Recently, strongly correlated materials based on 3d transition metal monoxides have attracted a lot of attention because they exhibit rich and intriguing physical properties such as superconductivity and magnetism. The transition metal monoxides CoO and NiO are antiferromagnetic (AFM)  in nature below their respective Neel temperature, and they have interesting structural and magnetic properties. The optical band gaps in the AFM state of NiO and CoO were experimentally found to be around 4 and 2.6 eV, respectively . Terakura et al.  have successfully predicted the AFM state of NiO, but the calculated band gap and the magnetic moment values were less than the experimental values. Huang et al.  have computed the band gap of CoO as 1.8 eV by the local density approximation (LDA)+dynamical mean field theory (DMFT) method. A number of approaches such as local spin density approximation (LSDA)+U  and generalised gradient approximation (GGA)+U  have been developed to account the electron correlation in these oxides. Tran et al.  analysed the electronic structure of transition metal monoxides MnO, FeO, CoO, and NiO. They have also computed the lattice constant, bulk modulus, and spin magnetic moment using hybrid exchange-correlation energy functional. The magnetic exchange interactions of transition metal monoxides CoO and NiO were studied by Fisher et al. . The AFM moments in MnO, FeO, CoO, and NiO have been investigated by powder neutron diffraction experiments . Cheetham and Hope  analysed the AFM phase of MnO and NiO using high-resolution powder diffraction experiments.
The quasi-particle band structures of the transition metal oxides CoO and NiO were calculated within the Green’s function (G) and the screened Coulomb Kernel (W) GW approximation, and the magnetic moments are reported as 2.4 μB for CoO and 1.3 μB for NiO using GGA calculation, while 2.6 μB for CoO and 1.5 μB for NiO using GGA + U calculation . Sumino et al.  have measured elastic constants of CoO by rectangular parallelopiped resonance (RPR) method as C11=262.4 GPa, C12=147.2 GPa, and C44=83.6 GPa. The elastic constants of NiO were determined by ultrasonic measuring technique .
In this article, the ground-state properties, electronic structure, magnetic and mechanical properties of transition metal oxides TMO(TM=Co and Ni) are investigated with NaCl structure.
2 Computational Details
First-principles calculations are performed using density functional theory within the GGA parameterised by Perdew–Burke–Ernzerhof (GGA-PBE) [14, 15] as implemented in the Vienna Ab initio Simulating Package (VASP) code [16–18]. The interaction between the ion and electron is described by the projector-augmented wave method . Ground-state geometries are determined by minimising the stresses and Hellmann–Feynman forces using conjugate-gradient algorithm with force convergence less than 10–3 eV Å–1. The Kohn–Sham orbitals are expanded using the plane wave energy cutoff of 400 eV for CoO and 350 eV for NiO. The Brillouin zone integrations are carried out using Monkhorst–Pack K-point mesh  with a grid size of 4×4×4 for total energy calculation. The cobalt 3d74s2, the nickel 3d84s2, and the oxygen 2s22p4 orbitals are treated as valence electrons. The strong on-site Coulomb repulsion between the 3d electrons is described using GGA-PBE/GGA-PBE+U formalisms. The GGA-PBE+U calculations are based on the rotationally invariant scheme proposed by Dudarev et al. . This approach combines the Hubbard “U” and Hund’s “J” in such a way that Ueff=U–J is used in the total energy calculations. In this study, these parameters are fixed as U=6 eV and J=0.95 eV for CoO and U=8 eV and J=0.95 eV for NiO, in order to get comparable band gap values with those observed in experiments. Also, the calculated Hubbard Ueff parameter is henceforth labelled as U for simplicity. The small tetragonal and rhombohedral distortions in the AFM phase of CoO and NiO are ignored in the present calculations.
3 Results and Discussion
3.1 Ground-State Properties
The total energies are computed for various cell volumes for the non-magnetic (NM), ferromagnetic (FM), and the AFM phases of CoO and NiO with NaCl structure. The AFM phase is found to be the most stable state when compared to FM and NM phases for both the oxides (Fig. 1a and b). Therefore, in this study, all calculations are performed for the AFM phase only. The optimised lattice constants are computed for the AFM phase for the volume corresponding to the minimum energy value (Fig. 1a and b). The computed total energies are fitted to the universal Birch–Murnaghan equation of state  to find the bulk modulus B0 and its pressure derivative B0′. All these computed parameters are given in Table 1, together with the experimental and other theoretical works [7–10] for comparison. The computed lattice constants are 4.213 Å for CoO and 4.16 Å for NiO with GGA-PBE scheme, which are smaller by 0.87% and 0.24%, respectively, while the lattice parameters obtained with GGA-PBE+U formalism are 4.23 Å for CoO and 4.24 Å for NiO. Thus, the computed lattice parameters are in good agreement with the experimental values. It is observed that GGA-PBE+U scheme improves the magnetic moment values of both cobalt and nickel ions.
3.2 Electronic Structure
The band structure calculations are executed to get information about the physical properties of the materials. The electronic band structures of CoO and NiO computed by GGA-PBE and GGA-PBE+U methods are shown in Figures 2a,b and 3a,b. The Fermi levels are indicated by dotted horizontal lines. It is found that the nature of the band gap is indirect for both the oxides. From the GGA-PBE calculation, it is seen that, for CoO, the valence band maximum is located at the W point and while in NiO it is at the L point. The oxygen 2s state gives rise to very weak dispersive bands around –15 eV in both the oxides. The highest valence band of these oxides is dominated by the transition metal (Co, Ni) 3d states. Below these bands, around 0 eV, the bands are due to very weak hybridisation of metal 3d states and oxygen 2p states. The conduction band minimum is found between L and K in both the oxides. Just above this, the parabola-shaped conduction band features the transition metal 4s state, and this band exhibits a strong dispersion. The indirect band gap Eg values of CoO and NiO are listed in Table 2 along with experimental and theoretical values [2, 11]. It is noted that the band gap Eg values calculated by the GGA-PBE method are far less than the experimental value, whereas GGA-PBE+U calculation improves the band gap values to a greater extent. The computed band gaps for CoO and NiO are 2.33 and 3.47 eV, respectively, which are close to the experimental values. The valence bands near the Fermi level are pulled down due to the interaction between the metal d-states and oxygen p-states, which in turn increases the band gap. The conduction band minimum is also shifted to the gamma point. That is, the Hubbard parameter U induces change in the hybridisation between the oxygen 2p orbital and the transition metal (Co, Ni) 3d orbital. The spin-dependent total density of states and partial density of states found using GGA-PBE and GGA-PBE+U methods are represented in Figures 4a,b, 5a,b, 6a,b and 7a,b. For both CoO and NiO, the total density of states is completely symmetric in majority and minority spins, which indicates that the electrons occupy the majority and minority spin states equally resulting in zero total spin moment.
In the partial density of states plots, the region can be divided into three parts: a strong peak around –15 eV below the Fermi level is due to the oxygen 2s states, the strong peaks just below the Fermi level is mainly due to the metal (Co, Ni) 3d states with a small contribution from the oxygen 2p states, and the conduction bands are mainly due to the unoccupied metal 3d states with very little hybridisation of the oxygen 2p states. Also from Figure 7a and b, it is observed that U correction enhances the oxygen 2p character in the valence band states. This in turn modifies the nature of the states at the top of the valence band. Moreover, there is an increase in the separation between the peaks on either side of the Fermi level.
The bonding nature of metal and O atoms can be confirmed by the charge density distribution shown in Figure 8. It is clearly seen that charges accumulate between the metal and O atoms, which means that a directional bonding exists between the metal and O atoms in CoO and NiO. The bonding nature of these materials is found to be a mixture of ionic and covalent-like due to the weak hybridisation of O and metal atoms as found in the density of states.
3.3 Magnetic Properties
The magnetic properties of CoO and NiO are due to the magnetic moments of cobalt and nickel ions, associated with the localised d electrons. The computed values of the magnetic moments obtained by the GGA-PBE and GGA-PBE+U methods, together with the experimental and other theoretical values [8–10], are listed in Table 1. It is found that the GGA-PBE+U method predicts a larger value for the magnetic moment than the GGA-PBE method for the two oxides; because GGA-PBE+U formalism reduces the GGA-PBE delocalisation error of the d electrons. Non-spin- and spin-polarised calculations are performed using the GGA-PBE method to obtain the total energies of CoO and NiO in their NM, FM, and AFM phases for various pressures. The variation in the total energies with pressure is shown in Figure 9a and b. It is observed that the magnetic phase transition from AFM to FM phase occurs only in NiO at 84 GPa, and the anti-ferromagnetism is quenched.
3.4 Mechanical Properties
The elastic constants provide information about the mechanical behaviour of the materials and their stability. In order to obtain the mechanical stability description of these systems, a set of elastic constants are obtained from the resulting change in the total energy deformation. The elastic constants Cij are calculated within the total energy method, where the unit cell is subjected to a number of finite size strains along several strain directions . The cubic lattices have three independent elastic constants (C11, C12, C44). Mechanical stability criteria for cubic crystal  at ambient conditions is given by C44>0, C11>|C12|, C11+2C12>0.
The mechanical properties such as bulk modulus, Young’s modulus (E), and shear modulus (G) are important physical quantities, especially for engineering applications. These parameters are determined using the Voigt–Reuss–Hill averaging scheme [25–28]. The elastic constants for CoO and NiO computed by the GGA-PBE scheme, together with the available experimental data [11, 12, 29], are given in Table 3. The bulk modulus is found to be inversely proportional to the volume as expected. The computed values of the bulk modulus are reasonably in agreement with the experimental values. The bulk modulus of CoO and NiO with U=0 are 196 and 177 GPa, respectively, which are only 6.5% and 17% more than the experimental values. Young’s modulus is a measure of the stiffness of a solid. The computed results reveal that NiO is a stiffer material than CoO. When the U parameter is included, there is a change in the elastic constant values. The value of C11 is observed to soften, while the values of the other two elastic constants increase. Therefore, the bulk modulus value is found to increase, while Young’s modulus and shear modulus decrease for both the oxides. Moreover, CoO and NiO obey elastic stability criteria suggesting that they are mechanically stable at the normal pressure.
The structural, electronic, and magnetic properties of transition metal oxides, CoO and NiO, are studied using GGA-PBE and GGA-PBE+U formalisms. The AFM phase is found to be the most stable state when compared to the FM and the NM phases. The calculated lattice parameters for the AFM phase are in agreement with the experimental results. The indirect band gap Eg values of CoO and NiO indicate that the GGA-PBE calculation underestimates the band gap values, whereas the GGA-PBE+U calculation improves the band gap values to a greater extent. A pressure-driven magnetic transition from the AFM phase to FM phase is observed in NiO at a pressure of 84 GPa. Both the oxides are found to be mechanically stable at normal pressure.
We thank our college management for their sustained support and encouragement. The financial assistance from UGC, India, under a research award scheme (No. F: 30-36/2011 SA-II) is acknowledged.
B. H. Brandow, Adv. Phys. 26, 651 (1977).Google Scholar
E. Z. Kurmaev, R. G. Wilks, A. Moewes, L. D. Fenkelstein, S. N. Shamin, et al. Phy. Rev. B 77, 165127 (2008).Google Scholar
L. Huang, Y. Wang, and X. Dai, Phys. Rev. B 85, 245110 (2012).Google Scholar
O. Bengone, M. Alouani, P. Blochl, and J. Hugel, Phys. Rev. B 69, 075413 (2004).Google Scholar
W. B. Zang, Y. L. Hu, K. L. Han, and B. Y. Tang, Phys. Rev. 74, 054421 (2006).Google Scholar
F. Tran, P. Blaha, K. Schwarz, and P. Novak, Phy. Rev. B 74, 155108 (2006).Google Scholar
W. L. Roth, Phys. Rev. 110, 1333 (1958).Google Scholar
C. Rodl, F. Fuchs, J. Furthmuller, and F. Bechstedt, Phys. Rev. B 79, 235114 (2009).Google Scholar
P. D. V. DuPleiss, S. J. Van tonder, and S. J. Alberts, J. Phys. C 4, 1983 (1971).Google Scholar
J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77, 3865 (1996).Google Scholar
J. P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 78, 1396 (1997).Google Scholar
G. Kresse and J. Hafner, Phys. Rev. B 47, 558 (1993).Google Scholar
G. Kresse and J. Furthmuller, Comput. Mater. Sci. 6, 15 (1996).Google Scholar
G. Kresse and J. Furthmüller, Phys. Rev. B 54, 11169 (1996).Google Scholar
S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys, and A. P. Sutton, Phys. Rev. B 57, 1505 (1998).Google Scholar
J. F. Nye, Physical Properties of Crystals, Oxford University Press, Oxford 1985.Google Scholar
M. Born and K. Huang, Dynamical Theory of Crystal Lattices, Clarendon, Oxford 1956.Google Scholar
Y. Lee and B. H. Harmon, J. Alloy Compd. 338, 242 (2002).Google Scholar
R. Hearmon, Adv. Phys. 5, 323 (1956).Google Scholar
K. S. Aleksandrov, L. A. Shabanova, and L. M. Reshchikova, Sov. Phys. Solid State 10, 1316 (1968).Google Scholar
About the article
Published Online: 2015-08-04
Published in Print: 2015-10-01