Research Article  Open Access
D. P. Rai, R. K. Thapa, "A First Principle Calculation of FullHeusler Alloy Co_{ 2 }TiAl: LSDA+ Method", International Scholarly Research Notices, vol. 2012, Article ID 410326, 5 pages, 2012. https://doi.org/10.5402/2012/410326
A First Principle Calculation of FullHeusler Alloy Co_{2}TiAl: LSDA+ Method
Abstract
We performed the structure optimization of Co_{2}TiAl based on the generalized gradient approximation (GGA) and linearized augmented plane wave (LAPW) method. The calculation of electronic structure was based on the fullpotential linear augmented plane wave (FPLAPW) method and local spin density approximation exchange correlation LSDA+. We also studied the impact of the Hubbard potential or onsite Coulomb repulsion () on electronic structure; the values are varied within reasonable limits to study the resulting effect on the physical properties of Co_{2}TiAl system. The calculated density of states (DOS) shows that halfmetallicity of Co_{2}TiAl decreases with the increase in values.
1. Introduction
SemiHeusler compound NiMnSb was the first found halfmetal ferromagnets (HMFs) by using first principle calculation based on density functional theory [1]. Co_{2}TiAl is a ferromagnetic halfmetal with an integral magnetic moment of 1โฮผB/atom [2]. It has been widely used in magnetic recording tapes, spin valves, giant magnetoresistance (GMR), and so forth. In recent years, it attracts substantial interests because of the halfmetallic property and the applicable potential for future spintronics. In halfmetal, one spin channel is metallic and the other is insulating with 100% spin polarization at the Fermi level [3, 4]. The electronic and magnetic properties of Co_{2}MnAl [5] and Co_{2}CrSi [6] using local spin density approximation (LSDA) show the halfmetallicity at the ground state. Rai and Thapa have also investigated the electronic structure and magnetic properties of  ( = Co, = Mn, = Ge, Sn) type Heusler compounds by using a first principle study and reported HMFs [7]. Rai et al. (2012) also studied the electronic and magnetic properties of Co_{2}CrAl and Co_{2}CrGa using both LSDA and LSDA+ and reported the increase in band gap, hybridization of dd orbitals as well as dp orbitals when treated with LSDA+ [8]. The Fermi level lies in the partially filled band of the majority spin, whereas in the minority spin, the Fermi energy falls in an exchangesplit gap between the occupied band and the unoccupied band. Since the magnetic properties are highly spin polarized near the Fermi energy, it is therefore interesting to investigate the orbital contributions of the individual atoms to the magnetic moment of Co_{2}TiAl. The LDA+ approach in which a Hubbard repulsion term is added to the LDA is functional for strong correlation of d or f electrons. Indeed, it provides a good description of the electronic properties of a range of exotic magnetic materials, such as the Mott insulator KCuF_{3} [9] and the metallic oxide LaNiO_{2} [10]. Two main LDA+ schemes are in widespread use today: The Dudarev [11] approach in which an isotropic screened onsite Coulomb interaction is added and the Liechtenstein [9] approach in which the and exchange () parameters are treated separately. Both the choice of LDA+ schemes on the orbital occupation and subsequent properties [12], as well as the dependence of the magnetic properties on the value of [13], has recently been analyzed. It goes without saying that the Hubbard model [14] is of seminal importance in the study of modern condensed matter theory. It is believed that the Hubbard model can describe many properties of strongly correlated electronic systems. The discovery of high temperature superconductivity has enhanced the interest in a set of Hubbardlike models that are used to describe the strongly correlated electronic structure of transition metal oxides [15].
2. Crystal Structure and Calculation
2.1. Crystal Structure
Heusler compounds crystallize in the cubic L_{21} structure (space group ) [16]. Co (green) atoms are at the (1/4, 1/4, 1/4) and (3/4, 3/4, 3/4), Ti (red) at (1/2, 1/2, 1/2), and Al (blue) atoms at (0, 0, 0). The cubic L2_{1} structure consists of four interpenetrating fcc sublattices, two of which are equally occupied by Co. The two Cosite fcc sublattices combine to form a simple cubic sublattice as shown in Figure 1.
2.2. Method
In this work, we have performed the fullpotential linearized augmented plane wave (FPLAPW) method accomplished by using the WIEN2K code [17] within LSDA and LSDA+ [9] schemes. We have calculated onsite Coulomb repulsion () based on Hubbard model. The standard Hubbard Hamiltonian [18] is of the form: where and creates (annihilates) an electron on site with spin orโ. A nearest neighbor is denoted by . is the onsite Coulomb repulsion between two electrons on the same site. The hybridization between nearest neighbor orbitals is denoted by , allowing the particles to hop to adjacent sites. The onsite energies are taken to be zero. Considering that the atoms are embedded in a polarizable surrounding, is the energy required to move an electron from one atom to another, far away, in that case. is equal to the difference of ionization potential () and electron affinity () of the solid. Removing an electron from a site will polarize its surroundings thereby lowering the ground state energy of the () electron system [19, 20]. Thus where are the ground state energy of () electron system.
To explore the effects of the onsite Coulomb energy on the electronic structures and the magnetic moments, different from 0.00โRy up to 0.29โRy for Co and 0.053โRy for Ti were used in the LSDA+ calculations.
3. Results and Discussions
We have studied Co_{2}TiAl using simple LSDA; that is, โRy as shown in Figure 2. The Fermi energy () is situated close to the valence band; there exists a small gap of 0.400โeV which is lower than the previously reported value of energy gap 0.456โeV [2] which was calculated by using GGA as given in Table 1. The robustness of halfmetallicity in Co_{2}TiAl can be explained by the impact of on the DOS which is taken into consideration in LSDA+. We have plotted DOS for each value of which is shown in Figure 2, and it is seen that the majorityspin bands shift towards low energy and the minorityspin bands shift toward high energy side. In the minorityspin of valence and conduction bands, the maximum contribution to DOS is from the Co atoms. The DOS in majorityspin of conduction band is minimum for Co atoms. For a large , the minorityspin band of Co extends across the Fermi level and gap disappears in . As a result, the DOS is no longer halfmetallic. The use of the LSDA+ method increases the width of the energy gap with increase in substantially up to some extent. The respective energy gaps for each value of are for โRy โeV, for โRy โeV, for โRy โeV, and for โRy and โRy โeV. Kandpal et al. calculated the energy gap of Co_{2}TiAl, 1.12โeV, using LDA+ [2]. In LSDA, the transition metal states are well separated from the states, whereas the LSDA+ method increases the energetic overlap between these states. In all cases, the gap is between the occupied and unoccupied transition metal states [21]. It can be seen that the bandwidth of the bands for the Co site is indeed smaller than for the Ti site as shown in Figures 2(a) and 2(b). The states on the Co sites are more localized and one can expect a larger onsite Coulomb interaction than that on the Ti site, which is in agreement with [22]. However, the halfmetallicity is retained till some value of as shown in Figure 2. Therefore, the dependency of DOS on implies that the halfmetallicity is robust sensitive to .
(a)
(b)
3.1. Magnetic Properties
The calculated partial and total magnetic moments are summarized in Table 2. For โRy, LSDA+ gives the partial moment of 1.0605โฮผB/atom for Co, โ0.71433โฮผB/atom for Ti, and the total moment was 0.9999โฮผB/atom. Similarly, LSDA (โRy) gives the orbital moment of 0.76070 for Co, โ0.31578โฮผB/atom for Ti, and 0.99999โฮผB/atom for total system being in good agreement with the previously calculated orbital moment 1.00โฮผB/atom reported by Kandpal [2]. The opposite signs of spin moments between Co and Ti indicate charge transfer from the Ti anion to the Co cation. With the increase of , the total magnetic moment as well as the moment of Co increases and the moment of Ti decreases as shown in Figures 3(a) and 3(b). The increase in magnetic moment is due the double occupancy which is a decreasing function of reported by F. Mancini and F. P. Mancini [23].

(a)
(b)
4. Conclusion
In conclusion, we have performed FPLAPW selfconsistent calculations for ferromagnetic halfmetal Co2TiAl within the LSDA and the LSDA+ schemes. The spinorbit coupling included in the selfconsistent calculations; the orbital magnetic moments are obtained from both the LSDA and the LSDA+ methods. It is found that the onsite Coulomb interaction dramatically enhanced the orbital moments. For โRy and โRy, the calculated total orbital moments are 0.99999โฮผB/atom and 0.999โฮผB/atom, respectively, being in good agreement with the previously reported result 1.00โฮผB/atom [2]. The calculated energy gap was found to be 0.84โeV for โRy. It also appears that decreases double occupancy and hence increases local moments. Our calculated results of for Co and Ti are 0.29โRy and 0.053โRy; respectively, the corresponding magnetic moments is not the integral value (HM) that is, 2.258โฮผ_{B}. Also Figure 2 shows that does not lie at the middle of the gap at โRy and ;โRy thus the half metallicity does not exist. By using LSDA+, we have found that Co2TiAl is possible halfmetal candidate having magnetic moment 0.99994โu_{B} at โRy. This value of integral magnetic moment supports the condition of halfmetallicity. Due to these characteristics like integer value of magnetic moment, 100% spin polarization at and the energy gap at the Fermi level in spindown channel make application of halfmetallic ferromagnets very important. The Cobased Heusler alloys Co_{2} ( is transition elements and is the elements) are the most prospective candidates for the application in spintronics. This is due to a high Curie temperature beyond room temperature and the simple fabrication process such as dcmagnetron sputtering in Co_{2}.
Acknowledgments
D. P. RAI acknowledges DST inspire research fellowship and R. K. Thapa a research grant from UGC (New Delhi), India.
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Copyright
Copyright © 2012 D. P. Rai and R. K. Thapa. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.