Mechanism of magnesium transport in spinel chalcogenides

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.


Introduction
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 LIBs 1,2 that motivate the search for alternative battery chemistries. 3,4 As a promising alternative, magnesium has been proposed [5][6][7][8] as an active element with a much higher earthabundance of 13.9% compared to 7×10 −4 % of Li. The ionic radii 1 of Mg 2+ , 0.86Å and Li + , 0.90Å 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 formation [11][12][13][14][15] and a high melting point.
A high multi-valent ionic conductivity of 1-10 mS cm −1 has 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 temperatures, Chevrel phases and layered and spinel TiS 2 structures have been studied in detail. 24 A Mg-ion migration barrier of about 550 meV was found in cubic Ti 2 S 4 using Galvanostatic 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 Mg 2+ and host oxygen lattice 23 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 voltage 26,27 and thus causes a reduction of the energy densities of chalcogenide materials.
Recently, MgSc 2 Se 4 has been found to be a super-ionic conductor exhibiting a high Mgion conductivity of 0.1 mScm −1 at room temperature. 25 This high ion mobility does not only make MgSc 2 Se 4 to 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 MgSc 2 Se 4 is also a good electron conductor.
Doping MgSc 2 Se 4 by Ti and Ce leading to Ti 4+ -and Ce 4+ 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 the lanthanoids. 29 Note that spinel structures including transition metal ions such as Ti, Mn, Fe and Co exhibit magnetic properties due to the unfilled 3d shell. The 3d electrons in the spinel compounds cause significant distortions of the crystal lattice, namely trigonal distortion. Since the physical and chemical properties of these compounds strongly depend on the d electrons, 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 octahedral 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 conductivity 33 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 Mgion mobility in MgB 2 X 4 spinel 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 k 64 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.

Computational details
First-principles calculations have been carried out in the framework of density-functional theory (DFT) 34,35 in order to determine the properties of MgB 2 X 4 (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 Simulation Package. [38][39][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 −5 eV 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 eVÅ −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 10Å between them has been chosen.
The Mg intercalation energy E inter in the spinel structure with respect to a metallic magnesium anode is given by where E(Mg y B 2 X 4 ) is the total energy of the spinel with a Mg concentration y in the unit cell, and E(Mg) is the cohesive energy of Mg bulk in the metal phase. The corresponding open circuit voltage (V OC ) is then given by where F is the Faraday constant and z corresponds to the elementary charges that are transferred upon the discharging reaction with z = 2 for Mg-ion batteries. When E inter is expressed in eV, then V OC in volts is simply given by E inter /2 for Mg-ion batteries.

Results and Discussion
Among the complex transition-metal (B) oxides and chalcogenides, spinel structure with the composition Mg 2+ B 3+ 2 X 2− 4 correspond to the most promising Mg-ion conductors. 28,42,43 The spinel structure, illustrated in Fig the effective radii r(Mg) and r(B) of the Mg and metal cations, respectively, according to 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 u = 3 8 , an ideal spinel structure without any trigonal distortion results. u > 3 8 is associated 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  distortion of the octahedron further divides the threefold degenerate t 2g states into a lower a 1g state and a twofold degenerate e g states as illustrated in Fig. 2a. It should be noted that the representation of the a 1g state is 1 √ 3 (xy + yz + zx), pointing towards the center of the B-lattice tetrahedron. The e g states are different from the doubly degenerate e g states and they are perpendicular to the a 1g state. At low temperatures, 46 alternatively a tetragonal distortion often occurs which splits the threefold degenerate t 2g states into a higher xy state and the twofold degenerate yz/zx lower states. The tetragonal distortion divides the doubly degenerate e g states as well into x 2 − y 2 and 3z 2 − r 2 states.
Besides the additional cystal field splitting, the trigonal distortions also modify the bonding distances, as mentioned in the previous paragraph. This can be quantified by explicitly looking at the Mg-X distances d(cn 4 ) and d(cn 6 ) 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 parameter u as, 45 Using Eq. 4, the ratio k 64 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 k 64 = 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  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 E a is influenced by the anion species and the size of the triangle. Oxide cathode materials typically exhibit sluggish Mg 2+ migration kinetics and also limited cycle lifes. The magnitude of the Mg 2+ 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 Mg 2+ 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 Mg 2+ migration barriers of some selected sulfide spinels. All compounds represent Mg-ion migration energy smaller than 0.7 eV, confirming the relatively good Mg 2+ conductivity in these spinel structures. MgTi 2 S 4 is identified as a suitable Mgion conductor, however, this compound is found to be unstable in the spinel structure and showed electronic conductivity. 49 Sulfide spinels enhance the p-d hybridization compared to oxides and tend to be more conducting. The various transition metal ions with d 1 up to the d 10 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 MgSc 2 S 4 spinel compound represents a balance between small Mg 2+ migration energies and sufficient structural stability. Thus, in the following we will only focus on MgB 2 X 4 compounds 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 MgB 2 X 4 spinels 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 t 2g (green) and e g (yellow) manifolds. In both compounds with Sc and Y cations, respectively, the valence bands are dominated by S-and Se-p bands, respectively, in the energy range from −4 eV to 0 eV. For both systems, the DOS of the t 2g and e g states 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  Table 1 lists calculated properties of the considered spinel systems. These include structural properties of Mg(Sc/Y) 2 (S/Se) 4 spinels, the Mg migration barrier, the Mg intercalation Table 1: Mg-X, B-X, B-B, and Mg-Mg bond lengths inÅ for spinel compounds. B and X denote transition-metal (Sc, Y) and anion (S, Se) respectively. Calculated relative barrier energy E a , intercalation energy E high inter (E low inter ) (Eq. 1) for high (low) Mg concentration in eV, and corresponding open-circuit voltage V high OC (V high OC ) in V. The volume changes with respect to the structure without Mg is indicated by ∆V /V .  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 MgY 2 (S/Se) 4 compounds exhibit smaller volume changes upon the addition of Mg atoms than the MgSc 2 (S/Se) 4 compounds.
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) 4 compounds 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 Mg 2+ tends to occupy the octahedral sites in the MgMn 2 S 4 spinel 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 MgB 2 X 4 (B = Sc, Ti, V, Cr, Mn, Fe, Co, Ni, Y, Al and X = S, Se).
In order to see this, we focus on the ratio k 64 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 MgAl 2 S 4 system characterized by a ratio of about k 64 = 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 k 64 . 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 k 64 the octahedral becomes increasingly more stable with respect to the tetrahedral site. Note that the ratio k 64 = 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 k 64 , as for example in MgMn 2 S 4 with k 64 = 1.05, the octahedral site is energetically more favorable whereas for larger values of k 64 as in MgSc 2 S 4 with k 64 = 1.10, the tetrahedral site becomes preferred (see Fig. 6a).
A similar reasoning has recently been presented in order to understand the Mg tetrahedral 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 implicitly assumes that the interaction is purely ionic between non-polarizable atomic charges  so that the ionic interaction is additive. Let us make a simple estimate about the stability of the tetrahedral Mg-X 4 site vs. the octahedral Mg-X 6 site assuming that only the direct interaction between the Mg 2+ cation and the neighbouring chalgonide X 2− anions contribute to the interaction. For non-polarizable, sperically symmetric and non-overlapping charges, the binding energies E(Mg-X 4 ) and E(Mg-X 6 ) in the tetrahedral and the octahedral arrangement, respectively, are given by 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-X 4 ) = E(Mg-X 6 ), for a ratio of First of all note that this ratio of 1.5 is much larger than the value of k 64 = 1.08 at which there is an equilbrium between tetrahedral site and octahedral site in MgAl 2 S 4 . In addition, the fact whether a spinel exhibits a tetrahedral or an octahedral site preference does not only depend on the ratio k 64 , but also on the anion parameter u. In Fig. 6b, we again show the ratio k 64 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 k div 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 k 64 .
In order to understand this trend, one should first note that according to Eq. 4 both distances d(cn 4 ) and d(cn 6 ) become larger with increasing u in 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 chalgonidederived states shown in Fig. 4. For covalent and metallic interactions, the strengths of single bonds typically decreases with increasing coordination 15 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 k 64 = d(cn 6 )/d(cn 4 ) 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 preferences in spinel structures. Similar to the Goldschmidt tolerance factor t 54 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 ABX 3 , we use a geometrical analysis to assess the relative stability of the Mg 2+ 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. 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.