Theoretical study on the structural, electronic and physical properties of layered alkaline-earth-group-4 transition-metal nitrides AEMN₂

Abstract

Thermodynamic, structural, and electronic properties of the layered ternary nitrides AEMN₂ (AE = alkaline-earth; M = group 4 transition metal) both with the KCoO₂ and α-NaFeO₂ structure-types are examined within density-functional theory. The AE:M atomic (or ionic) radius ratio seems to be the most important criterion in determining one structural arrangement over the other. We find that the majority of compounds are more stable with the KCoO₂ structure-type where M is coordinated to five nitrogen atoms in a distorted square-based pyramidal geometry. Strong interactions occur in both arrangements not only between nitrogen and transition metal atoms, but also between nitrogen and alkaline-earth metal atoms within and between the layers. Calculations show that all the AEMN₂ compounds with the tetragonal structure-type KCoO₂ are semiconducting with band gaps of approximately 1 eV. However, small band gap conductor and even semi-metallic behavior are computed for compounds with the alternative hexagonal α-NaFeO₂ structure-type.

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Introduction

The solid-state chemistry of ternary nitridometalates has rapidly emerged and matured in recent years, both in the context of their many varied physical properties and of their potentially useful applications.^1–14 These compounds display a wide range of varying and often unique crystal structures in which the metals – often transition metals – exhibit unusual oxidation states and/or coordination to nitrogen. Examples include the families of the ternary phases AE₃MN₃ (AE = alkaline earth metal; M = V, Cr, Mn, Fe)^15–17 and Ca₆MN₅ (M = Fe, Ga, Mn),^18,19 with structures unique to nitrides in which M³⁺ cations are coordinated to three nitrogen atoms to form sheets of carbonate-like [MN₃]⁶⁻ trigonal planar units separated by AE cations. Ternary nitridometalates can also adopt structure types encountered in oxide and carbide chemistry. CaNiN for instance adopts the YCoC structure and contains infinite chains of linear [NiN]²⁻ units,²⁰ whereas Ba₂VN₃ crystallizes in the Rb₂TiO₃ structure with chains of corner-sharing [VN₃]⁴⁻ tetrahedra.²¹ Another important family of ternary nitridometalates is that of AMN₂ compounds, which adopt structure types found in oxide and chalcogenide chemistry. A subset of these compounds, which embrace both alkali (alkaline-earth) metal-group 6 (group 5) transition metal and mixed transition metal systems, are layered compounds containing metal atoms coordinated by nitrogen in either octahedral or trigonal prismatic geometry.^6,8,22 The LiMN₂ compounds (M = Mo, W) were the first metallic layered lithium ternary nitrides to be discovered.²² They are hexagonal layered compounds of structure type P3 (isostructural with Na_0.6CoO₂, space group R3, no. 146) in which the Mo or W atoms are octahedrally surrounded by nitrogen atoms which themselves sit in a trigonal prismatic environment. The Li⁺ cations are located in trigonal prismatic holes between the [MoN₂]⁻ (or [WN₂]⁻) layers. These metallic layered compounds are also paramagnetic in character.²³ The mixed transition metal compound CrWN₂ is isostructural to LiMoN₂ with alternating layers of tungsten and chromium in a trigonal prismatic and octahedral coordination respectively with nitrogen.²⁴ CaMN₂, (M = Nb, Ta)²² and SrMN₂ (M = Zr, Hf)²⁵ adopt another layered arrangement; the α-NaFeO₂ structure-type (hexagonal space group R3̄, no. 166 with Z = 3). This arrangement, depicted in Fig. 1 for SrZrN₂, differs from the P3 structure in that the s-block metal cations are octahedrally coordinated by nitrogen. Interestingly, a certain number of AEMN₂ compounds (where M is a group 4 metal) including BaHf_1−xZr_xN₂ (x = 0–1)²⁶ and SrTiN₂ (Fig. 2)^27,28 are isotypic with the KCoO₂ structure type (tetragonal space group P4/nmm, no. 129, Z = 2). Unlike the hexagonal P3 and α-NaFeO₂ compounds above, the transition metal atoms in the KCoO₂-type are coordinated by five nitrogen atoms in a distorted square-based pyramidal geometry (with four long basal M–N bonds and one short apical M–N bond), forming layers of edge-sharing pyramids, which stack along the c direction. The alkaline-earth atoms AE are situated between the [MN₂]²⁻ layers and are also surrounded by five nitrogen atoms in a square-pyramidal arrangement.

Fig. 1 Crystal structure of SrZrN₂ (with the α-NaFeO₂ structure-type). Large blue, medium grey and small yellow spheres represent Sr, Zr and N atoms, respectively.

Fig. 2 Crystal structure of SrTiN₂ (with the KCoO₂ structure-type). Large blue, medium grey and small yellow spheres represent Sr, Ti and N atoms, respectively. The unit cell contains two independent nitrogen atoms (N1 (3/4, 1/4, 1/2) and N2 (1/4, 1/4, z)).

Surprisingly, with the usual nominal oxidation states of 3− and 2+ for N and the AE metals respectively, the formal d-electron count for the group 4 transition metal in the above AEMN₂ compounds is d⁰, irrespective of whether the α-NaFeO₂ or KCoO₂ structure is adopted. A glance at Table 1,²⁹ which contains both the existing and as-yet unknown AEMN₂ compounds, seems to suggest that both the electron count and the AE:M radius ratio govern the preferred structural arrangement.

Table 1 Relation between the experimentally observed structures and the AE:M atomic and ionic (in brackets) radius ratio for the AEMN₂ compounds studied²⁹

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Theoretical studies have been performed previously on some of the compounds given in Table 1 but these are few in number and much remains to be understood. The unexpected magnetic properties of SrTiN₂ were tentatively elucidated.²⁸ Electronic and vibrational properties of BaHfN₂ were studied showing semiconducting behavior with an indirect band gap.³⁰ However, no comprehensive study has yet been performed to rationalize the observed structures, bonding and physical properties in these AEMN₂ compounds. This is the modest ambition of this paper using density-functional theory (DFT) calculations.

Results and discussion

Thermodynamics and structure

Relative energies for the different AEMN₂ compounds in the two structure-types are compared in Fig. 3. Among the 9 compounds considered, 6 are computed to be thermodynamically more stable with the KCoO₂ tetragonal structural arrangement. This is in agreement with experiment in the case of SrTiN₂, BaZrN₂ and BaHfN₂.^26,27,31 In fact, it is apparent that all Ba compounds would prefer to adopt the KCoO₂ structure since the hypothetical compound BaTiN₂ is computed to be more thermodynamically stable when adopting the KCoO₂ structure than with the α-NaFeO₂ alternative by an appreciable margin. SrZrN₂ is computed to be slightly more stable (ca. 0.1 eV per formula unit (f. u.)) with the KCoO₂ structure, contrary to experiment, which demonstrates that the α-NaFeO₂ structure is formed by the nitridozirconate.²⁵ The computational result may suggest that both arrangements could be observed experimentally under appropriate synthetic conditions. SrHfN₂ is computed to be more stable with the α-NaFeO₂ structure, as is observed experimentally,²⁵ but the KCoO₂ is predicted to be only 0.2 eV per f.u. less stable. No group 4 transition metal CaMN₂ compounds have been reported so far. Calculations indicate that these hypothetical nitrides would preferentially crystallize in the α-NaFeO₂ structure. This is especially true for CaZrN₂ and CaHfN₂ where the difference in thermodynamic stability with the KCoO₂ structure type is large.

Fig. 3 Relative energy difference between KCoO₂ (pink) and α-NaFeO₂ (blue) structure types for AEMN₂ compounds. Compounds with formulae in pink and blue experimentally adopt the KCoO₂ and α-NaFeO₂ structure types, whereas those in black have yet to be discovered.

Given that all the above AEMN₂ compounds are isoelectronic, we can postulate that the AE:M atomic (or ionic) radius ratio may play a deciding role in imposing one structure over the other. Although the total number of characterized compounds is not large, it seems that for an AE:M atomic radius ratio of around 1.3, the α-NaFeO₂ structure type will be preferred, whereas with a ratio of approximately 1.4, the KCoO₂ structure prevails. On this basis, CaTiN₂ should be observed with the α-NaFeO₂ structure type, although, calculations indicate a very small thermodynamic preference for the KCoO₂ structure.

Tables 2 and 3 summarize the cell parameters, cell volumes, bulk moduli, and pertinent interatomic distances obtained from geometry optimization of the nine AEMN₂ compounds in each of the α-NaFeO₂ and KCoO₂ structure types, respectively. Existing experimental and computational data are included for comparison where available. The good agreement between the computed and experimental structures (deviation <2% overall) inspires confidence in the prediction of structures for those compounds that have yet to be synthesized and characterized. As expected, for a given AE element, a significant increase in both a and c parameters is computed on moving from Ti to Zr and Hf compounds leading to an expansion of more than 12% in the unit cell volumes for the α-NaFeO₂ structures. For AE = Ca and Sr, this increase is even larger for compounds adopting the KCoO₂ structure type (ca. 20% vs. ca. 13% for BaMN₂ compounds).

Table 2 DFT optimized (normal type) and experimental (italics) cell parameters, AE–N and M–N bond distances, unit cell volume V, bulk modulus B₀, and band gap of AEMN₂ compounds adopting the α-NaFeO₂ structure

Structure	a/Å	c/Å	A–N/Å	M–N/Å	V/Å^3	B0/GPa	Band Gap/eV	Ref.
CaTiN2	3.138	16.404	2.424	2.132	139.9	173	0
CaZrN2	3.318	16.843	2.494	2.266	160.6	161	0.45
CaHfN2	3.292	16.811	2.486	2.247	157.8	171	0.45
SrTiN2	3.211	17.335	2.571	2.159	154.8	139	0
SrZrN2	3.382	17.781	2.629	2.281	176.1	129	0.40
	3.373	17.676	2.609	2.292	174.1			25
SrHfN2	3.353	17.684	2.607	2.277	172.2	166	0.40
	3.345	17.678	2.602	2.273	171.3			25
BaTiN2	3.279	18.451	2.749	2.181	171.8	115	0
BaZrN2	3.446	18.921	2.818	2.302	194.6	123	0
BaHfN2	3.418	18.888	2.806	2.286	191.1	122	0

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Table 3 DFT optimized (normal type) and experimental (italics) cell parameters, AE–N and M–N bond distances, unit cell volume V, bulk modulus B₀, and band gap of AEMN₂ compounds adopting the KCoO₂ Structure

Structure	a/Å	c/Å	A–N/Å	M–N/Å	V/Å^3	B0/GPa	Band Gap/eV	Ref.
a Ref. 30 – range of band gaps values obtained from PBE calculations using different pseudopotentials.
CaTiN2	3.807	7.517	2.253 (×1)	1.844 (×1)	108.9	127	1.00
			2.694 (×4)	2.024 (×4)
CaZrN2	3.996	8.131	2.471 (×1)	2.009 (×1)	129.8	130	1.60
			2.826 (×4)	2.160 (×4)
CaHfN2	3.960	8.150	2.306 (×1)	2.016 (×1)	127.8	128	1.70
			2.807 (×4)	2.138 (×4)
SrTiN2	3.802	7.776	2.596 (×1)	1.836 (×1)	112.4	123	0.80
			2.691 (×4)	2.056 (×4)
	3.882	7.701	2.611 (×1)	1.827 (×1)	116.1			28
			2.747 (×4)	2.053 (×4)
	3.880	7.699	2.581 (×1)	1.839 (×1)	115.9			27
			2.745 (×4)	2.056 (×4)
SrZrN2	4.069	8.296	2.523 (×1)	2.018 (×1)	137.3	122	1.15
			2.882 (×4)	2.179 (×4)
SrHfN2	4.043	8.278	2.539 (×1)	2.008 (×1)	135.3	123	1.20
			2.865 (×4)	2.162 (×4)
BaTiN2	4.014	8.068	2.856 (×1)	1.827 (×1)	130.0	101	0.80
			2.939 (×4)	2.106 (×4)
BaZrN2	4.188	8.502	2.858 (×1)	2.000 (×1)	149.1	124	1.10
			2.932 (×4)	2.228 (×4)
	4.161	8.392	2.771 (×1)	2.011 (×1)	145.3			26
			2.948 (×4)	2.202 (×4)
BaHfN2	4.152	8.491	2.819 (×1)	1.993 (×1)	146.4	106	1.10
			2.946 (×4)	2.198 (×4)
	4.128	8.382	2.680 (×1)	2.050 (×1)	142.8		[0.95–1.25]^a	26
			2.922 (×4)	2.186 (×4)

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It is noteworthy that M–N distances differ substantially in α-NaFeO₂ and KCoO₂ structure types. Six rather long M–N distances are encountered around the octahedrally coordinated metal in the former, whereas four shorter (5–8%) M–N_basal contacts and a significantly shorter (by 12–15%) M–N_apical contact are computed for the latter.

Resistance to volume change was analyzed by computing the bulk modulus, i.e. the second derivative of the total energy with respect to the volume, for all AEMN₂ compounds. Values reported in Tables 2 and 3 show that compounds with the α-NaFeO₂ structure-type are less compressible than those with the KCoO₂ structure-type. This is particularly the case for the hypothetical CaMN₂ compounds with the former structure-type, the bulk moduli of which are computed to be comparable to that of steel and ca. 30% larger than compounds with the KCoO₂ structure-type.

Electronic and physical properties

Band structures and the partial and total density of states (PDOS and DOS) of the different AEMN₂ compounds were examined in order to gain insight regarding their electronic properties. For brevity, we will limit our discussion to the electronic structure of SrTiN₂ as a representative example of the series.

The calculated PDOS and DOS of SrTiN₂ when adopting the experimentally observed KCoO₂ structure type are depicted in Fig. 4. The total DOS divides into three main regions. The first is located between −15 and −10 eV and is derived from the N 2s orbitals. The second region that is centered close to −2.5 eV below the Fermi level (ε_F), is somewhat broader and is mostly due to N 2p N orbitals with a substantial admixture of Sr and Ti valence (mostly d-type) orbitals. Interestingly, PDOS for basal N1 and apical N2 atoms differ slightly, with those of the former being broader and lying at somewhat lower energy than those of the latter. The large DOS above the Fermi level shows the dominance of the Ti orbitals, which are combined to a small extent with Sr and N orbitals. The metal orbital participation in the valence band in the vicinity of ε_F, coupled with the nitrogen orbital contribution to the conduction band which is otherwise predominantly comprised of metal states reflect the rather strong metal–nitrogen covalent interactions. Contrary to previous LMTO results,²⁸ SrTiN₂ is computed to be diamagnetic. The simplified description of the Kohn–Sham potential in the LMTO method may be at the origin of this discrepancy.

Fig. 4 PDOS and DOS of SrTiN₂ with the KCoO₂ structure-type. The Fermi energy level is set at 0 eV.

The band structure shown in Fig. 5 indicates a semiconducting behavior with a calculated (direct) band gap of ca. 0.8 eV. Since LDA and GGA functionals usually underestimate band gaps, the true band gap is likely to be larger than 1 eV. For comparison, all the computed band gaps for the compounds with the KCoO₂ structure are given in Table 3. The small dispersion of bands in Γ → Z and A → M show the weak interaction between MN₂ nets. The lowest conduction band has primarily Ti 3d_xy character. The large dispersion of this band is due to Ti–Ti interactions within the layers. The band structure sketched in Fig. 5 agrees with a recent theoretical study carried out for some of these nitrides in order to evaluate their thermoelectric performance.³²

Fig. 5 Band structure of SrTiN₂ with the KCoO₂ structure-type. Γ = (0, 0, 0), X = (0, 1/2, 0), M = (1/2, 1/2, 0), Z = (0, 0, 1/2), R = (0, 1/2, 1/2), and A = (1/2, 1/2, 1/2) in units of the reciprocal lattice vectors. The amount of Ti 3d_xy character is shown in the band structure by the width of the lines. The Fermi energy level is set at 0 eV.

Total and partial DOS for SrTiN₂ in the alternative α-NaFeO₂ structure-type are shown in Fig. 6. As is the case for the KCoO₂ structure-type, the total DOS also divides into three main regions but the low lying band is notably narrower for the α-NaFeO₂ structure. The valence band is once more dominated by N 2s and 2p orbitals with some contribution from the metal atoms, whereas the conduction band derives mostly from the metal orbitals with some admixture of N orbitals. It is noteworthy that the tails of the valence and conduction bands are in contact at the Fermi level. This semi-metallic character is confirmed by an analysis of the band structure shown in Fig. 7. It is important to note that the (indirect) band gap is computed as almost 0.6 eV using the mBJ functional proposed by Tran and Blaha that yields band gaps with accuracy similar to hybrid functional or GW methods.³³ The band dispersion along the T → Γ direction that is perpendicular to the MN₂ layers is weaker. It is noteworthy that the band structure portrayed in Fig. 7 is qualitatively similar to the one computed for the isostructural compound CaTaN₂.³⁴ In this latter, an additional electron per formula unit occupies the bottom of the conduction band shown in Fig. 7. Therefore 2D-metallic properties are envisioned for CaTaN₂ whereas the AEMN₂ (M = Ti, Zr, Hf) compounds in the α-NaFeO₂ structure-type are expected to behave as semiconductors or semimetals according to DFT calculations. The alkaline-earth contribution to the occupied levels is very small in both cases. This suggests that a stronger more covalent interaction occurs in bonds within the MN₂ layers but that ionic bonding exists between the MN₂ layers and the Sr sheets (i.e. Sr²⁺(Ca²⁺)[MN₂]²⁻). Similar band structures are observed for the α-NaFeO₂-type variants of BaMN₂ (M = Ti, Zr, Hf) and CaTiN₂, suggesting that the bonding in these compounds is similar to that computed for the hypothetical SrTiN₂ phase.

Fig. 6 PDOS and DOS of SrTiN₂ with the hypothetical α-NaFeO₂ structure-type. The Fermi energy level is set at 0 eV.

Fig. 7 Band structure of SrTiN₂ with the hypothetical α-NaFeO₂ structure-type. Γ = (0, 0, 0), L = (0, 1/2, 0), FB = (0, 1/2, 1/2), and T = (1/2, 1/2, 1/2) in units of the reciprocal lattice vectors. The Fermi energy level is set at 0 eV.

Conclusions

We have examined the thermodynamics, structural features and electronic properties of the layered group 4 metal ternary nitrides AEMN₂ when adopting both the KCoO₂ and α-NaFeO₂ structure-types within density-functional theory. Considering that compounds with both structure-types are isoelectronic, evidently the AE:M atomic (ionic) radius ratio is the most important criterion in one structure being imposed over the other. We find that a majority is more stable with the KCoO₂ structure-type in which M is coordinated by nitrogen in a square-based pyramidal geometry.

Strong interactions occur in both arrangements not only between nitrogen and transition metal atoms, but also between nitrogen and alkaline-earth metal atoms. Calculations show that AEMN₂ compounds adopting the tetragonal structure-type KCoO₂ are semiconducting with band gaps of approximately 1 eV. By contrast, small band gap conducting and even semi-metallic behaviors are computed for the equivalent compounds when adopting the hexagonal α-NaFeO₂ structure-type.

Experimental section

DFT calculations were performed using the WIEN2k program package.³⁵ The exchange–correlation interaction was treated within the generalized gradient approximation (GGA) of Perdew, Burke, and Ernzerhof.³⁶ The muffin tin radii (R_MT) were chosen small enough to avoid overlapping during the optimization process. A plane wave cutoff corresponding to R_MTK_max = 8 was used, and the radial wavefunctions inside the non-overlapping muffin-tin spheres surrounding the atoms were expanded up to l_max = 12. The charge density was Fourier expanded up to G_max = 16 Å⁻¹. Total energy convergence was achieved with respect to the Brillouin zone (BZ) integration using a mesh of 500 k-points. This generated 30 and 44 k-points in the irreducible Brillouin zone (IRBZ) for compounds in the KCoO₂ and α-NaFeO₂ structure-type respectively.

Combining AE = Ca, Sr, Ba and M = Ti, Zr, Hf generates 9 ternary AEMN₂ compounds, the structures of which were optimized and analyzed both with the KCoO₂ and α-NaFeO₂ structure-types. Optimization was obtained by minimizing the total energies of their primitive cells as a function of volume according to Birch–Murnaghan's equation of state:^37,38

E (V) = E₀ + 9V₀B₀/16[(X² − 1)³B^′₀ + (X² − 1)²(6 – 4X²)]

where V₀ represents the equilibrium volume, B₀ the bulk modulus, B^′₀ the bulk modulus pressure derivative, E₀ the equilibrium energy and X = (V₀/V)^1/3.

Acknowledgements

The FP7 Marie Curie Initial Training Network FUnctional Nitrides for Energy Applications (FUNEA, European Project no. 264873) is gratefully acknowledged for financial support of this research and a doctoral fellowship for O. E. This work was granted access to the HPC resources of CINES under allocation 2011-[scr6170] made by GENCI (Grand Equipement National de Calcul Intensif).

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