points at much lower probabilities(⇠0.1%) in Figure 2(b).

2.2 Point Defect Energetics and Electronic Structure

First-principles point defect calculations is used to compute theformation energies of native defects as functions of the Fermi en-ergy in each of the 4 structures. We calculate the defect forma-tion energies in LGPS using density functional theory (DFT) anda standard supercell approach.36 Within the supercell approach,the formation energy (DED,q) of a point defect D in charge stateq is calculated as:

DED,q = (ED,q �EH)+Âi

niµi +qEF +Ecorr (1)

where EH and ED,q are the total energies of the defect-free,charge-neutral host supercell (EH) and the supercell containingdefect D in charge state q, respectively. The chemical potentialof element i is denoted by µi and ni is the number of atoms ofelement i added (ni<0) or removed (ni>0) from the supercell.EF is the Fermi energy. The term qEF is the characteristic en-ergy of exchanging charge between the defect and the reservoirof charge (Fermi sea). The supercell approach to calculating de-fect energetics suffers from artifacts arising due to finite size ef-fects. Additional artifacts are introduced due to the limitations ofDFT, most notably, the underestimation of the band gap with stan-dard functionals such as GGA-PBE.34 Various correction schemesare available to correct for the finite size artefacts and inaccurateelectronic structure; these corrections,36 are represented by theterm Ecorr in Eq. 1.

In total, we consider up to 31 different native defects compris-ing vacancies (VLi, VGe, VP, VS), anti-sites (GeP, PGe, PS, SP), andinterstitials (Lii), with each unique Wyckoff site treated as a dif-ferent defect. For each defect, charge states q = -3, -2, -1, 0, 1, 2,3 are considered; for some defects, such as VP, additional chargestates q = -5, -4, 4, 5 are also considered. The possible sites forLi interstitials are determined by a Voronoi tessellation schemeas implemented in the software, pylada-defects.37 In each struc-ture, the energetically most favorable interstitial configuration isassessed by relaxing up to 50 different possible interstitial config-urations.

The total energies of the supercells are calculated using thegeneralized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE)34 within the projector augmented wave (PAW)formalism as implemented in the VASP code.38 The total ener-gies are calculated with a plane-wave energy cutoff of 340 eVand a G-centered 4⇥4⇥2 Monkhorst pack k-point grid to samplethe Brillouin zone. The positions of the ions in the defect su-percells are relaxed following a similar procedure used in Refs.15,39. The elemental chemical potentials µi are expressed rel-ative to those of the elements in reference elemental phases asµi = µ0

i +Dµi, where µ0i is the reference chemical potential under

standard conditions and Dµi is the deviation from the reference.Dµi = 0 corresponds to i-rich conditions. For example, DµS = 0 (S-rich) corresponds to the equilibrium between LGPS and solid S.The reference chemical potentials (µ0

i ) are fitted to a set of mea-sured formation enthalpies of compounds, as implemented in theFERE approach.40

The finite-size corrections included in Ecorr, following themethodology in Ref. 36, are: (1) image charge correction forcharged defects, (2) potential alignment correction for chargeddefects, (3) band filling correction for shallow defects, and (4)correction of band edges for shallow acceptors/donors. The cal-culations are organized and the results are analyzed using oursoftware package, pylada-defects, for automation of point defectcalculations.37

The underestimation of the band gap in DFT is remedied byapplying individual valence and conduction band edge shifts (rel-ative to the DFT-computed band edges) as determined from GWquasi-particle energy calculations.36 We use DFT wave functionsas input to the GW calculations. The GW eigen-energies are iter-

1–9 | 3

structure 1 structure 2 structure 3 structure 4

Li Ge

c

P S

Fig. 3 Four representative LGPS structures selected for defect calculations. All four structures exhibit the so-called Z configuration of the GeS4 andP1S4 tetrahedra. The structures differ in the Li site occupations and have slightly different volumes per atom (19.74, 19.84, 19.94, 20.02 Å3/atom).

ated to self-consistency to remove the dependence on the single-particle energies of the initial DFT calculation. The input DFTwave functions are kept constant during the GW calculations,which allows the interpretation of the GW quasi-particle energiesin terms of energy shifts relative to the DFT Kohn-Sham energies.The GW quasi-particle energies are calculated for the 50-atomcells using a 4⇥4⇥2 k-point grid.

Under a given growth condition, the equilibrium EF is deter-mined by solving the charge neutrality condition. The concen-tration of defects are determined using Boltzmann distribution,such that

⇥Dq

⇤= Nsexp

��DHD,q/kBT

�, where

⇥Dq

⇤is the defect

concentration, Ns is the concentration of lattice sites where thedefect can be formed, kB is the Boltzmann constant, and T is thetemperature. At a given T, The concentrations of electrons andholes are functions of EF . To establish charge neutrality, the totalpositive charges should equal the negative charges. In this equa-tion, EF is the only free parameter. By solving charge neutralitycondition self-consitently, we can determine the equilibrium EF

and the relevant defect formation energies and concentrations.

2.3 Phase Stability

The phase stability of LGPS relative to decomposition into com-peting phases determines the bounds on the values of Dµi, de-scribed in the Section 2.2. To establish the phase stability re-gion of LGPS in the quaternary Li-Ge-P-S chemical potential phasespace, we consider all known compounds in this chemical spacereported in the Inorganic Crystal Structure Database (ICSD).41

We also consider additional LixPySz compounds suggested in Refs.30,31, which were compiled by Holzwarth et al.42 We find thatthe phase stability region of LGPS in the chemical potential spaceis bounded by 10 four-phase corners – at each corner LGPS is inequilibrium with 3 other phases. Overall, we find that LGPS is inequilibrium with S, Li3PS4, Li4GeS4, Li2S, GeS2, GeP3, and LiP7,which is in excellent agreement with the experimental phase di-agrams of LGPS.43,44 In Ref. 26, LGPS is predicted to be in equi-librium with S, Li2S, Li2GeS3, and Li2PS3, which does agree wellwith the experimental phase diagrams. More details of our phase

stability calculation and the values of µ0i and Dµi of Li, Ge, P, and

S at the 10 four-phase corners (labelled P-1 through P-10) areprovided in the supplementary information.

2.4 Ionic Conductivity

To calculate the Li ion conductivity in disordered LGPS, we per-formed ab initio molecular dynamics (AIMD) with VASP softwarepackage.38 The plane-wave energy cutoff was set to 260 eV anda k-point grid of 1⇥1⇥1 was used. The initial temperature in theAIMD simulation was set to 100 K and was gradually increased to500K with a heating rate of 0.5 K/fs using the Nosé-Hoover ther-mostat.45,46 The equilibrated system was then heated to 550 K,600 K, 650 K, 700 K, 800K . . .1300 K for the 200 ps AIMD simula-tion. The diffusion coefficients of LGPS at different temperaturesare calculated using the simulated trajectories as,

D =1

6Nt< ri(t)� ri(t = 0)> (2)

where D is the diffusion coefficient, N is the total number of Li+

ions, t is the simulation time, and ri are the positions of the Li+

ions. The diffusion coefficients obtained at different temperaturesare fitted to Arrehenius equation:

ln(D) =� EA

kBT+C (3)

where EA is the activation barrier, T is the temperature, kB isthe Boltzmann constant, and C is the intercept. The diffusioncoefficient at 300K (D300K) is estimated using the fitted equation(Eq. 3) with T = 300K. Finally, the ionic conductivity at 300K(s300K) is calculated using,

s300K =rz2F2D300K

RT, (4)

where r is the density of Li+ ions, z is the charge of Li+ ions, F isthe Faraday constant, and R is the gas constant.

4 | 1–9

3 Results

Figure 3 shows the four LGPS structures that were selected fromthe electrostatic energy minimization criterion followed by en-semble statistics, as described in Section 2.1. All four structuresexhibit the so-called Z configuration26 of the GeS4 and P1S4 tetra-hedra. The structures differ in the occupation of the Li sites, mostnoticeably in the c-axis channels. The four structures exhibitslightly different cell volumes (Section 2.1). For each of thesefour structures, we computed their accurate electronic structurewith GW-based methods and native defect energetics with first-principles defect calculations, which are discussed next.

3.1 Electronic Structure and Defect Chemistry of LGPS

Defect formation energies, and therefore, defect and electroniccarrier concentrations are sensitive to the electronic structure,particularly the band gap. We calculated the band gap of the fourstructures (Figure 3) and found them to range between 4.69-4.72eV. Our GW-calculated band gaps are larger compared to the bandgap calculated with hybrid DFT functional HSE06 (3.8 eV).26

Defect energetics are typically presented in the form of “defectdiagrams” with defect formation energies (DED,q) plotted as func-tions of the Fermi energy (EF ), as shown in Figure 4. Defects withpositive slopes are donors and with negative slopes are acceptors.The defect formation energies are also functions of the elementalchemical potentials µi (see Eq. 1). In other words, DED,q also de-pends on Dµi of each element in the Li-Ge-P-S quaternary phasespace, where the values of Dµi are bound by the condition of LGPSphase stability (Section 2.3). As such, the defect diagrams are afunction of the elemental chemical potentials.

Let us first examine the calculated defect energetics of one ofthe structures, namely structure 1 (Figure 4). The defect energet-ics of structures 2-4 are qualitatively similar to that of structure1. For the sake of simplicity, the defect energetics correspondingto the chemical potentials at two 4-phase corners (P-1 and P-10)are plotted in Figures 4(a), and 4(b), respectively. The valuesof Dµi (i = Li, Ge, P, S) corresponding to the corners P-1 and P-10 are tabulated in the supplementary information. The defectswith the lowest formation energies at the equilibrium Fermi en-ergy (EF,eq) are the dominant defects. In Figures 4(a), and 4(b),we find that the dominant defects (denoted by solid lines) are Livacancies (VLi), Li interstitials (Lii), and P1/Ge anti-sites (PGe).The Li vacancy with lowest formation energy forms at a Li site inthe c-axis channel, as opposed to in the a-b plane bridging sites.The formation of PGe anti-site defects are more favorable thanthe formation of GeP anti-sites. Therefore, we predict that LGPSis naturally off-stoichiometric (slightly P-rich) compared to theideal stoichiometry of P:Ge = 2:1. The predicted EF,eq is pinnedaround 2.56 (3.06) eV above the valence band maximum (EF = 0eV) at corner P-1 (P-10). Owing to the large band gap and Fermienergy pinning far from the band edges, the predicted free carrierconcentrations are very low, consistent with the fact that LGPS iselectrically insulating.

The defect energetics of the dominant defects for structures 1-4 at corners of the phase stability region corresponding to P-1and P-10 are summarized in Figure 5. In all cases, the dominant

(a)

(b)P-10

P-1EF,eq

EF,eq

LiiVLiPGeGePVPVGeVSSPPS

E D,q

(eV)

0

1

2

3

LiiVLiPGeGePVPVGeVSSPPS

E D,q

(eV)

0

1

2

3

EF (eV)

0 1 2 3 4

Fig. 4 Formation energies of native point defects (DED,q) inLi10GeP2S12 as functions of Fermi energy (EF ) at elemental chemicalpotentials corresponding to (a) P-1, and (b) P-10 (see Table S1). EF isreferenced to the valence band maximum. The upper limit of EF shownis the conduction band minimum such that EF values range from 0 to theband gap. Multiple lines of the same color represent the same defecttype at different Wyckoff sites. Lowest formation energy defects aredenoted by solid lines while those with higher formation energies withdotted lines. The equilibrium Fermi energy (EF,eq) marked by verticaldashed line is calculated at 300 K.

defects are VLi, Lii, and PGe; higher energy anti-site defects GeP(at site P1) are also shown. While the formation energies of thedominant defects in a given structure do not vary significantly be-tween corners P-1 and P-10, there is appreciable differences in the

1–9 | 5

VLi

GeP(a) P-1

Lii

structure 1 structure 2 structure 3 structure 4

PGe

E D,q

(eV)

0

0.2

0.4

0.6

0.8

1.0

(b) P-10

E D,q

(eV)

0

0.2

0.4

0.6

0.8

1.0

EF (eV)2.0 2.5 3.0 3.5 4.0

(c) P-1

(d) P-10

EF (eV)2.0 2.5 3.0 3.5 4.0

(e) P-1

(f) P-10

EF (eV)2.0 2.5 3.0 3.5 4.0

(g) P-1

(h) P-10

EF (eV)2.0 2.5 3.0 3.5 4.0

VLi

GeP

Lii

PGe

Fig. 5 Formation energies of native point defects (VLi, Lii, PGe, and GeP) as functions of Fermi energy (EF ) in LGPS structures 1-4 (Figure 3) atelemental chemical potentials corresponding to P-1 (a, c, e, g), and P-10 (b, d, f, h). EF is referenced to the valence band maximum. Multiple lines ofthe same color represent the same defect type at different Wyckoff sites.

formation energetics between different structures. The variationin the formation energy of defects between different representa-tive structures (structures 1-4) provides a quantitative measure ofthe statistical distribution of defect formation energies in the dis-ordered LGPS phase. The variation in defect formation energies isfurther dicussed in Section 4.

3.2 Ion conductivity in LGPS

The average Li ion conductivity at room temperature (s300K) andthe effective migration barrier (EA) are calculated for pristineLGPS (without native defects) and LGPS containing PGe defectsin each of the 4 structures (Figure 3). The computed values ofs300K and EA are tabulated in Table 1.

For the pristine structures, we obtain an average conductivityat 300K of 35 mS/cm (35 mS/cm with range 12-87 mS/cm) andan effective migration barrier of 0.19±0.02 eV. Similar computedvalues for room-temperature conductivities and migration barri-ers are reported in the literature (34 mS/cm and 0.19±0.01 eVin Ref. 26; 9 mS/cm (with range 2-40 mS/cm) and 0.21 eV inRef. 31). Our calculated effective migration barrier is also in fairagreement with experimentally measured barrier of 0.25 eV.3

For the structures containing PGe anti-site defects, we calcu-lated the average room-temperature conductivity to be 95 mS/cmwith range 41-183 mS/cm and the migration barrier 0.16±0.02eV. It is somewhat surprising that introduction of PGe anti-site de-fects enhance the average Li ion conductivity in LGPS. As also sug-gested in Ref. 26, the introduction of PGe anti-site defects seem to

structure conductivity eff. activation barriers300K (mS/cm) EA (eV)

pristine

structure 1 30+36�16 0.19±0.02

structure 2 39+33�18 0.18±0.01

structure 3 48+87�31 0.17±0.02

structure 4 22+20�11 0.21±0.02

with PGe

structure 1 112+88�49 0.15±0.01

structure 2 55+35�22 0.17±0.01

structure 3 80+76�39 0.16±0.02

structure 4 132+112�61 0.14±0.01

Table 1 Computed room-temperature (300 K) Li ion conductivity andeffective activation barriers (EA) in the pristine LGPS structures andLGPS structures containing PGe anti-site defects. The subscript andsuperscript for conductivity values are the asymmetric error bars.

6 | 1–9

(a) pristine

(b) with PGe defects

T (K)3005001000

structure 1structure 2structure 3structure 4

log 1

0 D

(cm

2 /s)

7

6

5

4

log 1

0 D

(cm

2 /s)

7

6

5

4

1000/T (1/K)0.5 1.0 1.5 2.0 2.5 3.0 3.5

Fig. 6 Arrhenius plots of computed diffusivities for the four structures inFigure 3, labelled structures 1-4. Diffusivities calculated in (a)defect-free, prisitine LGPS structures, and (b) structures containinganti-site PGe defect. Vertical dashed line represents room temperature(300 K). Room-temperature diffusivities (D300K) are calculated from theArrhenius fit by extrapolating to 300 K (solid lines).

flatten the energy landscape (lowering of migration barrier) forLi ion diffusion and thereby, promote ion migration. The conduc-tivity enhancement in LGPS with the introduction of PGe anti-sitedefects is observed uniformly across all 4 structures; in fact, theenhancement does not depend on the starting structure. For ex-ample, for structure 4, s300K increases 22 ! 132 mS/cm and forstructure 2, 39 ! 55 mS/cm.

4 Discussion

Thus far, we have shown that (1) the dominant defects in LGPSare VLi, Lii, and PGe, (2) there is a distribution of defect formationenergies within the ensemble of representative structures, and(3) PGe anti-site defects promote Li ion conductivity. In thissection, we discuss the fundamental and practical implications ofthese findings for solid-state battery electrolytes.

Dominant Defects and Distribution of Defect Energetics: The domi-nant defects in LGPS do not block the path of Li ion diffusion inthe c-axis channels. In addition to a soft anion sub-lattice29 andcation site disorder,13 the absence of path-blocking defects couldbe another reason for the remarkably high Li ion conductivites inLGPS and related compounds.3,28 This is unlike certain cathodematerials, such as LiFePO4, where path-blocking, anti-site defectsFeLi are present in high concentrations.21

Figure 7 shows the distribution of the formation energies (atthe equilibrium Fermi energy) of the relevant defects in LGPS.For a given defect type, the spread in the formation energiesarises from two sources: (1) variation with elemental chemicalpotentials µi (or Dµi), and (2) variations between structures 1-4.We observe that the spread in the defect formation energies rangefrom ⇠60 meV (PGe) to upto 140 meV (Lii). At the synthesis tem-perature (823 K), the spread in the formation energies translateinto a spread in the defect concentrations of 1.2⇥1020-2.7⇥1020

cm�3 for PGe and 8.8⇥1019-3.9⇥1020 cm�3 for Lii. Therefore,in the disordered phase of LGPS, one can expect the defectconcentrations to be an average over the corresponding defectconcentrations in the representative structures. However, giventhe higher probability of structures 1 and 2 (Figure 2, comparedto structures 3 and 4, the defect energetics from s1 and s2 canbe expected to be statistically more significant. Nonetheless,the spread in the defect formation energies in Figure 7 issignificant, which emphasizes the need to consider an ensembleof probable structures to estimate the defect energetics in thedisordered phase (as opposed to a single ordered representation).

Conductivity Enhacement by Anti-site Defects: Since PGe anti-sitedefects are found to promote Li ion conductivity in LGPS, it wouldbe prudent to synthesize LGPS under growth conditions that max-imize the concentration of these anti-site defects. To this end, weidentify the chemical potentials (within the phase stability region)corresponding to the most P-rich/Ge-poor conditions such thatthe defect formation energy of PGe is minimized. This is achievedat the chemical potentials corresponding to the 4-phase corner P-9 (see Table S1 in supplementary information). At P-9, LGPS isin equilibrium with Li3PS4, Li2S, and LiP7. In practice, this canbe achieved in experiments by performing phase boundary map-

1–9 | 7

defect type

s1s3s4

s2

native defect energetics: Li10GeP2S12

VLi Lii PGe GeP

E D,q

(eV)

0

0.1

0.2

0.3

0.4

0.5

Fig. 7 Range of formation energies of native defects (VLi, Lii, PGe),GeP) in the ensemble of four representative LGPS structures (s1–s4).

ping.39 For instance, synthesizing LGPS such that trace amountsof Li3PS4, Li2S, and LiP7 are present will ensure that chemicalpotentials during growth are at corner P-9.

5 Conclusions

To understand and find ways to improve ionic conductivity ofsolid-state electrolytes, it is important to fully investigate the de-fect energetics and the effect of defects on ionic conductivity.Computing the defect energetics in disordered phases is partic-ularly challenging. Here, we modeled the native defect chemistryof a disordered solid-state electrolyte, Li10GeP2S12, by employinga unique methodology. The results provide insights into the fun-damental understanding of defect properties and experimentalguidance to improve ionic conductivity in LGPS. To summarize,

1. The dominant defects in LGPS are VLi, Lii, and PGe. Anti-siteGeP defects are present in concentrations lower than PGe,suggesting LGPS is natively off-stoichiometric (P-rich).

2. Anti-site PGe defects enhance Li ion conductivity due to theflattening of the energy landscape for diffusion.

3. Synthesis of LGPS under P-rich/Ge-poor growth conditionswill maximize concentration of PGe, thereby improving ionicconductivity. LGPS grown in equilibrium with Li3PS4, Li2S,and LiP7 will be most P-rich/Ge-poor.

The calculation methodology presented in this work lays thegroundwork to investigate the defect properties of other well-known and emerging disordered solid-state electrolytes.

Conflicts of Interest

There are no conflicts to declare.

Acknowledgements

The work was authored by the National Renewable EnergyLaboratory (NREL), operated by Alliance for Sustainable Energy,

LLC, for the U.S. Department of Energy (DOE) under ContractNo. DE-AC36-08GO28308. Funding provided by the LaboratoryDirected Research and Development (LDRD) program at NREL.The research was performed using computational resourcessponsored by the Department of Energy’s Office of EnergyEfficiency and Renewable Energy and located at the NationalRenewable Energy Laboratory. The views expressed in this articledo not necessarily represent the views of the DOE or the U.S.Government.

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Electronic Supplementary Information

“Defect Chemistry of Disordered Solid-State Electrolyte

Li10GeP2S12”

Prashun Gorai,⇤,†,‡ Hai Long,‡ Eric Jones,† Shriram Santhanagopalan,‡ and Vladan

Stevanovic⇤,†,‡

†Colorado School of Mines, Golden, CO, USA

‡National Renewable Energy Laboratory, Golden, Colorado 80401, USA

E-mail: pgorai@mines.edu; vstevano@mines.edu

1

Phase Stability of Li10GeP2S12 in Chemical Potential Space

The phase stability region of Li10GeP2S12 is calculated from the DFT total energies of competing phases

and reference chemical potentials µ0Li = -1.58 eV, µ0

Ge = -4.34 eV, µ0P = -5.14 eV, and µ0

S = -4.01 eV.

Table S1: Phase stability region of Li10GeP2S12 in the quaternary Li-Ge-P-S chemical potential space.

Corner �µLi �µGe �µP �µS Phases in Equlibrium

(eV) (eV) (eV) (eV) with Li10GeP2S12

P-1 -2.528 -1.509 -1.634 0.000 S, GeS2, Li4GeS4

P-2 -2.530 -1.509 -1.627 0.000 S, GeS2, Li3PeS4

P-3 -2.396 -2.046 -2.030 0.000 S, Li2S, Li3PS4

P-4 -2.396 -2.040 -2.033 0.000 S, Li2S, Li4GeS4

P-5 -2.171 -0.427 -0.186 -0.629 LiP7, GeP3, Li3PS4

P-6 -2.166 -0.424 -0.187 -0.634 LiP7, GeP3, Li4GeS4

P-7 -2.245 -0.371 -0.204 -0.569 GeS2, GeP3, Li3PS4

P-8 -2.243 -0.366 -0.206 -0.571 GeS2, GeP3, Li4GeS4

P-9 -2.031 -0.587 -0.206 -0.729 Li2S, LiP7, Li3PS4

P-10 -2.030 -0.579 -0.206 -0.731 Li2S, LiP7, Li4GeS4

2

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