A DFT Study of Electronic and Magnetic Properties of Cr 2 O 3 using Spin-Polarized Approach

: The study of Cr 2 O 3 magneto-electric material, particularly, the profound understanding of its antiferro-magnetic, is vital for its spintronics applications. In this paper, we present a study on electronic and magnetic properties of Cr 2 O 3 crystal using the first-principles calculations based on the density functional theory. For more accurate results, additional Hubbard (U) parameter has been employed to GGA as well. Our calculated results are homogeneous with available experimental measurements. Results show the effects produced by GGA+U method upon the electronic and magnetic features of the material. We prove that, the spherically symmetric GGA+U approach for exchange correlation approximation portray is a superior depiction of electronic and magnetic properties of Cr 2 O 3 .


Introduction
The Cr2O3 is an important material among the corundum family crystals such as A12O3, Fe2O3, Ti2O3, and V2O3 [1]. The significance of Cr2O3 is due to its applications such as fuel cell electrodes, heterogeneous catalysis, gas sensors, energy collectors, high-performance industrial alloys, the primary constituent of passive films protecting stainless steels coatings of hightemperature materials, black matrix film, ceramics and thermal barrier coatings etcetera [2][3][4][5]. Due to its important industrial applications, Cr2O3 is under extensive investigations for a long time, both experimentally and theoretically [6][7][8]. Cr2O3 favors anti-ferromagnetic coupling at ground state that may play an important role for it spintronic applications. However, it's partially filled spin-polarized 3d shell that led Cr2O3 as a strongly correlated material, rather offer a challenge to electronic band theory. Hence, the role of strong electronic correlation in determining the physical properties of Cr2O3 is of common interest. On the other hand, density functional theory based computational investigations of transition-metal oxides, particularly the Cr2O3, correlation effects can't be reasonably treated using with common exchange-correlation potentials at a local spin density approximation (LSDA) or generalized-gradient approximation (GGA) [9]. These exchange correlation potentials, as a known fact, are unable to reproduce the electronic and magnetic properties of bulk Cr2O3 [10][11][12]. However, several efforts have been made in the past years to overcome this shortcoming of standard DFT calculations. The most common and one of the effective approach adapted in this regard is the addition Hubbard (U) parameter to LDA or GGA. For instance, Rohrbach et al [13] obtained more reasonable results for the electronic band structure of Cr2O3 by performing GGA+U where the simplified (spherically averaged) U − J a correction was used [9]. Shi et al. [5] found a good agreement between the experimental and their LSDA+U calculations for structural, spectral, and magnetic properties with the same values of U and J which are close to those found from the constrained occupation method within DFT. Although, several efforts are in practice for the correct description of the physical properties of strongly correlated Cr2O3 using DFT, further investigations are thoroughly in need to understand the real picture of its electronic structure to exploit and engineer its properties for further applications. In this work, we calculate the magnetic properties, the density of states (DOS) and the band structure of Cr2O3 using full potential linearized augmented plane wave(FP-LAPW) method at the level GGA and GGA+U for precise description of its results. To understand the electronic properties of Cr2O3, spin polarized total and partial density of states are calculated. The paper is designed such that section 2 presents the computational methods used. The results obtained in the present work with their discussion are presented in section 3. Where in section 4, conclusion based on the present study is given.

Computational Methods
The LAPW method is considered as the most accurate approach for electronic structure calculations of the crystals [1,14]. LAPW solves the Kohn-Sham equations for self consistently for the ground state density, total energy, and (Kohn-Sham) eigenvalues (energy bands) of a many-electron system (here a crystal) by introducing a LAPW basis set. The present first principles calculations have been performed using the state of the art FP-LAPW method [15][16] as incorporated in the WIEN2k code [17]. In this methodology/computational code, the unit cell is divided into (I) non-overlapping atomic spheres (centered at the atomic sites) and (II) an interstitial region. In the two types of regions different basis sets are used.
For these calculations, the structure of Cr2O3 have been built by taking the experimental data of the lattice constants and atomic positions from the reference [18]. For the exchange correlation potential treatment, we used the Perdew et al proposed GGA which can better reproduce the exchange potential [19]. Additional U-parameter has been used for the treatment of correlation effects. The Monkhorst-Pack scheme [20] based on the (8x8x8) k-points grid was employed for the Brillouin-zone integrals using the tetrahedron method with Blöch corrections [21]. To achieve energy eigenvalues convergence, the wave functions in the interstitial regions were expanded in plane waves with a cut-off Kmax = 9/RMT, where RMT denotes the smallest atomic sphere radius and Kmax gives the magnitude of the largest K vector in the plane wave expansion. The muffin-tin radii were assumed to be (1.84) atomic unit (a.u.) for Cr, and 1.66 a.u for O. The valence wave functions inside the spheres are expanded up to lmax =10, while the charge density was Fourier expanded up to Gmax =14 (a.u.) -1 . Self-consistency is obtained using k-points in the irreducible Brillouin zone (IBZ). The BZ integration was carried out using the tetrahedron method [22][23][24]. The band structure and density of states (DOS) are calculated. The self-consistent calculations are considered to be converging if the total energy of the system is stable.

Result and Discussion
The electronic structure and magnetic properties of Cr2O3 are investigated by means of the FP-LAPW method [25][26] within GGA and GGA+U. It is known that these PDOS contributions could be used to predict the behavior of different atoms in possible transition from nonmagnetic state to magnetic state. The coulomb repulsion of electrons can prevent formation of band states, stabilizing localized electron states. The d-levels in most of the transition metal oxides are partially filled; therefore the band theory predicts electron delocalization. The delocalization is resulted in the higher binding energies of d-electrons compared to its real value. In such cases, the bandgap is mostly affected by the presence of a delocalized d-orbital with higher binding energies in the valence band. To understand the electronic properties of Cr2O3, we determined the spin polarized total and partial density of states (DOS) as plotted in Figure (1).GGA+U.
The total DOS above in Figure (1a) are dominated by the Cr 3d, Cr dx2y2dxy, Cr p and O p. The DOS profile shows that a comparatively narrower at the deep valence band centered at ~ -18.8 eV is contributed by the Cr-s and O-p states. The partial DOS of Cr2O3 clearly shows a strong hybridization among O 2p and Cr 3d electrons in the energy regime ~-8 to -3 eV in the valence band. Moreover, the DOS for majority spin components in this energy regime is quantitatively nearly equivalent minority spin components that apparently do not contribute to the magnetic moment in Cr2O3. .The reduced magnetic moment in Cr2O3 due to the hybridization of Cr 3d and O 2p states has also been reported in [27]. Cr-3d PDOS contributions to the density of states in Cr2O3 are shown in Figure (1d) .The DOS shows a significant and broadened structure for majority spin component in the vicinity of Fermi level. However, the DOS at the corresponding energy is negligible for minority spin states. The unbalanced majority and minority spin DOS in the vicinity may be the reason of the observed magnetic moment in Cr2O3. Moreover, the electronic structures reflect significantly wider energy band gap for minority spin component compared to majority spins. Table 1

Conclusion
A quantum-mechanical study of chromium-oxide crystal has been carried out using the firstprinciples DFT methodology within the GGA and GGA+U approximation. We investigated the structural and electronic properties of the corundum (Cr2O3) by means of the FLAPW method. Results give a better idea of effects produced by GGA+U method upon the structural, electronic, electrical and magnetic features of the material. Calculated magnetic moments of Cr2O3 demonstrate the importance of GGA+U method. Local magnetic moment of (2.55µB) is calculated for Cr atom in Cr2O3 compounds. We prove that, the anti-bonding character in the Cr 3d-O 2p hybridization appearing in the energy band gap plays an important role in the phase and structural stability.