# Intra-atomic Hund’s exchange interaction determines spin states and
energetics of Li-rich layered sulfides for battery applications

Jae-Hoon Sim  Centre de Physique Théorique, Ecole Polytechnique, CNRS,
Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France  D. D. Sarma
Solid State and Structural Chemistry Unit, Indian Institute of Science,
Bengaluru 560012, India  Jean-Marie Tarascon  Collège de France, 11 place
Marcelin Berthelot, 75005 Paris, France  Sorbonne Université, Paris, France
Réseau sur le Stockage Electrochimique de l’Energie (RS2E), Amiens, France
Silke Biermann  Centre de Physique Théorique, Ecole Polytechnique, CNRS,
Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France  Collège de
France, 11 place Marcelin Berthelot, 75005 Paris, France  Department of
Physics, Division of Mathematical Physics, Lund University, Professorsgatan 1,
22363 Lund, Sweden  European Theoretical Spectroscopy Facility, 91128
Palaiseau, France, Europe

(August 28, 2024)

###### Abstract

Motivated by experimental suggestions of anionic redox processes helping to
design higher energy lithium ion-battery cathode materials, we investigate
this effect using first-principles electronic structure calculations for Li-
rich layered sulfides. We identify the determination of the energetic
contribution of intra-atomic Hund’s exchange coupling as a major obstacle to a
reliable theoretical description. We overcome this challenge by developing a
particularly efficient flavor of charge-self-consistent combined density
functional + dynamical mean-field theory (DFT+DMFT) calculations. Our scheme
allows us to describe the spin ground states of the transition metal $d$
shell, the electronic structure of the materials, and its energetics. As a
result of the high-spin to low-spin transition the average intercalation
voltage shows intriguing non-monotonic behavior. We rationalize these findings
by an analysis of the fluctuations of spin and charge degrees of freedom. Our
work demonstrates the relevance of most recent insights into correlated
electron materials for the physics of functional materials such as Li-ion
battery compounds.

##  I Introduction

Many modern technologies such as mobile devices or electric cars hinge upon
the development of high-energy density batteries. Finding cathode materials
fulfilling the mandatory criteria of being safe and inexpensive with high
capacity and intercalation voltage is a bottleneck in the field. A
traditional, extensively studied and commercially successful example is LiCoO2
Mizushima  _et al._ (1980). While for many years, the main strategy for
cathode materials in Li-ion batteries purely relied on cationic redox,
lithium-rich layered manganese oxides have recently attracted great interest,
involving both cationic and anionic redox processes Lu  _et al._ (2001); Assat
and Tarascon (2018); Li and Xia (2017). The lithium-rich layered manganese
oxide Li2MnO3 is the parent compound of the currently used
Li[LixMnyNizCo1-x-y-z]O2, which are famous for their reversible high
capacities exceeding 250 mAh/g Li and Xia (2017); Assat and Tarascon (2018);
Thackeray  _et al._ (2007). Undesired consequences of anionic redox processes
are, however, potential capacity loss and structural degradation, as well as
hysteresis.

Recently, new Li-rich layered sulfides Lix[Li0.33-2y/3Ti0.67-y/3Fey]S2 have
been reported Saha  _et al._ (2019). Its negligible cycle irreversibility,
mitigated voltage fade upon long cycling, low voltage hysteresis, and fast
kinetics suggest a new direction to alleviate the practical limitations of
using the anionic redox mechanism. Motivated by their experimental
realization, we perform first-principles calculations for the above-mentioned
Li-rich layered sulfides, using as prototypes the fully lithiated and
delithiated materials. We focus on the $y=1/3$ case, i.e.,
Lix+0.11Ti0.56Fe0.33S2 (LTFSx). Note that the value $y=1/3$ is close to the
optimal value reported in experiment, namely $y=0.3$ Saha  _et al._ (2019).

First-principles electronic structure calculations based on density functional
theory (DFT) within the local density approximation (LDA) and generalized
gradient approximation (GGA) have developed into a tremendously useful tool
for addressing materials properties and even helping design functional
materials. Nevertheless, these approximations have known limitations in
describing materials containing open $d$\- or $f$-shells with sizable Coulomb
interactions. Indeed, the standard LDA/GGA description based on the
delocalized electron gas picture is not an optimal starting point for the
rather localized behavior of $d$\- or $f$-electrons. In the materials of
interest here, as we will see below, a particular challenge is the necessity
of capturing the spin-state of these localized electrons correctly, since the
energetics of the materials depend on the corresponding exchange contribution.
A second limitation arises for properties that involve excited states, which
are in principle inaccessible to standard DFT, even if the exact ground state
energy functional was available. Over the last decades, combinations of
LDA/GGA with many-body techniques have evolved into extremely useful and well-
established tools overcoming these limitations. Among these, DFT plus
dynamical mean-field theory (DFT+DMFT) Metzner and Vollhardt (1989); Georges
and Kotliar (1992); Zhang  _et al._ (1993); Anisimov  _et al._ (1997);
Lichtenstein and Katsnelson (1998) and DFT$+U$ Anisimov  _et al._ (1991);
Liechtenstein  _et al._ (1995) stand out as methods of choice.

Here, we suggest a new flavor of the charge-self-consistency in DFT+DMFT
methods, to describe the observed electronic ground states and the competing
phases of our target materials, namely a high-spin (HS) Mott insulator, a
correlated metal, and a low-spin (LS) band insulator phase. We show that the
energetics of the cathode materials largely depends on the spin ground state,
revealing that the impact of the Hund’s coupling is not only of theoretical
interest but has practical implications for battery materials. These findings
place our target materials in the context of the nowadays celebrated “Hund’s
materials”, where intra-atomic exchange amplifies the consequences of strong
local Coulomb interactions de’ Medici  _et al._ (2011); Georges  _et al._
(2013). They should be viewed as strongly correlated electron systems, the
properties of which can be described in the language of modern first-
principles many-body techniques.

The paper is organized as follows. In Sec. II, we present details of our
calculational techniques. In particular, we have developed a new flavor for
DFT+DMFT that takes into account charge redistributions due to correlation
effects. In Sec. III, the results of our calculations are presented and
comparisons between static (DFT$+U$) and dynamic (DFT+DMFT) mean field
approximations to electronic Coulomb correlations are made. Finally, a summary
and some conclusions are given in Sec. IV.

##  II Methods and formalism: charge-self-consistent DMFT scheme

Within the DFT+DMFT framework, the charge-density $\rho(r)$ and the local
Green’s function $G_{\rm loc}(i\omega_{n})$ are determined by the stationary
condition of the free energy functional $\Gamma[\rho,G_{\rm loc}]$ Georges
_et al._ (1996). Single-shot (i.e. non-charge-self-consistent) DFT+DMFT
calculations are performed in many works in the literature, where $\rho$ is
given from LDA/GGA calculations. In the case of the LTFS, however, due to the
larger covalency between S-$p$ and Fe-$d$ compared to oxides, it is important
to take into account the effect of the charge redistribution due to
correlation effects. The importance of the charge redistribution is
demonstrated by the fact that conventional one-shot DFT+DMFT with fixed DFT
charge density predicts a significantly underestimated voltage as we will
discuss in Sec. III. On the other hand, given the large unit cells with low
symmetry, fully charge-self-consistent DFT+DMFT calculations are hardly
accessible due to the computational cost.

In this paper, we use an efficient way to calculate the total energy given as

$\displaystyle E_{{\rm DFT}+{\rm DMFT}}=$ $\displaystyle E_{{\rm
GGA}}[\rho_{U}]-\braket{H_{{\rm
KS}}^{GGA}[\rho_{U}]}_{\rho_{U}}+\operatorname{Tr}(H_{{\rm KS}}[\rho_{U}]G)$
$\displaystyle+\frac{1}{2}\operatorname{Tr}\Sigma G-E_{{\rm DC}}^{{\rm
FLLnS}},$ (1)

where the GGA energy functional $E_{{\rm GGA}}[\rho_{U}]$ and
$H_{KS}^{GGA}[\rho_{U}]=\frac{\delta E_{{\rm
GGA}}}{\delta\rho(r)}\bigr{|}_{\rho=\rho_{U}}$ are evaluated at the ground
state density $\rho_{U}(r)$ obtained from GGA$+U$ calculations. The double
counting $E_{{\rm DC}}^{{\rm FLLnS}}$ is taken to be the spin-independent
"fully localized limit" double counting using the spin-averaged occupations
Anisimov  _et al._ (1993), dubbed "FLL-nS" in Ref. Ylvisaker  _et al._ (2009).

We stress that $\rho_{U}$ differs from the ground state density $\rho_{0}$ of
$H_{{\rm KS}}[\rho_{U}]$ due to the absence of the $+U$ contribution in
$E_{{\rm GGA}}$ and $H_{KS}^{GGA}$. One can understand our scheme as follows:
i) First, charge-self-consistent DFT+DMFT calculations are performed with a
Hartree-Fock impurity solver. ii) Given the resulting charge density, we
produce the Hamiltonian $H_{{\rm HK}}+H_{{\rm int}}$ on which the DMFT
calculations are performed with a more sophisticated impurity solver, namely
continuous-time quantum Monte Carlo (CT-QMC) simulations Werner  _et al._
(2006); Werner and Millis (2006). This approach is not only computationally
efficient, but also useful for comparing the results to the DFT$+U$
calculations. Because both DFT$+U$ and +DMFT use the same charge density
$\rho_{U}$, it is easier to investigate the correction stemming from the
dynamical self-energy, without ambiguity. Once we use the HF solver in step
ii), for example, $E_{{\rm DFT+DMFT}}$ exactly reduces to the DFT$+U$ results.

In Eq. (1), we have a adopted charge-only-dependent exchange-correlation GGA
functional and FLL-nS double counting. The FLL-nS double counting scheme has
been introduced within the DFT$+U$ functional Ylvisaker  _et al._ (2009) (cFLL
in the notation of Ref. Ryee and Han, 2018):

$E_{{\rm DFT}+U}[\rho_{U}]=E_{{\rm DFT}}[\rho_{U}]+E_{{\rm
int}}^{U}[n_{\alpha\beta}^{\sigma}]-E_{{\rm DC}}^{{\rm FLLnS}}[N_{d}],$ (2)

where $E_{{\rm int}}^{U}$, and $E_{{\rm DC}}^{{\rm FLLnS}}$ are the Hartree-
Fock energy of the local Coulomb interaction, and the double counting
contribution, respectively. We note that the $E_{{\rm GGA}}$ and $E_{{\rm
DC}}^{{\rm FLLnS}}$ contributions to the total energy depend only on the
charge density $\rho=\rho_{\uparrow}+\rho_{\downarrow}$ and the occupation of
the $d$-shell $N_{d}=\sum_{\sigma,\alpha}n_{\alpha\alpha}^{\sigma}$ ,
respectively. The spin density contributions to the total energy are
controlled solely by the interaction term $E^{U}_{{\rm int}}$ or by the self-
energy $\Sigma^{\sigma}$ in the DMFT formalism. For more details, see Appendix
B.

Benchmarks of different DFT$+U$ formalisms Anisimov  _et al._ (1991, 1993);
Dudarev  _et al._ (1998); Liechtenstein  _et al._ (1995) have been performed
in previous studies Ylvisaker  _et al._ (2009); Ryee and Han (2018). The spin-
dependent DFT$+U$ extensions exhibit seemingly unphysical behavior, including
an unreasonable $J$ dependence of structural parameters in nickelates Pavarini
(2011) and the energy difference between high-spin/low-spin states in 3$d$ and
4$d$ transition-metal compounds Ryee and Han (2018). Similar arguments were
also given in the DFT+DMFT context Chen  _et al._ (2015); Park  _et al._
(2014); Chen and Millis (2016).

##  III Results and discussion

###  III.1 Electronic structure of the layered sulfides within GGA and GGA$+U$

Figure 1:  (a) Crystal structure of Lix[Li0.33-2y/3Ti0.67-y/3Fey]S2. Green,
blue, and yellow spheres represent Li ions, a mixture of Li/Ti/Fe ions, and S
ions, respectively. Visualization was conducted using VESTA Momma and Izumi
(2011) (b) Schematic electronic structure near the Fermi level for the $y=1/3$
case. |S-p denotes non-bonding S $3p$ states. The oxidation states are
estimated from simple charge counting. (c, d) The projected density of states
(PDOS) obtained from GGA calculations. Red dashed lines represent the S $3p$
contained in the octahedron with Li as a central atom, capturing the non-
bonding states.

The Li-rich layered Li1.33-2y/3Ti0.67-y/3FeyS2 can be thought of as a
Li1.33Ti0.67S2 compound, where some of the Ti4+ have been substituted by Fe2+
with a charge compensation by adjusting the Li content. The crystal structure
is shown in Fig. 1(a) Saha  _et al._ (2019). It consists of stacks of
transition metal disulfide layers crystallizing in a honeycomb lattice
structure. Upon lithiation, Li-ions intercalate in the interlayer space or
within the layers. At the DFT level, the Fermi surface is dominated by Fe2+
$d$-derived bands, which are strongly hybridized with S2- $p$ orbitals. The
Ti4+ ions are electronically inactive because of their $3d^{0}$ configuration.
Upon removal of Li, both cationic Fe2+ and anionic S2- contribute to the
corresponding redox processes Saha  _et al._ (2019). As shown in Ref. Saha
_et al._ , 2019, the experimentally observed capacity of 245 mAhg-1 for
Li1.13Ti0.57 Fe0.3S2 ($y=0.3$), which corresponds to the removal of $\sim
1.06$ Li ions per formula unit, cannot be reached by cationic redox alone,
even when assuming a multi-electron oxidation from Fe2+ to Fe4+. Hence anionic
redox processes, leaving unoccupied sulfur $p$ states near the Fermi level in
Fig. 1(d), have to be invoked to understand the high capacity of these
materials. From a simple charge count, we can estimate the oxidation states
Fe3+ ($d^{5}$) and S1.67- ($p^{5.67}$) for fully charged (i.e., fully
oxidized) LTFS0 (see Fig. 1(b)).

Fig. 1(c) and (d) show the projected density of states (PDOS) corresponding to
the non-spin-polarized Kohn-Sham band structure obtained from GGA
calculations. Within the GGA, fully discharged LTFS1 is a band insulator with
fully occupied $t_{2g}$ orbitals, while LTFS0 is metallic with nominal
$t_{2g}^{5}e_{g}^{0}$ configuration. Although the FeS6 octahedra are slightly
distorted due to the different size of the Li, Ti, and Fe ions in the metal
layer, we will refer to the three lower-lying orbitals as $t_{2g}$ and to the
two higher-lying ones as $e_{g}$. Due to the large hybridization between the
Fe-$d$ and S-$p$ states in LTFS0, larger ligand field splittings are expected
from different hopping strengths for $t_{2g}$ and $e_{g}$ orbitals. To
calculate the ligand field splittings, projective Wannier functions within an
energy window of $W=[-1,3]$ eV were constructed. The on-site energy level
differences between the three lower- and two higher-lying orbitals, denoting
the crystal-field splitting, amount to 1.04 eV and 1.78 eV for LTFS1 and
LTFS0, respectively. Here the smaller crystal-field splitting for LTFS1
compared to LTFS0 also implies a redox-driven spin-state transition and is
consistent with the experimentally observed high-spin and low-spin states for
LTFS1 and LTFS0, respectively Watanabe  _et al._ (2019); Saha  _et al._
(2019).

The red dashed lines in Fig. 1 (c, d) represent the S $3p$ states of the S
atoms forming the LiS6 octahedra, which are of non-bonding nature. We
calculated $r_{nb}$, the fraction of the partial charge residing in the “non-
bonding” state by integrating the relevant partial DOS (red dashed and solid
green lines) shown in Fig. 1 (c) and (d) within a given energy window. For
LTFS1, $r_{nb}$ is 0.80 in the energy window $[-2,0]$, while 0.46 for
$[-6,-2]$. The larger $r_{nb}$ near the Fermi level clearly shows the non-
bonding contribution in the above bonding states as presented in Fig. 1(b).

Figure 2: The projected density of states (PDOS) calculated from (a,b) spin-
polarized GGA (SGGA), and (c, d) GGA$+U$. The upper and lower panels in the
PDOS plots represent up and down spin components, respectively.

Fig. 2(a) and (b) present the PDOS with spin-polarized GGA (SGGA). LTFS1 is
metallic with HS Fe-$d$ configuration, while the Fe-$d$ shell in the LTFS0 is
not fully polarized with partially unoccupied minority spin components. The
spin-configuration is consistent with the previously reported assignments
based on Mössbauer spectroscopy experiments, where the fitted spectra show
finite quadrupole splitting for both Fe3+ and Fe2+ oxidation states Saha  _et
al._ (2019). Interestingly, when creating a hypothetical oxide compound by
replacing S by O, one can artificially tune the ligand splitting due to the
corresponding smaller overlap between O-$p$ and Fe-$d$ orbitals as compared to
the one of S-$p$ and Fe-$d$ (not shown). The result is a HS configuration for
both end-member compounds with large spin magnetic moments on the Fe sites,
$M_{s}=3.80$ and $3.49$ $\mu_{B}$, respectively. The appearance of these
different spin configurations implies that a change of the anion from O to S
can significantly affect battery properties such as the capacity and more so
the operating voltageSaha  _et al._ (2019); Watanabe  _et al._ (2019).

The GGA$+U$ results for LTFS1 and LTFS0 are shown in Fig. 2(c, d),
respectively. The effect of the Coulomb interaction between the localized
$d$-electrons is clearly seen from the Mott insulating gap for LTFS1 and from
the suppression of the Fe-$d$ contribution near the Fermi-level. Here, the
insulating gap opening is driven by a reduction in the occupancy fluctuations
over the three non-degenerate low-lying orbitals, referred to as $t_{2g}$ for
convenience, due to the energy penalty of the
$\frac{1}{2}(U-J)\sum_{\alpha\sigma}(n_{\alpha\sigma}-n_{\alpha\sigma}^{2})$
term in $\Delta E^{U}$ (See Appendix B). From the SGGA calculation, for
example, the eigenvalues of the density matrix for the $t_{2g}$ subspace of
the minority spin component are given as 0.27, 0.30, and 0.57, while GGA$+U$
provides 0.26, 0.25, and 0.83. For LTFS0, however, the spin magnetic moments,
$M_{S}=2.30\>\mu_{B}$ are enhanced from the result of GGA,
$M_{S}=1.84\>\mu_{B}$, due to the $-\frac{J}{4}M_{S}^{2}$ contribution in
$\Delta E^{U}$. Considering the experimental assignment of the LS state to
Fe${}^{3+}(d^{5})$ ions in the fully charged sample and the ideal magnetic
moments $\sim 1$ $\mu_{B}$ of the $d^{5}$ in the LS state Saha  _et al._
(2019), the predicted magnetic moments by the DFT+$U$ calculations are
significantly overestimated, stressing the limitations of the static mean
field approximation done in GGA+$U$. This motivated us to go beyond GGA+$U$
and perform GGA+DMFT calculations for these same materials. Detailed
comparisons are presented in the next section.

###  III.2 Electronic structure of the layered sulfides within GGA+DMFT

Figure 3: (a, b) The spectral function from GGA+DMFT. Probability of the
atomic states in Fe-$d$ shell projected onto the (c, d) spin $S_{z}$ and (e,
f) number of electrons $N$. The solid and dashed vertical lines indicate the
averaged value for $J_{H}=0.9$ and $0.0$ eV, within GGA+DMFT and the dotted
line from the GGA$+U$ calculations.

To capture the electronic ground state or quasi-particle electronic structure
of the HS Mott insulating LTFS1 and the LS correlated metallic LTFS0, in this
section, we have employed GGA+DMFT. Specifically, we study the effect of
dynamic correlations between the Fe-$d$ electrons on the electronic structure
and the magnetic properties using the Hubbard interaction $U$ and the Hund’s
exchange coupling $J_{H}$ as parameters. The paramagnetic GGA+DMFT
calculations are performed at an inverse temperature of $\beta=40$ eV-1
corresponding to about 290 K. Since the occupied valence orbitals are
drastically modulated by spin-state transition, the correlation between spin
states and redox properties is expected.

In Fig. 3(a) and (b), we present orbital-resolved spectral functions from
paramagnetic GGA+DMFT calculations. The difference of these results to the DOS
from the static GGA$+U$ approximation is significant, in particular for LTFS0
near the Fermi-level, underlining the importance of the dynamic correlation
effects. Within GGA$+U$, the Fe-$d$ contribution to the DOS near the Fermi
level is small, contrasting to the large Fe-$d$ character in the GGA+DMFT
result. The suppression of the Fe-$d$ contribution in the GGA$+U$ calculation
originates from the suppression of charge fluctuations by the Coulomb
interactions (see energy penalty term $\Delta E^{U}$ in Appendix B), favoring
integer occupation.

The reason why GGA$+U$ struggles to capture the electronic structure of the
correlated metallic phase can be traced back to the absence of dynamical
fluctuations in this method. This can be seen from the probability
distribution of the atomic configurations. Indeed, solving the DMFT by Monte
Carlo sampling has the advantage to give direct access to the charge and spin
fluctuations: Fig. 3(c-f) shows the probabilities of the different atomic
configurations in terms of the magnitude of the spin component and the orbital
occupancies. As expected, a sharp peak at around $\lvert S_{z}\rvert=2$ is
observed in Fig. 3(c) indicating the HS configuration. For metallic LTFS0, the
spin fluctuations $\delta\braket{}{S_{z}}{}^{2}=0.33$ with mean value of spin
magnetic moments $\braket{\lvert S_{z}\rvert}_{{\rm DMFT}}=0.82$ are lager
than that of LTFS1 ($\delta\braket{}{S_{z}}{}^{2}=0.15$ and $\braket{\lvert
S_{z}\rvert}_{{\rm DMFT}}=1.67$). We note that these fluctuations of the spin
magnetic moments on the Fe sites cannot be captured in a static-mean-field
theory such as DFT+$U$. The calculated magnetic moments in metallic LTFS0
compounds, $M_{S}=2\braket{\lvert S_{z}\rvert}_{\rm DMFT}\sim 1.65$ $\mu_{B}$,
are in better agreement with the experimental assignment of the LS state Saha
_et al._ (2019) than GGA+$U$ (2.30 $\mu_{B}$).

The probability distribution projected onto the number of electrons is also
presented in Fig. 3(e, f). Both compounds show non-negligible charge
fluctuations with $\delta\braket{N}^{2}=0.48$ and $0.93$ for LTFS1 and LTFS0,
respectively. As expected, the metallic LTFS0 shows larger fluctuations than
insulating LTFS1. Due to the strong hybridization with the S-$p$ orbitals the
mean occupation is slightly larger than nominally expected. In particular, the
electron occupancy of 6.63 of the Fe-$d$ orbitals for LTFS0 is larger than
that of LTFS1, 6.52, which is consistent with stronger $d$-$p$ hybridization
as noticed in Fig.2(b). On the other hand, 0.67 electrons are depleted from
the sulfur ion coordinated with four Li and two Fe/Ti ions, while 0.52
electrons come from other sulfur ions surrounded by six transition-metal ions.
The strong intermixing of the Fe-$d$ and S-$2p$ states visible from Fig. 3(b),
corresponding to holes created in the S-p states, reveals the covalent nature
of Fe-S bonding. The different numbers of holes in the states associated with
the different sulfur ions are a proxy for the different chemical environments,
revealing distinct mechanisms occuring when extracting electrons from
unhybridized S $2p$ non-bonding states sitting in Li–S–Li configurations Seo
_et al._ (2016); Saha  _et al._ (2019).

Figure 4: (a) Quasiparticle weight $Z$ of the $t_{2g}$ and $e_{g}$ orbitals
are shown as functions of Hund’s coupling $J_{H}$ for LTFS0 with $U=3.0$
(solid line) and $3.5$ eV (dashed line). The system gradually evolves to a
more strongly correlated state with increasing $J_{H}$, while it seems weakly
correlated at $J_{H}=0$ with $Z_{t_{2g}}(Z_{e_{g}})=0.75(0.8$7) (dash-dot and
dotted line, respectively) (b) Spectral functions projected onto the Fe-$d$
orbitals calculated for different temperatures.

In metallic systems, dynamic correlation effects can be quantified by means of
the quasi-particle renormalization factor $Z\approx\left[{\rm
1-Im}\Sigma(i\omega_{0})/\omega_{0}\right]^{-1}$, where $\Sigma(i\omega_{0})$
is the electronic many-body self-energy calculated from DMFT at the first
Matsubara frequency. This factor is close to unity for a weakly correlated
system, while a small $Z$ indicates a strongly correlated phase. In our case,
LTFS1 is a Mott insulator, albeit with a quite characteristic many-body
behavior. For LTFS0, the calculated value of $Z$ is 0.26 for the $t_{2g}$
orbitals, demonstrating the presence of rather strong correlations even in the
metallic phase of this system. This rather strong renormalization of the
quasi-particle states in this system stems from the intra-atomic Hund’s
exchange coupling $J_{H}$, making LTFS0 a realization of what in the
literature is sometimes called a “Hund’s metal” de’ Medici  _et al._ (2011);
Georges  _et al._ (2013), as demonstrated in Fig. 4(a). In this figure, the
quasiparticle weight $Z$ as a function of $J_{H}$ is shown for two different
interaction strengths, $U=3.0$ and $3.5$ eV. We observe that $Z$ diminishes
gradually as $J_{H}$ or $U$ increase with a more pronounced dependency on
$J_{H}$. The fact that neither GGA nor GGA$+U$ can capture the electronic
ground state or quasi-particle electronic structure and the strong
quasiparticle renormalization evidenced by the small $Z$ factor in the
metallic LTFS0 phase allows us to attribute the system to the class of
strongly correlated materials. To investigate the nature of the correlations
in LTFS0, we study in more detail the spectra projected onto the Fe-$d$
orbitals, up to high temperature (for fixed lattice constant and atomic
positions) Deng  _et al._ (2019). The spectra shown in Fig. 4(b) are
characterized by two distinct peaks, namely $t_{2g}$ states near the Fermi
level and $e_{g}$ states around an energy of 1.5 eV. As temperature increases,
the $t_{2g}$ bands gradually broaden, while their overall shape remains
unchanged. This is a distinctive feature from the traditional correlated metal
which is near the Mott transition, where two broad side peaks, the so-called
Hubbard bands, with gap features develop at high temperatures Deng  _et al._
(2019).

###  III.3 Average intercalation voltage

Figure 5: (a) Calculated intercalation voltage as a function of $J_{H}$
obtained from GGA+DMFT (green). The dashed and dotted horizontal lines
indicate the calculated voltage via GGA and spin-polarized GGA. (inset)
Estimated voltage in atomic limit. See main text for more details. (b) Voltage
decomposed into Coulomb interaction energy and kinetic energy contribution.

We now turn to another key quantity of a redox couple of battery materials,
the intercalation voltage, which for our systems has been determined
experimentally to be around 2.5 eV Saha  _et al._ (2019). In theoretical
calculations, the average battery voltage can be evaluated as $V=\left(E({\rm
LTFS_{0}})+E({\rm Li})-E({\rm LTFS_{1}})\right)/e$, where the three terms are
the total energies of the delithiated system, elemental Li and the lithiated
system. We have calculated this quantity using different total energy
functionals. The conventional GGA results, 2.11 V, underestimate the
experimental value by around $0.37$ eV Saha  _et al._ (2019), while the SGGA
result $V=2.34$ V is in better agreement with experiment. An accurate
description of the spin state of the Fe-$d$ shell is important not only for
the electronic excitations, but also for the energetics, as seen from the
difference between GGA and SGGA. The relatively small error of SGGA (without
$U$) for this layered sulfide indicates that the strong local Coulomb
interactions among the Fe-$d$ electrons are largely screened by the S-$p$
orbitals, having covalent character.

Let us now discuss the impact of the local correlations that govern the
behavior of the Fe-$d$ orbitals. Within GGA+DMFT, the calculated voltage is
2.31 V, which is close to the conventional SGGA result. It exhibits only a
weak dependence on the Hubbard $U$; for example, $V=$ 2.34 V within GGA+DMFT
calculations with $U=3.5$, confirming that the on-site Coulomb interactions
are largely screened due to the large Fe-$d$ and S-$p$ orbital hybridisations.
However, we will see that the spin state of the valence orbitals, which is
determined by $J_{H}$ and the ligand field splitting, are important to
describe the properties of the cathode material.

We now turn to the effect of Hund’s coupling $J_{H}$ and the spin-states on
the energetics e.g., the operating voltage of the battery. As can be seen from
Fig. 5(a), the voltage curve as a function of Hund’s $J_{H}$, while keeping
$U=3.0$ eV fixed, shows a V-shape behavior. For small values of the Hund’s
exchange $J_{H}$, where LTFS1 is in the LS state, the voltage decreases at a
rate of 0.01 V per 0.1 eV change in $J_{H}$, while in the regime of large
$J_{H}$ where LTFS1 is in the HS phase, the voltage starts to increase by 0.07
V per 0.1 eV $J_{H}$. This behavior can be explained in a simple manner by
considering the atomic limit. Assuming an isolated Fe-$d$ shell with nominal
electron configuration, one easily obtains (see Appendix B) the V-shape
voltage curve as a function of $J_{H}$:

$\displaystyle\delta E_{\rm atom}$ $\displaystyle=E(d^{6})-E(d^{5})$
$\displaystyle\propto\begin{cases}4.89J_{H}-2\Delta,&\text{for
}J_{H}/\Delta>0.35\\\ -0.82J_{H},&\text{for }J_{H}/\Delta<0.35.\end{cases}$
(3)

As shown in the inset of Fig. 5(a), the overall trends, including the V-shape
behavior and the larger (smaller) slope in the HS (LS) region, are well
described in this simple atomic model. Further details can be found in
Appendix B. Qualitative differences are observed in the voltage for the LS and
HS cases. Fig. 5(b) shows the operating voltage decomposed into the Coulomb
interaction $V_{\rm pot}$, where the Hartree and exchange-correlation energies
corrected by $+U$ terms are taken into account, and the kinetic energy $V_{\rm
kin}$ contribution. For $J_{H}>0.7$ it is clearly seen that the increase in
voltage is attributed to the $V_{\rm pot}$ part. In other words, a proper
theoretical description of the energy gain from Hund’s interactions in the HS
state is a decisive element to capture the operating voltage.

The predictive power of the DFT+DMFT approach is, obviously, limited by the
underlying approximations, namely the neglect of non-local correlations –
blamed for an underestimation of the voltage within DFT+DMFT in the literature
Isaacs and Marianetti (2020) – and ambiguities concerning the double counting
corrections and the quality of the charge density used for generating the non-
interacting part of the Hamiltonian. For example, GGA+DMFT calculations
performed using a Hamiltonian generated with the GGA charge density (or Kohn-
Sham Hamiltonian) without charge self-consistency predict a voltage of 1.49 V,
greatly underestimating the experimental value. In a previous study, it was
noted that non-charge-self-consistent DFT+DMFT calculations could worsen the
predicted voltage compared to conventional DFT Isaacs and Marianetti (2020).
Nevertheless, our results suggest that the overall energetics needed for a
reasonable description of the voltage is captured in our efficient charge-
self-consistent DFT+DMFT scheme. In future work, we plan to further
investigate the relative effects of non-local correlation vs charge-
redistribution using more sophisticated methods such as $GW$+DMFT, which allow
for progress both concerning a better description of the quasi-particle band
structures and the double counting issue.

##  IV Conclusion

We have performed DFT+DMFT calculations using an efficient charge-self-
consistent scheme to investigate the electronic structure, local properties,
and intercalation voltage of the Li-rich layered sulfides
Lix[Li0.33-2y/3Ti0.67-y/3Fey]S2. A careful comparison between results using
different methods, including spin-averaged and spin-polarized GGA, and
GGA+DMFT was made. Both of the end members, namely the fully discharged
Li1.11Ti0.56Fe0.33S2 and the fully charged compound Li0.11Ti0.56Fe0.33S2, are
strongly correlated systems, identified as high-spin Mott-insulator and low-
spin correlated metal, respectively. We have shown that dynamical correlations
originating from Hund’s exchange coupling $J_{H}$ are important to describe
this class of materials, while the effective local Hubbard $U$ is largely
screened. The impact of $J_{H}$ on the intercalation voltage may suggest new
pathes for designing higher-energy lithium ion-battery cathode materials. To
our knowledge, this is the first demonstration of Hund’s physics playing a
crucial role in the electrochemcial properties of real-life battery materials.
A deeper understanding of these effects will hopefully contribute to paving
the way to better battery materials in the future.

## ACKNOWLEDGMENTS

This work was supported by IDRIS/GENCI Orsay under projet number
No.A0110901393. We thank the computer team at CPHT for support. D.D.S.
acknowledges funding from Science and Engineering Research Board, Department
of Science and Technology, Government of India and Jamsetji Tata Trust. D.D.S.
is also thankful to the Foundation of Ecole Polytechnique for the Gaspard
Monge Visiting Professorship.

##  Appendix A Computational details

By using DFT within the VASP Kresse and Furthmüller (1996) code and DFT+DMFT
implemented in the DMFTpack software combined with OpenMX _DM ; ope ; Sim and
Han (2019) and the impurity solver implemented in Ref. Haule, 2007, we have
studied the electronic and magnetic properties of
Lix[Li0.33-2y/3Ti0.67-y/3Fey]S2. The DFT calculations were performed within
the generalized gradient approximation as parameterized by Perdew, Burke and
Ernzerhof (GGA-PBE) Perdew  _et al._ (1996). The D3 method of Grimme et al.
was used for van der Waals corrections Grimme  _et al._ (2010, 2011) Atomic
positions were relaxed with a force criterion of 1 meV/Å. The lattice
constants are fixed to the experimental values: $a=b=3.54$ Å and $a=b=3.35$ Å
for the discharged ($x=1$) and fully charged ($x=0$) phases, respectively Saha
_et al._ (2019). $13\times 13\times 8$ k-points were used in the Brillouin
zone for the momentum space integrations.

To describe electronic correlation effects, the GGA$+U$ Anisimov  _et al._
(1991, 1993) and single-site paramagnetic GGA+DMFT Anisimov  _et al._ (1997);
Lichtenstein and Katsnelson (1998) have been employed. The interaction part of
the Hamiltonian for the d-shell reads:

$\displaystyle H_{{\rm int}}=$
$\displaystyle\frac{1}{2}\sum_{\sigma,m,m^{\prime}}U_{mm^{\prime}}n_{m\sigma}n_{m^{\prime}\bar{\sigma}}$
$\displaystyle+\frac{1}{2}\sum_{\sigma,m\neq
m^{\prime}}(U_{mm^{\prime}}-J_{mm^{\prime}})n_{m\sigma}n_{m^{\prime}\sigma}.$
(4)

Here the direct and exchange interaction parameters, $U_{mm^{\prime}}$ and
$J_{mm^{\prime}}$, are parameterized by the Slater integrals of the $d$-shell,
namely $F_{0},$ $F_{2}$, and $F_{4}$ Pavarini (2011). We present our results
in terms of the $U=F_{0}$ and $J_{H}=(F_{2}+F_{4})/14$, within the assumption
$F_{4}/F_{2}=0.625$. Unless otherwise stated, calculations were performed with
$U=3.0$, $J_{H}=0.9$, and FLLnS double counting : $\Sigma_{DC}^{\rm
FLLnS}=U(N-\frac{1}{2})-J(\frac{N}{2}-\frac{1}{2})$ Ylvisaker  _et al._
(2009); Ryee and Han (2018). The charge-density-only GGA$+U$ should be
distinguished from other DFT$+U$ flavors such as spin-polarized GGA$+U$
Liechtenstein  _et al._ (1995) and the Dudarev’s simplified DFT$+U$ method
Dudarev  _et al._ (1998); Han  _et al._ (2006). The interaction parameters,
$U$ and $J_{H}$, used in this work are consistent with that used in the
previous study, $U_{\text{eff}}=U-J_{H}=2.0$ eV Saha  _et al._ (2019) and in
reasonable range compared to other iron-based compounds van Roekeghem  _et
al._ (2016). The ferromagnetic ground state, with a lower energy of 3 meV than
the antiferromagnetic state at $U=3.0$ and $J_{H}=0.9$ eV, is assumed for
various $U$ and $J_{H}$ parameters.

In our single site DMFT calculation, a natural atomic orbitals projector onto
to Fe-$d$ orbitals with an energy window of $W=[-6,5]$ eV containing Fe-$d$
and S-$p$ orbitals has been employed Sim and Han (2019). The self-energy is
decomposed into three $10\times 10$ matrices corresponding to three
inequivalent Fe atoms in the unit cell, i.e.,
$\Sigma(i\omega_{n})=\oplus_{i=1}^{3}\Sigma_{{\rm Fe(i)}}(i\omega_{n})$.
$\Sigma_{{\rm Fe(i)}}(i\omega_{n})$ is determined from the fictitious impurity
problem with self-consistency condition. The impurity problems are solved by
employing a hybridization expansion continuous-time quantum Monte Carlo (CT-
QMC) Werner  _et al._ (2006); Werner and Millis (2006) algorithm implemented
in Ref. Haule (2007). The self-energy in the real frequency domain is obtained
from the Matsubara self-energy by analytic continuation using the maximum
quantum entropy method Sim and Han (2018), extending the maximum entropy
method to matrix valued Green’s functions Jarrell and Gubernatis (1996);
Gunnarsson  _et al._ (2010).

##  Appendix B On-site Coulomb energy in DFT+$U$ and DFT+DMFT

In this section we calculate the total energy contribution from the on-site
interaction $\Delta E_{U}=\braket{H_{{\rm int}}}-E_{{\rm DC}}$. For
simplicity, in this section the interaction $H_{{\rm int}}$ is assumed to be
of Slater-Kanamori form with $U_{SK}=U+8J_{H}/7,$ $J_{SK}=5J_{H}/7$, and
$U^{\prime}_{SK}=U_{SK}-2J_{SK}$ Pavarini (2011); Isaacs and Marianetti
(2020).

#### DFT$+U$

$E_{{\rm
int}}^{U}=\frac{U}{2}\sum_{(\alpha\sigma)\neq(\beta\sigma\prime)}n_{\alpha\sigma}n_{\beta\sigma^{\prime}}-\frac{J_{H}}{2}\sum_{\sigma}\sum_{\alpha\neq\beta}n_{\alpha\sigma}n_{\beta\sigma}$
and the double counting term is $E^{{\rm DC}}_{{\rm
FLLnS}}=\frac{U}{2}N(N-1)-\frac{J}{2}\sum_{\sigma}\frac{N}{2}(\frac{N}{2}-1)$,
where $n_{\alpha\sigma}$ is an eigenvalue of the occupation number matrix with
orbital index $\alpha$ and spin $\sigma$.
$N=\sum_{\sigma}N^{\sigma}=\sum_{\sigma}\sum_{\alpha}n_{\alpha\sigma}$. The
Coulomb energy correction $\Delta E^{U}=E_{int}^{U}-E_{{\rm DC}}$ can be
written as follows:

$\Delta
E^{U}=\frac{1}{2}(U-J)\sum_{\alpha\sigma}(n_{\alpha\sigma}-n_{\alpha\sigma}^{2})-\frac{J}{4}M^{2},$

where $M=N_{\uparrow}-N_{\downarrow}$ is the magnetic moment of the localized
$d$-orbitals. We note that the first term is obtained from the double counting
energy $E_{{\rm DC}}^{{\rm FLL}}$ Han  _et al._ (2006). The second term
imposes an energy gain due to the finite magnetic moment $M$, leading to a
magnetic polarization, which is not included in the non-spin polarized GGA
exchange-correlation functional.

In DMFT, the Coulomb interaction energy is beyond the HF approximation.
$E_{{\rm int}}^{{\rm
DMFT}}=\frac{U}{2}\braket{\hat{N}^{2}-\hat{N}}-\frac{J_{H}}{2}\sum_{\sigma}\braket{\hat{N_{\sigma}}^{2}-\hat{N_{\sigma}}}$.
After some algebra, using $E_{\text{DC}}^{\text{FLLnS}}$, we obtain

$\Delta E^{{\rm DMFT}}=\frac{1}{2}(U-J)\sum_{\alpha\sigma}(\braket{\delta
N^{2}})-\frac{J}{4}M_{{\rm eff}}^{2},$

where $\braket{\delta N^{2}}=\braket{N^{2}}-\braket{N}^{2}$ and $M_{{\rm
eff}}=(N^{2}-4\braket{N_{\uparrow}N_{\downarrow}})^{1/2}$. We can see that
within the HF approximation, $\braket{N_{\uparrow}N_{\downarrow}}\approx
N_{\uparrow}N_{\downarrow}$, $M_{{\rm eff}}$ is equivalent to the conventional
magnetization, namely $M_{{\rm eff}}=N_{\uparrow}-N_{\downarrow}$.

#### Atomic limit

Three different configurations, namely, $d^{6}$(HS), $d^{6}$(LS), and
$d^{5}$(LS) are concerned. Here, the $d^{6}$(HS) and $d^{5}$(LS)
configurations are expected as ground state configurations of Fe in LTFS1 and
LTFS0, respectively. Assuming FLL-nS double counting, one can see that $\Delta
E_{U}$ can be expressed as:

$\displaystyle\Delta
E_{U}\approx\begin{cases}-4J_{H}+2\bar{\Delta}&d^{6}(HS)\\\
1.71J_{H}&d^{6}(LS)\\\ 0.89J_{H}&d^{5}(LS)\end{cases}.$ (5)

Given these expressions, we can estimate $J_{H}$ dependence of the Coulomb
energy contribution to the voltage for different atomic configurations.
Interestingly, our DFT+DMFT results show similar trends for the transition
from the $d^{6}$ to the $d^{5}$(LS) configuration:

$\displaystyle V_{LS}=E(d^{6})-E(d^{5})\propto$
$\displaystyle\begin{cases}4.89J_{H},&\text{for }J_{H}/\Delta>0.35\\\
-0.82J_{H},&\text{otherwise}.\end{cases}$ (6)

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