An ab initio Study on the Mechanical Stability, Spin-Dependent Electronic Properties, Molecular Orbital Predictions, and Optical Features of Antiperovskite A3InN (A = Co, Ni).

Structural, mechanical, spin-dependent electronic, magnetic, and optical properties of antiperovskite nitrides A3InN (A = Co, Ni) along with molecular orbital diagram are investigated here by using an ab initio density functional theory (DFT). The mechanical stability, deformation, damage tolerance and ductile nature of A3InN are confirmed from elastic calculations. Different mechanical anisotropy factors are also discussed in detail. The spin dependent electronic properties such as the band structure and density of states (DOS) of A3InN are studied and, the dispersion curves and DOS at Fermi level are different for up and down spins only in case of Co3InN. These calculations also suggest that Co3InN and Ni3InN behave as ferromagnetic and nonmagnetic, respectively. The induced total magnetic moment of Co3InN is found 2.735 μB/cell in our calculation. Mulliken bond population analysis shows that the atomic bonds of A3InN are contributed by both ionic and covalent bonds. Molecular orbital diagrams of A3InN antiperovskites are proposed by analyzing orbital projected band structures. The formation of a molecular orbital energy diagram for Co3InN is similar to Ni3InN with respect to hybridization and orbital sequencing. However, the orbital positions with respect to the Fermi level (E F) and separations between them are different. The Fermi surface of A3InN is composed of multiple nonspherical electron and hole type sheets in which Co3InN displays a spin-dependent Fermi surface. The various ground-state optical functions such as real and imaginary parts of the dielectric constant, optical conductivity, reflectivity, refractive index, absorption coefficient, and loss function of A3InN are studied with implications. The reflectivity spectra reveal that A3InN reflects >45% of incident electromagnetic radiations in both the visible and ultraviolet region, which is an ideal feature of coating material for avoiding solar heating.

Introduction

Antiperovskite transition-metal nitrides (ATMNs) have become familiar to scientists and researchers in the last several decades.[1−3] ATMNs demonstrate a broad range of attractive and tunable physical properties such as the Invar-like effect, itinerant antiferromagnetism, giant magneto-resistance, superconductivity, damage tolerance, spin-glass-like activities, strong spin–lattice coupling characteristics, and topological electronic behavior.[1,4−8] Perovskite compounds are denoted by the formula ABX3, where A and B are cations and X is an anion. In antiperovskite compounds, the general formula is reversed, so that the X sites are occupied by a cation, while A and B sites are occupied by different types of anions. Typical ATMNs adopt the crystal structure A3BX with space group Pm3̅m (no. 221), where A is a transition metal; B is a divalent or trivalent element; and X is nitrogen.[4,5] B. V. Beznosikov predicted more than 80 nitride compounds with antiperovskite structure.[9] The generation of large negative thermal expansion in Ge-doped antiperovskite manganese nitrides Mn3XN (X = Cu, Zn, Ga) is reported by Takagi et al.[10] Ferro- and paramagnetic orders are observed in ATMNs when the transition elements are substituted by group 13 metals.[11,12] It is reported that the nonstoichiometry affects the magnetic properties in Ni-rich antiperovskite carbides depending on Ni/C atomic ratios.[13−15] However, the nonmagnetic ground state is found in many cases for Ni-based ternary carbides due to the reduced Stoner factor and C–Ni bonding nature.[16−18] Although there are predictions about some highly stable two-dimensional (2D) hypercoordinate materials such as Cu–Si and Cu–Ge alloy films, Ni2Ge and Ni2Si monolayers, aluminum boride (AlB6) nanosheets with interesting physical properties, and diverse applications,[19−26] here we consider only the bulk antiperovskite A3InN in our calculation.

The nickel (Ni)-based antiperovskite nitrides gained considerable interest due to the discovery of superconductivity in Ni3CuN and Ni3ZnN with TC around 3 K.[27,28] Ni3CdN exhibits very soft and weak ferromagnetism, and Ni3InN exhibits a spin–glass-like behavior.[29,30] Different groups synthesized Co3InN and Ni3InN by different methods.[1−4] Due to the diversity of the physical properties of the antiperovskite materials and limited knowledge on the cobalt- and Ni-based nitrides, first-principle calculations for Co3InN and Ni3InN were carried out in order to study shear, Young’s moduli, Poisson’s ratio, and spin-polarized band structure and projected density of states (pDOS) only.[3] To extend these previous works, we thus focus here on exposing different physical properties of Co- and Ni-based antiperovskite A3InN (A = Co, Ni) nitrides using an ab initio density functional theory (DFT) method. The spin-polarized structural, elastic, and mechanical stability and the effect of spin in electronic behavior, magnetic nature, bonding character, and optical properties of antiperovskites A3InN are elucidated in detail. In addition, to expose the strength and extent of anisotropy of A3InN, we focused on the Vickers hardness and different anisotropy indices, respectively. The anisotropic mechanical behavior and elastic responses of A3InN are presented by two- and three-dimensional graphical presentations. To take full advantage of the electronic properties of A3InN in addition to spin-polarized bands and pDOS, we represent here the results of magnetic moments (total and individual atoms), charge density distribution mapping, and spin-polarized Fermi surface topologies. The Fermi surface can provide information for predicting the physical properties of a metal.

The molecular orbital (MO) theory is regarded as an effective tool to determine the molecular structure and bonding nature of a compound. It takes the idea of overlapping between different atomic orbitals. The distribution of different MO energy levels is represented by a MO diagram. It also gives insight about bonding, bond order, and change of molecular behavior with ionization. A lot of reports are published on the MO diagram for ternary X2YZ Heusler compounds,[31,32] where X and Y are transition metals and Z is a main group element. Compared to X2YZ Heusler compounds, the study of d-orbital hybridization and bonding order for A3BX-based antiperovskite has not been explored yet to the best of our knowledge.

Moreover, we predicted the orbital hybridization among different atoms based on orbital-projected band diagrams and unfolding orbital degeneracy of A3InN which may help to better understand the magnetic behavior of these compounds.

Finally, the interaction of A3InN with electro-magnetic radiation is expressed by means of some nonlinear optical constants such as reflectance, absorption coefficient, refractive index, complex dielectric constants, optical conductivity, and loss function. As far as we know, there are no experimental or theoretical data available on the Vickers hardness, anisotropic elastic indices, Fermi surface topology, and nonlinear optical properties of A3InN, which are focused here. We strongly believe this study will be useful for using A3InN in the arena of engineering applications.

Methods of Calculations

The ground-state physical properties of the A3InN (A = Co and Ni) are simulated using ab initio density functional theory (DFT) via the CASTEP code.[33] The Vanderbilt-type ultrasoft pseudopotential method is employed to treat the outermost electrons as valence electrons for each atom in A3InN. The Perdew–Burke–Ernzerhof (PBE) parametrization within the generalized gradient approximation (GGA) is applied to treat the exchange and correlation interactions.[34] The plane-wave cutoff energy of 500 eV is used to expand the Eigen functions of the valence and nearly valence electrons. The Monkhorst–Pack grid of 18 × 18 × 18 k-points is used to investigate the Brillouin zone. The Broyden–Fletcher–Goldferb–Shanno (BFGS) algorithm is employed to optimize the geometry optimization through minimizing the total energy and the internal forces.[35] To optimize the geometry conditions, the tolerances for total energy, maximum force, maximum stress, and maximum atomic displacement are set to less than 5 × 10–6 eV per atom, 0.01 eV/Å, 0.02 GPa, and 5 × 10–4 Å, respectively. The single-crystal elastic tensors (C) are calculated by the “stress–strain” method embedded into the CASTEP code in which a set of finite identical deformations is applied. The resultant stress is calculated after optimization of the internal degrees of freedom, and a maximum amplitude of 0.003 Å is chosen for each strain having four steps. The atomic population analysis is performed by means of the conventional Mulliken formalism. The electron charge density difference and Fermi surface are calculated by setting the k-point separation to less than 0.01 Å–1 with 26 × 26 × 26 grids. The optical properties of A3InN are calculated using the same code. It is important to note that spin polarization is considered in all calculations. Orbital-projected band diagrams of these compounds A3InN (A = Co and Ni) are also evaluated using the Quantum ESPRESSO[36] code with the above motioned parameters.

Results and Discussion

Structural Properties

Bulk A3InN (A = Co and Ni) is crystallized as a cubic system in space group Pm3̅m (no. 221).[1] The crystal structure of antiperovskite A3InN nitrides in the above-mentioned space group is shown in Figure . The structural unit cell of A3InN, which adopts an octahedral coordinate, consists of six A atoms, one In atom, and one N atom. The Wyckoff positions are at (0, 1/2, 1/2), (0, 0, 0), and (1/2, 1/2, 1/2) for A, In, and N atoms, respectively. The equilibrium crystal structures of A3InN are optimized at the minimum total energy. The calculated lattice parameter and unit cell volume of A3InN are tabulated in Table . It is observed that the lattice constants of A3InN which are very close (<1.07%) compared to the experimental values[1,4] indicate the highest level of accuracy of our study.

Figure 1

Crystal structure of A3InN (A = Co and Ni) ternary antiperovskites.

Table 1

Comparison between the Theoretically Calculated and Experimental Values of Unit Cell Parameters of A3InN (A = Co, Ni)a

phase	a (Å)	V (Å)3	method	reference
Co3InN	3.877	58.287	DFT (GGA)	this work
Ni3InN	3.901	59.361	DFT (GGA)	this work
Co3InN	3.855	57.289	DFT (GGA)	(3)
	3.753	52.861	DFT (LDA)
Ni3InN	3.882	58.501	DFT (GGA)	(3)
	3.784	54.181	DFT (LDA)
Co3InN	3.8518	57.146	electrochemical bulk synthesis	(1)
Ni3InN	3.8599	57.509	electrochemical bulk synthesis	(1)
Co3InN	3.8541	57.249	solid–gas reactions	(4)
Ni3InN	3.8445	56.822	solid–gas reactions	(4)

GGA = generalized gradient approximation. LDA = local density approximation. FM = ferromagnetic. NM = Nonmagnetic.

Mechanical Properties

The mechanical properties are important because they describe different characteristics such as elasticity, plasticity, strength, hardness, ductility, and brittleness of a material. Among these properties elastic constants correlate the dynamical behavior of a material with its mechanical properties. The single-crystal elastic tensors (C) of newly synthesized A3InN (A = Co, Ni) are calculated to provide a deep insight into the mechanical stability and stiffness of these materials. In general, materials with cubic symmetry have three adiabatic elastic stiffness constants, namely, C11, C12, and C44. The constant C11 is related to tensile, whereas C12 and C44 are associated to share. The values of C11, C12, and C44 are listed in Table . The elastic constants of A3InN satisfy the Born stability[37] criteria C11 > 0, C11 – C12 > 0, C44 > 0, and (C11 + 2C12) > 0 signifies the mechanical stability for these cubic crystals. It has also been seen that the constants follow C11 > C12 > C44, which indicates the anisotropic nature, whereas the low value of C44 is allied to the shear deformation and damage-tolerant nature of A3InN. The different elastic moduli (B, G, Y, and ν) are also calculated from individual elastic constants. The bulk modulus (B) of A3InN is given byThe shear modulus (G) of a crystalline solid varies with direction and is defined by the Voigt–Reuss–Hill (VRH) equation aswhere GV and GR are the Voigt and Reuss bonds, respectively. The arithmetic average of GV and GR gives the shear modulus of A3InN.The isotropic polycrystalline aggregate values for Young’s modulus (E) and Poisson’s ratio (v) are obtained using the following equations asThe values of B, G, Y, and v are tabulated in Table . In general, the hardness of a compound is measured by B and G. The values of B (G) are found to be 197 (93) GPa and 182 (63) GPa for Co3InN and Ni3InN, respectively, indicating higher hardness of Co3InN. Moreover, the high value of the bulk modulus with low shear modulus reveals the damage-tolerant, quasi-ductile, easily machinable, and stiff nature of A3InN. On the other hand, Co3InN is found to be stiffer than Ni3InN due to the higher value of Y. Another three important factors, Poisson’s ratio (v), Pugh’s ratio (G/B), and Cauchy pressure (C12 – C44), have also been calculated and shown in Table . From Table , it is seen that v is greater than Frantsevich’s criterion[38] value of 0.26 and G/B < 0.57. These parameters suggest that A3InN is ductile. The elastic nature of A3InN is also verified by the Cauchy pressure.[39] The negative value of Cauchy pressure indicates the brittle nature of the compound, while a positive value endorses the quasi-ductile nature. In addition, the values of B, G, Y, v, G/B, and (C12 – C44) are compared with other antiperovskites and found to be consistent, which is shown in Table .

Table 2

Comparison of Elastic Properties between A3InN (A = Co, Ni) and Other Antiperovskites

compound	C11	C12	C44	B	G	Y	v	G/B	C12 – C44	ref
Co3InN	333	129	87	197	93	241	0.29	0.472	42	this work
Co3InN	317	126	95	190	95	224	0.28	0.50	31	(3)a
Ni3InN	283	131	55	182	63	170	0.34	0.346	76	this work
Ni3InN	274	131	60	178	64	172	0.34	0.359	71	(3)a
Ti3TlN	196	131	52	153	43	118	0.40	0.276	79	(8)a
Ni3SnN	266	132	41	177	50	137	0.37	0.282	91	(8)a
Co3AlC	451	119	86	230	112	290	0.29	0.487	38	(8)a

Theoretical.

The Vickers hardness (HV) is highly related to the elastic constants of a material. We have calculated the HV of A3InN using the different approximations proposed by Teter et al.,[40] (HV)Teter = 0.151 G, Tian et al.,[41] (HV)Tian = 0.92(G/B)1.137G0.708, and Chen et al.,[42] (HV)Chen = 2[(G/B)2G]0.585 – 3. The calculated values of HV are presented in Table . It is noticeably seen that Co3InN has higher HV than Ni3InN which may be due to a higher elastic moduli and bond strength.

Table 3

Calculated Vickers Hardness (HV) of A3InN (A = Co, Ni)a

compound	(HV)Teter	(HV)Tian	(HV)Chen	HGao	ref
Co3InN	14.043	9.701	8.781		this work
Ni3InN	9.513	5.174	3.525		this work
Ti3TlN				3.6	(8)
Ni3SnN				3.58	(8)
Co3AlC				4.0	(8)
Ti3AlC				7.8	(43) [exptl]

The results of other antiperovskite compounds are listed for comparison.

Mechanical anisotropy is closely related to different essential physical processes of a material. To visualize the level of anisotropic mechanical behavior and elastic responses of A3InN, theYoung’s modulus (Y), linear compressibility (β), shear modulus (G), and Poisson’s ratio (v) are plotted in contour plots of three- (3D) and two-dimensional (2D) presentations (Figures and 3) using the ELATE code.[44] From Figures and 3, it is demonstrated that all the constants except linear compressibility are deviated from the complete spherical (3D) and circular (2D) shapes, indicating the anisotropic nature of these compounds. However, the deviation from the spherical and circular shapes is the same in all directions (xy, xz, and yz planes) but a bit higher for Co3InN compared to Ni3InN. The maximum and minimum values of Y, β, G, and ν are enlisted in Table .

Figure 2

Three- and two-dimensional contour diagrams of (a) Young’s modulus (Y), (b) linear compressibility (β), (c) shear modulus (G), and (d) Poisson’s (v) ratio for the Co3InN antiperovskite.

Figure 3

Three and two-dimensional contour diagrams of (a) Young’s modulus (Y), (b) linear compressibility (β), (c) shear modulus (G), and (d) Poisson’s (v) ratio for Ni3InN antiperovskite.

Table 4

Minimum and Maximum Limit of Y, β, G, and ν for A3InN (A = Co, Ni)a

	Young’s modulus (GPa)		linear compressibility (TPa–1)		shear modulus (GPa)		Poisson’s ratio
compound	Ymax	Ymin	βmax	βmin	Gmax	Gmin	vmax	vmin	ref
Co3InN	228.79	260.99	1.6893	1.6893	87.543	101.99	0.25286	0.34833	this work
Ni3InN	151.05	200.93	1.8309	1.8309	55.462	76.338	0.25331	0.45183	this work
Ti3TlN	91.04	140.09			32.5	52	0.19	0.54	(8)
Ni3SnN	114.17	178.44			41	67	0.23	0.53	(8)
Co3AlC	229.37	401.31			86	166	0.13	0.49	(8)
Ti3AlC		140			83		0.25		(43)
Ni3MgC		151			58		0.3		(45)

The results of other antiperovskite compounds are listed for comparison.

The elastic anisotropy of cubic A3InN can be expressed using the Zener anisotropy factor[46]AZ asThe Zener anisotropy factor is related to share anisotropy, and AZ = 1 for a perfect elastically isotropic material.

An alternative measure of the elastic anisotropy (AG) for the cubic compound is proposed by Chung and Buessem[47] asThe anisotropy factor AG is expressed in percentage, and AG = 0 indicates the isotropic nature of a material.

The universal elastic anisotropy index (AU)[48] involves all the individual components of the elasticity tensor of a material which may be defined for a cubic crystal asThe logarithmic universal Euclidean anisotropy (AL)[49] for a cubic compound is related to the Voigt and Reuss bounds on the bulk modulus simplifies asThe orientation of Poisson’s ratio is specified in two directions; the lateral contraction/extension in one direction and the corresponding extension/contraction in a normal direction. In general, the extreme values of Poisson’s ratio may occur due to strain along ⟨110⟩ and the corresponding orthogonal strain along ⟨001⟩ and ⟨110⟩ for a cubic compound.[50] The extrema of Poisson’s ratio in these particular directions is expressed asThe values of AZ, AU, and AL, ν(110, 001) and ν(110, 11̅0), are listed in Table . From the table it is clear that the values of different anisotropy parameters except AZ are higher for Co3InN than Ni3InN. The zero value of AU and AL signifies the higher isotropic nature of Co3InN than Ni3InN. However, these anisotropy parameters confirm that Co3InN is more anisotropic in compression, whereas Ni3InN is more anisotropic in share. The extrema of Poisson’s ratio is found, ν(110, 11̅0) > ν(110, 001), which reveals that the orthogonal strain may occur along this particular direction for A3InN.

Table 5

Anisotropy Parameters and Extrema of Poisson’s Ratio for A3InN (A = Co, Ni)a

compound	AZ	AG	AU	AL	ν(110, 001)	ν(110, 11̅0)	ref
Co3InN	0.85	0.003	0.00	0.00	0.250	0.352	this work
Ni3InN	0.72	0.129	0.161	0.071	0.247	0.456	this work
Ti3AlC		0.015	0.154				(51)
Fe3AlC		0.093	1.032				(51)
Sm3AlC		0.375	6.00				(51)

The results of other antiperovskite compounds are listed for comparison.

Band Structure and Density of States (DOS)

The electronic band structure and density of states (DOS) play an imperative role to visualize several optoelectrical and magnetic properties of a crystal at the microscopic level. The spin-polarized electronic band structures of A3InN (A = Co, Ni) are calculated for spin-up (↑) and spin-down (↓) using high-symmetry k-points along the path X–R–M–G–R in the first Brillouin zone. The results are displayed in Figure (a)–(c). From Figure , it is seen that the dispersion curves for up and down spins are different for Co3InN. Two bands for up and two bands for down spin cross the EF which is shown in Figures (a) and 4(b). On the contrary, the dispersion curves for both spins are the same for Ni3InN. Two bands cross the EF, as seen in Figure (c). The dispersion curve for down spin is not shown due to the spin-symmetric nature of Ni3InN. Both Co3InN and Ni3InN exhibit metallic nature, as the conduction and valence bands notably overlap at the Fermi level.

Figure 4

Electronic band structures with (a) spin-up and (b) spin-down channels for Co3InN and (c) the band structure for only the spin-up channel for Ni3InN ternary antiperovskites. The pattern of the band structure of Ni3InN for the down-spin channel is identical with that for the spin-up channel and is not shown here.

To explore the origin of metallic and magnetic nature, the spin-polarized total density of states (DOS) and orbital-resolved partial DOS (pDOS) are calculated for A3InN. The spin-dependent DOS and pDOS of Co3InN are shown in Figure . It is clearly seen that the DOS at the Fermi level is nonzero for both spin channels [Figure (a)], revealing the metallic nature of Co3InN with zero energy gap. The observed DOS values are found at 1.176 states/eV and −2.893 states/eV at EF for the spin-up and spin-down channel, respectively. Thus, the DOS of Co3InN is contributed by both spins but significantly ∼2.5 times higher for the spin-down channel. The contribution of individual atoms in Co3InN that occupied the electronic states per unit energy is determined by the pDOS shown in Figure (b)–(d).

Figure 5

(a) Total density of states (DOS) and (b–d) partial density of states (pDOS) of the Co3InN antiperovskite.

It is seen that the valence band of Co3InN is roughly separated into three sub-bands, ranging from (−9.1 to −5) eV, (−5 to −0.5) eV, and (−0.5 to EF) eV, showing some sharp peaks. These peaks arise due to the hybridization of different electronic states. The lower energy band (−9.1 to −5) eV is contributed by Co-3d, Co-3p, In-5s, In-5p, and N-2p orbitals, whereas the rest of the band is donated by Co-3d, In-5p, and N-2p orbitals. The electrons in the Co-3d orbital contribute strongly in the conduction mechanism of the Co3InN phase for both spin channels. However, the contribution of Co-3p, Co-4s, In-5s, In-5p, and N-2p orbitals is minor. In addition, the DOS and pDOS peaks are seen to be blue-shifted for the spin-down channel, which indicates the presence of strong spin–orbit coupling (SOC) in Co3InN.

The spin-dependent DOS of Ni3InN is shown in Figure (a)–(d). The nonzero values of DOS [Figure (a)] for both spin channels indicate the metallic nature of Ni3InN. It is notable that the DOS and pDOS patterns of Ni3InN are identical for up and down channels in contrast to the case of Co3InN.

Figure 6

(a) Total density of state (DOS) and (b–d) partial density of states (pDOS) of Ni3InN antiperovskite.

The calculated value of the DOS at the Fermi level is 1.04 states/eV for both spins in Ni3InN. The DOS of Ni3InN is mainly contributed by Ni-3d orbitals, whereas the Ni-3p, In-5s, In-5p, and N-2p orbitals contribute faintly as seen from Figure (b)–(d). The valence band of Ni3InN is roughly separated into three sub-bands as Co3InN. The lower (−9.1 to −4.80) eV, middle (−4.80 to −0.792) eV, and top sub-bands that cross EF result from the hybridization of Ni-3d, Ni-3p, In-5s, and N-2p orbitals, Ni-3d, Ni-3p, In-5p, and N-2p orbitals, and Ni-3d, Ni-3p, In-5s, In-5p, and N-2p orbitals, respectively. From the pDOS [Figures (c) and 6(c)], it is clearly seen that the contribution of an In atom is feeble at EF, which indicates the bare minimum involvement of the In atom in metal-like conductions of both Co3InN and Ni3InN.

Magnetic Properties and Molecular Orbital Analysis

The magnetic moments for total and individual atoms are calculated from the integrated DOS at the Fermi level. The magnetic moment in Co3InN may arise due to shifts of the pDOS for spin-up and -down channels. The induced total magnetic moment of Co3InN is 2.735 μB/cell, which is contributed by Co, N, and In atoms. The local magnetic moment is calculated as 0.9163 μB, 0.0919 μB, and −0.1651 μB for Co, N, and In atoms, respectively. On the contrary, the total magnetic moment for Ni3InN is calculated as 0.526 × 10–8 μB/cell, which is negligible. It is noteworthy that we have also considered the magnetic unit cell with both ferromagnetic and antiferromagnetic ordering of Co(Ni) atoms to find out the nature of A3InN. Here it should be mentioned that a single unit cell of A3InN (cf. Figure ) is enough for both ferromagnetic (where all A atoms are in the same spin orientation) and antiferromagnetic settings (A atoms in alternate layers are with opposite spin orientations) and considered as a magnetic unit cell. Our calculations suggest that Co3InN (Ni3InN) is ferromagnetic (nonmagnetic) in nature.

In order to further understand the magnetic nature and the contribution of s, p, and d orbitals fully in the band structure of A3InN, we pictured the orbital hybridization among different atoms based on orbital-projected band diagrams. Comparing Figures (b) and 6(b), it is clear that for empty states in the pDOS Co-3d states have a significant contribution only for the down-spin channel but a nominal contribution for the up spin of Co-3d and for both spin channels of Ni-3d. These findings of pDOS characteristics are similar to an earlier study.[3] For the next part, we consider only the down-spin channel.

Orbital-projected band diagrams along Γ–X–R–M–Γ–R directions for Co3InN and Ni3InN are evaluated using the Quantum ESPRESSO[36] code and presented in fat band representation in Figures and 8, respectively. In these figures, the vertical width of the belts (dispersion lines) represents spectral weights of (a) Co/Ni-eg (d, d), (b) Co/Ni-t2g (d, d, d), (c) In-t1u (p, p, p), (d) N-t1u (p, p, p), and (e) In a1g(s) orbitals, respectively. For better clarity, orbital-projected bands are not superimposed on normal band dispersion curves. Figures and 8 represent normal band dispersion curves where the orbital degeneracy at different energy levels is identified by red color orbital symbols on the vertical line along the high-symmetry point Γ. The Fermi energy is set to zero and shown by a dashed black line. The inset of Figure (a) is for two separate calculations considering Co-d and Co-d orbitals, respectively, and the vertical width of the belts confirms that the 2-fold degeneracy of eg is broken, and it splits in two orbitals d (B1g) and d (A1g), respectively, where the energy of d is greater than that of d. The findings are similar for Ni3InN where 2-fold degeneracy of eg is also broken with d > d and not shown in the inset of Figure (a).

Figure 7

Orbital projected band structure of Co3InN for spin-down channels in a fat band representation along Γ–X–R–M–Γ–R directions. Vertical width of the belts (dispersion curves) represents spectral weights of (a) Co-eg, (b) Co-t2g, (c) In-t1u, (d) N-t1u, and (e) In-a1g orbitals, respectively. For better clarity, orbital projected bands are not superimposed on normal band dispersion curves. (f) Represents normal band dispersion curves where the orbital degeneracy at different energy levels is identified by red color symbols on the vertical line along the high-symmetry point Γ. The Fermi energy is set to zero and shown by a dashed black line. The inset of (a) is for two separate calculations considering Co-d and Co-d orbitals, respectively, and the vertical width of the belt confirms that the 2-fold degeneracy of eg is broken, and it splits in two orbitals d (B1g) and d (A1g), respectively, where the energy of d is greater than that of d.

Figure 8

Orbital-projected band structure of Ni3InN for the spin-down channel in fat band representation along Γ–X–R–M–Γ–R directions. Vertical width of the belts (dispersion curves) represents spectral weights of (a) Ni-e, (b) Ni-t2g, (c) In-t1u, (d) N-t1u, and (e) In-a1g orbitals, respectively. For better clarity, orbital-projected bands are not superimposed on normal band dispersion curves. (f) Represents normal band dispersion curves where the orbital degeneracy at different energy levels is identified by red color symbols on the vertical line along high symmetry point Γ. The Fermi energy is set to zero and shown by a dashed black line. The 2-fold degeneracy of eg is broken, and it splits in two orbitals d (B1g) and d (A1g), respectively, as marked in (f), confirmed by two separate calculations (not shown here) considering Ni-d and Ni-d orbitals, respectively (for details see text).

From Figures and 8, it is seen that the band structure of Co3InN (Ni3InN) is projected into Co (Ni)-eg (d, d) and t2g (d, d, d), In/N-t1u (p, p, p), and a1g (s) orbitals. The Eigen states at the Γ point in the Brillouin zone are considered to explain the orbital hybridization process. Figure (b) and 9(c) represents the crystal field splitting and the possible hybridization between d-orbitals of Co–Co/Ni-Ni and s- and p-orbitals of In/N atoms. These hybridizations arise due to interactions of the bonding and the antibonding states of the orbitals.

Figure 9

Schematic molecular orbital diagram between spin-down orbitals sitting at different sites in the case of the (b) Co3InN and (c) Ni3InN antiperovskites. First, the hybridization between the two different A (A = Co/Ni) atoms with D4 site symmetry are considered (a). Afterward A–A and an N atom (with O site symmetry) are hybridized with another A and an In (with O site symmetry) atom. The Fermi energy is shown by a dashed black line. The 1-, 2-, and 3-fold degenerate states are shown by green, red, and blue colors, respectively. Here NB refers to the nonbonding and (*) antibonding state. The 2-fold degenerate state eg splits into two NB orbitals d (B1g) and d (A1g), respectively.

Antiperovskite A3InN (A = Co, Ni) compounds with Hermann–Mauguin notation m3̅m belong to octahedral (O) symmetry.[1] Within Co3InN (Ni3InN), N and In atoms are coordinated by 6 and 12 Co (Ni) atoms, respectively, forming a Co6N (Ni6N) octahedron and Co12In (Ni12In) cuboctahedron, respectively.[1] In A3InN (A = Co, Ni), In and N atoms occupy Wyckoff positions a (0, 0, 0) and b (1/2, 1/2, 1/2) with the site symmetry of O, and 3 A atoms occupy Wyckoff positions c (0, 1/2, 1/2) with the site symmetry D4, respectively.[52]

Here we used eg (d, d), t2g (d, d, d), t1u (p, p, p), and a1g (s) symbols of O symmetry for orbitals shown in the first bracket. We also used A1g (d), B1g (d), B2g (d), and Eg (d, d) symbols of D4 symmetry for orbitals shown in the first bracket in the molecular orbital diagram for considering A–A hybridization between 3d orbitals of two A atoms with local site symmetry D4 [cf. Figure (a)].

Based on the orbital-projected band diagrams (Figures and 8), we predicted a possible hybridization scheme which is explained below. Figure (b) and 9(c) is molecular orbital diagrams between spin-down orbitals sitting at different sites in the case of the Co3InN and Ni3InN antiperovskites, respectively. Here we used a similar approach as used in earlier studies to construct the MO diagrams of other perovskites with different crystal structures (such as X2YZ).[32,53,54] Here the hybridizations between the two different A (A = Co, Ni) atoms with D4 site symmetry are considered first [cf. Figure (a)]; afterward A–A and an N (site symmetry O) atoms are hybridized with another A and an In (site symmetry O) atoms, respectively. The site-symmetry approach establishes symmetry relations between the localized atomic electron states and crystalline states; then the localized states transform following the overall crystal symmetry.

Here, the hybridization and combination of double-degenerated Eg (d, d) and single-degenerated B2g (d) of A–A and A with symmetry D4 transformed according to O symmetry to produce triple-degenerated bonding t2g orbitals and antibonding t2g* orbitals. On the other hand, the hybridization and combination of two single-degenerated A1g (d) and B1g (d) of A–A and A with symmetry D4h transformed according to O symmetry to produce double-degenerated bonding eg orbitals and antibonding eg* orbitals. Due to the unavailability of more sites with D4 symmetry to hybridize the antibonding states of A–A: Eg* (d, d), A1g* (d), B2g* (d), and B1g* (d) remain as nonbonding in the resulting MO orbital diagram [cf. Figures (b) and 9(c)]. However, double-degenerated Eg (d, d) and single-degenerated B2g (d) of A–A with symmetry D4h combined and transformed according to O symmetry to produce the nonbonding triple-degenerated t2g state. Interestingly, two remaining nonbonding orbitals A1g (d) and B1g (d) do not combine to create a double-degenerated nonbonding eg state.

As a whole for Co3InN, we postulate in addition to bonding (a, t, t, and e) and antibonding (eg*, t2g*, and t1u*) states the hybridizations of Co–Co [Figure (a)] and N with another Co, and an In atom also produces one nonbonding 3-fold t2g and one nonbonding eg state with broken 2-fold degeneracy [cf. Figure (b)]. In the molecular orbital diagram, it is also clear that the eg state splits in two orbitals d (B1g) and d (A1g), respectively, where d > d, as confirmed by separate calculations considering Co-d and Co-d orbitals separately, compared with the inset of Figure (a). Such splitting might be due to local Jahn–Teller distortion, which we believe is a local z-out distortion of O complexes which results d > d.[55] In the molecular orbital diagram of Co3InN, one antibonding t2g* state is above EF, followed by nonbonding d and d states toward higher energy.

The p orbitals of N and In atoms hybridize and form bonding and antibonding states of t1u orbitals, whereas s orbitals hybridize and form bonding and antibonding a1g orbitals. The antibonding state a1g* is above the considered energy window and does not appear in this diagram. The triple-degenerated nonbonding Co (t2g) state dominates the highest orbital state in the valence band of Co3InN.

The formation of a molecular orbital energy diagram for Ni3InN [Figure (c)] is similar to Co3InN with respect to hybridization and orbital sequencing. However, the orbital positions with respect to EF and separations [compare Figure (b) with Figure (c)] between them are different. An earlier DFT study postulated that Ni-based ternary nitrides with cubic antiperovskite structure are nonmagnetic due to hybridization between Ni-3d and N-2p states.[3] Our results presented above do not support such a hybridization scheme. Interestingly, in the molecular orbital diagram for Ni3InN [Figure (c)], EF lies between two nonbonding d and d states. These two nonbonding states in the immediate vicinity of EF might be the reason for almost zero magnetism in Ni3InN.

Mulliken Populations Analysis

The nature of chemical bonding via the charge transfer mechanism in a compound can be predicted using the Mulliken atomic population analysis. The Mulliken atomic and bond overlap populations of A3InN are tabulated in Table .

Table 6

Mulliken Atomic and Bond Overlap Populations of A3InN (A = Co, Ni)a

		Mulliken atomic population				Mulliken bond overlap population
phase	atoms	s	p	d	total	charge (e)	bond	nμ	Pμ	dμ (Å)
Co3InN	N	1.68	4.02	0.0	5.71	–0.71	N–Co	03	2.01	1.94
	Co	1.29	2.01	23.43	26.73	0.27	Co–In	03	0.21	2.74
	In	0.56	2.02	9.97	12.55	0.45	Co–Co	03	–1.17	2.74
Ni3InN	N	1.65	4.00	0.0	5.65	–0.65	N–Ni	03	1.92	1.95
	Ni	1.20	2.16	26.43	29.79	0.21	Ni–In	03	–0.03	2.76
	In	0.65	1.93	9.97	12.56	0.44	Ni–Ni	03	–0.57	2.76

nμ, Pμ, and dμ denote the bond number, bond overlap population, and bond length, respectively.

From Table , it is seen that N atoms contain negative Mulliken charge, while it is positive for A (Co, Ni) and In atoms. Hence, charges transferred from A/In to N atoms, which indicates the presence of ionic bonding in A3InN. The ionic nature of A3InN can be restricted by the electronegativities of the atomic species. The distribution of carrier density in different bonds and a quantitative measure of bonding and antibonding strengths in A3InN are described by the Mulliken bond overlap population (BOP) analysis. In general, the positive and negative BOP stands for the bonding and antibonding nature of atoms, respectively. The high and low values of BOP point out the increase of covalent and ionic bond nature, respectively, between two atoms.[56] In this study, the N–A bonds possess a higher degree of covalency with bonding nature. The Co–In bond is also covalent, whereas A–A bonds show ionic nature.

Charge Density Distribution Mapping and the Fermi Surface

The bonding nature of A3InN (A = Co, Ni) is investigated by the charge density difference of unlike atomic sites for further insights. The charge density mapping of A3InN is shown in Figure on the (001) crystallographic plane. The high and low charge density of electrons is indicated by red and blue colors of the adjacent scale bar, respectively. In general, the greater accumulation of charges (positive value) is favorable for the formation of covalent bonds between two atoms, while the negative value or the lower accretion of charges indicates the formation of ionic binding. The contour maps of electron charge density show higher accumulation of charges in (A, In) than (A, A) atoms, which is consistent with the Mulliken bond population analysis. Moreover, the spherical charge distributions around A atoms indicate the ionic nature of chemical bonds in A3InN.

Figure 10

Electronic charge density mappings of (a) Co3InN and (b) Ni3InN ternary antiperovskites on the (001) crystallographic plane.

The conception of Fermi surface is regarded as the heart for understanding the state of a compound with metallic nature. The dynamical properties of a material largely depend on the position and shape of the Fermi surface with respect to the Brillouin zone (BZ).[57,58] The Fermi surface topology of the ternary nitride perovskite A3InN is evaluated in the equilibrium structure with spin-polarized conditions at zero pressure. It is important to note that the effect of spin channels is absent on the Fermi surface of Ni3InN. This spin-independent nature of the Fermi surface for Ni3InN is also supported by band dispersion study. The Fermi surface topologies for spin-up, spin-down, total channel of Co3InN, and total channel of Ni3InN are shown in Figure (a)–(d), respectively, and explained separately below.

Figure 11

Fermi surface topologies for (a) spin-up, (b) spin-down, (c) total channel of Co3InN, and (d) total channel of Ni3InN.

Co3InN with spin-up channel: (1) Half of an oblate-spheroid-shaped open surface between high-symmetry points X, and R [marked by H1 in Figure (a)] within the BZ is a hole pocket. A total of 24 such hole pockets are visible at the 6 zone faces. The conventional equation of the oblate spheroid is written aswhere x-, y-, and z-axes intercepts are a, a, and c, respectively, and a > c. One hole pocket at the right face of the BZ cube is marked to show the x, y, z directions and their a, a, c intercepts. From Figure (a), it is evident that the z-direction is considered along the high-symmetry direction X–R. The ratio a:c is calculated as ∼3.25:2 (3.25c = 2a). The center of the hole pocket is situated exactly at the middle of the two high-symmetry points X and R. The open surface of a hole pocket that cuts the BZ face creates approximately an elliptical shape with the major axis along the y-direction. The elongated 4 of such a major axis on a single BZ face forms a perfect square which is 45° rotated [shown by dotted line, Figure (a)] and inscribed within the square of the BZ face. Here it should be mentioned that these hole pockets are not exactly ideal oblate-spheroid-shaped. (2) In addition, at the zone edges, the open surfaces centered at M seen within the BZ are electron pockets [marked by E1 in Figure (a)]. They enclose approximately rectangular areas at the adjacent sides of zone faces. It seems if two such surfaces are attached, it will create a dome shape with an approximately square base. A total of 12 such electron pockets are visible in Figure (a). Co3InN with spin-down channel: (3) Closed surface around the G (gamma) point which very nearly resembles an octahedron with curved edges as observed in Figure (b), marked by H2 as a hole pocket. (4) In addition, one fourth of an oblate-spheroid-shaped (a > c) open surface centered at M observed at the BZ edge is an electron pocket, marked by E2 in Figure (b). The open surfaces of an electron pocket that cuts the adjacent sides of zone faces create half (cut along the minor axis) of an elliptical shape (with equal size, at both sides) with the major axis along a high-symmetry MX direction. One electron pocket at the left-upper edge of BZ is marked to show the x, y, z directions and their a, a, c intercepts of an oblate spheroid. Here, the ratio a/c is calculated as ∼3:2 (3c = 2a). A total of 12 such electron pockets are visible in Figure (b).

Co3InN with total-spin channel: As both the electron pockets [explained in (2) and (4)] for both up- and down-spin channels are centered at same point M, the Fermi surface nesting is expected as evident in Figure (c), where the open surface of an electron pocket (originated from up spin) in turn is nested within another one-fourth portion of a larger oblate-spheroid-shaped open surface of another electron pocket (originated from down spin).

Ni3InN with total-spin channel: Here a similar explanation holds as Co3InN with the spin-up channel explained earlier and shown in Figure (d). However, here a major difference is that the volume of hole (electron) pockets is smaller (larger) compared to Co3InN with a spin-up channel. The observations are consistent with the band diagram analysis.

In conclusion, the Fermi surface of A3InN is composed of multiple electron- and hole-type sheets. The nonspherical shape of Fermi sheets indicates the metallic conductivity of both Co3InN and Ni3InN. The multi Fermi sheets of A3InN are formed due to low dispersion of A-3d, In-5p, and N-2p states, as revealed by pDOS studies. The observed Fermi surface nesting may have an effect on the magnetic order and phonon softening in Co3InN.[59]

Optical Properties

The optical properties of A3InN are calculated as a function of incident photons along the [100] direction. A phenomenological damping constant of 0.05 eV, the free-electron plasma frequency of 10 eV, and Gaussian smearing of 0.5 eV are used for the inclusion of intraband transitions for nonlinear optical constant calculations. The absorption coefficient (α) determines how far the light of a certain wavelength can penetrate into a material before being absorbed. The variation of the absorption coefficient α with photon energy is shown in Figure (a). From this figure, it is clearly seen that the optical absorption starts from zero photon energy, which confirms the absence of optical band gap in A3InN, complementing the metallic nature obtained in the electronic structure calculations. The absorption coefficient increases sharply in the visible spectrum region (after ∼2.5 eV) and displays a high value for a wide energy range (∼18 eV). Hence, A3InN could be used as a promising absorber of electromagnetic radiation for both visible and ultraviolet (UV) regions.

Figure 12

Energy-dependent different nonlinear optical constants along the [100] electric field polarization: (a) absorption coefficient, (b) reflectivity, (c) photoconductivity, (d) loss function, (e) refractive index, (f) extinction coefficient, (g) real part of the dielectric function, and (h) imaginary part of the dielectric function of Co3InN and Ni3InN ternary antiperovskites.

The reflectivity spectra (R) of A3InN are shown in Figure (b). It is seen that A3InN reflects more that 98% of light at zero photon energy. After that, the reflectivity decreases gradually with increasing photon energy. However, the average reflectivity of A3InN is >45% in the visible and UV spectral region, which indicates the suitability of A3InN as a coating material to avoid solar heating.[60]

The optical conductivity (σ) starts at zero photon energy [Figure (c)], which reconfirms the metallic nature of A3InN. The response of σ with photon energy is slightly blue-shifted for Co3InN compared to Ni3InN. There is a sharp peak at ∼(4.2–4.5) eV which might have arisen due to the interband transitions of charge carriers from occupied to unoccupied orbitals in A3InN.

The loss function of a material is interconnected with absorption and reflection. It describes the energy loss of charge carrier traveling in a material. The energy loss function (δ) is calculated for A3InN and shown in Figure (d). There is no loss peak observed in the energy range of (0–15) eV due to large absorption of electromagnetic radiation. The highest loss peaks appeared for Co3InN and Ni3InN at ∼22.80 eV and ∼22.94 eV, respectively, which represent the plasma resonance owing to collective charge excitation. The corresponding frequency at which the highest peak of the loss spectrum occurs is called the plasma frequency (ωp) of the material. Below ωp, A3InN is expected to reflect the electric field screening by electrons; otherwise, it is transparent to the incident photon.

The appearance of ωp in the UV range makes A3InN reflective in the visible spectral range. The refractive index is a complex parameter, composed of the real and imaginary parts of the complex index of refraction. The real part of the refractive index (n) demonstrates the phase velocity of the electromagnetic wave inside a material. The frequency dependence of the refractive index is shown in Figure (e). It is seen that n is higher at low photon energy (<5 eV), which decreases with the increasing photon energy. Both Co3InN and Ni3InN display a sharp peak at ∼3.78 eV and ∼4.06 eV, respectively, due to the intraband transition of electrons.

The imaginary part of the complex index of refraction is known as the extinction coefficient (k), which measures the amount of attenuation of the incident light when traveling through the material. The extinction coefficient of A3InN is plotted in Figure (f). It is seen that A3InN exhibits a large static value of k, which decreases gradually with higher photon energy. The high static value of k is indicative of metallic conduction in A3InN.

The macroscopic electronic response of a material is explained by means of the complex dielectric constant as ε(ω) = ε1(ω) + iε2(ω), where, ε1(ω) and ε2(ω) are the real and imaginary part of the dielectric function ε(ω), respectively. The response of the real and imaginary part of the dielectric constant with photon energy is plotted in Figure (g) and 12(h), respectively. From Figure (g), it is seen that A3InN displays negative values of ε1 at very low energies, after which ε1 vanishes. The response of ε1 at low energy is shown in the inset of Figure (g). The negative values of ε1 at low photon energies signify the Drude-like behavior of A3InN. On the other hand, A3InN displays positive values of ε2 and drops down to zero sharply at very low energies [Figure (h)]. The variation of ε2 at low energies is shown in the inset of Figure (h). Both ε1 and ε2 disappear at very low photon energy, which signifies the high metallic nature of A3InN.

Conclusions

In summary, we have investigated the structural, mechanical, spin-dependent electronic and molecular orbital predictions, and magnetic and optical properties of antiperovskites nitrides A3InN (A = Co, Ni) using the first-principles DFT calculations combined with the spin polarization effect. The calculated values of different elastic moduli show the damage-tolerant, quasi-ductile, easily machinable, and stiff nature of A3InN. Poisson’s ratio, Pugh’s ratio, and Cauchy pressure calculations suggest that A3InN is ductile in nature. The values of elastic anisotropy indices reveal that Co3InN is more anisotropic in compression, while Ni3InN is more anisotropic in share. The effect of spin channels is observed in the band structure, DOS, and Fermi surface of Co3InN only. The Fermi surface of A3InN is composed of multiple nonspherical electron and hole type sheets, and the observed Fermi surface nesting in Co3InN might contribute to charge density wave formation and also to enhance the effect of electron–phonon coupling. The induced total magnetic moment of Co3InN is predicted as 2.735 μB/cell, whereas Ni3InN shows nominal magnetic moments. Molecular orbital diagrams of A3InN antiperovskites are drawn by analyzing orbital-projected band structures. It is exciting to note that the hybridizations of different energy states are identical for both Co3InN and Ni3InN. Therefore, it is expected that it will follow the similar trend for A3BX compounds.

The bonding nature between atoms of A3InN is thoroughly explained with the aid of Mulliken atomic populations and charge density calculations. The above-mentioned properties reveal that these compounds display the combined bonding nature like ionic and covalent. The variations of the various optical constants such as real and imaginary parts of the dielectric constant, refractive indices, reflectivity, absorption coefficient, and loss function with the energy of incident radiation show metallic behavior, complementing the outcomes of electronic band structure and DOS calculations. The reflectivity spectra show that A3InN displays superior reflectivity in the visible and UV spectral region, which indicates that A3InN compounds have potential significance to be used as coating materials to evade the solar heat. We hope this work will provide helpful data for the family of antiperovskite materials for further investigation through experiments and theories.