# Electrolytes for Li-ion all-solid-state batteries: a first-principles
comparative study of Li10GeP2O12 and Li10GeP2S12 in the LISICON and LGPS
phases

Giuliana Materzanini  Theory and Simulations of Materials (THEOS) and National
Centre for Computational Design and Discovery of Novel Materials (MARVEL),
École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
Leonid Kahle  Theory and Simulations of Materials (THEOS) and National Centre
for Computational Design and Discovery of Novel Materials (MARVEL), École
Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland  Aris
Marcolongo  IBM RSM Zurich Research Laboratory, Zurich, Switzerland  Nicola
Marzari  Theory and Simulations of Materials (THEOS) and National Centre for
Computational Design and Discovery of Novel Materials (MARVEL), École
Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland

###### Abstract

In this work we address Li-ion diffusion in thio-LISICON materials and in
their oxide counterparts, exploring both the orthorhombic and tetragonal
phases of Li10GeP2S12 (LGPS) and Li10GeP2O12 (LGPO) through extended Car-
Parrinello molecular dynamics in the canonical and isobaric-isothermal
ensemble. The (quasi-)orthorhombic and tetragonal phases are studied both for
the oxide and for the sulfide, with the aim of comparing their conductivity
with the same approach; out of these four case studies, tetragonal LGPO has
not been reported and, while dynamically stable, it sits (0.04 Ha/formula
unit) above orthorhombic LGPO. We calculate activation energies for diffusion
of 0.18 eV and 0.23 eV for tetragonal and orthorhombic LGPS, and of 0.22 eV
and 0.34 eV for tetragonal and orthorhombic LGPO. In line with experiments, we
find orthorhombic LGPO orders of magnitude less conductive, at room
temperature, than the two sulfide systems. However, this is not the case for
tetragonal LGPO, which, although less stable than its orthorhombic allotrope,
shows at room temperature a conductivity comparable to orthorhombic and
tetragonal LGPS, and, if synthesized, could make a very attractive Li-ion
conductor.

###### pacs:

Valid PACS appear here

##  I Introduction

More than two centuries after the beginning of the industrial revolution, a
major societal challenge is to limit its reliance on fossil fuels by switching
to renewable and green energy sources Arunachalam and Fleischer (2008), a goal
that cannot be achieved unless an efficient way to store electrical energy is
provided Kyriakopoulos and Arabatzis (2016). Electrochemical storage is
particularly suited to meet the needs of electrical grids powered by sun, wind
or tides Yang  _et al._ (2010, 2011) and can also enable the electrification
of transportation, providing the energy to automotive vehicles that are less
dependent on fossil fuels, i.e. hybrid electrical, hybrid plug-in electrical
and pure electrical vehicles Tie and Tan (2013), as well as the
electrification of residential services, as for example in the popular case of
the Tesla Home Battery Powerwall tes .
Among the existing electrochemical technologies, about 30 years after their
first introduction in the market by Sony in 1991 Yoshino  _et al._ (1991);
Nagaura (1990); Blomgren (2017), Li-ion batteries offer some of the best
performance in terms of energy density, memory effects, near-reversibility and
longer life cycles Poizot and Dolhem (2011); Goodenough and Park (2013) and
have enabled the rise of portable electronics and furnished the power supply
for safety systems, personal transportation, and digital technologies Winter
and Brodd (2004). Smart-grid integration and mass-market full electrification
are the next frontiers, and for that the complex trade-off between energy
density (and so range), power density, cost and safety of batteries needs to
be addressed Larcher and Tarascon (2015); Kwade  _et al._ (2018), in addition
to the availability of the raw materials Olivetti and Cullen (2018).
A Li-ion battery cell exploits the redox activities of the anode and cathode
to generate electricity outside the cell while reversibly intercalating Li
ions between the two electrodes through an ionic conducting electrolyte
Goodenough and Park (2013). Due to dendrite formation and explosive failure
with Li metal anodes Whittingham (1976); Goodenough and Kim (2009); Blomgren
(2017), the anode is also an insertion material as the cathode Goriparti  _et
al._ (2014); Rozier and Tarascon (2015), and non-aqueous electrolytes are
usually exploited, due to their wider electrochemical window compared to
water-based electrolytes ($\sim$ 4.2 V against $\sim$ 1.2 V Fong  _et al._
(1990); Xu (2004)).
Energy density has been considered so far the preminent figure-of-merit in Li-
ion batteries, and huge efforts have been put on the design of cathodes with
high voltages and capacities Melot and Tarascon (2013); Whittingham (2004),
focusing on three main classes of cathodes, based respectively on the LiCoO2
intercalation structure Mizushima  _et al._ (1980), the LiMn2O4 spinel
structure Liu  _et al._ (2014) and the polyanion olivine structure Padhi  _et
al._ (1997). For example, by partially substituting the Co cations with Ni, Mn
or Al in LiCoO2 and successively increasing the Li concentration in the
material, the capacity of LiCoO2 was almost doubled Thackeray  _et al._
(2007); Wang  _et al._ (2007); Rozier and Tarascon (2015), and partially
replacing the Mn3+/Mn4+ couple with Ni2+/Ni4+, Cr3+/Cr4+, Fe3+/Fe4+ and
Co3+/Co4+ in LiMn2O4 has given rise to new generations of high-voltage (5 V)
cathodes Liu  _et al._ (2014); Shigemura  _et al._ (2002).
However, the effort to design high-energy cathodes cannot come without a
parallel effort to address the safety and efficiency challenges that arise
when the interaction between the current electrolytes and the electrodes is
taken into account. Examples are: the combustion reactions of high-capacity
cathodes (Li0.5CoO2 and its derivatives) in the presence of ethylene or
propylene carbonate electrolytes at temperatures above 180oC MacNeil and Dahn
(2002); Wang  _et al._ (2007), the Mn dissolution of LiMn2O4 in the
electrolyte with subsequent oxidation and degradation of the electrolyte and
capacity loss of the cell Amatucci  _et al._ (1997), and in general the
oxidations/reductions of the electrolyte if the electrochemical potentials of
the cathode/anode are outside the electrolyte stability window Goodenough and
Park (2013). Although the latter issue is spontaneously fixed by the formation
of the solid electrolyte interface (SEI) that widens the electrochemical
window of the electrolyte Xu (2004), this interface is usually also
responsible for the low ionic transport from the electrolyte to the electrode
and vice-versa. Usually, electrolyte decomposition occurring at more than 4.2
V vs. Li+/Li is considered the main reason for the capacity fading upon
cycling Dedryvere  _et al._ (2010). Finally, when the passivating interface is
destroyed at high temperatures (even 80oC), liquid organic electrolytes in
contact with the positive material can form highly toxic fluoro-organic
compounds Hammami  _et al._ (2003). All these issues make the use of state-of-
the-art Li-ion batteries (where the considerable size and weight of the
batteries make these much more exposed to a fire risk during short circuits) a
technology still in need of safety and performance improvements, to the point
that their original scope, i.e. addressing some core environmental problems,
is still a matter of debate Larcher and Tarascon (2015).
Some safety challenges can in principle be addressed through an all-solid-
state battery strategy, by substituting liquid organic with solid-state
(crystalline, glassy or amorphous) inorganic electrolytes Knauth and Tuller
(2002); Knauth (2009); Quartarone and Mustarelli (2011); Kim  _et al._ (2015);
Li  _et al._ (2015); Hu (2016); Manthiram  _et al._ (2017); Braga  _et al._
(2017); Jiang  _et al._ (2017); Gao  _et al._ (2018). In addition to solving
several safety issues and the problem of leakage presented by the currently
used organic electrolytes, these materials have a lithium transport number
very close to one Quartarone and Mustarelli (2011), better thermal stability
and wider electrochemical windows, and can enable the use of Li metal anodes
Bates  _et al._ (1993); Braga  _et al._ (2017). Historically, the drawbacks
have been a generally lower ionic conductivity with respect to organic
electrolytes and a complex engineering of the electrolyte/electrode interface
Goodenough and Kim (2009).
In this respect, sulfides proved to be an interesting choice, presenting a
superior ionic conductivity with respect to many other materials. Moreover,
their softness reduces grain boundary resistance and facilitates the
manifacture of electrode/electrolyte interface Hayashi  _et al._ (2016);
Sakuda  _et al._ (2013).
The LISICON family of superionics, thoroughly investigated in the ’60s, ’70s
and ’80s Shannon  _et al._ (1977); Hong (1978); Bruce and West (1980, 1982);
Rodger  _et al._ (1985); Abrahams  _et al._ (1988, 1989); Bruce  _et al._
(1990), was originally composed of oxide materials, solid solutions between
hcp-based $\gamma$-Li3PO4-structure crystals (wurtzite-based edge-sharing
tetrahedra with cations at the center and anions at the corners Zemann (1960);
West (1975); West and Bruce (1982); O’keeffe and Hyde (1978)) and tetragonal
close-packed Li4YO4 crystals (with Y = Ge, Si, Ti) Völlenkle  _et al._ (1968);
Dittmar and Schäfer (1976); Tranqui  _et al._ (1979); Baur (1981). Through
X-ray and neutron diffraction techniques the structures were determined to be
monoclinic Tranqui  _et al._ (1979) or orthorhombic, both in the solid
solutions and in the parent phases West (1975); O’keeffe and Hyde (1978);
Abrahams and Bruce (1991). Forming solid solutions such as Li(3+x)YxP(1-x)O4
or Li(2+2x)Zn(1-x)GeO4, i.e. tuning stoichiometries as Li3PO4 West (1975) or
Li2ZnGeO4 (where the number of cations and anions is the same) Bruce and West
(1982) toward stoichiometries as Li4YO4 West (1973), introduces Li
interstitials in the pristine wurtzite structure West (1975); Bruce and West
(1982) and favours ionic conductivity Hong (1978), that usually reaches a
maximum at intermediate values of $x$ in the solid solutions (see, e.g.,
Rodger  _et al._ (1985)). However, it became clear that replacing oxygen with
sulphur, with a larger ionic radius and more polarizable character Murayama
_et al._ (2002a), could improve the conductivity. The thio-LISICON
Li(3+x)P(1-x)Ge(Si)xS4 family, a solid solution between the parent phases
Li3PS4 Mercier  _et al._ (1982); Tachez  _et al._ (1984); Homma  _et al._
(2011) and Li4GeS4 Kanno  _et al._ (2000), was studied for glass, glass-
ceramic Mizuno  _et al._ (2006); Minami  _et al._ (2006); Ohtomo  _et al._
(2013a, b); Tatsumisago  _et al._ (2013) and crystalline systems Kanno and
Murayama (2001); Murayama  _et al._ (2002b, 2004), showing superior
conductivity with respect to the oxides, with the best conductivity at room
temperature ($2\times 10^{-3}$ Scm-1) reached for $x=0.75$ Kanno and Murayama
(2001).
In an additional effort to improve conductivity, a new phase (tetragonal,
space group $P4_{2}/nmc$, $\\#$137) of thio-LISICONs at the composition
$x=0.67$ (named thereafter LGPS) was discovered in 2011 Kamaya  _et al._
(2011), with a record conductivity of $12\times 10^{-3}$ Scm-1. This inspired
a new wave of efforts, both experimental Kuhn  _et al._ (2013a, b); Bron  _et
al._ (2013); Kuhn  _et al._ (2014); Kato  _et al._ (2014); Seino  _et al._
(2014); Whiteley  _et al._ (2014); Hori  _et al._ (2015a, b, c); Kwon  _et
al._ (2015); Yang  _et al._ (2015); Kato  _et al._ (2016); Weber  _et al._
(2016) and theoretical Mo  _et al._ (2011); Adams and Rao (2012); Xu  _et al._
(2012); Ong  _et al._ (2013); Lepley  _et al._ (2013); Zhu  _et al._ (2015);
Wang  _et al._ (2015); Zhu  _et al._ (2016); Han  _et al._ (2016), aiming to
understand conduction mechanisms and to push the ionic conductivity even
further. By tuning lithium, germanium and phosphorus compositions, a room-
temperature conductivity of $14.2\times 10^{-3}$ Scm-1 was reached Kwon  _et
al._ (2015), and by substituting germanium with silicon and simultaneously
partially replacing sulphur with chlorine a value of $25\times 10^{-3}$ Scm-1
was obtained Kato  _et al._ (2016). For an extensive up-to-date overview of
LGPS and LGPS-like materials we refer the reader to the recent review by Kanno
and coworkers Kato  _et al._ (2020).
However, apart from the high ionic conductivity that places these sulfide
conductors at the same level of liquid ionic electrolytes, there are important
drawbacks that cannot be disregarded when trying to deploy them as
electrolytes for all-solid-state batteries. First of all, LGPS and LGPS-like
sulfides have very narrow thermodynamical electrochemical stability windows.
Results from simulations Zhu  _et al._ (2015); Mo  _et al._ (2011) and
impedance spectroscopy Sakuma  _et al._ (2016); Wenzel  _et al._ (2016) show
that LGPS is chemically unstable below 1.71V vs Li+/Li as a consequence of Ge
and P cations reduction at Li metal Wenzel  _et al._ (2016) or Li alloy Sakuma
_et al._ (2016) anodes, giving lithiation products Li15Ge4, Li3P and Li2S; it
is also unstable above 2.14V due to the sulphur oxidation at the cathode, with
delithiation products P2S5, GeS2 and S Han  _et al._ (2015). Both calculations
and experiments agreed on the electrochemical stability of LGPS and pointed
out its claimed wide (5V) electrochemical window Kamaya  _et al._ (2011) to be
essentially kinetic Wenzel  _et al._ (2016); Zhu  _et al._ (2015). Although
this interfacial instability can in principle be dealt with coating oxide
layers Zhu  _et al._ (2016), the solid electrolyte interface, mainly formed
due to the chemical instability at the anode of Ge in the LGPS structure
Sakuma  _et al._ (2016), shows high resistance (4.6 $\Omega$cm2 after one year
Wenzel  _et al._ (2016)) and the requirement for high Li-ion conductivity
across the interface remains a key challenge to be addressed Lotsch and Maier
(2017). In principle neither a Li metal anode nor high-voltage cathodes (both
targeting improved energy density) can be used with LGPS unless coatings are
exploited, as is the case for the liquid electrolytes. Moreover, sulfides show
a well-known hygroscopicity as well as instability in air in the potential
range of normal Li battery operation Tatsumisago  _et al._ (2013); Jung  _et
al._ (2015); Hayashi  _et al._ (2016); Ma  _et al._ (2018), they can provoke
corrosion of the vacuum chamber Kim  _et al._ (2009) and a careful suppression
of hydrolysis is mandatory in order to avoid that these materials decompose
and generate harmful (and lethal) H2S.
All these issues would suggest the need to turn to a safer electrolyte that
does not require the engineering of a coating film protecting the electrodes
(which is also the case for liquid electrolytes), while evaluating if this
would hamper ionic conductivity. Safety, non-toxicity and no hazard-risks are
also the main reasons justifying the interest in solid-state electrolytes, and
when seeking for improved materials’ performance one shouldn’t forget the
original goal of large-scale electrochemical storage, i.e. establishing a
nontoxic, sustainable-energy economy. For the above reasons, oxygen-
substituted Li3PS4 Suzuki  _et al._ (2016); Wang  _et al._ (2016); Takada  _et
al._ (2005); Neveu  _et al._ (2020); Minami  _et al._ (2008) and LGPS-like
conductors Sun  _et al._ (2016); Hori  _et al._ (2016); Kim and Martin (2019),
and in general oxysulfides Wang  _et al._ (2017), as well as non sulphur-
substituted LISICON materials Song  _et al._ (2018); Gilardi  _et al._ (2020),
have been recently considered in the literature. Oxides are expected to be
more electrochemical stable on the cathode side (for example, the cathode
coating for LGPS should be in principle an oxide Zhu  _et al._ (2016); Lotsch
and Maier (2017)), and the chemical stability of LGPS at the anode side is
expected to improve by substituting sulphur with oxygen, which creates
stronger covalent Ge-O bonds Sun  _et al._ (2016). Indeed, chlorine-
substituted LGPS-like oxides show large electrochemical windows up to 9 V Song
_et al._ (2015), and oxysulfides show a wide electrochemical window of 5 V
Suzuki  _et al._ (2016), as predicted by theory Ong  _et al._ (2013). In
addition, partially-oxygen-substituted LGPS shows room-temperature
conductivity only slightly inferior to LGPS, and almost the same activation
energy Sun  _et al._ (2016). The latter result is at variance with
calculations, that for the oxide analogue of LGPS, Li10GeP2O12 (LGPO), predict
almost twice the activation barrier as for LGPS Ong  _et al._ (2013). So far
there is no experimental evidence for this phase of LGPO, but only for a
LISICON material obtained as solid solution of $\gamma$-Li3PO4 and Li4GeO4
Rodger  _et al._ (1985) and having the same composition as LGPO, i.e.
Li4-xGe1-xPxO4 with $x$ = 0.67, whose structure, reported in Ref. Rabadanov
_et al._ (2003), belongs to the same space group as $\gamma$-Li3PO4 $Pnma$,
$\\#$62) and for which the moderate ionic conductivity of $1.8\times 10^{-6}$
Scm-1 at 40oC is reported in Ref. Ivanov-Shitz and Kireev (2003) (see also
Refs. Song  _et al._ (2018); Gilardi  _et al._ (2020)). Thus, an in-depth
investigation on structure and conductivity of the oxygen-substituted LGPS
(tetragonal) and of its experimentally reported orthorhombic phase becomes
important to investigate the feasibility of oxygen substitution to solve the
above safety and instability problems of LGPS. In addition, whereas many
simulations exist for the tetragonal (LGPS) phase Marcolongo and Marzari
(2017); Ong  _et al._ (2013); Mo  _et al._ (2011); Adams and Rao (2012), so
far there have been no theoretical studies on the less conductive, monoclinic
(quasi orthorhombic) thio-LISICON phase Kanno and Murayama (2001). Comparing
oxides and sulfides in different phases would be also important in order to
relate structure and anionic substitution to conductivity in these
superionics.
In this paper we address Li-ion diffusion in Li10GeP2O12 and Li10GeP2S12, both
in the LGPS-like structure (tetragonal, space group $P4_{2}/nmc$, $\\#$137)
and in the thio-LISICON structure (orthorhombic, space group $Pnma$, $\\#$62)
by means of first-principles molecular dynamics simulations. Diffusion
coefficients for the four systems are extracted from long ($\sim$200 ps)
trajectories, generated by CPMD within the NVE, NVT and NPT ensembles. In the
remainder of this paper we will refer to Li10GeP2S/O12 in the tetragonal phase
(reported in Kamaya  _et al._ (2011); Kuhn  _et al._ (2013b) for the sulfide
material) as t-LGPS and t-LGPO respectively, and to Li10GeP2S/O12 in the
orthorhombic phase (reported in Kanno and Murayama (2001) for the sulfide and
in Rabadanov  _et al._ (2003) for the oxide material) as o-LGPS and o-LGPO
respectively.
The paper is organised as follows: in Section II we present the details of the
computational methods together with the results for the equilibrium geometries
and structure relaxation, and details of the electronic structure for the
different systems. In Section III we give results for the diffusion from
molecular dynamics simulations at fixed cell and constant temperature (NVT).
In Section IV results from the variable-cell simulations (NPT) at the chosen
temperature of 600 K are reported. A general discussion and the conclusions
follow in Sec. V and Sec. VI, respectively.

##  II Methods and static calculations

###  II.1 Methods

We use Car-Parrinello (CP) molecular dynamics Car and Parrinello (1985), based
on Kohn-Sham density-functional theory (DFT) Hohenberg and Kohn (1964); Kohn
and Sham (1965) in the plane-wave pseudopotential formalism Payne  _et al._
(1992); Galli and Pasquarello (1993), as implemented in the cp code of the
Quantum ESPRESSO distribution Giannozzi  _et al._ (2009).
Instead of solving self-consistently the Kohn-Sham equations at each MD step
and following a trajectory on the Kohn-Sham Born-Oppenheimer (BO) energy
surface $E^{KS}[\mathbf{R}]$ Kohn and Sham (1965), in CP one follows a
trajectory on the fictitious energy surface of the coupled electron-ion
Lagrangian, that is a functional of both ionic degrees of freedom and
electronic wavefunctions Car and Parrinello (1985):

$\mathcal{L}^{CP}=T_{ions}+T^{fict}_{el}-E^{CP}[\\{\psi_{j}\\},{\mathbf{R}}]-\sum_{jk}\Lambda_{jk}\big{(}\int
d\mathbf{r}{\psi}_{j}(\mathbf{r}){\psi_{k}}(\mathbf{r})-\delta_{jk}\big{)}.$
(1)

Compared to the Lagrangian of the physical system, the Car-Parrinello
Lagrangian (reported in Eq. (1) for the NVE case) contains
$E^{CP}[\\{\psi_{j}\\},{\mathbf{R}}]$ in place of $E^{KS}[\mathbf{R}]$ (i.e.
the “instantaneous” Kohn-Sham energy for wavefunctions $\psi_{j}$ not
necessarily on the Born-Oppenheimer surface), the Lagrange multipliers
$\\{\Lambda_{jk}\\}$ to ensure orthonormality of the electronic wavefunctions,
and the fictitious kinetic energy $T^{fict}_{el}$ of the electronic
wavefunctions

$T^{fict}_{el}=\mu\sum_{i}\int{\dot{\bigl{|}{\psi}_{i}\bigr{|}}}^{2}d\mathbf{r},$
(2)

that has no relation to the physical quantum kinetic energy of the electrons
Remler and Madden (1990), but allows for a dynamical evolution of the Kohn-
Sham states following the ionic motion. The parameter $\mu$, whose
dimensionality is [$m\times l^{2}$], or [$E\times t^{2}$], needs to be chosen
sufficiently small in order to ensure that $T^{fict}_{el}$ is small compared
to the kinetic energy of the ions
$T_{ions}=\displaystyle\sum_{i}^{N_{ions}}\frac{1}{2}m_{i}\mathbf{v}_{i}^{2}$,
so as to avoid an irreversible transfer of energy from the “hot” ions to the
fictitious degrees of freedom, that would gain kinetic energy and move away
from the Kohn-Sham surface during the dynamics Galli and Pasquarello (1993);
Pastore  _et al._ (1991). However, a smaller $\mu$ implies a smaller time step
in the integration of the equations of motion, to ensure accuracy and to keep
numerically correct the constant of motion during the dynamics. In the
Supplemental Material we report these quantities for the NVE simulations of
t-LGPO, and we show that the choice of $\mu=500$ a.u. and time step $dt=4$
a.u. satisfies the criteria discussed above. A similar analysis for t-LGPS is
reported in the Supplemental Material of Ref. Marcolongo and Marzari (2017),
giving the same values for $\mu$ and $dt$. In analogy with Ref. Marcolongo and
Marzari (2017) for t-LGPS, we employ for t-LGPO norm-conserving
pseudopotentials Hamann (2013); Schlipf and Gygi (2015) with a plane-wave
cutoff E${}_{cut}=80$ Ry. Thanks to the release of version 1.1 of the SSSP
library Prandini  _et al._ (2018); Lejaeghere  _et al._ (2016), we could
choose a lower cutoff E${}_{cut}=50$ Ry and ultrasoft pseudopotentials (with
400 Ry cutoff for the electron density) Garrity  _et al._ (2014) using the
Standard Solid State Pseudopotential (SSSP) Efficiency library 1.0 Prandini
_et al._ (2018) (GBRV Garrity  _et al._ (2014) for Li, O and S, PSLib Dal
Corso (2014) for Ge and P) for o-LGPS and o-LGPO. Brillouin zone integrations
are performed using the Gamma point, and the exchange-correlation functional
is PBE Perdew  _et al._ (1996).
In CP Car and Parrinello (1985); Galli and Pasquarello (1993) the electronic
ground state at a given ionic geometry is first reached by performing damped
dynamics Verlet (1967); Frenkel and Smit (2001) for the electronic
wavefunctions up to a certain kinetic energy threshold, that we choose as
1$\times$10-11a.u.. Successively, damped ionic Verlet (1967); Frenkel and Smit
(2001) and cell Parrinello and Rahman (1981) dynamics follows to identify the
relaxed minimum for the ionic positions and cell parameters.

###  II.2 Supercells and relaxed structures

For this work we select two supercells, one for the orthorhombic and one for
the tetragonal case, and adapt each of these to the oxide and to the sulfide
materials. For the orthorhombic structures we build a 100-atom ($1\times
1\times 3$) supercell from the unitary cell (orthorhombic Pnma) reported in
Ref. Rabadanov  _et al._ (2003) for o-LGPO (Fig. 1a), whereas for the
tetragonal structures we use the 50-atom supercell already used in previous
studies Marcolongo and Marzari (2017); Ong  _et al._ (2013), starting from the
crystallographic positions for the tetragonal cell ($P4_{2}/nmc$) of t-LGPS
Kuhn  _et al._ (2013b); Kamaya  _et al._ (2011) (Fig. 1b). We couldn’t find in
the literature (see also Adams and Rao (2012)) space group and atomic
positions of the monoclinic thio-LISICON structure reported in Fig. 2 and
Table I of Ref. Kanno and Murayama (2001) for the solid solution
Li4-xGe1-xPxS4, but we note that this structure is very similar to the
orthorhombic Pnma structure reported for o-LGPO in Ref. Rabadanov  _et al._
(2003) (Fig. 1a). We thus adapt the o-LGPO structure Rabadanov  _et al._
(2003) to the thio-LISICON volume at $x$ = 0.67 Kanno and Murayama (2001), to
build the o-LGPS supercell. Furthermore, we use the volume ratio between the
two orthorhombic cells Rabadanov  _et al._ (2003); Kanno and Murayama (2001)
to determine the t-LGPO volume from the experimental t-LGPS volume Kuhn  _et
al._ (2013b); Kamaya  _et al._ (2011).
In Table 1 we report the calculated lattice parameters and angles for LGPO and
LGPS in the tetragonal and orthorhombic supercells considered in this work,
obtained through CP damped-dynamics relaxations (cfr. Sec. II.1). The
convergence criteria used here are an energy difference between two
consecutive steps below $5\times 10^{-7}$ a.u. ($\simeq 1.4\times 10^{-5}$ eV)
and forces on the ions below $3\times 10^{-5}$ a.u. ($\simeq 1.5\times
10^{-3}$ eV/Å). Since the CP method is restricted to $\Gamma$-only sampling,
we also provide the optimized geometry obtained through variable-cell
relaxation with the Broyden–Fletcher–Goldfarb–Shanno algorithm (pw code
Giannozzi  _et al._ (2009)) with converged grids of $\mathbf{k}$ points (444
or 343) and the same parameters (pseudopotentials and cutoff energy) used in
the CP calculations. The available experimental results Kamaya  _et al._
(2011); Kanno and Murayama (2001); Rabadanov  _et al._ (2003) are also listed,
for comparison. The agreement with the experimental results is in general very
good for the the $\mathbf{k}$-point-converged relaxed cell geometries, but is
also satisfactory for the $\Gamma$-sampled CP-relaxed cell geometries. For
o-LGPS, we point out that the experimental parameters of Ref. Kanno and
Murayama (2001) refer to a (1$\times$3$\times$3) monoclinic cell with
$\theta=0.65$ (here $\theta=0.67$), which might be slightly different from the
(1$\times$1$\times$3) orthorhombic supercell of o-LGPO from Ref. Rabadanov
_et al._ (2003), that we have adapted here to o-LGPS. Finally, in our
calculations no symmetry restrictions are imposed to the cell during the
optimization, leading to quasi-orthorhombic and quasi-tetragonal cells, with
angles very close, but not necessarily equal to $None$.

###  II.3 Density of states and band gaps

The band gap of a superionic material is a useful property that gives an
estimate of its electrochemical window of stability as a solid-state
electrolyte Goodenough and Park (2013). We calculate band structures and
densities of states (DOS) at the Kohn-Sham PBE level (thus, only qualitatively
accurate) and evaluate band gaps for the different structures considered in
this work, with the aim of shedding light on their relative electrochemical
stabilities as electrolytes in a battery. In Figs. 2 and 3 the projected DOS
(p-DOS, i.e. the DOS projected over the different n, l, m components and then
summed for any given atom) are displayed for LGPO and LGPS respectively, both
in the tetragonal and orthorhombic phases. The p-DOS displayed are taken from
the converged SCF density at the relaxed geometry with full
$\mathbf{k}$-points sampling (see Table 1), but very similar results were
obtained from the $\Gamma$-point relaxed geometry (band gaps converged within
0.1 eV). A number of bands $\sim 20\%$ larger than the number of filled bands
is used (302 and 151 bands for the orthorhombic and tetragonal structures,
respectively). Our calculations show the sulfides, having about half the band
gap with respect to the oxides, are thus expected to have significantly
smaller electrochemical windows. Our findings are supported by the
aforementioned experimental results reporting a 9 V electrochemical stability
window for the chlorine-substituted LISICON Li10Si1.5P1.5Cl0.5O11.5 Song  _et
al._ (2015) and a 5 V electrochemical stability window for the oxysulfide
Li3+5xP1-xS4-zOz Suzuki  _et al._ (2016) and also by previous calculations
comparing t-LGPO and t-LGPS Ong  _et al._ (2013).

##  III Fixed-cell NVT and NVE Molecular Dynamics

###  III.1  Details of the simulations

We perform Car-Parrinello molecular dynamics simulations at fixed volume, in
cells defined by the experimental crystallographic data for LGPO and LGPS in
the tetragonal and orthorhombic structures Rabadanov  _et al._ (2003); Kamaya
_et al._ (2011), subsequently relaxed (cfr. Sec. II.2). Simulations are
carried out in the NVT ensemble, with a time step of 4 a.u. ($\sim$ 0.1 fs,
see Sec. II.1). All atoms are free to move and a Nose-Hoover thermostat
Martyna  _et al._ (1992) is attached to the system for at least the first 160
ps.
In the Supplemetal Material we discuss the thermostat parameters (frequency
and atomic-type specificity) and we test the effect of the thermostat on the
simulated diffusion properties for t-LGPO. We choose a species-specific
thermostat working at a frequency of 17 THz and we show that the thermostat
used here doesn’t perturb the dynamics to a significant degree. We thus run
NVT simulations for t-LGPO at 8 temperatures between 600 K and 1200 K (600,
635, 720, 771, 830, 900, 1080, 1200 K), for o-LGPO at 6 temperatures between
600 K and 1200 K (600, 720, 900, 1000, 1100, 1200 K) and for o-LGPS at 4
temperatures between 600 K and 1200 K (600, 720, 900, 1200 K). For t-LGPS we
also compare the NVE data (after 160ps thermostatting) obtained by two of us
with the same method and published in Ref. Marcolongo and Marzari (2017), the
t-LGPS simulations being done at 9 temperatures between 520 and 1200 K.

###  III.2 Tracer diffusion

The mean square displacement of a given ionic species is a measure of its
mobility in a material and can be put in direct relationship with the self-
diffusion coefficient. The latter is also known as (and will be referred to
from now on) the tracer diffusion ($D_{tr}$) coefficient, as it can be
compared to pulsed-field gradient nuclear magnetic resonance experiments Price
(1997). According to the Einstein relation for diffusion Einstein (1905);
Chandler (1987); Frenkel and Smit (2001), the tracer diffusion coefficient in
3 dimensions is:

$D_{tr}^{Li}=\lim_{t\to\infty}\frac{1}{6}\frac{d}{dt}MSD_{tr}^{Li}(t),$ (3)

where the Li-ion tracer mean square displacement $MSD_{tr}^{Li}(t)$ in a
system of $N_{Li}$ ions over a sufficiently long time $t$ (no ballistic
regime) is Chandler (1987)

$MSD_{tr}^{Li}(t)=\frac{1}{N_{Li}}\displaystyle\sum_{i}^{N_{Li}}\big{\langle}\big{|}\mathbf{R}_{i}(t^{\prime}+t)]-\mathbf{R}_{i}(t^{\prime})\big{|}^{2}\big{\rangle}$
(4)

with $\mathbf{R}_{i}$ being the instantaneous position of the $i$-th Li ion
and $\big{\langle}...\big{\rangle}$ an average over the times $t^{\prime}$.
From the NVT simulations (Sec. III.1) we calculate $MSD_{tr}^{Li}(t)$ (Eq.
(4)) and $D_{tr}^{Li}$ (Eq. (3)) at all temperatures considered. For the
method employed to calculate ${MSD}_{tr}^{Li}(t)$, $D_{tr}^{Li}$ and their
statistical uncertainties He  _et al._ (2018) we refer the reader to the
Supplemental Material Frenkel and Smit (2001); Allen and Tildesley (2017). In
Fig. 4 we report ${MSD}_{tr}^{Li}(t)$ and the corresponding $D_{tr}^{Li}$ for
the four structures considered, at the representative temperature of 900 K. At
this temperature we note that diffusion is equally fast, approximately, in
t-LGPO as t-LGPS and o-LGPS, while being much slower in o-LGPO. Results for
the other temperatures (not shown here) display a similar trend. We consider
diffusion here as an activated process, that obeys an Arrhenius law Atkins and
De Paula (2013)

$lnD_{tr}(T)=lnA-\frac{E_{a_{D_{tr}}}}{k_{B}T}$ (5)

where the constant $A$ is related to the attempt frequency whereas the
activation energy $E_{a_{D_{tr}}}$ is the energy barrier for the diffusion. We
report the Arrhenius plot for $D_{tr}^{Li}$, in the widely used form
$logD_{tr}(1000/T)$, in Fig. 5. As observed in Fig. 4 for diffusion at one
temperature, also the temperature dependence of the diffusion coefficients
(Fig. 5) reveals basically two different cases: highly diffusive systems
(t-LGPS, o-LGPS and t-LGPO), and a less diffusive one (o-LGPO). This latter
finding is in qualitative agreement with results from impedance spectroscopy
measurements for o-LGPO, reporting a rather high activation energy of $\sim
0.54$ eV Ivanov-Shitz and Kireev (2003); Gilardi  _et al._ (2020).
From the tracer coefficient $D_{\mathrm{tr}}$ the ionic conductivity $\sigma$
(entirely ascribed to the Li ions) can be calculated according to the Nernst-
Einstein equation:

$\sigma(T)=\frac{N_{Li}Z_{Li}^{2}e^{2}}{Vk_{B}T}D_{tr}=\frac{N_{Li}e^{2}}{Vk_{B}T}D_{tr}.$
(6)

In Eq. (6) $N_{Li}/V$ is the density of the charge carriers (i.e. the Li
ions), $Z_{Li}e=e$ their charge (being $e$ the elementary charge and assuming
that the average Born effective charge is +1, see also Ref. Grasselli and
Baroni (2019)), and $D_{tr}$ is the self-diffusion coefficient from Eq. (3).
From Eq. (6), assuming negligible the change of volume with temperature, the
temperature dependence for $D_{tr}$ is the same as for $\sigma T$.

###  III.3 Collective diffusion

As pointed out in Refs. Marcolongo and Marzari (2017); French  _et al._
(2011); Harris (2010), the tracer diffusion coefficient of Eq. (3) assumes
independent uncorrelated contributions from all the ions. The so-called charge
diffusion coefficient $D_{\sigma}$ France-Lanord and Grossman (2019); Afandak
and Eslami (2017) provides a more realistic estimate, as it accounts for ion-
ion correlated diffusion. For a Li ion in a system of $N_{Li}$ ions, it can be
written France-Lanord and Grossman (2019); Afandak and Eslami (2017):

$D_{\sigma}^{Li}=\lim_{t\to\infty}\frac{MSD_{\sigma}^{Li}(t)}{6t},$ (7)

where

$MSD_{\sigma}^{Li}(t)=\frac{1}{N_{Li}}\displaystyle\sum_{i,j}^{N_{Li}}\big{\langle}\big{[}\mathbf{R}_{i}(t^{\prime}+t)-\mathbf{R}_{i}(t^{\prime})\big{]}\big{[}\mathbf{R}_{j}(t^{\prime}+t)-\mathbf{R}_{j}(t^{\prime})\big{]}\rangle$
(8)

so that the corrected Nernst-Einstein equation Marcolongo and Marzari (2017)
includes the collective $MSD_{\sigma}(t)$ (cfr. Eq. (8)) through $D_{\sigma}$
(cfr.Eq. (7)):

$\sigma(T)=\frac{N_{Li}e^{2}}{Vk_{B}T}D_{\mathrm{\sigma}}.$ (9)

It is worthwhile to mention that $D_{tr}$ and $D_{\sigma}$ have been here
derived in an Einstein form (i.e. from the mean square displacement), whereas
in Ref.Marcolongo and Marzari (2017) the same quantities are reported in the
Green-Kubo formalism Kubo (1957).
The Arrhenius plots for Li-ion tracer (Eq. 3) and charge (Eq. 7) diffusion in
the four systems are displayed in Fig. 6. In Table 2 the activation energies
(Eq. (5)) from $D_{tr}$ and $D_{\sigma}$ are reported, together with the
available experimental results for $\sigma T$ Kamaya  _et al._ (2011); Kanno
and Murayama (2001); Ivanov-Shitz and Kireev (2003) and the results from
first-principles calculations in the literature Ong  _et al._ (2013).
As already noted in Ref. Marcolongo and Marzari (2017) for t-LGPS, also for
t-LGPO, o-LGPS and o-LGPO the activation energy remains approximately
unchanged when including ion-ion correlations, i.e. comparing $E_{a_{D_{tr}}}$
with $E_{a_{D_{\sigma}}}$ (Fig. 6 and Table 2). We also note that the
statistical errors for $D_{\sigma}$ are systematically higher than for
$D_{tr}$, due to the well known slower convergence of the former in the time
averages Marcolongo and Marzari (2017). Fig. 6 and Table 2 confirm that o-LGPO
has moderate conducting properties, in qualitative agreement with experiments
Ivanov-Shitz and Kireev (2003) that place this material among the candidate
solid electrolytes for microbatteries Gilardi  _et al._ (2020). The
diffusivity of tetragonal LGPO, however, is some orders of magnitude superior
with respect to its orthorhombic allotrope Rabadanov  _et al._ (2003); Ivanov-
Shitz and Kireev (2003) at all temperatures, at variance with the results
reported in Ref. Ong  _et al._ (2013) and comparable to its more studied
sulfide analogues Kamaya  _et al._ (2011); Kanno and Murayama (2001), as
partially anticipated by a previous computational study exploiting an ab-
initio trained frozen-lattice potential Kahle  _et al._ (2018). The latter
finding could hopefully open the way to experimental attempts aimed at
synthesizing LGPO in the same phase as t-LGPS Kamaya  _et al._ (2011); Kuhn
_et al._ (2013a).

##  IV Variable$-$cell NPT Molecular Dynamics

###  IV.1 Details of the simulations

Based on the results for diffusion, we aim here at getting more insight into
the relative stability of the oxide and the sulfide systems in the
orthorhombic and tetragonal structures considered in this work. For this
purpose, we set up variable-cell molecular dynamics (VC-MD) simulations for
each of the four systems, allowing both size and shape of the cell to change
during the dynamics, within the Parrinello-Rahman approach, to sample the
isobaric-isothermal ensemble Parrinello and Rahman (1980). The supercell is
coupled to a barostat that keeps constant the pressure of the cell through an
interaction term $-p_{ext}V$ (where $p_{ext}$ is the desired external
pressure, here = 0, and $V$ is the volume of the cell), while adding to the
Lagrangian a kinetic energy term that accounts for the cell motion Andersen
(1980); Parrinello and Rahman (1980). The atomic positions are expressed in
scaled coordinates in the basis of the time-dependent cell vectors Parrinello
and Rahman (1981). The mass of the barostat is chosen in such a way that the
frequency of the cell volume fluctuations multiplied by $V^{1/3}$ is of the
order of the free-particle sound velocity $\sqrt{(k_{B}T)/M}$ Parrinello and
Rahman (1981); Nosé and Klein (1983), where $M$ is the total mass of the atoms
in the cell, as suggested in the original work by Andersen Andersen (1980). In
addition, temperature is controlled during the simulations through a Nose-
Hoover thermostat Martyna  _et al._ (1992) already discussed in Section III.1;
we choose the temperature of 600 K for these simulations.

###  IV.2 Volumes and energies

The electronic total energy and volume fluctuations per formula unit during
the NPT simulations are displayed in Fig. 7 and Fig. 8 respectively, for the
four structures considered. One formula unit here is Li10GeP2O12 or
Li10GeP2S12, i.e. 25 atoms. In Table 3 we report the mean values for
geometries, electronic total energies and enthalpies. For all systems the
average internal pressure during the dynamics is of the order of $10^{-2}$ GPa
or less.
Each of the four structures is found to be stable during the NPT simulations,
and no hint of a phase transition is observed (Figs. 7 and 8). However,
whereas for the sulfide system the orthorhombic and tetragonal structures show
similar energetics (Fig. 7b), for the oxide system (Fig. 7a) the orthorhombic
structure is clearly lower, as also reported in Table 3 for
$\left<E_{TOT}\right>$ and $\left<H\right>$. The trend
$\left<E_{TOT}\right>$(o-LGPO)$<\left<E_{TOT}\right>$(t-LGPO) is observed
systematically also in the NVT simulations of Sec. III, but the NPT
simulations show in addition that this trend is unchanged, at least at 600 K,
even if the cell is let free to vary its volume and shape (of course, we
compare here $\left<E\right>$, but the thermodynamic functional is the Gibbs
free energy $G=E-TS+PV$). The stability of the orthorhombic structure for the
oxide can be seen as a further confirmation that LGPO in a tetragonal phase
would be less stable and that partial sulfur substitution by oxygen was
possible in tetragonal LGPS only up to $x=0.9$ in Li10GeP2S(12-x)Ox Sun  _et
al._ (2016).
For cell parameters (Table 3), in the case of the oxide the volume
fluctuations are larger for the tetragonal than for the orthorhombic
structure, whereas they have comparable amplitude in the sulfide. Also, for
the tetragonal phases one observes that the in-plane lattice parameters $a$
and $b$ display stronger oscillations in the oxide with respect to the
sulfide, and the former shows a larger fractional difference between
$\left<a\right>$ and $\left<b\right>$ than the latter (0.7% instead of 0.09%).
These results point to a slight deviation from tetragonality for the oxide,
with the simulations happy to switch between two slightly-non-tetragonal
equivalent forms. In Fig. 9 we show $a$, $b$ and $c$ for the 50-atom t-LGPO
supercell described above and for the 200-atom ($2\times 2\times 1$) t-LGPO
supercell that we also simulate (NPT, 100 ps) for this purpose: while $a$ and
$b$ often switch in the 50-atom supercell (Fig. 9(a)), in the 200-atom cell
simulation these fluctuations are suppressed (Fig. 9(b)).

###  IV.3 Stability of t-LGPO vs o-LGPO

The NVT results for diffusion (Sec. III and Figs. 5 and 6) show that one could
in principle obtain the same conductivity in t-LGPO as in o-LGPS or t-LGPS,
i.e. from the oxide analogue of t-LGPS. However, results from the NPT
simulations presented in this Section (Fig. 7 and Table 3) reiterate that this
phase, though being dynamically stable over 200 ps, is less favoured with
respect to its orthorhombic allotrope o-LGPO Rabadanov  _et al._ (2003); Song
_et al._ (2018). It thus becomes appealing to investigate the possibility of a
transition from the more stable and less conductive o-LGPO to the metastable
and more conductive t-LGPO. Neglecting entropy, a first step in such
investigation is to extract representative configurations from the NPT
dynamics of the two LGPO structures and construct equations of state
$E_{TOT}(V)$ by isotropically compressing and expanding the cells at each of
these configurations. The location of a possible crossing between a t-LGPO and
an o-LGPO $E_{TOT}(V)$ would suggest whether compression or expansion could
induce a transition from t-LGPO to o-LGPO or viceversa. A set of 16 such
equations of state (8 for t-LGPO and 8 for o-LGPO) is displayed in Fig. 10(a).
For each phase the 8 configurations are chosen uniformly in the range of
$E_{TOT}$ (Fig. 7a) and for each configuration the maximum
expansion/compression of the cell volume is $\pm$10%. In addition, we fit the
values of $E_{TOT}(V)$ in Fig. 10(a) to a Murnaghan equation of state Tyuterev
and Vast (2006); Murnaghan (1944); Giannozzi  _et al._ (2009), from which we
calculate the pressure $p$ and the enthalpy $H$. The enthalpy-pressure curves
are reported in Fig. 10(b). From both the energy-volume and enthalpy-pressure
curves (Fig. 10) it is not straightforward to get unambiguous information
about whether a cell compression or expansion could help a transition between
t-LGPO and o-LGPO: in particular in Fig. 10(b) one can see that in general the
enthalpies have a similar slope, as a function of pressure, in the two phases.
We can extract the same information directly from the VC-MD dynamics. In Fig.
11 we report the enthalpy-pressure data sampled at every 20 time steps of the
VC-MD simulations for the two phases, together with a linear fit for each of
them. It is clear that t-LGPO has univocally a higher enthalpy than o-LGPO for
an extended range of pressures. Fig. 11 might show that this trend could be
inverted at $p\ll 0$, so that an expansion should favour the tetragonal phase
more than a compression, but in general the enthalpy cost, if ascribed
entirely to the 10 Li atoms in the formula unit, is around 100 meV/atom (see
Table 3), i.e. affordable at temperatures of $\sim$1100K. Eventually,
considering the free energy $G=H-TS$, one should stress that the entropic
contribution, stronger for the more diffusive phase, could tilt the
thermodynamic balance in favour of t-LGPO, at large enough temperatures.

##  V  Discussion

Table 2 reports the activation energies for diffusion (Eq. (5)) in the four
LGPO and LGPS structures examined in this work, compared with the available
experiments Kanno and Murayama (2001); Kuhn  _et al._ (2013a); Ivanov-Shitz
and Kireev (2003); Gilardi  _et al._ (2020). It should be noted first that
many physical parameters can affect the bulk ionic conductivity in impedance
spectroscopy measurements Uddin and Cho (2018), that the polycrystalline
texture of the samples introduces many grain-boundary effects, and that on the
computational side the Nernst-Einstein equation (Eq. (6)), even including
Haven ratios Murch (1982) and the effects of the ionic correlations (Eq. (9),
Marcolongo and Marzari (2017)), assumes the oxidation number of the Li atoms
to be constant ($Z=+1$) during the dynamics French  _et al._ (2011); Grasselli
and Baroni (2019). Comparison with the experimental results should thus be
regarded as a guide rather than an ultimate benchmark. To better clarify this
comparison, we report in Fig. 12(a) the $\sigma T$ Arrhenius plots from our
calculations (Eqs. (5) and (6)) for t-LGPS, o-LGPS and o-LGPO together with
the corresponding experimental values Kanno and Murayama (2001); Kamaya  _et
al._ (2011); Ivanov-Shitz and Kireev (2003); Gilardi  _et al._ (2020), and the
computational one for t-LGPO. For t-LGPS and o-LGPO we note a similar
underestimation ($\sim 60-70\%$) of the activation barrier with respect to the
experiments Kamaya  _et al._ (2011); Ivanov-Shitz and Kireev (2003); Gilardi
_et al._ (2020), that might be related to the different ranges of temperatures
covered - our results being at 500 K - 1200 K and the experimental results at
150 K - 700 K. For o-LGPS the activation barrier for conductivity is in close
agreement with the experimental value, whereas the absolute values of
log($\sigma T$) are systematically higher (see Fig. 12(a)). The latter result
is likely related to the differences between the cell of Ref. Kanno and
Murayama (2001) and the cell that we have used for the simulations, that is
the LGPS-adapted o-LGPO cell from Refs. Rabadanov  _et al._ (2003); Ivanov-
Shitz and Kireev (2003), as details of the genuine II Thio-LISICON structure
Kanno and Murayama (2001) are not available (see also Adams and Rao (2012)).
We also recall that, as shown in Table 1, convergence of the lattice
parameters is achieved with a (343) $\mathbf{k}$-points mesh, whereas, due to
an already high numerical cost, we use for these simulations the
$\Gamma$-sampled 100-atom supercell. Nevertheless, extrapolating the t- and
o-LGPS and the o-LGPO results at room temperature, we still reproduce the
experimentally observed rank $\sigma$(o-LGPO)$\ll$ $\sigma$(o-LGPS) $<$
$\sigma$(t-LGPS) Ivanov-Shitz and Kireev (2003); Kanno and Murayama (2001);
Kamaya  _et al._ (2011); Gilardi  _et al._ (2020). We conclude that the
present results, in good agreement with the available experimental literature
for what concerns the absolute conductivities of each structure, reproduce
well the relative conductivity trends of the experimentally known structures
(Fig. 12(a) and Table 2).
The agreement of theoretical and experimental conductivities for the first
three cases can serve as the ground to discuss conductivity in the fourth
case, which is, among those studied in this paper, the only one that has not
been experimentally found so far, i.e. the oxide-analogue of t-LGPS (t-LGPO).
In Fig. 12(b) an enlargement of the CPMD results displayed in Fig. 12(a) is
reported. t-LGPO would be a very conductive material, with a $\sigma T$
Arrhenius behaviour very similar to the one of o-LGPS. This finding is at
variance with the diffusion results for t-LGPO from a previous first-
principles calculation Ong  _et al._ (2013), where Li-ion diffusivity in this
system is reported to be significantly lower than in t-LGPS, with a much
higher activation barrier of 0.36 eV (see also Table 2). Interestingly enough,
we find instead an activation barrier of 0.22 eV which is very close to the
theoretical activation barriers of 0.18 eV and 0.23 eV for t-LGPS and o-LGPS
respectively, and much smaller than the activation energy of 0.34 eV for
o-LGPO (see Table 2 and Fig. 12(b)).
t-LGPO, though remaining stable in a 200 ps simulation, is thermodynamically
unfavoured with respect to o-LGPO (Fig. 7), and in fact only o-LGPO has been
synthesized so far Ivanov-Shitz and Kireev (2003); Rabadanov  _et al._ (2003);
Song  _et al._ (2018); Gilardi  _et al._ (2020). A solid solution starting
from t-LGPS (Li10GeP2S(12-x)Ox) was shown to be stable only up to $x=0.9$ Sun
_et al._ (2016), showing that preparing t-LGPO via solid-solution synthesis
doesn’t seem a feasible method, due to the difference in ionic radii between
oxygen and sulphur (see also Fig. 8). In the case of the solid solution
Li3+5xP1-xS4-zOz, a highly conductive tetragonal structure has been obtained
up to $z=0.8$ thanks to rapid quenching of the molten mixture from T $>$ 700
∘C Suzuki  _et al._ (2016). A totally different procedure would be to start
from the experimental o-LGPO Ivanov-Shitz and Kireev (2003); Rabadanov  _et
al._ (2003); Song  _et al._ (2018); Gilardi  _et al._ (2020) and try to induce
a phase transition from orthorhombic to tetragonal by either imposing a
negative hydrostatic pressure or introducing a tensile in-plane strain via
epitaxial growth, as pointed out in Sec. IV.3 (see also Donner  _et al._
(2011)). Most importantly entropy, being in principle higher for the more
conductive phase t-LGPO, could tilt the thermodynamic balance between the two
phases. We leave the latter consideration to a further study.

##  VI  Summary and conclusions

Sulfide crystalline lithium ionic conductors, in particular within the thio-
LISICON family Kanno and Murayama (2001); Kamaya  _et al._ (2011), have shown
optimal conductive properties that are in general superior to the oxide
materials originally proposed within the same family (LISICONs Hong (1978);
Rodger  _et al._ (1985); Fujimura  _et al._ (2013)). They also display severe
practical hindrances Kerman  _et al._ (2017), starting from the need of
strictly controlling the atmosphere in which they are processed, in order to
avoid releasing toxic H2S Jung  _et al._ (2015) and consequent degradation in
conductivity Muramatsu  _et al._ (2011). Recent experimental work shows that
H2S generation is significantly suppressed after adding Li2O or P2O5 to the
Li2O-Li2S-P2S5 glass Ohtomo  _et al._ (2013a); Minami  _et al._ (2008). In
this respect, new investigations on the oxide-analogue crystalline materials,
potentially less conductive but displaying superior safety features, are
highly desirable.
The purpose of this paper is to compare ionic conductivities and relative
stability of crystalline LGPS with those of its oxide analogue, LGPO. LGPS has
been extensively studied so far, both experimentally Kuhn  _et al._ (2013a,
b); Kato  _et al._ (2014); Kwon  _et al._ (2015); Han  _et al._ (2016); Weber
_et al._ (2016); Sun  _et al._ (2017); Wenzel  _et al._ (2016); Yang  _et al._
(2015); Zhang  _et al._ (2017) and theoretically Adams and Rao (2012); Du  _et
al._ (2014); Hu  _et al._ (2014); Mo  _et al._ (2011); Ong  _et al._ (2013);
Wang  _et al._ (2015); Xu  _et al._ (2012); Zhu  _et al._ (2016); Marcolongo
and Marzari (2017), in its highly conductive tetragonal phase (space group
$P4_{2}/nmc$, $\sigma=1.2\times 10^{-2}$ $\mathrm{S}\mathrm{cm}^{-1}$ at room
temperature Kamaya  _et al._ (2011), here denoted t-LGPS), but it is known
from experiments to exist also in a monoclinic allotrope (space group
$P2_{1}/m$, here denoted o-LGPS), that also shows high conductivity
($\sigma\sim 10^{-3}$ $\mathrm{S}\mathrm{cm}^{-1}$ at room temperature) and
whose cell is related to the orthorhombic parent lattices Li4GeS4 and Li3PS4
Kanno and Murayama (2001). On the other hand, LGPO is known from experiments
to exist in its moderately conducting (1.8$\times 10^{-6}$
$\mathrm{S}\mathrm{cm}^{-1}$ at 40 oC Ivanov-Shitz and Kireev (2003); Gilardi
_et al._ (2020)) orthorhombic phase (space group $Pnma$, here denoted o-LGPO)
Rabadanov  _et al._ (2003); Song  _et al._ (2018); Rodger  _et al._ (1985);
Muy  _et al._ (2018), that has been also studied in simulations using
classical molecular dynamics Muy  _et al._ (2018) and machine-learning
techniques Fujimura  _et al._ (2013), but no experimental evidence of an
analogue tetragonal phase, that we call here t-LGPO, has been reported in the
literature so far.
We have calculated Li-ion diffusion and conductivity of t- and o-LGPS and t-
and o-LGPO via first-principles molecular dynamics (Car-Parrinello, plane
waves, ultrasoft pseudopotentials Giannozzi  _et al._ (2009)), with $\sim$200
ps NVT simulations for the four structures at 600, 720, 900, 1000, 1100 and
1200 K. We have also performed variable-cell NPT simulations at T = 600 K for
the four cases studied, with the aim of comparing their relative stabilities.
Results for the conductivity for the three experimentally studied structures
agree well with the experiments. t-LGPO, i.e. the oxide-analogue of t-LGPS
(that has not been synthesized so far) reveals itself to be a fast ionic
conductor, with an Arrhenius behaviour for Li-ion conductivity in this system
similar to the one in o-LGPS and a conductivity at room temperature comparable
to the conductivities of o- and t-LGPS calculated here. In addition, t-LGPO,
being an oxide, is not expected to carry the stability problems encountered in
sulfide electrolytes Kerman  _et al._ (2017) for solid-state batteries, and
could represent a relevant and important challenge for experimental synthesis
and stabilization.

##  VII Acknowledgments

This research was supported by the Swiss National Science Foundation, through
project 200021-159198 and the MARVEL NCCR. We acknowledge computational
support from the Swiss National Supercomputing Centre CSCS (projects s836 and
mr28). Fruitful discussions with Tommaso Chiarotti and Claire Villevieille are
gratefully acknowledged.

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(a)

(b)

Figure 1: The 100-atom and 50-atom supercells (side and top view) used in the
simulations for (a) o-LGPO and (b) t-LGPO from Refs. Rabadanov  _et al._
(2003) and Kamaya  _et al._ (2011), respectively (see text). Li atoms are
displayed in green, O atoms are in red, and Ge and P atoms are at the center
of the dark and light purple tetrahedra, respectively. The analogous LGPS
supercells have sulphur atoms replacing oxygen atoms.

(a)

(b)

Figure 2: Projected DOS for a) t-LGPO and b) o-LGPO at the PBE level. Energies
are reported with respect to the highest occupied level. A Kohn-Sham band gap
larger than 4 eV is present.

(a)

(b)

Figure 3: Same as Fig. 2 for a) t-LGPS and b) o-LGPS. Energies are reported
with respect to the highest occupied level. A Kohn-Sham band gap of $\sim$2 eV
is present.

(a) t-LGPS

(b) o-LGPS

(c) t-LGPO

(d) o-LGPO

Figure 4:  Li-ion tracer mean square displacements in the four structures
considered in this work, at 900K (NVE simulations for t-LGPS Marcolongo and
Marzari (2017), NVT simulations for t-LGPO, o-LGPS and o-LGPO).

Figure 5: Arrhenius plots and corresponding activation energies for Li-ion
tracer diffusion coefficients in the four structures considered in this work.

(a) t-LGPS

(b) o-LGPS

(c) t-LGPO

(d) o-LGPO

Figure 6: Arrhenius plots for Li-ion tracer and charge ($\sigma$) diffusion in
the four structures studied.

(a) t- and o-LGPO

(b) t- and o-LGPS

Figure 7: Electronic total energy per formula unit (25 atoms) from variable-
cell molecular dynamics calculations (NPT at 600 K) for the orthorhombic and
tetragonal phases (in red and green colour respectively) of LGPO (a) and LGPS
(b). See also Table 3.

(a) t- and o-LGPO

(b) t- and o-LGPS

Figure 8: Volume per formula unit (25 atoms) from variable-cell molecular
dynamics simulations (NPT at 600 K) for the orthorhombic and tetragonal phases
(in red and green colour respectively) of LGPO (a) and LGPS (b). See also
Table 3.

(a)

(b)

Figure 9: Lattice parameters for t-LGPO during the NPT simulations: a) for the
50-atom supercell (see also Table 3); b) for the 200-atom (2x2x1) supercell
(see text).

(a)

(b)

Figure 10: Energy-volume (a) and enthalpy-pressure (b) relations (see text)
for t-LGPO and o-LGPO, reported in green and red respectively.

Figure 11: Enthalpy-pressure relations as obtained from NPT simulations at 600
K. t-LGPO and o-LGPO results are reported in green and red, respectively.

(a)

(b)

Figure 12: (a) Arrhenius plots for $\sigma$T (Li-ion conductivity $\sigma$
multiplied by T) from Eqs. (5) and (6) in t-LGPS, o-LGPS, t-LGPO and o-LGPO
from NVT CPMD simulations, and compared with the experimental results for the
sulfides and o-LGPO. CPMD simulations (here) and experiments (Refs. Kuhn  _et
al._ (2013a); Kanno and Murayama (2001); Ivanov-Shitz and Kireev (2003);
Gilardi  _et al._ (2020)) are highlighted in the blue-gray and light-gray
ovals, respectively. (b) Enlargement of (a) for the CPMD simulations.

|  |  a |  b |  c |  $\alpha$ |  $\beta$ |  $\gamma$
---|---|---|---|---|---|---|---
t-LGPS |  $\Gamma$ |  8.764 |  8.781 |  12.943 |  91.17 |  91.20 |  90.42
|  (444) |  8.661 |  8.819 |  13.072 |  91.21 |  90.51 |  90.33
|  Kamaya  _et al._ (2011) |  8.718 |  8.718 |  12.634 |  90.00 |  90.00 |  90.00
t-LGPO |  $\Gamma$ |  6.836 |  7.191 |  10.433 |  89.43 |  90.56 |  89.97
|  (444) |  6.769 |  7.214 |  10.501 |  89.57 |  90.49 |  89.95
|  |  |  |  |  |  |
o-LGPS |  $\Gamma$ |  12.770 |  8.537 |  18.330 |  89.82 |  89.33 |  90.24
|  (343) |  13.604 |  7.888 |  18.684 |  90.27 |  90.33 |  90.17
|  Kanno and Murayama (2001) |  13.393 |  22.947 |  18.201 |  90.00 |  92.562 |  90.00
o-LGPO |  $\Gamma$ |  10.604 |  6.398 |  15.052 |  89.71 |  90.50 |  90.31
|  (343) |  10.788 |  6.223 |  15.147 |  89.69 |  90.63 |  90.15
|  Rabadanov  _et al._ (2003) |  10.690 |  6.195 |  5.027 |  90.00 |  90.00 |  90.00
Table 1: Cell geometries (lattice parameters and angles) for the four systems
studied in this work ($\Gamma$-sampling in CP and fully converged
$\mathbf{k}$-point sampling with a (444) and a (343) Monkhorst-Pack mesh for
the t- and o- structures, respectively) compared with the available
experimental values (see text). Lengths are in Å and angles in degrees. t-LGPS
and t-LGPO cells have 50 atoms, o-LGPS and o-LGPO cells have 100 atoms. See
text for Ref. Kanno and Murayama (2001).

|  $E_{a_{D_{tr}}}$ |  $E_{a_{D_{tr}}}$ |  $E_{a_{D_{\sigma}}}$ |  $E_{a_{\sigma T}}$
---|---|---|---|---
|  |  (Ref. Ong  _et al._ (2013)) |  |  (expt.)
t-LGPS |  0.18 $\pm$ 0.02 |  0.21 $\pm$ 0.04 |  0.17 $\pm$ 0.07 |  0.26 Kamaya  _et al._ (2011)
o-LGPS |  0.23 $\pm$ 0.02 |  |  0.21 $\pm$ 0.04 |  0.24 Kanno and Murayama (2001)
t-LGPO |  0.21 $\pm$ 0.02 |  0.36 $\pm$ 0.05 |  0.21 $\pm$ 0.06 |
o-LGPO |  0.34 $\pm$ 0.04 |  |  0.32 $\pm$ 0.07 |  0.54 Ivanov-Shitz and Kireev (2003)
Table 2: Activation energies for the tracer diffusion and charge diffusion (in
eV) for the four systems studied in this work (see text), compared with
experiments and previous calculations.

| $\left<a\right>$ | $\left<b\right>$ | $\left<c\right>$ | $\left<\alpha\right>$ | $\left<\beta\right>$ | $\left<\gamma\right>$ | $\left<V\right>$ | $\left<E_{TOT}\right>$ | $\left<H\right>$
---|---|---|---|---|---|---|---|---|---
t-LGPS | 9.01 $\pm$ 0.18 | 9.01 $\pm$ 0.18 | 13.12 $\pm$ 0.25 | 90.0 $\pm$ 2.3 | 89.9 $\pm$ 2.3 | 90.1 $\pm$ 1.7 | 531.5 $\pm$ 13.4 | -340.359 $\pm$ 0.010 | -340.361 $\pm$ 0.072
o-LGPS | 12.85 $\pm$ 0.20 | 8.86 $\pm$ 0.12 | 18.72 $\pm$ 0.27 | 90.1 $\pm$ 1.1 | 90.1 $\pm$ 1.8 | 90.1 $\pm$ 1.3 | 533.0 $\pm$ 8.8 | -340.356 $\pm$ 0.007 | -340.357 $\pm$ 0.053
t-LGPO | 7.15 $\pm$ 0.31 | 7.20 $\pm$ 0.30 | 10.72 $\pm$ 0.27 | 90.1 $\pm$ 2.2 | 90.0 $\pm$2.1 | 90.1 $\pm$ 1.3 | 275.0 $\pm$ 7.1 | -374.053 $\pm$ 0.010 | -374.055 $\pm$ 0.087
o-LGPO | 10.69 $\pm$ 0.12 | 6.50 $\pm$ 0.08 | 15.42 $\pm$ 0.18 | 90.0 $\pm$ 0.8 | 89.9 $\pm$ 1.0 | 90.1 $\pm$ 0.9 | 267.9 $\pm$ 3.5 | -374.092 $\pm$ 0.007 | -374.093 $\pm$ 0.059
Table 3: Average lattice parameters, angles, volumes, total energies and
enthalpies from NPT dynamics at T = 600 K for the four structures considered.
Lengths are in Å, angles in degrees and energies in Hartree per formula unit.
Formula units contain 25 atoms: Li10GeP2S12 for t- and o-LGPS and Li10GeP2O12
for t- and o-LGPO (in the calculations tetragonal supercells contain 50 atoms
and orthorhombic supercells contain 100 atoms). Errors are calculated as mean
standard deviations in a block analysis, as done for diffusion (see
Supplemental Material and Refs. Frenkel and Smit (2001); Allen and Tildesley
(2017)). See also Figs. 7 and 8.