Influence of Spin State and Cation Distribution on Stability and Electronic Properties of Ternary Transition-Metal Oxides.

This work is a systematic ab initio study of the influence of spin state and cation distribution on the stability, dielectric constant, electronic band gap, and density of states of ternary transition-metal oxides. As an example, the chemical family of spinel ferrites MFe2O4, with M = Mg, Sc-Zn is chosen. Dielectric constant and band gap are calculated for various spin states and cation configurations via dielectric-dependent self-consistent hybrid functionals and compared to available experimental data. When choosing the most stable spin state and cation configuration, the calculated electronic properties are in reasonable agreement with measured values. The nature of the excitation is investigated through projected density of states. A pronounced dependence of band gap energy and dielectric constant on the spin state and cation configuration is observed, which is a possible explanation for the large variation of the experimental results, in particular, if several states are energetically close.

Introduction

It has been shown previously that the quantum-chemical calculation of properties of open-shell transition-metal oxides represents a challenge, even for state-of-the-art computational methods.[1] Transition-metal oxides are interesting candidates for many applications because of their versatile chemistry, such as photoelectrochemical water splitting,[2] spintronics,[3] and many more. Theoretical modeling of these systems allows for the investigation of properties that are difficult to access experimentally, such as the nature of excitation or the effect of geometrical changes on the electronic structure. Furthermore, it is possible to investigate compounds that have not been synthesized yet, or the effect of ion distribution on the electronic structure, which is experimentally difficult to obtain. Thus, it is of general interest to find theoretical methods that are suitable to treat these compounds.

Standard density functional theory (DFT) functionals based on the localized-density approximation (LDA) or generalized-gradient approximation (GGA) have been successfully applied to a wide range of materials. However, properties of strongly correlated systems, particularly transition-metal oxides with a problematic electronic structure, are often not predicted reliably. Additionally, certain electronic properties such as the fundamental band gap are known to pose a problem in calculation. A major contribution to this error arises from the self-interaction error due to the approximate nature of the exchange term in LDA and GGA functionals. Most of the published computational studies of transition-metal oxides thus rely on the one-center Hubbard U correction (DFT + U)[4] because of its superior computational efficiency compared to hybrid methods. The DFT + U method has been applied frequently to calculate properties of the class of spinel ferrites.[5−30] The correction parameter U can in principle be calculated self-consistently from first principles,[31] but is in most cases adjusted empirically and has to be optimized individually for each compound and every (d-) element. Thus, the predictive power of this approach is limited. Therefore, we decided to employ a different theoretical framework. The calculation of the fundamental band gap is usually approximated by the energy difference between the lowest unoccupied and the highest occupied one-particle energy level. It has been demonstrated repeatedly that hybrid methods give more accurate ionization energies and electron affinities within the one-particle approximation than GGA functionals due to a reduction of the self-interaction error.[32] In principle, the amount of Fock exchange in the hybrid exchange functional (eq ) can also be treated as an empirical parameter.

In this work, we apply a technique where α is obtained from first principles. Previous work[33−36] showed that dielectric-dependent self-consistent hybrid (ddsch) methods[37,38] provide high accuracy for electronic band gaps and absolute band positions. In this work, we therefore applied ddsch methods, where the Fock exchange fraction α is self-consistently calculated from the static dielectric constant[38] as

In this work, we applied an implementation suggested by Erba,[38] which includes a fully automated self-consistent calculation of α from the dielectric constant by adopting a coupled-perturbed-Hartree–Fock/Kohn–Sham (CPHF/KS) approach. Using this procedure, every compound as well as every cation and spin distribution will be treated with an optimal fraction of exchange. As we will show in the following, cation and spin distribution have a strong effect on electronic properties of the ferrites, making this feature crucial.

However, it has to be clarified, that using the one-electron KS levels to deduce the band gap of magnetic insulators with highly localized 3d orbitals is only an approximation. Systems with strongly correlated electrons cannot be described accurately by the single-determinant Ansatz of DFT. Physically more sound results are obtained with the GW method[32,39] or the dynamical mean-field theory.[40] However, these approaches are computationally too demanding to investigate the large number of systems considered in this study. The aim of this work is not to predict or reproduce experimental band gaps, but to emphasize the influence of cation distribution and magnetic state on the electronic properties. Additionally, the multi-reference character of all MFe2O4 compounds with M being a d0 or d10 element is not very pronounced because of the rather stable high spin d5 configuration of Fe3+.

The investigated material class of spinel ferrites MFe2O4 (M = Mg, Sc–Zn) represents a systematic test set of transition-metal oxides including all d-elements of the 4th period and a d0 element (Mg). CaFe2O4, as the fourth-row “d0-ferrite”, crystallizes in an orthorhombic space group and was therefore not considered.

The selected ferrites, as far as they are experimentally investigated, crystallize in the cubic space group Fd3̅m (spinel structure). Oxygen forms a cubic close-packed lattice where M and Fe occupy one-eighth of tetrahedral (8a) and half of the octahedral (16d) states. The structural formula can be written as [M1–Fe]T[MFe2–]OO4, where the superscripts T and O denote the tetrahedral and octahedral sites, respectively. In this notation, the cation distribution parameter y is also called the degree of inversion. A spinel with y = 0 is considered perfectly normal, while y = 1 denotes a fully inverse spinel. The cation distribution depends on M as well as the synthesis conditions.[41] In real systems, the degree of inversion is neither exactly 1 nor exactly 0, so that 0 < y < 1. An example for a normal spinel is ZnFe2O4 for which a degree of inversion of y < 1% was reported,[42] but it was also shown that the degree of inversion is extremely dependent on the synthesis conditions.[43] An example for an inverse spinel is NiFe2O4 with a degree of inversion close to 100%.[44]

Not only the cation distribution, but also spin and charge distribution depend on the nature of M. It is generally accepted that spinel ferrites with M = Mg, Zn are antiferromagnets. If M is a d5–d9 element, the resulting spinel ferrite is a ferrimagnet, but also other spin configurations are possible.

Spinel ferrites with M = Sc, Ti, V, Cr have been scarcely investigated so far. In general cation, charge and spin distributions are therefore not known as well. Hence, all unique spin states and cation distributions with y = 0, 1 within the primitive unit cells (PUCs) were tested in the present work.

Results and Discussion

Investigated Configurations

The investigated spin states depend on the cation distribution as well as the oxidation state of M. While for the oxidation states II and IV, there are only two possible cation distributions in the PUC, where M is either in tetrahedral or in octahedral positions, the situation is more complicated for MIII. For the oxidation state MIII, FeII and FeIII ions are present, whose distribution has to be accounted for. All of the investigated occupancies are displayed in Table and denoted by the letters A to G. To decrease the computational effort, no partial inversion was accounted for, so that y = 0, 1 for every occupancy. Normal and inverse spinels correspond to configuration A and B, respectively. For y = 0 and 1, every unique cation distribution in the PUC was considered. Table shows the related spin distributions together with an acronym, where “fm” indicates ferromagnetism, “afm” indicates antiferromagnetism, and “fi” indicates ferrimagnetism. If M is a d0 or d10 element, no spin was considered and the respective assignment for M in Table can be ignored. The final nomenclature for the investigated states is composed of the variant of cation distribution from Table and the acronym of the spin state from Table . An overview of the resulting configurations can be found in the Supporting Information.

Table 1

Fractional Coordinates of Tetrahedral and Octahedral Positions in the Primitive Unit Cell and Their Possible Occupancies

	tet (8a)		oct (16d)
position	1	2	3	4	5	6
X	0.125	–0.125	–0.5	–0.5	0.0	–0.5
Y	0.125	–0.125	–0.5	–0.5	–0.5	0.0
Z	0.125	–0.125	–0.5	0.0	–0.5	–0.5

Table 2

Investigated Spin States and Their Acronyms

position
1	2	3	4	5	6	acronym
↑	↑	↑	↑	↑	↑	fm
↑	↓	↑	↑	↓	↓	afm1
↑	↓	↑	↓	↑	↓	afm2
↓	↓	↑	↑	↑	↑	fi1
↑	↑	↓	↓	↑	↑	fi2
↑	↑	↑	↑	↓	↓	fi3
↑	↑	↑	↓	↓	↓	fi4
↑	↑	↓	↓	↑	↓	fi5

Important Configurations

The cation and spin configurations introduced above were set up for the PUC of MFe2O4 with M = Mg, Sc–Zn. While keeping the cubic symmetry, full structural relaxations were carried out using PW1PW and effective core potential (ECP) basis sets. For the relaxed structures, ddsch calculations with ECP and also triple-ζ basis sets were performed. The complete set of the results can be found in the Supporting Information. From these results, it is evident that the calculated dielectric constant and thus the amount of Fock exchange are dependent on the basis set. While providing a stable SCF procedure, in particular during geometry optimization, the rather small ECP basis sets lead to strong overestimations of the band gaps. Therefore, the final calculation of the band gap was performed with the larger rev2-pob-TZVP basis sets. Hence, only the relative stabilities (calculated with standard PW1PW), dielectric constants (ddsch), and band gaps (ddsch) obtained with the triple-ζ basis set are listed in Tables and 4. The most relevant spin configurations were chosen by an energy difference of maximum 20 kJ/mol.

Table 3

Energetics, Dielectric Constant, and Band Gap of the Most Stable Spin Configurations for M = Mg, Mn–Zn

M	state	ΔE [kJ/mol]	ε	εoptexp	Eg [eV]	Egexp [eV]
Mg	B-fi4	0.0	4.96		3.57	1.85–2.43a
	A-fm	18.5	4.22		3.56
	B-afm2	19.7	4.76		3.45
Mn	A-fi1	0.0	6.48	(4.24b, 4.84b)	1.07	0.98–2.37c
Fe	B-fi4	0.0	10.94		0.91	0.12–1.92d
	A-fi2	2.8			0.00
	B-fi1	5.0			0.00
	B-fi2	15.3	7.88		0.00
Co	B-fi1	0.0	6.76	6.66e	1.28	1.17–2.62f
	A-fi1	4.1	5.71		2.06
	B-fi4	13.4	6.71		1.54
	A-afm1	19.8	5.85		2.02
Ni	B-fi1	0.0	6.42	5.52e	2.27	1.52–3.54g
	B-fi4	12.2	6.34		1.82
Cu	B-fi1	0.0	7.37	(4.37b, 5.15b)	0.95	1.37–1.61h
	B-fi3	18.9	6.72		1.06
Zn	A-fm	0.0	4.74	5.76e	3.00	1.78–3.25i
	A-afm1	2.2	4.89		2.89
	B-fi4	14.0	5.57		2.71

References.[47−52]

Reference (53).

References.[54−56]

References (57) and (58).

Reference (59).

References.[50,59−66]

References.[16,59,61,63,65,67−73]

References.[62,63,69,74,75]

References.[59,62,63,76−79]

Table 4

Energetics, Dielectric Constant, and Band Gap of the Most Stable Spin Configurations for M = Sc–Cr

M	state	ΔE [kJ/mol]	ε	Eg [eV]
Sc	E-fi2	0.0	5.06	2.77
	D-fi1	7.6	4.83	2.81
	D-afm2	15.4	5.07	1.85
Ti	F-afm1	0.0	4.44	3.18
	G-fm	16.2	5.48	0.00
V	D-fi2	0.0		0.00
Cr	D-fi1	0.0	5.52	2.25
	D-fi2	0.7	5.57	2.34
	D-fi3	10.5		0.00

Results for M = Mg, Mn–Zn

Results for configurations of spinels with M = Mg, Mn–Zn within this energy range are listed in Table together with optical dielectric constants and a range of experimental band gaps from the literature. A more detailed list of the experimental gaps, including measurement temperature, method, and particle shape and size, can be found in the Supporting Information, Table S4. Because all of the experimental gaps are obtained from optical measurements, the calculated fundamental gaps cannot be directly compared. The energy of excited states is always smaller than the electronic gap because of excitonic effects. Thus, the optical gap is usually smaller than the fundamental gap. Exceptions can occur because of selection rules, for example, when the lowest excited states are symmetry forbidden. The excitonic binding energy of transition-metal oxides varies between a few millielectron volts[45] and several tenth of an electron volts.[46]

As Table shows, all of the compounds, except MnFe2O4, feature several cation configurations and spin states that have to be considered.

In the most stable configuration of MgFe2O4, the magnesium atoms occupy octahedral states, and the iron atoms in tetrahedral positions couple with iron atoms in octahedral positions. For this configuration, a band gap of 3.57 eV was calculated. This overestimation of the highest experimental band gap by 1.14 eV cannot be explained by excitonic effects. The other configurations which have to be considered exhibit a different antiferromagnetic coupling with the same cation distribution as well as a ferromagnetic configuration with magnesium in tetrahedral positions. For those configurations, the band gaps are quite similar, 3.45 and 3.56 eV, respectively. Mössbauer spectra have shown that MgFe2O4 is a mixed spinel with preference for inverse cation distribution.[44] The required energy to shift a Mg2+ ion from a tetrahedral to an octahedral site was found as 0.14 eV = 13.5 kJ/mol. This is consistent with the calculated energy difference between the states B-fi4 and A-afm1 of 21.2 kJ/mol (see Supporting Information, Table S2). The magnetic moment was measured to be μB ≠ 0,[44] which is most probably because of the incomplete inversion of the material.

For MnFe2O4 only one configuration matches the selected energy criterion. Here, manganese atoms occupy tetrahedral positions and couple ferrimagnetically with the iron atoms in octahedral sites. With a calculated band gap of 1.07 eV, the resulting band gap is close to the lower limit of the experimental range. The present results are in accordance to neutron diffraction measurements, which show that MnFe2O4 exhibits a predominantly normal cation distribution with y ≈ 0.8 while coupling ferrimagnetically.[44]

For magnetite (Fe3O4), the experimental results for the electronic structure vary strongly, ranging from a conducting system up to a semiconductor with a band gap of 1.92 eV. Except for a ferrimagnetic state, where FeII atoms occupy octahedral positions, the considered configurations are all predicted to be conducting. Here, a band gap of 0.91 eV was obtained. While it is generally accepted that Fe3O4 is an inverse spinel, experimental results show indications for an average charge of +2.5 of iron in octahedral positions[80] as well as a charge separation to +2 and +3,[81] depending on the temperature. Because the present results rely on calculations at 0 K, a charge separation was to be expected but has only been observed for configuration B-fi2, which exhibits two symmetry-inequivalent iron atoms in the octahedral positions due to the spin alignment. In experiment, Fe3O4 shows ferrimagnetic coupling between tetrahedrally and octahedrally coordinated iron atoms.[81] This corresponds to configuration B-fi1 which only differs from the most stable configuration by 5 kJ/mol.

CoFe2O4 also displays four different combinations of cation and spin distribution within the chosen energy range. In the most stable one, cobalt atoms occupy octahedral sites. Iron and cobalt atoms in the octahedral positions couple ferrimagnetically with iron atoms in tetrahedral positions. A corresponding spin configuration with cobalt atoms in tetrahedral positions, as well as two states with a net magnetic moment of 0, either with cobalt in octahedral or tetrahedral positions, are considered. All of the calculated band gaps fit within the experimental range. However, as shown in Table S4 in the Supporting Information, there are two experimental ranges for the band gap, 1.17–1.58 and 2.44–2.62 eV, respectively. These results most probably correspond to the respective indirect and direct transition.[60] Hence, assuming excitonic effects to be small, only the calculated band gaps for B-fi1 and B-fi4 are in the correct range. Experimental results indicate that CoFe2O4 is largely inverse with y = 0.76–0.93 and shows ferrimagnetic coupling.[44] Moving a Co2+ ion from an octahedral to a tetrahedral site requires an energy of 0.19 eV = 18.3 kJ/mol. This differs from the calculated energy difference between configuration B-fi1 and A-fi1 (4.1 kJ/mol), which is lower, but the preference for inverse cation distribution as well as the ferrimagnetism is reproduced in accordance with the experiment.

For NiFe2O4, two energetically close configurations were identified, both with nickel atoms occupying octahedral sites and ferrimagnetic spin ordering. Mössbauer spectra determined the degree of inversion of NiFe2O4 to be close to 100% with a magnetic moment of 2.20μB per chemical formula unit, which indicates a ferrimagnetic coupling between tetrahedral and octahedral sites.[44] This corresponds to the most stable configuration B-fi1. Again, the measured band gaps show a high variation, but in contrast to CoFe2O4, the results cannot be divided in two groups (see Supporting Information, Table S4). Both of the calculated band gaps, 1.82 and 2.27 eV, are within the experimental range.

According to the calculations, CuFe2O4 has two relevant configurations, in which copper atoms occupy octahedral lattice sites. Both of them are ferrimagnetic (fi1 and fi3). In this case, the range of the calculated (0.11 eV) and experimental (0.24 eV) band gaps is rather small. The difference between measured and calculated values, 0.31–0.66 eV, is consistent with large excitonic effects. However, because the SCF stability for CuFe2O4 was not satisfactory, these results have to be considered with caution. Depending on the synthesis route, CuFe2O4 was shown to be mostly inverse with y = 0.74–1.[82] In a different experimental study, it was shown that CuFe2O4 exhibits a ferrimagnetic nature that can be changed to superparamagnetic behavior by the synthesis conditions.[83] Both inverse cation distribution and the ferrimagnetism are reproduced correctly by the presented results.

When surveying the experimental band gap of ZnFe2O4, two clusters of values, one in the range of 1.78–2.01 eV and another one in the range of 2.61–3.25 eV, can be found, similar to CoFe2O4. Yet, unlike to CoFe2O4, these results may not be assigned to the indirect and direct band gap whose energy difference is only ∼0.1 eV.[76] A possible explanation would be the presence of oxygen defects in some experiments, which are not covered by the stoichiometric models used in the calculations. This explanation is supported by the findings of Sultan and Singh.[78] Experimentally, ZnFe2O4 was shown to be an antiferromagnetic normal spinel with a degree of inversion below 1%;[42] however, the degree of inversion is extremely dependent on the synthesis conditions.[43] While the cation ordering is reproduced correctly by the presented results, the ferromagnetic solution is more stable than the antiferromagnetic one by 2.2 kJ/mol. The same result was gained by previous theoretical work[84] on hybrid-DFT level of theory (PBE0) with an energetic preference of 0.030 eV = 2.9 kJ/mol for the ferromagnetic state.

Apart from the band gap, the ddsch approach also delivers another physical property, the static dielectric constant ε (Table ). The experimental dielectric constants of MnFe2O4 (4.24 and 4.84) and CuFe2O4 (4.37 and 5.15) in the study by Vasuki and Balu[53] deviate significantly from the calculated dielectric values of MnFe2O4 A-fi1 with 6.48, and CuFe2O4 B-fi1 (B-fi3) with 7.37 (6.72). Yet, the optical band gaps obtained in this study, 5.98 eV for MnFe2O4 and 5.63 eV for CuFe2O4, vary strongly from other published values (see Supporting Information, Table S4), so the severity of this divergence should be considered cautiously. A qualitative result from ref (52) is that the measured value of the dielectric constant is very sensitive to the determination of the refraction index, which was done via the Moss formula[85] and relations suggested by Anani et al.[86] The results of Chand et al.[59] for CoFe2O4 (6.66), NiFe2O4 (5.52), and ZnFe2O4 (5.76) are in better agreement with the respective theoretical values of 6.76, 6.42, and 4.47, especially for CoFe2O4. However, it has to be mentioned that the experimental band gaps for NiFe2O4 (3.54 eV) and ZnFe2O4 (3.25 eV) in this study are again rather high.

Results for M = Sc–Cr

The same methodology was applied to predict the electronic properties of the remaining members of the first-row transition-metal ferrites MFe2O4. Table shows the relative stability, dielectric constant, and band gap for the most stable configurations with M = Sc–Cr.

The dielectric constant for the different configurations of ScFe2O4, 4.83–5.07, is relatively insensitive to the cation and spin distribution, but their impact on the band gap is significant. While the two ferrimagnetic configurations E-fi2 and D-fi1 have a relatively large band gap of 2.81 eV, the antiferromagnetic configuration has a band gap of only 1.85 eV.

In TiFe2O4, the most stable configurations F-afm1 and G-fm contain a tetravalent Ti and divalent Fe, which is in agreement with the literature.[87] According to our results, TiFe2O4 is either a conductor or it shows a pronounced band gap of 3.18 eV, depending on the cation and spin distribution. This result is supported by previous works, where an experimental band gap of 2.0 eV[88] is reported, while a theoretical study on the GGA level of theory by Liu et al. predicts TiFe2O4 to be a ferromagnetic half-metal.[89]

The most stable configuration of VFe2O4 is D-fi2 with trivalent vanadium ions. As can be seen in the Supporting Information, almost all of the investigated cation and spin distributions display conducting behavior. This is also the case for the most stable configuration listed in Table .

CrFe2O4 is most stable with FeII in tetrahedral positions and CrIII occupying octahedral sites. Three different ferrimagnetic configurations were found to be the most stable. For the configurations D-fi1 and D-fi2, a band gap of ∼2.3 eV was calculated, while the third configuration D-fi3 has a metallic ground state.

Projected Density of States

To investigate the influence of cation distribution and spin state on the nature of excitation, the projected density of states (PDOS) was calculated for all of the configurations listed in Tables and 4. In this section, the PDOS for ZnFe2O4 are displayed exemplarily, and PDOS for all of the other configurations listed in Tables and 4 can be found in the Supporting Information, Figures S1–S25.

Figure shows the PDOS for ZnFe2O4 A-fm. The band gap in the minority channel is an oxygen-to-iron transition. As can be seen from the figure as well as from Table S2 in the Supporting Information, the band gap energy in the majority channel is higher than in the minority channel. Furthermore, because all of the iron states near to the valence band maximum (VBM) are occupied by up-spin electrons, there are no iron states hybridizing into the VBM in the minority channel. On the other hand, these unoccupied down-spin states are lower in energy than unoccupied iron states from the majority channel and hence dominate the conduction band minimum (CBM). The asymmetry of spin up and spin down channel can also be seen in the spin densities on the oxygen atoms of ∼0.3μB, which is still small compared to the magnetic moment of iron, ∼4.3μB. The pronounced difference between majority and minority channel will have an impact on the optical properties of this configuration.

Figure 1

PDOS for ZnFe2O4 A-fm.

Figure shows the PDOS for ZnFe2O4 A-afm1. Again, the VBM is dominated by O 2p states and the CBM by Fe 3d states. The Zn 3d states seem to have an insignificant influence on the VBM and CBM.

Figure 2

PDOS for ZnFe2O4 A-afm1.

The PDOS for ZnFe2O4 B-fi4 is shown in Figure . Here, the channels are no longer symmetric because the iron atoms carrying up-spin electrons are located in tetrahedral positions, while those with down-spin electrons occupy octahedral sites. Again, the figure shows an oxygen-to-iron VBM–CBM transition in both channels.

Figure 3

PDOS for ZnFe2O4 B-fi4.

The composition of the VBM and CBM of the configurations listed in Tables and 4 can be seen in Table . The VBM of those spinels where M is a d0 or d10 element, which are ZnFe2O4 and MgFe2O4, is dominated by oxygen states, independent of the configuration. The same can be seen for NiFe2O4 B-fi4. Those spinels where M is a d5 to d9 element mostly exhibit a VBM consisting of hybridized O 2p and M 3d states. Exceptions are, as already mentioned, NiFe2O4 B-fi4, but also Fe3O4 B-fi4, which exhibits a VBM dominated by Fe 3d states, and CoFe2O4 B-fi1, whose VBM is dominated solely by M 3d states. The spinels with M = Sc, Ti, Cr show a VBM dominated by Fe 3d states. When the transition metals M and Fe contribute significantly to the VBM, the excitation might be a d-d transition.

Table 5

Composition of the VBM and CBM of the Configurations Listed in Tables and 4

main composition of VBM
O 2p	O 2p/M 3d	Fe 3d	M 3d
ZnFe2O4 A-fm	MnFe2O4 A-fi1	Fe3O4 B-fi4	CoFe2O4 B-fi1
ZnFe2O4 A-afm1	CoFe2O4 A-fi1	ScFe2O4 E-fi2
ZnFe2O4 B-fi4	CoFe2O4 B-fi4	ScFe2O4 D-fi1
MgFe2O4 B-fi4	CoFe2O4 A-afm1	ScFe2O4 D-afm2
MgFe2O4 A-fm	NiFe2O4 B-fi1	TiFe2O4 F-afm1
MgFe2O4 B-afm2	CuFe2O4 B-fi1	CrFe2O4 D-fi1
NiFe2O4 B-fi4	CuFe2O4 B-fi3	CrFe2O4 D-fi2

The CBM of the majority of configurations listed in Tables and 4 is dominated by Fe 3d states. Exceptions are CuFe2O4 B-fi1 and B-fi3, whose CBM consists of hybridized states of all elements, as well as Fe3O4 B-fi4 and TiFe2O4 F-afm1, whose CBM consists of hybridized Fe 3d and M 3d states.

Conclusions

In this study, electronic properties of transition-metal oxides depending on cation and spin distribution were investigated using a dielectric-dependent self-consistent hybrid density approach. The calculated optical dielectric constants and fundamental band gaps show a strong dependency on the cation and spin distribution. Not only the band gap energy, but also the nature of the lowest-energy excitation is affected. This result is a possible explanation for the great variability in experimental results where the materials might have different degrees of inversion and magnetic states because of different preparation routes. If it were possible to control cation and spin distribution, this would be an additional way to tune the materials.

Computational Details

The calculations were carried out with the CRYSTAL17[90] program package version 1.0.2. While keeping the cubic symmetry, full structural relaxations were carried out using the PW1PW[91] functional, ECP basis sets based on those by Heifets et al.[92] for O, by Heyd et al.[93] for Mg, and by Dolg et al.[94] for Sc–Zn (see the Supporting Information). ddsch calculations based on the GGA-functional PWGGA[95] were carried out with the optimized geometries and the ECP basis sets, and with the triple-ζ basis sets rev2-pob-TZVP.[96] A Monkhorst–Pack grid of 8 × 8 × 8 was sufficient for most of the calculations, yet for conducting states, a denser Gilat net of 16 × 16 × 16 was necessary to reach SCF convergence. Different spin states were generated by setting initial atomic magnetic moments (ATOMSPIN) and fixing the total magnetic moment during the calculation (SPINLOC2/BETALOCK). The truncation criteria for bielectronic integrals (TOLINTEG) were set to 10–7 for the overlap and penetration threshold for Coulomb integrals and for the overlap threshold for HF exchange integrals. For the pseudo overlap in the HF exchange series, the truncation criteria were set to 10–14 and 10–42. The SCF accuracy was set to 10–7 a.u. for geometry optimizations and to 10–6 a.u. for ddsch and single-point calculations, which are the default accuracies in CRYSTAL17.