Mechanism of magnesium transport in spinel
Mohsen Sotoudeh,†Manuel Dillenz,†and Axel Groß∗,†
†Institute of Theoretical Chemistry, Ulm University, Albert-Einstein-Allee 11, 89081 Ulm,
‡Helmholtz Institute Ulm (HIU) for Electrochemical Energy Storage, Helmholtzstraße 11,
89069 Ulm, Germany
In the area of sustainable energy storage, batteries based on multivalent ions such
as magnesium have been attracting considerable attention due to their potential for
high energy densities. Furthermore, they are typically also more abundant than, e.g.,
lithium. However, as a challenge their low ion mobility in electrode materials remains.
This study addresses the ionic conductivity of magnesium in spinel host materials
based on periodic density functional theory calculations in order to identify the critical
parameters which determine the mobility and insertion of ions. We will in particular
highlight the critical role that trigonal distortions of the spinel structure play for the
ion mobility. In detail, we will show that it is the competition between coordination
and bond length that governs the Mg site preference in ternary spinel compounds upon
trigonal distortions which can only be understood by also taking covalent interactions
into account. Based on our theoretical study, we rationalize the impact of the metal
distribution in the host material and the ion concentration on the diﬀusion process.
Furthermore, cathode-related challenges for practical devices will be addressed. Our
ﬁndings shed light on the fundamentional mechanisms underlying ionic conductivity in
solid hosts and thus may contribute to improve ion transport in battery electrodes.
The development of Li-ion batteries (LIBs) had a major impact on the wide-spread use
of portable electronic devices. However, there are safety and abundance issues associated
with LIBs1,2 that motivate the search for alternative battery chemistries.3,4 As a promising
alternative, magnesium has been proposed5–8 as an active element with a much higher earth-
abundance of 13.9% compared to 7×10−4% of Li. The ionic radii1of Mg2+, 0.86 ˚
Li+, 0.90 ˚
A are rather similar, but Mg has the advantage of being a bivalent ion which
leads to a higher volumetric capacity of Mg metal anodes compared to Li, 3833 mAhcm−3
vs 2062 mAhcm−3, and also to a low reduction potential of -2.37 V vs SHE compared to
-3.05 V of Li .9,10 Furthermore, Mg-ion batteries (MIBs) exhibit a low tendency for dendrite
formation11–15 and a high melting point.
A high multi-valent ionic conductivity of 1-10 mS cm−1has been achieved in MIBs at high
temperatures . 16,17 However, a major problem for MIBs lies in the sluggish kinetics during
intercalation at room temperature.2,18 It should be noted that the design of chemically stable
electrodes with high ionic conductivity is highly desirable,2,19–23 as a low ionic mobility can
severely limit the performance of batteries.
In order to address the slow migration of Mg-ion in cathode materials at low tempera-
tures, Chevrel phases and layered and spinel TiS2structures have been studied in detail.24
A Mg-ion migration barrier of about 550 meV was found in cubic Ti2S4using Galvanos-
tatic Intermittent Titration Technique measurements. Studies on the sulﬁde and selenide
spinel frameworks indicate low energy barriers for Mg-ion diﬀusion comparable to those
of LIBs.25 In contrast, oxide spinel cathode materials exhibit sluggish kinetics of Mg-ion
which is caused by the relatively strong Coulombic attraction between the guest Mg2+ and
host oxygen lattice23 which leads to a lower ion mobility. The smaller electronegativity of
sulfur and selenium lattices enlarges the lattice constant of these materials and thus also
their ion mobility as typically diﬀusion barriers become smaller for larger lattice constants.
Nevertheless, the increase of the ion mobility through the lowering of diﬀusion barriers is
also accompanied by lower Mg insertion energies into the spinel structures which lowers the
voltage26,27 and thus causes a reduction of the energy densities of chalcogenide materials.
Recently, MgSc2Se4has been found to be a super-ionic conductor exhibiting a high Mg-
ion conductivity of 0.1 mScm−1at room temperature.25 This high ion mobility does not only
make MgSc2Se4to a promising cathode material for MIBs, it also suggests that it could be
used as a solid electrolyte. However, solid electrolytes need to exhibit a very low electronic
conductivity whereas MgSc2Se4is also a good electron conductor.
Doping MgSc2Se4by Ti and Ce leading to Ti4+- and Ce4+ impurities, respectively, has
been considered as a means to lower and neutralize the electronic conductivity.22 Still, a
high electron conductivity has been observed in these materials which has been related to
the presence of defects or the phase deformation.25,28 Furthermore, it has been shown that
for chalcogenide spinels containing lanthanoids the Mg mobility increases with the size of
Note that spinel structures including transition metal ions such as Ti, Mn, Fe and Co
exhibit magnetic properties due to the unﬁlled 3dshell. The 3delectrons in the spinel com-
pounds cause signiﬁcant distortions of the crystal lattice, namely trigonal distortion. Since
the physical and chemical properties of these compounds strongly depend on the delectrons,
it is important to understand the role of electrons on the ionic ordering, lattice distortion,
and magnetic properties. Speciﬁcally, there are no convincing explanations with respect to
the factors that determine the spatial distribution of the cations over the tetrahedral or oc-
tahedral sites and also with regard to the dependence of the activation barriers for migration
on the doping level.30,31 Studies on concerted migration 32 and the impact of the structural
framework on the ionic conductivity33 were carried out to analyze the factors determining
the energy barriers for migration. However, there are still open questions regarding the
cation ordering within the lattice and ion mobility in the various concentrations.
In this paper we report ﬁrst-principles electronic structure calculations addressing the Mg-
ion mobility in MgB2X4spinel structures. We particularly focus on the electronic properties
determining ion migration in these materials. We ﬁnd a strong dependence of the stability
of the octahedral vs. the tetrahedral sites on the ion concentration which we explain by
an octahedral distortion and the corresponding changes in the lattice constants. Based on
geometric considerations, we identify the ratio of distances in the octahedron and tetrahedron
k64 as a descriptor for the stability of the cations within the octahedral and tetrahedral sites
in the spinel lattice. These insights also provide a framework for proposing promising spinel
materials with a high ion-mobility based on fundamental materials properties.
First-principles calculations have been carried out in the framework of density-functional
theory (DFT)34,35 in order to determine the properties of MgB2X4(B = Sc, Ti, V, Cr, Mn,
Fe, Co, Ni, Y, Al and X = S, Se) spinel with regard to Mg migration. Exchange-correlation
eﬀects are approximated within the generalized gradient approximation (GGA) using the
Perdew-Burke-Ernzerho (PBE) functional.36 The calculations are performed employing the
Projector Augmented Wave (PAW)37 method as implemented in the Vienna Ab-initio Sim-
ulation Package.38–40 The nudged elastic band (NEB)41 method is used to determine Mg-ion
migration barriers. A 2×2×2 supercell of the primitive spinel cell is constructed for the
NEB calculations, including 56 atoms. The total energy has been evaluated with a 2×2×2
k-point mesh. A plane wave cutoﬀ of 520 eV has been chosen in the expansion of the wave
functions, and total energies have been converged within 1 ×10−5eV per supercell.
Mg-ion migration in the chalcogenides has been studied in the low (one Mg vacancy per
supercell) and high (one Mg inside supercell) vacancy limit. The structures were fully relaxed
until the forces on the atoms were converged within 0.05 eV ˚
A−1. The NEB calculations have
been carried out with four distinct images between the tetrahedral and octahedral sites to
evaluate the Mg-ion migration trajectory. To minimize the interaction between the migrating
Mg ions across periodic boundaries, a distance of 10 ˚
A between them has been chosen.
The Mg intercalation energy Einter in the spinel structure with respect to a metallic
magnesium anode is given by
Einter(Mg) = E(Mgx+yB2X4)−(E(MgyB2X4) + xE(Mg)) ,(1)
where E(MgyB2X4) is the total energy of the spinel with a Mg concentration yin the unit
cell, and E(Mg) is the cohesive energy of Mg bulk in the metal phase. The corresponding
open circuit voltage (VOC ) is then given by
where Fis the Faraday constant and zcorresponds to the elementary charges that are
transferred upon the discharging reaction with z= 2 for Mg-ion batteries. When Einter is
expressed in eV, then VOC in volts is simply given by Einter/2 for Mg-ion batteries.
Results and Discussion
Among the complex transition-metal (B) oxides and chalcogenides, spinel structure with the
4correspond to the most promising Mg-ion conductors.28,42,43 The
spinel structure, illustrated in Fig. 1, consists of a face-centered cubic lattice of X anions
(X = O, S, Se) with two kinds of interstices between the sites of the fcc lattice: tetrahedral
interstices MgX4and octahedral interstices BX6. The BX6octahedra forms a network of
edge-sharing chains while the Mg ions are located in the tetrahedrally vacant spaces of X ions,
forming the MgX4units. The B sublattice of the spinel structure is known as the pyrochlore
lattice with a strong geometrical frustration eﬀects. The Mg sublattice forms a diamond
lattice. As far as the electronic structure of the transition metal spinels are concerned, the
d-orbitals split into the high-lying doubly degenerate egand low-lying triply degenerate t2g
orbitals caused by the crystal ﬁeld splitting of the regular BX6octahedron.
However, spinel structures often exhibit a trigonal distortion of the octahedra that cor-
responds to a displacement of the X-ions along the  direction and changes the Oh
octahedral symmetry to a D3hoctahedral symmetry but keeps the overall octahedral shape
unchanged (see Fig. 2a).44 The trigonal distortion can be characterized by a uanion param-
eter45 that reﬂects the displacement of the X-ions along the  direction in units of the
lattice constant a. Sickafus et al.45 showed that this parameter can be expressed through
Figure 1: (a) Rock-salt, (b) zinc-blende, and (c) spinel structure. The spinel lattice is an
ordered mixture of the zinc-blende and rock-salt structure. The A species (yellow) of AB2X4
occupy the tetrahedral sites, while the B species (blue) only occupy octahedral sites. The
red spheres denote the oxide and chalcogenide anions such as O2−, S2−, and Se2−.
the eﬀective radii r(Mg) and r(B) of the Mg and metal cations, respectively, according to
u= 0.3876 r(B)
Interestingly, the eﬀective radius of the X anions does not enter this expression which means
that the size of these anions obviously does not aﬀect the trigonal distortions. For a value of
8, an ideal spinel structure without any trigonal distortion results. u > 3
with a trigonal distortion of the octahedra through which the tetrahedrons are enlarged at
the expense of the octahedrons whereas it is the other way around for u < 3
8. The trigonal
0.385 0.386 0.387 0.388 0.389 0.390
Anion parameter u
Figure 2: (a) Illustration of the transition from an undistorted octahedron cage with a Oh
octahedral symmetry to a trigonally distorted cage with a D3hoctahedral symmetry and the
associated further crystal ﬁeld splitting of the d-states. (b) Dependence of the ratio k64 on the
anion parameter ucharacterizing the trigonal distortion for S-spinels. The black dots denote
the results for the original spinel structures with the Mg ion in a tetrahedral site and the
octahedral vacancy being empty, and the black line corresponds to the analytical expression
Eq. 5. The blue diamonds are determined for the relaxed spinels with the octahedral site
occupied by a Mg anion. The blue line is a linear regression of these results.
distortion of the octahedron further divides the threefold degenerate t2gstates into a lower
a1gstate and a twofold degenerate e0
gstates as illustrated in Fig. 2a. It should be noted that
the representation of the a1gstate is 1
√3(xy +yz +zx), pointing towards the center of the
B-lattice tetrahedron. The e0
gstates are diﬀerent from the doubly degenerate egstates and
they are perpendicular to the a1gstate. At low temperatures,46 alternatively a tetragonal
distortion often occurs which splits the threefold degenerate t2gstates into a higher xy state
and the twofold degenerate yz/zx lower states. The tetragonal distortion divides the doubly
degenerate egstates as well into x2−y2and 3z2−r2states.
Besides the additional cystal ﬁeld splitting, the trigonal distortions also modify the bond-
ing distances, as mentioned in the previous paragraph. This can be quantiﬁed by explicitly
looking at the Mg-X distances d(cn4) and d(cn6) in the tetrahedral and octahedral sites,
respectively. In the original spinel structures with the Mg ion in a tetrahedral site and the
octahedral vacancy being empty, these distances can be expressed as a function of the anion
d(cn6) = 2(u−3
Using Eq. 4, the ratio k64 between the bond lengths in the octahedral and the tetrahedral
sites is given by
Here we indicated that in the perfect crystal with u= 3/8=0.375 the ratio is k64 = 2/√3≈
1.15 which means that in this structure the Mg-X bond length in the octahedral sites is 1.15
times larger than the tetrahedral bond length.
In Fig. 2b, we have plotted the ratio k64 as a function of the anion parameter ufor a
number of ternary Mg spinels. The upper black circles correspond to the values for the Mg
ion in a tetrahedral site and the octahedral vacancy being empty. It is obvious that k64
decreases approximately linearly with uin the small considered interval of uvalues which
are larger than the value of 0.375 for the ideal structure, i.e., for all considered spinels the
size of the tetrahedra is enlarged at the expense of the octahedron.
Furthermore, it is important to note that in the presence of the Mg ions in the octahedral
vacancy, k64 is further reduced as illustrated by the blue symbols in Fig. 2b. Hence due to the
explicit interaction of Mg cations with the surrounding chalcogenide anions, the size of the
octahedrons further shrinks with respect to the tetrahedron. The dependence of k64 on uis
in general still linear, but there are outliers. This is particularly obvious for MgMn2S4where
the presence of Mn apparently leads to a signiﬁcant compression of the occupied octahedron.
The trigonal distortion roughly increases with higher d-states occupation of the transition
Figure 3: (a) Illustration of single-ion migration from tetrahedral site to the octahedral void
and then to the next tetrahedral site. The chalcogenide atoms such as S and Se are shown
by the red spheres, the migrating Mg ion is presented by the spheres inside the tetrahedrons
and the octahedron. (b) Calculated Mg migration barriers for several transition metal ions
in the sulﬁde-spinels.
metals, except for V and Co that do not exactly follow the order.
We now focus on the Mg mobility in the ternary spinel structures. The Mg-ion migration
occurs between two tetrahedral sites via the migration across the face-sharing octahedral
void which is shown in Fig. 3a. The transition state for the Mg migration is located in
the triangular face between the octahedral and tetrahedral sites. The magnitude of the
activation energy Eais inﬂuenced by the anion species and the size of the triangle. Oxide
cathode materials typically exhibit sluggish Mg2+ migration kinetics and also limited cycle
lifes. The magnitude of the Mg2+ migration barriers can be reduced by introducing a soft
anion (i.e. S, Se, Te) lattice.25,47,48 This leads to a weaker Coulombic attraction and a larger
lattice constant which also increases the distance between the guest Mg2+ and the host lattice,
thus enhancing ion mobility. However, an increase in the ion mobility is typically associated
with a reduction of energy density because low diﬀusion barriers are usually accompanied
by small intercalation energies.
Fig. 3b shows the calculated Mg2+ migration barriers of some selected sulﬁde spinels. All
compounds represent Mg-ion migration energy smaller than 0.7 eV, conﬁrming the relatively
good Mg2+ conductivity in these spinel structures. MgTi2S4is identiﬁed as a suitable Mg-
ion conductor, however, this compound is found to be unstable in the spinel structure and
showed electronic conductivity.49 Sulﬁde spinels enhance the p-dhybridization compared to
oxides and tend to be more conducting. The various transition metal ions with d1up to
the d10 conﬁgurations lead to magnetic structures that are caused by the strong Coulomb
repulsion within the d-orbitals.50 In addition, the d-orbital electron interaction decreases
the atomic distances and adds more trigonal distortion to the system as shown in Fig. 2b.
This obviously increases the Mg migration barriers. Hence transition metals with occupied
d-orbitals in general reduce the Mg-ion conductivity depending on the particular orbital
character. Transition metal ions such as Sc with empty d-orbitals, on the other hand,
lead to small migration barriers. In particular, the MgSc2S4spinel compound represents
a balance between small Mg2+ migration energies and suﬃcient structural stability. Thus,
in the following we will only focus on MgB2X4compounds with empty d-orbitals which
are characterized by a high Mg-ion mobility according to our calculations. It is interesting
to note that an analogous trend has been found in a recent computational study of Mg
migration in lanthanoid chalcogenide spinels.29 In these systems, apparently the height of
the Mg migration barriers increases with higher f-state occupancy.
In order to elucidate the inﬂuence of the electronic structure on the properties of the
spinels, we have plotted in Fig. 4 the density of states (DOS) of MgB2X4spinels with B =Sc
and Y, and X = S and Se that can be realized experimentally.51,52 Note that these spinel
structures also exhibit trigonal distortions, but they are smaller than those for the spinels
with later d-band metals, as shown in Fig. 2b. In Sc and Y, the d-orbitals are empty which
leads to unoccupied t2g(green) and eg(yellow) manifolds. In both compounds with Sc and
Y cations, respectively, the valence bands are dominated by S- and Se-pbands, respectively,
in the energy range from −4 eV to 0 eV. For both systems, the DOS of the t2gand egstates
is rather broad and overlaps with each other. The main eﬀect of replacing S ions by Se
ions is a reduction of the band gap by about 0.5 eV and a smaller crystal ﬁeld splitting
-4 -3 -2 -1 0 1 2 3 4 5 6
Figure 4: Density of states for MgSc2S4, MgSc2Se4, MgY2S4, and MgY2Se4from top to
bottom. The total DOS is given in gray. The projected DOS are shown in red for S and Se,
in green for t2gand yellow for egd-orbitals. The energy zero is set to the top of the valence
between anti-bonding egand non-bonding t2gstates. In the valence band depicted in Fig. 4,
d-derived states appear although Y and Sc in principle have no occupied d-states. These
states originate from the hybridization between the d-states of the transition metal and the
chalcogenides pbands,53 but they do not dominate the behavior of the valence band.
Table 1 lists calculated properties of the considered spinel systems. These include struc-
tural properties of Mg(Sc/Y)2(S/Se)4spinels, the Mg migration barrier, the Mg intercalation
Table 1: Mg-X, B-X, B-B, and Mg-Mg bond lengths in ˚
A for spinel compounds.
B and X denote transition-metal (Sc, Y) and anion (S, Se) respectively. Cal-
culated relative barrier energy Ea, intercalation energy Ehigh
inter) (Eq. 1) for
high (low) Mg concentration in eV, and corresponding open-circuit voltage Vhigh
OC ) in V. The volume changes with respect to the structure without Mg is
indicated by ∆V/V .
Compound Mg-X (˚
A) B-X (˚
A) B-B (˚
A) Mg-Mg (˚
A) Ea(eV) Ehigh
inter (eV) Vhigh
OC (V) Elow
inter (eV) Vlow
OC (V) ∆V /V (%)
MgSc2S42.464 2.593 3.784 4.634 0.415 -5.149 2.574 -5.165 2.582 -10
MgSc2Se42.587 2.725 3.974 4.868 0.375 -3.915 1.958 -4.114 2.057 -5
MgY2S42.510 2.740 3.949 4.836 0.360 -5.508 2.754 -5.561 2.780 +4.8
MgY2Se42.624 2.868 4.131 5.059 0.361 -4.329 2.165 -4.432 2.216 -2
0 10 20 30 40 50
Reaction path length %
0 10 20 30 40 50
Reaction path length %
Figure 5: The Mg2+ 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 corresponding to the S-spinels (black)
and the Se-spinels (blue) for low and high concentrations of Mg ions. Note that the full
migration path in principle includes the further migration from the octahedral to the tetra-
hedral site, but as the corresponding energies are symmetric with respect to the octahedral
site, this part is omitted.
energy and the open-circuit voltage in the high and low Mg concentration limit, and the vol-
ume change upon Mg intercalation. Based on the calculations, MgY2Se4is a favorable
candidate due to the combination of a small migration barrier, an suﬃciently large open-
circuit voltage, and a small volume change. MgSc2Se4and MgY2S4are also characterized by
parameters which make them suitable as Mg-ion conductors. However, the performance of
MgY2S4is detoriated, in spite of a high open-circuit voltage VOC , by an unfavorable volume
expansion of almost 5% after Mg-ion removal.
In order to further assess the ion mobility in these spinels structures, the energies along the
Mg migration paths for MgSc2(S/Se)4and MgY2(S/Se)4in the high and low concentration
of Mg-ion are plotted in Fig. 5. Note that in the high Mg-ion concentration limit, there are
7 Mg ions in the 2×2×2 supercell located in the tetrahedral sites one of which is migrating,
whereas in the high Mg-ion concentration limit, there is only one Mg ion in the supercell that
is also the migrating ion. The Mg-ion migration barriers of MgY2S4(∼360 meV), MgY2Se4
(∼361 meV), and MgSc2Se4(∼375 meV) in the high Mg concentration limit are rather
small leading to a high Mg mobility which is comparable to Li+in fast Li-conductors. This
suggests that S- and Se-spinel structures together with Sc and Y cations can act as excellent
Mg conductors. Furthermore, the band gaps of about 1.5 eV for the selenides and of about
2 eV for the sulﬁdes should lead to a relatively low electron conductivity. Hence in principle,
these materials might as well be considered as promising candidates for solid electrolytes in
Mg-ion batteries because of their high Mg-ion mobility. However, experiments still found a
high electron conductivity in these compounds,22 probably due to the presence of defects or
phase deformations,25,28 hindering their use as solid electrolytes.
In the low Mg concentration limit, the Mg-ion migrations barriers in the S- and Se-spinels
are increased compared to the high concentration limit, as shown in Fig. 5. Furthermore, in
Mg0.125Sc2(S/Se)4the Mg-ion prefers the six-fold coordination of the octahedral site, whereas
in Mg0.125Y2(S/Se)4the Mg-ion prefers the fourfold coordination of the tetrahedral site. Thus
in the S- and Se-spinels structures together with Sc the most favorable site for the Mg-ion
changes from the octahedral to the tetrahedral site upon increasing the Mg concentration.
This varying site preference, which is not the case for the Y cation, might be detrimental
for the performance of the Sc-containing cathodes upon charge/discharge. In addition, the
MgY2(S/Se)4compounds exhibit smaller volume changes upon the addition of Mg atoms
than the MgSc2(S/Se)4compounds.
Up to now, we have concentrated on the electronic properties, structural parameters,
and Mg migration paths. Of particular interest is that all Mg(Sc/Y)2(S/Se)4compounds
favor the tetrahedral sites for the Mg ions. However, in the low Mg concentration limit,
Mg ions prefer the octahedral site in the Sc spinels. In order to analyze this behavior, we
will ﬁrst concentrate on the high Mg concentration limit. Interestingly, according to our
calculations Mg2+ tends to occupy the octahedral sites in the MgMn2S4spinel in the high
Mg concentration limit. Here we will show that it is the competition between coordination
and bond length induced by the trigonal distortion that governs the Mg site preference in
ternary spinel compounds MgB2X4(B = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Y, Al and X = S,
In order to see this, we focus on the ratio k64 between the Mg-X bond length in the
occupied tetrahedral and octahedral sites, as shown for some ternary spinels by the blue
symbols in Fig. 2b. According to our calculations, for the MgAl2S4system characterized
by a ratio of about k64 = 1.08, the octahedral and the tetrahedral site become energetically
degenerate with regard to the Mg occupation, as illustrated in Fig. 6a. This can be explained
by a competition between bond length and coordination as a function of the ratio k64. The
octahedral site has the higher coordination than the tetrahedral site, but obviously in the
ideal structure the elongation of the Mg-X bond length by 1.15 with respect to the tetrahedral
sites makes the octahedral site energetically still less favorable. However, for decreasing ratio
k64 the octahedral becomes increasingly more stable with respect to the tetrahedral site. Note
that the ratio k64 = 1.08 is still larger than 1, but at this value the larger bond length is
compensated by the higher coordination of the octahedral site. For even smaller values of
k64, as for example in MgMn2S4with k64 = 1.05, the octahedral site is energetically more
favorable whereas for larger values of k64 as in MgSc2S4with k64 = 1.10, the tetrahedral site
becomes preferred (see Fig. 6a).
A similar reasoning has recently been presented in order to understand the Mg tetra-
hedral site preference in lanthanoid chalcogenide spinels,29 based on the concept that the
preference for coordination of a cation by an anion can be estimated by classic radii ratio
rules. This argumentation about the competition between bond length and coordination im-
plicitly assumes that the interaction is purely ionic between non-polarizable atomic charges
0.385 0.386 0.387 0.388 0.389
Anion parameter u
Ni Fe Co
0 10 20 30 40 50
Reaction path length %
Figure 6: (a) Mg-ion migration barriers for spinel compounds with diﬀerent trigonal dis-
tortions characterized by k64. (b) The ratio k64 as a function of the anion parameter ufor
selected spinel compounds. Blue diamonds denote high Mg concentration compounds and
red triangles low Mg concentration compounds. The black line represents a dividing line
between Mg tetrahedral and octahedral site preference.
so that the ionic interaction is additive. Let us make a simple estimate about the stability
of the tetrahedral Mg-X4site vs. the octahedral Mg-X6site assuming that only the direct
interaction between the Mg2+ cation and the neighbouring chalgonide X2−anions contribute
to the interaction. For non-polarizable, sperically symmetric and non-overlapping charges,
the binding energies E(Mg-X4) and E(Mg-X6) in the tetrahedral and the octahedral ar-
rangement, respectively, are given by
E(Mg −X4) = 4QMg2+ QX2−
E(Mg −X6) = 6QMg2+ QX2−
where we have used cgs units for the sake of simplicity. For this purely ionic interaction the
binding energies are the same, i.e., E(Mg-X4) = E(Mg-X6), for a ratio of
First of all note that this ratio of 1.5 is much larger than the value of k64 = 1.08 at which
there is an equilbrium between tetrahedral site and octahedral site in MgAl2S4. In addition,
the fact whether a spinel exhibits a tetrahedral or an octahedral site preference does not only
depend on the ratio k64, but also on the anion parameter u. In Fig. 6b, we again show the
ratio k64 as a function of the anion parameter u, but now we also include some additional
data points for the low Mg-concentration limit. In addition, we have inserted a dividing line
given by kdiv
64 = 4.78(1 −2u). In spinels above this line, the migrating Mg ions prefer the
tetrahedral site whereas in those below this line, the octahedral site is more stable. Thus
for larger values of u, the octahedral become more stable than the tetrahedral sites only for
smaller values of the ratio k64.
In order to understand this trend, one should ﬁrst note that according to Eq. 4 both
distances d(cn4) and d(cn6) become larger with increasing uin the parameter range that is
considered here. However, for purely ionic interactions between non-polarizable spherically
symmetric ions, the competition in the energetic stability between two diﬀerent structures
does not depend on the absolute distances, only on the ratio of distances, 29,54,55 as reﬂected
in the simple estimate Eq. 7. Consequently, these results can only be explained assuming
that the interaction is not purely ionic and that it falls oﬀ stronger than 1/d with distance d.
Or, in other words, covalent interactions contribute substantially to the stability of the Mg
atoms in the voids. The important role of covalent contributions in the interaction within
the spinels is also reﬂected in the signiﬁcant width in the density of states of the chalgonide-
derived states shown in Fig.4. For covalent and metallic interactions, the strengths of single
bonds typically decreases with increasing coordination15 based on bond-order conservation
arguments, so that the single bond becomes weaker for higher coordination. Furthermore,
these interactions scale with the overlap between atomic orbitals which falls oﬀ exponentially
for larger distances. Hence, the ratio k64 =d(cn6)/d(cn4) needs to become smaller for
absolute larger distances, i.e., for larger values of u, to make the octahedral more stable than
the tetrahedral site.
Our ﬁndings provide a simple picture of the key parameter underlying Mg-ion site pref-
erences in spinel structures. Similar to the Goldschmidt tolerance factor t54 which is used
to reﬂect the variance in the stability of perovskites based only on ratio of the atomic radii
of A, B, and X in ABX3, we use a geometrical analysis to assess the relative stability of
the Mg2+ sites in spinels. Our calculations and considerations of the structure of the spinel
compounds clearly indicate that it is the ratio together with the absolute values of the Mg-X
bond lengths in the octahedral and tetrahedral sites that determines the site preference and
thus also the Mg mobility.
Conclusions and Summary
Based on periodic density functional theory calculations, we have studied Mg ion mobility
in spinel chalcogenides which are promising candidates for cathodes in Mg-ion batteries.
Overall, we ﬁnd that trigonal distortions of the spinel structures play a critical role for
both the Mg site preference as well as for the Mg migration barriers. With respect to the
transition metal used in the spinels we ﬁnd that an increasing d-band occupancy leads to
smaller lattice constants and larger trigonal distortions which both lead to larger migration
barriers and thus decreasing diﬀusitivities. Hence we concentrated on spinel chalcogenide
compounds with the early d-band metals Sc and Y together with the soft ion chalcogenide
S and Se.
Indeed, all these four considered spinels exhibit small diﬀusion barriers of about 400
meV and smaller. In addition, these materials allow open-circuit potentials with respect to
metallic Mg of about 2.5 V for the sulﬁdes and of about 2.0 V for the selenides. This makes
them well-suited as cathode materials for Mg-ion batteries. On the other hand, the low
diﬀusion barriers together with the band gap of about 1.5 eV for the selenides and of about
2 eV for the sulﬁdes limiting their electronic conductivity suggests that these materials could
also be used as solid electrolytes in Mg-ion batteries because of their high Mg-ion mobility.
In many spinel structures studied so far, the tetrahedral sites exhibits a higher stability
than the octrahedral sites for Mg insertion. Interestingly, we ﬁnd that in the Sc-based spinels
this stability is reversed in the low Mg concentration limit. Our detailed analysis reveals
that the varying site preference is a consequence of the competition between coordination
and bond length induced by trigonal distortions and absolute changes in the bond distances
demonstrating the important role of covalent contributions to the chemical interaction within
the spinels. In general, our results and the analysis based on electronic and geometric
factors provide a conceptual framework to understand fast ion conductivity in spinel electrode
materials that will also be beneﬁcial for the understanding and improvement of ion mobility
in other materials classes.
M.S. thanks Sung Sakong and Mohnish Pandey for fruitful discussions. Financial sup-
port from the Cluster of Excellence POLiS (EXC-2154, project 422053626) of the Deutsche
Forschungsgemeinschaft (DFG) and computer time provided by the state of Baden-W¨urttemberg
through bwHPC and the German Research Foundation (DFG) through grant no INST
40/467-1 FUGG (JUSTUS cluster) are gratefully acknowledged.This work contributes to the
research performed at CELEST (Center for Electrochemical Energy Storage Ulm-Karlsruhe).
Conﬂict of Interest
The authors declare that the research was conducted in the absence of any commercial or
ﬁnancial relationships that could be construed as a potential conﬂict of interest.
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