# A First Principle Calculation of Full-Heusler Alloy Co2TiAl: LSDA+U Method

Authors: D. P. Rai; R. K. Thapa
Journal: ISRN Condensed Matter Physics
 (2012)

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## Abstract

We performed the structure optimization of Co2TiAl based on the generalized gradient approximation (GGA) and linearized augmented plane wave (LAPW) method. The calculation of electronic structure was based on the full-potential linear augmented plane wave (FP-LAPW) method and local spin density approximation exchange correlation LSDA+U. We also studied the impact of the Hubbard potential or onsite Coulomb repulsion (U) on electronic structure; the values are varied within reasonable limits to study the resulting effect on the physical properties of Co2TiAl system. The calculated density of states (DOS) shows that half-metallicity of Co2TiAl decreases with the increase in U values.


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## Body

## 1. Introduction

Semi-Heusler compound NiMnSb was the first found half-metal ferromagnets (HMFs) by using first principle calculation based on density functional theory [1]. Co2TiAl is a ferromagnetic half-metal with an integral magnetic moment of 1 μB/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 half-metallic property and the applicable potential for future spintronics. In half-metal, one spin channel is metallic and the other is insulating with 100% spin polarization at the Fermi level EF [3, 4]. The electronic and magnetic properties of Co2MnAl [5] and Co2CrSi [6] using local spin density approximation (LSDA) show the half-metallicity at the ground state. Rai and Thapa have also investigated the electronic structure and magnetic properties of X2YZ- (X = Co, Y = Mn, Z = 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 Co2CrAl and Co2CrGa using both LSDA and LSDA+U and reported the increase in band gap, hybridization of d-d orbitals as well as d-p orbitals when treated with LSDA+U [8]. The Fermi level lies in the partially filled 3d band of the majority spin, whereas in the minority spin, the Fermi energy falls in an exchange-split gap between the occupied band and the unoccupied 3d 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 Co2TiAl. The LDA+U approach in which a Hubbard U 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 KCuF3 [9] and the metallic oxide LaNiO2 [10]. Two main LDA+U schemes are in widespread use today: The Dudarev [11] approach in which an isotropic screened on-site Coulomb interaction U is added and the Liechtenstein [9] approach in which the U and exchange (J) parameters are treated separately. Both the choice of LDA+U schemes on the orbital occupation and subsequent properties [12], as well as the dependence of the magnetic properties on the value of U [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 Hubbard-like 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

X
2
Y
Z Heusler compounds crystallize in the cubic L21 structure (space group Fm3-m) [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 L21 structure consists of four interpenetrating fcc sublattices, two of which are equally occupied by Co. The two Co-site fcc sub-lattices combine to form a simple cubic sub-lattice as shown in Figure 1.Figure 1
Structure of Co2TiAl.


### 2.2. Method

In this work, we have performed the full-potential linearized augmented plane wave (FP-LAPW) method accomplished by using the WIEN2K code [17] within LSDA and LSDA+U [9] schemes. We have calculated onsite Coulomb repulsion (U) based on Hubbard model. The standard Hubbard Hamiltonian [18] is of the form:
(1)H=-t∑〈ij〉,σciσ†cjσ+U∑ni↑ni↓,
where niσ=ciσ†ciσ and ciσ†(ciσ) creates (annihilates) an electron on site i with spin σ=↑ or ↓. A nearest neighbor is denoted by 〈ij〉. U is the onsite Coulomb repulsion between two electrons on the same site. The hybridization between nearest neighbor orbitals is denoted by t, allowing the particles to hop to adjacent sites. The on-site energies are taken to be zero. Considering that the atoms are embedded in a polarizable surrounding, U is the energy required to move an electron from one atom to another, far away, in that case. U is equal to the difference of ionization potential (EI) and electron affinity (EA) of the solid. Removing an electron from a site will polarize its surroundings thereby lowering the ground state energy of the (N-1) electron system [19, 20]. Thus
(2)EI=EN-1-EN,EA=EN-EN+1U=(EN-1-EN)-(EN-EN+1),
where EN(±1) are the ground state energy of (N±1) electron system.To explore the effects of the on-site Coulomb energyU on the electronic structures and the magnetic moments, different U from 0.00 Ry up to 0.29 Ry for Co and 0.053 Ry for Ti were used in the LSDA+U calculations.


## 2.1. Crystal Structure

X
2
Y
Z Heusler compounds crystallize in the cubic L21 structure (space group Fm3-m) [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 L21 structure consists of four interpenetrating fcc sublattices, two of which are equally occupied by Co. The two Co-site fcc sub-lattices combine to form a simple cubic sub-lattice as shown in Figure 1.Figure 1
Structure of Co2TiAl.


## 2.2. Method

In this work, we have performed the full-potential linearized augmented plane wave (FP-LAPW) method accomplished by using the WIEN2K code [17] within LSDA and LSDA+U [9] schemes. We have calculated onsite Coulomb repulsion (U) based on Hubbard model. The standard Hubbard Hamiltonian [18] is of the form:
(1)H=-t∑〈ij〉,σciσ†cjσ+U∑ni↑ni↓,
where niσ=ciσ†ciσ and ciσ†(ciσ) creates (annihilates) an electron on site i with spin σ=↑ or ↓. A nearest neighbor is denoted by 〈ij〉. U is the onsite Coulomb repulsion between two electrons on the same site. The hybridization between nearest neighbor orbitals is denoted by t, allowing the particles to hop to adjacent sites. The on-site energies are taken to be zero. Considering that the atoms are embedded in a polarizable surrounding, U is the energy required to move an electron from one atom to another, far away, in that case. U is equal to the difference of ionization potential (EI) and electron affinity (EA) of the solid. Removing an electron from a site will polarize its surroundings thereby lowering the ground state energy of the (N-1) electron system [19, 20]. Thus
(2)EI=EN-1-EN,EA=EN-EN+1U=(EN-1-EN)-(EN-EN+1),
where EN(±1) are the ground state energy of (N±1) electron system.To explore the effects of the on-site Coulomb energyU on the electronic structures and the magnetic moments, different U from 0.00 Ry up to 0.29 Ry for Co and 0.053 Ry for Ti were used in the LSDA+U calculations.


## 3. Results and Discussions

We have studied Co2TiAl using simple LSDA; that is, U=0.00 Ry as shown in Figure 2. The Fermi energy (EF) is situated close to the valence band; there exists a small gap of 0.400 eV which is lower than the previously reported value of energy gap 0.456 eV [2] which was calculated by using GGA as given in Table 1. The robustness of half-metallicity in Co2TiAl can be explained by the impact of U on the DOS which is taken into consideration in LSDA+U. We have plotted DOS for each value of U which is shown in Figure 2, and it is seen that the majority-spin bands shift towards low energy and the minority-spin bands shift toward high energy side. In the minority-spin of valence and conduction bands, the maximum contribution to DOS is from the Co atoms. The DOS in majority-spin of conduction band is minimum for Co atoms. For a large U, the minority-spin band of Co extends across the Fermi level and gap disappears in EF. As a result, the DOS is no longer half-metallic. The use of the LSDA+U method increases the width of the energy gap with increase in U substantially up to some extent. The respective energy gaps for each value of U are for U=0.00 Ry Eg=0.40 eV, for U=0.10 Ry Eg=0.84 eV, for U=0.20 Ry Eg=0.50 eV, and for UCo=0.29 Ry and UTi=0.053 Ry Eg=0.32 eV. Kandpal et al. calculated the energy gap of Co2TiAl, 1.12 eV, using LDA+U [2]. In LSDA, the transition metal d states are well separated from the sp states, whereas the LSDA+U method increases the energetic overlap between these states. In all cases, the gap is between the occupied and unoccupied transition metal d states [21]. It can be seen that the bandwidth of the d bands for the Co site is indeed smaller than for the Ti site as shown in Figures 2(a) and 2(b). The d states on the Co sites are more localized and one can expect a larger on-site Coulomb interaction than that on the Ti site, which is in agreement with UTi<UCo [22]. However, the half-metallicity is retained till some value of U as shown in Figure 2. Therefore, the dependency of DOS on U implies that the half-metallicity is robust sensitive to U.Table 1
The calculated lattice parameters and magnetic moments are compared with the previous results.
Compounds
Lattice constantao (Å)
Magnetic momentμB (LSDA)
Energy gapEg (eV)
Previous
Our calculation
Previous
Our calculationMcal
Previous
Our result
Co2TiAl
5.828 [2]
6.210
1.00 [2]
0.999
0.456 [2]
0.400(a) Red line denoted the total DOS and blue line is denoted the partial DOS of Co atoms. (b) Red line denoted the total DOS and blue line denoted the partial DOS of Co atoms.
(a)
(b)


### 3.1. Magnetic Properties

The calculated partial and total magnetic moments are summarized in Table2. For U=0.10 Ry, LSDA+U gives the partial moment of 1.0605 μB/atom for Co, −0.71433 μB/atom for Ti, and the total moment was 0.9999 μB/atom. Similarly, LSDA (U=0.00 Ry) gives the orbital moment of 0.76070 for Co, −0.31578 μB/atom for Ti, and 0.99999 μB/atom for total system being in good agreement with the previously calculated orbital moment 1.00 μB/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 U, 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 U reported by F. Mancini and F. P. Mancini [23].Table 2
The calculated partial and total magnetic moments versusU values.
Coulomb repulsion
Magnetic momentμB of Co2TiAl
Energy gapEg (eV)
(U) Ry
M
Co
M
Ti
M
Al
M
tot
0.000
0.761
−0.316
−0.031
0.999
0.40
0.100
1.061
−0.714
−0.064
0.999
0.84
0.200
1.71625
−0.939
−0.043
2.150
0.50
U
Co
=
0.290
U
Ti
=
0.053
1.579
−0.585
−0.057
2.258
0.32(a) Plot ofMTi versus U Ry. (b) Plot of (Mtot and MCo) versus U Ry.
(a)
(b)


## 3.1. Magnetic Properties

The calculated partial and total magnetic moments are summarized in Table2. For U=0.10 Ry, LSDA+U gives the partial moment of 1.0605 μB/atom for Co, −0.71433 μB/atom for Ti, and the total moment was 0.9999 μB/atom. Similarly, LSDA (U=0.00 Ry) gives the orbital moment of 0.76070 for Co, −0.31578 μB/atom for Ti, and 0.99999 μB/atom for total system being in good agreement with the previously calculated orbital moment 1.00 μB/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 U, 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 U reported by F. Mancini and F. P. Mancini [23].Table 2
The calculated partial and total magnetic moments versusU values.
Coulomb repulsion
Magnetic momentμB of Co2TiAl
Energy gapEg (eV)
(U) Ry
M
Co
M
Ti
M
Al
M
tot
0.000
0.761
−0.316
−0.031
0.999
0.40
0.100
1.061
−0.714
−0.064
0.999
0.84
0.200
1.71625
−0.939
−0.043
2.150
0.50
U
Co
=
0.290
U
Ti
=
0.053
1.579
−0.585
−0.057
2.258
0.32(a) Plot ofMTi versus U Ry. (b) Plot of (Mtot and MCo) versus U Ry.
(a)
(b)


## 4. Conclusion

In conclusion, we have performed FP-LAPW self-consistent calculations for ferromagnetic half-metal Co2TiAl within the LSDA and the LSDA+U schemes. The spin-orbit coupling included in the self-consistent calculations; the orbital magnetic moments are obtained from both the LSDA and the LSDA+U methods. It is found that the on-site Coulomb interaction U dramatically enhanced the orbital moments. For U=0.00 Ry and U=0.10 Ry, the calculated total orbital moments are 0.99999 μB/atom and 0.999 μB/atom, respectively, being in good agreement with the previously reported result 1.00 μB/atom [2]. The calculated energy gap was found to be 0.84 eV for U=0.10 Ry. It also appears that U decreases double occupancy and hence increases local moments. Our calculated results of U for Co and Ti are 0.29 Ry and 0.053 Ry; respectively, the corresponding magnetic moments is not the integral value (HM) that is, 2.258 μB. Also Figure 2 shows that EF does not lie at the middle of the gap at UCo=0.29 Ry and UTi=0.053; Ry thus the half metallicity does not exist. By using LSDA+U, we have found that Co2TiAl is possible half-metal candidate having magnetic moment 0.99994 uB at U=0.10 Ry. This value of integral magnetic moment supports the condition of half-metallicity. Due to these characteristics like integer value of magnetic moment, 100% spin polarization at EF and the energy gap at the Fermi level in spin-down channel make application of half-metallic ferromagnets very important. The Co-based Heusler alloys Co2YZ (Y is transition elements and Z is the sp 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 dc-magnetron sputtering in Co2YZ.



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*Source: 410326-2012-08-16.xml*