Helmholtz Institute Ulm (HIU) for Electrochemical Energy Storage,
Helmholtzstraße 11, 89069 Ulm, Germany

# Oxide spinels with superior Mg conductivity

Mohsen Sotoudeh<EMAIL_ADDRESS>Institute of Theoretical
Chemistry, Ulm University, Oberberghof 7, 89081 Ulm, Germany  Manuel Dillenz
Institute of Theoretical Chemistry, Ulm University, Oberberghof 7, 89081 Ulm,
Germany  Johannes Döhn  Institute of Theoretical Chemistry, Ulm University,
Oberberghof 7, 89081 Ulm, Germany  Julian Hansen  Institute for Applied
Materials, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1,
76344 Eggenstein-Leopoldshafen, Germany  Sonia Dsoke  Institute for Applied
Materials, Karlsruhe Institute of Technology, Hermann-von-Helmholtz-Platz 1,
76344 Eggenstein-Leopoldshafen, Germany  Axel Groß<EMAIL_ADDRESS>Institute of Theoretical Chemistry, Ulm University, Oberberghof 7, 89081 Ulm,
Germany

###### Abstract

Mg batteries with oxide cathodes have the potential to significantly surpass
existing Li-ion technologies in terms of sustainability, abundance, and energy
density. However, Mg intercalation at the cathode is often severely hampered
by the sluggish kinetics of Mg2+ migration within oxides. Here we report a
combined theoretical and experimental study addressing routes to identify
cathode materials with an improved Mg-ion mobility. Using periodic density
functional theory calculations, Mg2+ migration in oxide spinels has been
studied, revealing key features that influence the activation energy for Mg2+
migration. Furthermore, the electronic and geometrical properties of the oxide
spinels as well as their stability have been analyzed for a series of
different transition metals in the spinels. We find that electronegative
transition metals enable a high Mg-ion mobility in the oxide spinel frameworks
and thus a favorable cathode functionality. Based on the theoretical findings,
some promising candidates have been identified, prepared and structurally
characterized. Our combined theoretical and experimental findings open up an
avenue toward the utilization of functional cathode materials with improved
Mg2+ transport properties for Mg-metal batteries.

###### keywords:

ionic conductivity, density functional theory, magnesium batteries, ternary
oxide spinels

##  1 Introduction

The field of post-lithium-ion batteries using, e.g., Na-ions 1, 2 or Mg-ions
3, 4, 5, 6 has recently received substantial attention in the quest for more
sustainable and abundant batteries without compromising performance parameters
such as energy density and capacity. In particular the concept of pairing a
metallic anode with a high voltage cathode promises to achieve higher energy
density 7, 8 than current Li-ion technologies. Among them, batteries based on
the bivalent metal magnesium theoretically reach higher volumetric energy
densities compared to monovalent-based batteries. 7, 8 However, among the
materials typically used in Mg batteries only few combine an acceptable ionic
conductivity with a high operating voltage. For example, MgMo6S8 4 and MgTi2S4
9 sulfides allow an effective reversible Mg intercalation. Still, their
capacities and operation potentials are too low to enable devices with a
competitive energy density.

In contrast to materials based on soft anions such as sulfides, oxide
materials combined with a metallic Mg anode offer high voltages and
capacities. 10. As an example, the ion-exchanged olivine-type MgFeSiO4 cathode
exhibits a relatively high potential of 2.4 V vs. Mg2+/Mg and a capacity of
300 mAh/g. 11 This material represents a low-cost, safe and corrosion free
cathode which can be applied in rechargeable Mg batteries. Furthermore,
MgMnSiO4 has been considered as a potential cathode candidate material for Mg
batteries. 12 However, spinel structures typcially show a lower migration
barrier compared to the olivine structures. Finding spinel materials with a
high ionic conductivity has therefore become an active topic of research in
the field of Mg batteries. 13, 14, 15, 16, 17, 18, 19, 20

As already mentioned above, inorganic oxides in the spinel structure 14, 21
exhibit a favorable combination of capacity and voltage, as illustrated in Fig
1. Recently, experimental Mg2+ migration barriers of about 600 meV have been
reported in Cr- and Mn-spinel oxides, in agreement with density functional
theory (DFT) calculations 21. It has been found that Mg-ion migration is
influenced by structural disorder in the spinel lattice, which can be
controlled during synthesis 21. However, the strong Coulombic interactions of
the Mg cations with the oxide host lattice generally hamper the reversibility
of Mg intercalation and lead to high Mg migration barriers 22. Recently, we
have been able to identify a descriptor for the ion mobility in crystalline
materials based on the ionic radii, the oxidation states and the difference in
the electronegativities of the migrating cations and the anions of the host
lattice 19. Furthermore, we explored the effect of the transition metal
cations in spinel compounds on their transport properties, suggesting that the
transition metals can signficantly influence the Mg2+ migration barriers in
these compounds 19, 20. Applying this concept, we found that oxide spinels
combined with large electronegative transition metal cations promise to
exhibit a rather high Mg2+ mobility 19.

Figure 1: Total volumetric capacity versus average voltage for MgB2S4 sulfide
spinels (orange), MgB2O4 oxide spinels (blue) as well as MgB2O5 pentoxides
(brown), derived from DFT calculations within the PBE+$U$ approach. The
colored areas represent the estimated range of the corresponding compounds.

It should be mentioned that utilizing 4$d$ and 5$d$ transition metals instead
of 3$d$ transition metals can lead theoretically to high capacity electrode
materials for rechargable batteries 23, 24. The 4$d$ and 5$d$ transition
metals had been overlooked for a long time due to their unfavorable weight.
However, employing anionic redox mechanisms can enable large reversible
capacities 25. As an example, the rock salt compound Li1.3Nb0.3Mn0.4O2 with
the 4$d$ transition metal Nb was found to exhibit a high reversible capacity
of 300 mAh g–1 in Li-ion batteries 23. Even for chalcogenide spinel lattices,
it was recently shown that their Mg-ion mobility can be enhanced by selecting
the heavy transition metal lanthanoid. 24 Therefore, the use of 4$d$ and 5$d$
transition metals in cathode materials can not only lead to high voltages but
also to promising ion mobilities.

In this work, based on our descriptor concept for the ion mobility 19, we will
take the next step towards the realisation of battery materials with improved
migration properties. Performing electronic first-principles density
functional theory (DFT) calculations, we focus on 3$d$ oxide spinels and their
Mg2+ mobility by considering 4$d$ and 5$d$ metals as dopants in our
calculations in order to improve the properties of the spinels. Based on the
theoretical results, the synthesis of materials with predicted improved
properties was initiated, leading to the synthesis of promising cathode
materials at different doping levels. As the next step, the properties of
these materials as battery cathodes will be experimentally tested. This work
thus reports the first steps in the identification and utilization of
promising battery materials based on an understanding of the fundamental
principles underlying battery operation at the atomic level.

##  2 Methods

###  2.1 Computational details

First-principles calculations in the framework of density-functional theory
(DFT) 26, 27, 28 were performed for MgB2O4 spinel structures. The exchange-
correlation effects were described in the generalized gradient approximation
(GGA) using the Perdew-Burke-Ernzerhof (PBE) functional 29. The calculations
were carried out with the Projector Augmented Wave (PAW) 30 method as
implemented in the Vienna Ab-initio Simulation Package 31, 32, 33. The PAW
pseudopotentials include the 2$p$ and 3$s$ orbitals of Mg, the 2$s$, 2$p$ of
O, the 3$d$, 4$s$ orbitals of Ti, Co, Ir, and the 3$p$, 3$d$, 4$s$ orbitals
for other transition metals.

A plane wave cutoff of 520 eV has been chosen for the wave function, and the
convergence criterion for total energies was set to 1 $\times$ 10-5 eV per
supercell. A 2$\times$2$\times$2 k-point mesh has been used for the unit cell
containing eight formula units with 56 atoms. The partial occupancies for each
orbital are set with the tetrahedron method 34, 35 and the so-called Blöchl
corrections 36. The atomic positions and volume are relaxed without any
restriction.

In the simulations, to properly describe the localized (strongly correlated)
$d$-electrons, Hubbard $U$ corrections 37 have been used. For the 3$d$
orbitals of the transition metals, they have been set to $U_{\mathrm{Ti}}$ =
3.00 eV, $U_{\mathrm{V}}$ = 3.25 eV, $U_{\mathrm{Cr}}$ = 3.70 eV,
$U_{\mathrm{Mn}}$ = 3.90 eV, $U_{\mathrm{Fe}}$ = 5.30 eV, $U_{\mathrm{Co}}$ =
3.32 eV, and $U_{\mathrm{Ni}}$ = 6.20 eV. For Ir and Rh, only the conventional
GGA-PBE functional is used without any $U$ parameter.

The electronic structure in oxide spinels strongly depends on the type of
magnetic order in the material. Both antiferromagnetic and ferromagnetic
orders are probed to identify the ground state of the systems. Our
calculations show that in general the ferromagnetic structures are
energetically less stable than the antiferromagnetic configurations that have
a zero overall magnetic moment for the 3$d$ transition metals. Thus, the
electronic structure has been set to collinear antiferromagnetic spin
arrangements for 3$d$ transition metals, while for Co, Rh, and Ir metals, a
non-magnetic order has been assumed.

In order to identify the Mg-ion migration barriers, the nudged elastic band
(NEB) 38 method was used. In order to model the Mg migration, one Mg atom was
removed within the supercell, resulting in a Mg0.875B2O4 stoichiometry. It
should be noted that the diffusion in the considered compounds is referred to
as interstitial diffusion in the literature 39. During the NEB calculations,
the total energies were evaluated with the PBE functional and PBE+$U$
corrections in order to validate the calculated migration barrier energies.
All of the NEB calculations have been carried out with four distinct images
between the tetrahedral and octahedral sites to evaluate the Mg-ion migration
trajectory. The NEB calculations were fully relaxed until the forces on the
atoms were converged within 0.05 eV Å-1. As Fig. 2 demonstrates, including $U$
corrections severely alters the calculated migration barrier heights, for
early transition metals the barriers are increased by up to 0.5 eV, whereas
for the late transition metals, decreased by up to 0.5 eV. We consider the
migration barriers determined with the $U$ correction to be less reliable for
the following reasons: despite the correct prediction of the ground state
configurations, including all distortions, the PBE+$U$ functionals provide
inaccurate energy differences between the tetrahedral site and octahedral
void, namely inaccurate site preference energies, as demonstrated by comparing
DFT+$U$ calculations with DFT calculations including exact exchange. For
example the site preference energy in the MgV2O4 is about 0.7 eV using the PBE
functional and about 0.5 eV using the HSE06 functional 40, while the PBE+$U$
functional yields an energy of about -0.1 eV. Therefore, obviously the PBE
functional yields a better estimate of the migration barrier energy for the
considered materials in the present work than the PBE+$U$ functional.

Figure 2: Calculated migration barrier energies using the PBE and PBE+$U$
functionals for the 3$d$ transition metal oxides in the spinel framework.

The open circuit voltage ($V_{OC}$) of the spinel materials is given by

$V_{OC}=-\frac{E_{inter}}{zFx},$ (1)

where $F$ is the Faraday constant, and $z$ corresponds to the elementary
charges that are transferred between anode and cathode upon the process

$\displaystyle\mathrm{Mg}_{1+y}\mathrm{B}_{2}\mathrm{O}_{4}\rightleftharpoons\mathrm{Mg}_{y}\mathrm{B}_{2}\mathrm{O}_{4}+\mathrm{Mg},$
(2)

with $z=2$ for Mg-ion batteries. $E_{inter}$ is the Mg intercalation energy in
the spinel structure with respect to metallic magnesium which has been
calculated, neglecting the zero-point energies and entropic contributions,
according to

$E_{inter}(\mathrm{Mg})=E(\mathrm{Mg_{1+y}B_{2}X_{4})}-(E(\mathrm{Mg_{y}B_{2}X_{4}})+xE(\mathrm{Mg}))\
,$ (3)

where $E(\mathrm{Mg_{y}B_{2}X_{4}})$ is the total energy of the spinel with a
Mg concentration $y$ in the unit cell, and $E(\mathrm{Mg})$ is the cohesive
energy of Mg bulk metal. The open circuit voltage $V_{OC}$ in volts is given
by $E_{inter}/2$ for Mg-metal batteries when $E_{inter}$ is given in eV.

Thermodynamic stability has been used to address the theoretical chemical
stability and synthesizability 41, comparing a compound’s energy with all
possible configurations with the same stoichiometry. The existing Materials
Project 42 databases have been utilized to evaluate all possible decomposition
products. The difference to the configuration with the lowest energy, commonly
termed as the energy above hull $E_{hull}$, is provided here as a measure for
stability. In order to correctly calculate $E_{hull}$, the following
procedures have been taken into account: i) We have carefully checked that the
formation energies of all considered compounds agree well with the
corresponding values given by the Materials Project. ii) The well-known
problems due to the overbinding of dioxygen in GGA calculations are accounted
by a correction suggested by Wang et al. 43. iii) We do not consider the
decomposition of the considered compounds into elemental metals, preventing
the problem of comparing the GGA+U energies of the oxide phases with the GGA
calculations of the elemental metallic phases 44. In the discussion of the
stability of the considered structures given below we consider structures to
be stable if their energy above hull is below 0.05 eV/atom, first of all due
to the fact that the accuracy of our DFT calculations is in this range, and
second, because structures that are only slightly unstable with respect to the
thermodynamically most stable structure might still be kinetically rather
stable.

##  3 Results and Discussion

###  3.1 Spinel structure

The oxide spinel structure which the chemical formula AB2O4 crystallizes with
a cubic close-packed oxygen lattice. In the normal spinel crystal structure, A
cations occupy $1/8$ of the tetrahedral sites, while B cations fill $1/2$ of
the octahedral sites as shown in Fig. 3a. B and O ions form a network of edge-
sharing BO6 octahedra, whereas A and O ions form a three-dimensional network
of corner-sharing tetrahedra. In the oxides studied here, the size of the
A-type ions is sufficiently small making their occupation of the tetrahedral
site senergetically more preferable than the occupation of the octahedral
sites 18. The network made by the B ions is known as the pyrochlore lattice,
as shown in Fig. 3b. The pyrochlore lattice gives rise to very strong
geometrical frustration effects. Two types of alternate stacking planes,
namely a two-dimensional triangular and a kagomé lattice, can form along the
$[111]$ direction, as shown in Fig. 3c. The A sublattice of the spinel
structure assumes the shape of a diamond lattice, which is shown in Fig. 3d.
The presence of several competing factors, such as the crystal field
stabilization of the transition metals (B cations) or entropic contributions,
can result in a variety of disordered or inverted configurations 45. A
fraction of A cations (such as Mg) can change sites with the transition metal
B cations, resulting in partially inverted lattices and charge disproportion
of the B cations (2B3+ $\rightarrow$ B2+ \+ B4+).

Figure 3: (a) Spinel structure, emphasizing the two basic structural units of
AO4 tetrahedra and BO6 octahedra. (b) B sublattice of spinel structure, which
defines a pyrochlore lattice, and the lowest-energy spin states obtained by
PBE+$U$. The B-cations are colored in blue. Green and yellow arrows represent
the two spin directions. (c) B-sublattice structure (pyrochlore lattice)
viewed from the $\langle 111\rangle$ direction. (d) A sublattice (diamond
lattice).

Many spinel oxides have cubic symmetry at high temperatures, but this does not
rule out the possibility of distorted BO6 octahedra. The octahedra undergo
compression or elongation along the $[111]$ direction, pointing towards the
centers of the BO4 tetrahedra, as illustrated in Fig. 4. This gives rise to a
trigonal distortion of the octahedra, changing the $O_{h}$ octahedral symmetry
to a $D_{3h}$ octahedral symmetry 46. The trigonal distortion can be
parameterized by the anion parameter $u$ 47 which reflects the displacement of
the O-ions within the cubic cell. In ideal octahedra $u=3/8$, whereas
$u<0.375$ indicates a compression of the octahedra and $u>0.375$ corresponds
to their elongation. Sickafus et al. 47 showed that the anion parameter $u$
can be expressed as a function of the effective radii $r$ of Mg and of the
transition metal B according to

$u=0.3876\left(\frac{r(\mathrm{B})}{r(\mathrm{Mg})}\right)^{-0.07054}\ .$ (4)

This distortion is an important factor influencing the electronic
configuration of the transition metals in the spinel structure. The resulting
values for the considered oxide spinels are given in Tab. 1. All compounds
show $u$ values higher than 0.375 confirming the existence of trigonal
distortion for all cases.

Figure 4: Schematic representation of the crystal field splitting of $d$-orbitals at a tetrahedral site ($T_{d}$) and at octahedral sites without ($O_{h}$) and with the trigonal distortion ($D_{3h}$). The central B cation is assumed to be coordinated by oxygen atoms. Table 1:  Calculated anion parameter $u$ indicating the local trigonal distortion of the BO6 octahedra, using Eq. 4 and the oxygen radius of 1.36 Å. The ionic radii of the B cations are taken from Shannon 48 for a sixfold coordinated configuration and the corresponding ionic charge of 3+ present in this list. Compound |  |  $r$(B)/Å | Anion parameter
---|---|---|---
MgTi2O4 |  | 0.67 | 0.4074
MgV2O4 |  | 0.64 | 0.4088
MgCr2O4 |  | 0.62 | 0.4097
MgMn2O4 |  | 0.65 | 0.4083
MgFe2O4 |  | 0.65 | 0.4083
MgCo2O4 |  | 0.61 | 0.4102
MgNi2O4 |  | 0.60 | 0.4106
MgIr2O4 |  | 0.67 | 0.4074
MgRh2O4 |  | 0.68 | 0.4070

###  3.2 Basic electronic configuration

In Fig. 5, we have plotted the calculated density of states whose gross
features we will now summarize. Most spinel oxides containing $d$-series
transition-metal ions exhibit magnetic insulator configurations due to the
strong Coulomb repulsion within the $d$-orbitals. In rare cases, a spin-
singlet state forms, and the system tends to be non-magnetic. Compared to
sulfide spinels, oxides are less conducting due to their decreased bandwidth,
which weakens $p$-$d$ hybridization 49.

Figure 5:  DOS of MgM2O4 (B = Ti, V, Cr, Mn, Fe, Co, Ni, and Rh) spinel
structures in the stable antiferromagnetic order for 3$d$ metals and non-
magnetic order for Rh. The graphs show the total DOS (grey) and the projected
DOS for O-$p$ (red), B-$t_{2g}$ (green), B-$e_{g}$ (yellow). The projected DOS
is considered for the B ions with both majority- and minority-spin directions.

Crystal field splitting divides the transition-metal $d$-orbitals, located at
the B site of a regular BO6 octahedron, into high-lying, doubly degenerate
$e_{g}$ and low-lying triply degenerate $t_{2g}$ orbitals, as illustrated in
Fig. 4. Note that the crystal-field splitting originates from the covalent
interaction with the oxygen neighbors. The stronger $\sigma$ bond of the
$e_{g}$ states in comparison to the $\pi$ bonds formed by the $t_{2g}$ states
shifts the antibonding $e_{g}$ states energetically above the $t_{2g}$ states.
In addition to the crystal field splitting, the local trigonal distortion
further divides the $t_{2g}$ orbitals into an $a_{1g}$ orbital, and doubly
degenerate $e_{g}^{\prime}$ orbitals. At low temperatures, alternatively a
tetragonal distortion splits the $t_{2g}$ orbitals into $xy$ and the doubly
degenerate $yz/zx$ pair and the doubly degenerate $eg$ orbitals into non-
degenerate $x^{2}-y^{2}$ and $3z^{2}-r^{2}$ orbitals.

If a transition-metal ion locates in the tetrahedral A site, the transition-
metal $d$-orbitals split into the high-lying $t_{2g}$ and low-lying $e_{g}$
orbitals, opposite to the octahedral B-site. The B-ions in spinels usually
have a large magnetic moment, leading to a Hund’s rule-splitting between
transition-metal $d$ levels in the majority and the minority spin direction
and changes in orbital character.

###  3.3 Jahn-Teller distortions

In continuation of the preceding considerations, we now further analyze the
different electronic configurations occurring in oxide spinels. The local
structure around the transition metal ion will be affected by the electron
filling of the 3$d$-orbitals, namely the Jahn–Teller effect. A prominent
Jahn–Teller configuration is $3t_{2g}^{3}e_{g}^{1}$ (Mn3+). The high-spin
configuration of the Mn3+ ion introduces a single electron in the Mn-$e_{g}$
state. The system lowers its energy through a tetragonal distortion which
splits the doubly degenerate $e_{g}$ level into the singly filled
$d_{3z^{z}-r^{2}}$ orbital and the higher lying empty $d_{x^{2}-y^{2}}$
orbital. In contrast, for $d^{3}$ and $d^{5}$ configurations, the $e_{g}$
states are either empty or filled, respectively, and therefore, no Jahn-Teller
distortion occurs. The 3-fold degenerate $t_{2g}$-states form metal-oxygen
$\pi$-bonds, which are weaker than the $e_{g}$ $\sigma$-bonds, and therefore,
the Jahn-Teller effect is smaller compared to the $e_{g}$-states. Hence, weak
distortions are expected fo $3d^{1}$ (Ti3+), $3d^{2}$ (V3+), and $3d^{3}$
(Cr3+) as well as the $3t_{2g}^{4}e_{g}^{2}$ (Co3+) and $3t_{2g}^{5}e_{g}^{2}$
(Ni3+) configuration. The Rh- and Ir-spinels show no Jahn-Teller distortions
due to the predominant ferromagnetic order of these spinels which enhances the
$p$-$d$ hybridization.

###  3.4 Selected oxide spinels

After reviewing the gross features of the atomic and the electronic structure
of the oxide spinels, we will now discuss how transition metals affect the Mg
migration in the series of magnesium oxide spinels considered in this study.
We concentrate on normal spinel phases that crystallize in cubic single-phase
lattices (space group $Fd\overline{3}m$, see Fig. 3).

####  3.4.1 MgTi2O4

The first member of the class of oxide spinels considered in this study is
MgTi2O4. The ions are in the formal oxidation states
Mg2+Ti${}_{2}^{3+}$O${}_{4}^{2-}$. Thus, an electron is inserted into the
$t_{2g}$ states of the Ti sites. The 3$d^{1}$ electronic configuration
together with the total spin $S=1/2$ make this system to a strong candidate
for the occurrence of dimerization in a single-valence system. In fact,
MgTi2O4 undergoes a metal-insulator (M-I) transition when cooled below
$T_{\mathrm{M-I}}=260$ K, accompanied by a strong decrease of the magnetic
susceptibility and a transition to a tetragonal structure. In the tetragonal
structure of MgTi2O4, Ti moves away from the center of the TiO6 octahedron,
and the six nearest-neighbor Ti-Ti distances become inequivalent.

The calculated DOS of MgTi2O4 is shown in Fig. 5. The filled valence band,
which extends from -7 eV to 0 eV, consists of predominantly O-$p$ character
from -7 eV to -3V with some contribution of Ti-$d$ orbitals from -1 eV to 0
eV. The unfilled Ti-$d$ states are located 1-5 eV above the valence band. The
Ti-based spinel has the lowest voltage with respect to a Mg metal anode among
all the Mg spinel compounds based on other redox-active cations. The Ti-based
compound has an energy of 0.013 eV/atom above hull which shows the stability
of the compound and the possibility to be synthesized. Compared to an ideal
octahedron that possesses Ti-O distances with equal length, two Ti-O bonds are
compressed by 6% in this system. Thus, the octahedron has both tetragonal and
trigonal distortions, within the corresponding point group $D_{3h}$.

The Ti oxide spinel is associated with a low open-circuit voltage with respect
to magnesium metal as well as rather high Mg migration barrier, i.e. a low ion
mobility, as detailed in Tab. 2. The inverse spinel, Mg2TiO4, is more stable,
which makes Mg-ion migration even more difficult. Thus this compound is not
suited as a cathode material for Mg batteries.

Table 2: Mg-O, and B-O bond lengths in Å for spinel compounds. B denotes the
transition-metal. Calculated relative barrier energy $E_{a}$, energy above
hull $E_{hull}$, intercalation energy $E_{inter}^{high}$ ($E_{inter}^{low}$)
(Eq. 3) for high (low) Mg concentration in eV, and corresponding open-circuit
voltage $V_{OC}^{high}$ ($V_{OC}^{low}$) with respect to a Mg metal anode in
V. The volume changes with respect to the structure without Mg is indicated by
$\Delta V/V$.

Compound | Mg-O (Å) | B-O (Å) |  $E_{a}^{PBE}$(eV) |  $E_{hull}$(eV/atom) |  $E_{inter}^{high}$ (eV) |  $V_{OC}^{high}$ (V) |  $E_{inter}^{low}$ (eV) |  $V_{OC}^{low}$ (V) |  $\Delta V/V$ (%)
---|---|---|---|---|---|---|---|---|---
MgTi2O4 | 2.027 | 2.025/2.147 | 0.791 | 0.013 | -2.258 | 1.129 | -3.805 | 1.903 | -3.5
MgV2O4 | 1.993 | 2.040/2.072 | 0.783 | 0.013 | -5.464 | 2.732 | -5.090 | 2.545 | -1.4
MgCr2O4 | 1.994 | 2.031/2.033 | 0.497 | 0.037 | -7.318 | 3.659 | -6.994 | 3.497 | -1.5
MgMn2O4 | 1.998 | 1.961/2.312 | 0.672 | 0.036 | -5.958 | 2.979 | -4.817 | 2.409 | -0.9
MgFe2O4 | 2.002 | 2.049/2.053 | 0.940 | 0.000 | -7.696 | 3.848 | -6.959 | 3.479 | 0.8
MgCo2O4 | 1.965 | 1.931 | 0.513 | 0.039 | -6.754 | 3.377 | -8.688 | 4.344 | 3.2
MgNi2O4 | 1.973 | 1.878/2.030 | 0.680 | 0.064 | -7.188 | 3.594 | -4.311 | 2.155 | -2.0
MgIr2O4 | 2.064 | 2.096 | 0.319 | 0.287 | -3.956 | 1.978 | -4.557 | 2.278 | 3.6
MgRh2O4 | 2.027 | 2.069 | 0.429 | 0.000 | -4.390 | 2.195 | -5.871 | 2.935 | 2.4

####  3.4.2 MgV2O4

Passing from MgTi2O4 to MgV2O4, the octahedral site is now filled by V3+ which
accommodates two electrons into the triply degenerate $t_{2g}$ orbitals. The
orbital degrees of freedom of these electrons allow a more complex interplay
of frustrated spins and orbitals. The trigonal distortion plays a significant
role in this compound while the tetragonal distortion is not pronounced. The
Vi-O bonds are compressed by 1%, splitting the $t_{2g}$ orbitals.

MgV2O4 is a S = 1 Mott insulator with a pyrochlore lattice. The DOS of MgV2O4
shown in Fig. 5 reveals that filled V-$d$ states are located above the oxygen
valence band in the range from -1 eV to 0 eV. The empty V-$d$ states are
located in the range from 1.5 eV to 6 eV. Compared to the Ti spinel, the
$t_{2g}$ orbitals are filled to a larger degree and are shifted downward close
to the O-$p$ states. Similar to the Ti spinel, this compound has an energy of
0.013 eV/atom above hull indicating that this material could be synthesized
and it may be stable.

As Tab. 2 shows, the calculated open-circuit voltage for the V spinel is about
2.7 V, however, the height of the migration barrier for Mg is 0.783 eV,
indicative of a low ion mobility. The normal spinel, MgV2O4, is more stable
than the inverse one. However, inspite of the high potential, the low ion
mobility prohibits the use of this compound as cathode material.

####  3.4.3 MgCr2O4

MgCr2O4 is an S = 3/2 Mott insulator with trivalent Cr. The three electrons
are accommodated in the triply degenerate $t_{2g}$ orbitals of Cr-$d$, which
all three degenerate states singly occupied, and cooperative distortions of
the CrO6 octahedra are absent with no orbital degrees of freedom. This
compound exhibits a geometrical frustration associated with the pyrochlore
structure of the Cr-sublattice.

The calculated DOS of MgCr2O4 shown in Fig. 5 exhibits filled Cr-$t_{2g}$
orbitals located in the upper part of the O-$p$ band in the range from -1 eV
to 0 eV. In contrast to the Ti and V spinels, the filled $t_{2g}$ orbitals are
merged with the upper part of the oxygen valence band. The bottom part of the
conduction band consists of Cr-$e_{g}$ orbitals in the majority spin
orientation. Structurally the Cr-O bonds are almost equal in the CrO6
octahedra, and solely trigonal distortions play a role. This compound has an
energy above hull of 0.037 eV/atom which indicates that the structure is
possibly stable. Compared to the other 3$d$ oxide spinel, it has relatively
small Mg migration barriers.

The small migration barrier of 0.497 eV and open-circuit voltage of 3.6 V
demonstrate that this compound can potentially be used as a cathode material
(see Tab. 2). The reported experimental value of 0.62$\pm$0.10 eV for the Mg
migration barrier in MgCr2O4 at high temperature 21 is consistent with our
calculation.

####  3.4.4 MgMn2O4

MgMn2O4 is an S = 2 Mott insulator with four electrons in the Mn-$d$ orbitals.
The $t_{2g}$ orbitals are filled by three electrons, and an additional
electron is inserted into the anti-bonding $e_{g}$ states of the Mn sites. The
presence of the electron in the $e_{g}$ orbitals causes the formation of Jahn-
Teller polarons. Structurally, two Mn-O bonds in the MnO6 octahedra are
elongated by 18% and cause a splitting of the doubly degenerate $e_{g}$
orbitals. Thus, the bandgap in MgMn2O4 is affected by this Jahn-Teller effect.
Note that the weakening of the Mn-O antibonding orbitals with neighboring
oxygen atoms upon the increase of the corresponding Mn-O bond distances causes
a stabilization of the occupied states.

As shown in Fig. 5, the filled valence band in MgMn2O4, which extends from -6
eV to 0 eV, is predominantly of O-$p$ character with some contribution from
the Mn-$d$ orbitals. The filled Mn-$e_{g}$ states in the majority spin
directions are located in the upper part of the oxygen valence band in the
range from -1 eV to 0 eV. The empty Mn-$e_{g}$ states in the majority spin
direction are in the range from 1.5 eV to 2 eV. In this compound, both
tetragonal and trigonal distortions play a signficant role. The energy above
hull of 0.036 eV/atom indicates possible stability of the structure.

Furthermore, we obtained a migration barrier of 0.672 eV in agreement with the
experimental value of 0.690$\pm$0.090 eV 21.Still, compared to the other 3$d$
oxide spinel, this only allows a relatively low mobility for the Mg ions due
to the high migration barrier. The open-circuit voltage is 2.98 V.
Nonetheless, the rather low stability makes the formation of the Mg-Mn
antisite defects upon inversion and consequently an even larger migration
barrier very likely. It should be mentioned that MgMn2O4 stabilizes in the
tetragonal crystal structure at low temperature. Thus a phase transition
occurs from the cubic structure to the tetragonal one. Still, compared to the
other 3$d$ oxide spinel, the Mn spinel shows relatively high voltage and
stability, which makes it a potential candidate for cathode usage.

####  3.4.5 MgFe2O4

MgFe2O4 is an S = 5/2 Mott insulator with half-filled Fe-$d$ orbitals. As Fe3+
accommodates five electrons in the $d$ orbitals within the high spin
configuration, this system exhibits a geometrical frustration of the Fe-
sublattice with no orbital degree of freedom. The band gap lies between filled
O-$p$ and the empty $t_{2g}$ states on the Fe site in the minority-spin
direction which determines all important features of the transport properties
in the Fe-spinel. Structurally, the Fe-O bonds in the FeO6 octahedra are
elongated by 0.5%, which, however, causes no Jahn-Teller active mode due to
the small distortion.

In contrast to the other 3$d$ spinel oxides, the top of the valence band has
an O-$p$ character, and the Fe-$d$ orbitals in the majority spin direction
split off below the O-$p$ valence band from -6.5 eV to -8 eV, as shown in Fig.
5. The conduction band consists of Fe-$d$ states in the minority-spin
direction in the range of 2-4 eV. The bandgap is about 2 eV predominantly in
majority-spin direction. All Fe-O bond lengths are equal showing that only
trigonal distortions play a role in this compound. The small energy above hull
of 0.000 eV/atom indicates a high stability which means that one should be
able to prepare this material. However, this compound shows the highest Mg
migration barrier and thus the lowest Mg-ion mobility compared to the other
considerd 3$d$ oxide spinels. Note that the Fe spinel lattice has a half-
closed shell which presumably makes it challenging to remove electrons.

The Fe spinel possesses the highest open-circuit voltages among the considered
oxide spinels, as detailed in Tab. 2. However, despite the high voltage, this
compound suffers from low ion mobility.

####  3.4.6 MgCo2O4

The Co spinel oxide is a non-magnetic Mott insulator in which its electrons in
the Co-$d$-orbitals are paired. This compound accommodates six electrons in
the $d$-orbitals, introducing three electrons with opposite signs in the
$t_{2g}$ states. Thus, this system exhibits no geometrical frustration in the
Co-sublattice, orbital, and spin degrees of freedom. Structurally, the Co-O
bonds in the CoO6 octahedra are identical, and only the trigonal distortion
plays an important role.

The calculated DOS of MgCo2O4 (see Fig. 5) shows a filled valence band which
extends from -7 eV to 0 eV, predominantly of O-$p$ character with the filled
contribution of Co-$t_{2g}$ orbitals in the upper part of the valence band.
The conduction band dominantly consists of the empty Co-$e_{g}$ contributions
along with oxygen bands. In this compound, the trigonal distortion is strong
compared to the other considered oxide spinel and causes a broadening of the
Co-$t_{2g}$ orbitals.

This Co oxide exhibits a good ionic conductivity reflected in the relatively
low Mg migration barrier of 0.513 eV, hence it might be a good candidate as a
cathode material. The energy above hull of 0.039 eV/atom shows that the spinel
framework is may be the stable phase for Co. The Co spinel shows a relatively
high open-circuit voltage of 3.38 eV which makes it a promising candidate as
electrode material. However, among the considered oxide spinels, this material
has a relatively large volume change of 3% upon charging/discharging which is,
however, still acceptable (see below).

####  3.4.7 MgNi2O4

In contrast to the other normal spinels, MgNi2O4 is weakly metallic with S =
3/2. This compound accommodates seven electrons in the $d$-orbitals,
introducing two electrons in the minority-spin direction of the $t_{2g}$
states. There is an interplay of frustrated spins and orbital degrees of
freedom in this system. Structurally, two Ni-O bonds are elongated by 8% in
the NiO6 octahedra which leads to the presence of distortion modes.

The calculated DOS of MgNi2O4 in Fig. 5 shows a continuously filled valence
band from -7 eV to 0 eV, predominantly of O-$p$ and Ni-$d$ character. At the
valence band top, the Ni-$t_{2g}$state in the minority-spin direction splits
into filled and empty orbitals with a small gap. Thus, the small bandgap in
MgNi2O4 is due to the splitting of Ni-$t_{2g}$ in the minority-spin direction.
This compound shows a strong trigonal distortion compared to the other
considered oxide spinel, but also tetragonal distortions play a role.

The energy above hull of 0.064 eV/atom indicates a low possibility for
synthetization and stability of the material. However, this compound has a
rather large Mg migration barrier indicating a low ionic conductivity. The
calculated open-circuit voltage is about 3.6 eV, which is relatively high. Our
calculations demonstrate a drop in potential to about 2.1 eV for low Mg
concentrations. Furthermore, the tetragonal crystal structure is more stable,
as Mg prefers the octahedral sites in this compound.

####  3.4.8 Mg(Rh/Ir)2O4

We turn now to oxide spinels with 4$d$ and 5$d$ transition metals located in
the B sites. These compounds often show a ferromagnetic ordering of the spins.
The identical magnetic direction of the spins does not induce any distortion
and the B-O bonds show no elongations or compressions. Electronically they are
not Jahn-Teller active.

It should be noted that in spinel oxides, the B-O bonds are predominantly
covalent. We have previously shown 19 that the migration barrier depends
sensitively on the B-O bond length and the electronegativity difference
between the B and O atoms.

We found a minimum in the height of the migration barriers for
electronegativity difference values of $\Delta{\chi}^{2}$ $\approx$ 2\. Based
on the identification of this pronounced minimum and the corresponding
matching properties of Rh and Ir, we identify MgIr2O4 as a super-ionic
conductor with high Mg mobility with a migration barrier of only 0.319 eV, as
shown in Fig. 6. Rh with a similar electronegativity as Ir also leads to a
high Mg mobility in the spinel oxide framework with a migration barrier of
only 0.429 eV (see Fig. 6).

The electronic structure of these two compounds is similar to each other. They
are non-magnetic oxides, solely with trigonal distortions. The calculated DOS
of MgRh2O4, Fig. 5 shows a filled valence band from -7.5 eV to -2.5 eV,
dominantly of O-$p$ character, followed by Rh-$t_{2g}$ states from -2.5 to 0
eV. The bandgap originates from the crystal field splitting of the $d$
orbitals of about 1 eV. The conduction band is made of empty $e_{g}$ orbitals.
This compound shows relatively small trigonal distortions compared to the
other oxide spinels.

While the energy above hull of 0.287 eV/atom for MgIr2O4 indicates instability
of the spinel structure, the high stability of the MgRh2O4 spinel oxide has
been calculated through the energy above hull of 0.000 eV/atom. The calculated
open-circuit voltages are 2.2 eV and 1.9 eV for the Rh and Ir spinels,
respectively. Our calculations yield an increase of the open-circuit voltage
for low Mg concentrations in both systems. Therefore, they are ideal
candidates for cathode materials in Mg batteries with medium voltages.

Figure 6: Mg migration energy barriers (in eV) as a function of the reaction
path coordinate derived from periodic DFT calculations combined with NEB for
the single-ion migration from tetrahedral site to the octahedral void and then
to the next tetrahedral site corresponding to the MgIr2O4 spinels with the
black color and the MgRh2O4 spinels with the blue color. The minimum energy is
set to zero.

Note that iridium and rhodium exhibit a low abundance in the earth’s crust,
making both iridium and rhodium to expensive raw materials. Battery materials
should be sustainable and associated with a high security of supply. However,
there is always a compromise between gravimetric and volumetric energy
density, abundance, cost and performance. Our calculations demonstrate that
there are spinel oxides that promise a high Mg mobility as an electrode
material. Possibly by doping less expensive oxide spinels with either iridium
or rhodium a compromise between cost and performance can be achieved.

###  3.5 Performance evaluation of the considered spinels

After having separately discussed the considered spinel materials, we now
summarize the results with respect to the ion mobility, and in the following
we do the same with respect to the electrochemical and structural properties
of the oxide spinels. Mn2O4 spinels have been successfully commercialized for
use in Li-ion batteries. However, for this particular spinel, there is a high
Mg2+ diffusion barrier of 0.67 eV, which is higher than the Li+ ion migration
barriers in all M2O4 (M = Mn, Co, Ni, Cr) spinels which lie in the range
0.40-0.60 eV. Among the considered 3$d$ transition metals, Fe (0.94 eV) and V
(0.78 eV) have the highest barriers for Mg2+ ion diffusion, while Cr and Co
are better conductors (0.51 eV). It should be noted that the migration
barriers at low Mg concentrations are usually higher than those at high Mg
concentrations. Thus, the Mg mobility becomes lower as the spinels change from
the Mg-rich to the Mg-deficient limit.

Our calculations reveal that the tetrahedral sites are more stable for Mg than
the octahedral sites for early transition metals in fully occupied Mg-spinels,
while late transition metals such as Fe, Co, and Ni lead to a higher Mg
stability in the octahedral sites.

Going from the 3$d$ transition metals to the 4$d$ and 5$d$ metals such as Rh
and Ir, the migration barriers obtained for Mg2+ in these oxide spinels lie
within 0.30-0.50 eV in the high Mg concentration limit. Accordingly, whereas
Li+ intercalates easily into various spinel oxides, Mg2+ intercalation can
benefit from more electronegative transition metals. Furthermore, these
elements suppress the structural distortions coming from the antiferromagnetic
order of the 3$d$-orbitals. We also propose that investigating doped systems
with two different transition metals might lead to the idenfication of further
oxide spinel materials with favorable properties as cathode materials for Mg
batteries.

An additional essential parameter for the performance of a battery system is
the open-circuit voltage. After understanding the electronic and structural
properties, we address the principles that determine the crucial properties of
electrochemical cells, voltages, and capacities.

The open-circuit voltage, $V_{OC}$, is obtained through the Mg intercalation
energy $E_{inter}$ with respect to a metallic magnesium anode. In the Mg
intercalation energy, the Gibbs free energy $G$ is approximated by the total
energy $E$, neglecting the influence of zero-point energies and entropic
contributions at finite temperatures.

Figure 7: The calculated voltage range for oxide spinel compounds as a
function of the redox-active transition metals. The voltage range is obtained
as the range between the calculated voltages for the low and high Mg
concentrations of the considered compounds.

The calculated open-circuit voltages in the high magnesium concentrations
indicate that the spinel oxides based on Cr, Mn, Fe, Co, and Ni exhibit high
theoretical voltages for Mg intercalation. It should be noted that the
voltages are referenced to the bulk metal phase of Mg metal. The Mg-based
spinel oxides exhibit lower potentials for Mg intercalation (less than 4 V)
compared with Li spinels that operate at potentials for Li intercation
voltages 4 to 5 V. Except for Ti which has an Mg intercalation voltage lower
than 2 V, most Mg intercalation potentials range between 2 and 4 V. However,
the additional charge of the bivalent Mg2+ causes a higher energy density in
spinel cathodes compared to the monovalent Li+. Hence, the considered Mg oxide
spinels are viable candidates from the point of view of theoretical capacity.

The maximum theoretical specific energy (MTSE) corresponding to the energy per
unit weight is given by

$\displaystyle\mathrm{MTSE}=(xV/W_{t})F$ (5)

where $W_{t}$ is the sum of the molecular weights of the reactants engaged in
the intercalating reaction with a Mg concentration $x$, and $V$ stands for the
average voltage of this reaction. $F$ is the Faraday constant.

Theoretically, most spinel oxides exhibit an average voltage of 2-4 V and
correspondingly a high MTSE. Therefore, oxide spinels are currently the most
promising candidates for high-performance Mg battery cathodes.

It should be mentioned that in these materials cooperative Jahn–Teller
distortions are the cause of volume contraction/expansion. In Tab. 2, we also
listed the calculated volume change $\Delta V/V$ upon Mg removal. Most of
these materials exhibit volume contraction and expansion after demagnesiation
of less than 3.7 % compared to the structures full of Mg2+. In the commercial
LiCoO2 cathode for Li-ion batteries, the cell volume shrinks by about 5% after
delithiation. 50 Therefore, the volume changes for magnesium intercalation are
relatively low for oxide spinel cathodes, indicating a good cathode
functionality with respect to volume change.

###  3.6 Synthesizability of Ir3+ and Rh3+ substituted spinel compounds

Our calculations have demonstrated that Mg2+ spinel oxides including 4$d$ and
5$d$ transition metals have promising properties as cathode materials for Mg
batteries, in particular also with regard to ion mobility. Nevertheless, the
feasibility of synthesizing pure compounds as well as solid solutions under
laboratory conditions needs to be guaranteed. Literature research confirms the
synthesisability of Rh3+ containing spinel oxides as pure end-member compounds
of Mg spinels like MgRh2O4 51, 52 or in solid solutions like
Co0.2Zn0.8Fe2-xRhxO4 53 or CuCr2-xRhxO4 54. Therefore, a prosperous
incorporation of Rh3+ in these compounds seems promising and should be
subjected to further investigation to find prospective candidates for
optimised ionic conductors in the field of Mg-oxide spinel compounds.

In contrast to Rh, the 5$d$ equivalent Iridium show a different behavior. Just
few oxidic compounds with Ir3+, like in layered K0.75Na0.25IrO2 55, are
reported, as most oxides show Ir in oxidation states $\geq$ 4+. Miao and
Seshadri have addressed this problem explaining it by relativistic effects 56.
These relativistic effects lead to a distinct stabilization in oxidation
states between those typical for Rh and Ir oxides. In contrast to Rh, which is
stable as sesquioxide, the iridates exhibit a higher stability in dioxides, or
at lower chemical potentials a decomposition to Ir and O2 appear.

These results are confirmed by our lab experiments in which we investigated
possible solid solutions in the MgCr2-xMxO4 (M: Rh, Ir) system. Different
attempts to include Ir3+ in the Magnesiochromite lattice failed, leading to
the formation of either IrO2 or Ir as second phases.

Figure 8: XRD patterns at room temperature for MgCr2-xRhxO4 with x = 0-0.5 (a)
full pattern; (b) section from 15-20∘ 2th showing the shift of peak positions.

In contrast to that, initial results of our group suggest that the
incorporation of Rh3+ into the Magnesiochromite lattice is possible. Samples
with different amounts of rhodium have been synthesized by solid state
reaction of MgO (ABCR 99,5%), Cr2O3 (Strem 99%) and Rh2O3 (Santa Cruz
Biotechnology). Stoichiometric amounts of the oxides were mixed in an agate
mortar in acetone. Afterward the mixture was pressed into pellets and
calcinated in alumina boats in a muffle oven (Nabertherm) for one week at
1000∘C with one intermediate grinding. The resulting brown powder was analysed
by X-ray diffraction on a STOE StadiP diffractometer with monochromatic
MoK$\alpha_{1}$ radiation. The XRD patterns of compounds with Rh content of x
= 0-0.5 are depicted in Fig. 8a. An expected shift of the peak position can be
clearly observed (Fig. 8b) as well as a continuous change in the intensities.
Rietveld refinements were performed with Fullprof 57. The results are given in
Tab. 3 where the crystallographic information is listed.

Beside the main phase, small amounts of 0.2 – 4.2% of impurities are visible
in the diffractograms. These consist of Cr2-xRhxO3 and Rh. The lattice
parameters follow Vegard’s law, with a small deviation (see Fig. 9). This
sample shows the highest amount of impurity and differs by its color. In
contrast to the others, this one is slightly green. The formation of a solid
solution is therefore obvious, but must be investigated further, as well as
the effects of the substitution on the physical properties. This will be part
of a future work.

Table 3: Crystallographic information obtained by rietveld refinement. xRh | Cell-parameter $a$ | Thermal displacement | Occupancy | Impurities | Chi2
---|---|---|---|---|---
by weight | in Å | parameter $u$ in Å2 | OccRh | in wt% |
0.0 | 8.34376 | 0.38601 | 0 | - | 3.00
0.1 | 8.35497 | 0.38626 | 0.00416 | 1.0% Cr2O3 | 1.70
0.2 | 8.35589 | 0.38628 | 0.00248 | 4.2% Cr2O3 | 2.25
0.3 | 8.36755 | 0.38612 | 0.00539 | 0.2% Rh | 2.00
0.4 | 8.37596 | 0.38605 | 0.01020 | 0.4% Rh | 1.51
0.5 | 8.38555 | 0.38584 | 0.01316 | 0.3% Rh | 1.80
Figure 9: linear fit of lattice parameter with rising content of xRh (grey:
excluded sample); standard deviation is indistinguishable from symbols.

##  4 Conclusions and Summary

In this work, we have combined first-principles eletronic structure
calculations and experimental preparation efforts to identify promising oxide
spinel materials for Mg battery cathodes with superior Mg mobility properties.
According to our calculations, oxide spinels with the electronegative $4d$ and
$5d$ transition metals Rh and Ir allow such superior transport properties
together with an acceptable open circuit voltage with respect to a Mg metal
anode. The specific non-magnetic electronic structure of these oxide spinels
make the compounds rather stable showing only relatively small trigonal
distortions. Furthermore, we performed first attempts to synthesize these
materials obtaining promising results, in particular with respect to Ir
compounds. In general, our results and the analysis based on a combined
experimental and theoretical study provide a conceptual framework to
understand fast ion conductivity in oxide spinel electrode materials that will
also be beneficial for the understanding and improvement of ion mobility in
other materials classes.

## Acknowledgements

This work contributes to the research performed at CELEST (Center for
Electrochemical Energy Storage Ulm-Karlsruhe) and was funded by the German
Research Foundation (DFG) under Project ID 390874152 (POLiS Cluster of
Excellence) and the Dr. Barbara Mez-Starck foundation. The authors acknowledge
computer time provided by the state of Baden-Württemberg through bwHPC and the
German Research Foundation (DFG) through grant no INST 40/575-1 FUGG (JUSTUS 2
cluster).

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