study of half-metallic properties in co2crsb using

International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online) An Online International Journal Available at http://www.cibtech.org/jpms.htm 2012 Vol. 2 (1) January-March, pp.221-228/ Rai et al. Research Article

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STUDY OF HALF-METALLIC PROPERTIES IN CO2CrSb USING GGA AND LSDA

D. P. Rai, A. Shankar, Sandeep and *R. K. Thapa

Department of Physics, Mizoram University, Aizawl, India-796004 *Author for correspondence

ABSTRACT Based on the density functional theory (DFT) calculations, the electronic and magnetic properties of Heusler compound Co2CrSb were investigated. The system Co2CrSb were treated with generalized gradient approximation (GGA) as well as local spin density approximation (LSDA). Co2CrSb gives the 100% spin polarization at EF. Co2CrSb is the most stable Half-Metallic Ferromagnets (HMF). But the energy gap when treated with GGA is much larger as compared with the system treated with LSDA. We have also found that the increase in the total magnetic moments in GGA. Based on the calculated results we have predicted the half-metal ferromagnetic character for Co2CrSb. Key Words: GGA, LSDA, DOS, band structure, HMF, spin polarization. PACS No: 70, 71.5.-m, 71.15.Mb, 71.20.Be INTRODUCTION In 1983, de Groot discovered half-metallic ferromagnetism in semi-Heusler compound NiMnSb [de Groot et al. (1983)] by using first-principle calculation based on density functional theory. After that, half-metallicity attracted much attention [Katsnelson et al. (2008)] because of its prospective applications in spintronics [Zutic et al. (2004)]. Recently rapid development of magneto-electronics intensified the research on ferromagnetic materials that are suitable for spin injection into a semiconductor [Ohno (1998)]. One of the promising classes of materials is the half-metallic ferrimagnets (HMF), i.e., compounds for which only one spin channel presents a gap at the Fermi level, while the other has a metallic character, leading to 100% carrier spin polarization at EF [Zutic et al. (2004), de Boeck et al. (2002)]. Ishida et al. [Ishida et al. (1982)] have also proposed that the full-Heusler alloy compounds of the type Co2MnZ, (Z=Ge, Sn), are half-metals. Heusler alloys have been particularly interesting systems because they exhibit much higher ferromagnetic Curie temperature than other half-metallic materials [Webster et al. (1988)]. Rai et al. [Rai et al. (2010), Rai et al. (2011)] investigated the ground state study of Co2MnAl and Co2CrSi using LDA+U and LSDA method respectively. The preparation and characterization of bulk Co2MnZ (Z=Si, Ge, Ga and Sn) to be used as targets for pulsed laser deposition (PLD) of magnetic contacts for spintronic devices [Manea et al. (2005)]. Rai and Thapa 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 [Rai and Thapa (2012)]. 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. In our present work, we have studied the structural, electronic and magnetic properties of Co2CrSb using full potential linearized augmented plane wave (FP-LAPW) method. COMPUTATION DETAIL A computational code (WIEN2K) [Blaha et al. (2001)] based on FP-LAPW method was applied for structure calculations of Co2CrSb. GGA was used for the exchange correlation potential. The multipole exapansion of the crystal potential and the electron density within muffin tin (MT) spheres was cut at l=10. Nonspherical contributions to the charge density and potential within the MT spheres were considered up yo lmax=6 . The cut-off parameter was RKmax=7. In the interstitial region the charge density


222

and the potential were expands as a Fourier series with wave vectors up to Gmax=12 a.u-1. The number of k-points used in the irreducible part of the brillouin zone is 286. The Muffin Tin sphere radii (RMT) for each atom are 2.45 a.u. for Co, 2.45 a.u. for Cr and 2.31 a.u. for Sb. CRYSTAL STRUCTURE Heusler alloy [Heusler (1903)] with chemical formula Co2CrZ (Z = Sb). The full Heusler structure consists of four penetrating fcc sublattices with atoms at Co1(1/4,1/4,1/4), Co2(3/4,3/4,3/4), Cr(1/2,1/2,1/2) and Z(0,0,0) positions which results in L21 crystal structure having space group Fm-3-m as shown in Fig. 1.

Figure 1: Unit cell Structure of Co2CrZ: Co(green), Cr(red) and Z(blue) generated by xCrysden package. RESULTS AND DISCUSSIONS Structural optimization for Co2CrSb Systematic calculations of the electronic and magnetic properties of the Heusler compounds Co2CrSb were carried out in this work. The results of the electronic properties calculations are compared to study the effect of the different kinds of atoms and valence electron concentration on the magnetic properties and in particular the band gap in the minority states. The electronic properties were calculated using GGA and LSDA respectively. The optimized lattice constant, isothermal bulk moduli, its pressure derivative are calculated by fitting the total energy to the Murnaghan’s equation of state [Murnaghan (1944)]. The optimized lattice parameters were slightly higher than the experimental lattice parameters, the change in lattice parameters are given by Δ(ao). It is confirmed that the ferromagnetic configuration is lower in energy in case of the systems Co2CrSb [Table 1]. The results of the structural optimization are shown in Fig. 1. The detail values of the optimized Lattice parameters and bulk moduli are given in Table 1. Spin Polarization and half-metallic ferromagnets The electron spin polarization (P) at Fermi energy (EF) of a material is defined by equation (1) [Soulen et al. (1998)].

F F

F F

E EP

E E

(1)


223

Where FE and FE are the spin dependent density of states at the EF. The ↑ and ↓ assigns the majority and the minority states respectively. P vanishes for paramagnetic or anti-ferromagnetic materials even below the magnetic transition temperature. It has a finite value in ferromagnetic materials below Curie temperature [Ozdogan et al. (2006)]. The electrons at EF are fully spin polarized (P=100%) when

FE or FE equals to zero. In present work, we have studied the system Co2CrSb shows 100% spin polarization at EF [Table 2]. According to our results, the compound Co2CrSb is interesting as it shows large DOS at the EF of FE =2.5 states/eV within both GGA as well as LSDA.

Figure 2: Volume optimization of Co2CrSb Table 1: Lattice parameters, Bulk modulus and Equilibrium energy. Compounds Lattice Constants ao (Å) Bulk Modulus Equilibrium Previous Calculated Δ(ao) B(GPa) Energy (Ry) Co2CrSb 6.011a 6.034 0.023 170.763 -20642.8466

aRef : M. Gilleßen (2009) Fig.3 summarizes the results of the DOS which were calculated using LSDA. As Compared to LSDA, GGA increases the exchange splitting between the occupied majority and the unoccupied minority states and thus to larger gap for Co2CrSb [Fig. 5]. According to Figs. (4, 6) the indirect band gap along the Γ-X symmetry for Co2CrSb are 0.25 eV and 0.45 eV using LSDA and GGA respectively. Incase of both LSDA and GGA the Fermi energy (EF) lies in the middle of the gap of the minority-spin states, determining the half-metal character [Figs. (5, 6)]. Table 2: Energy gap and Spin polarization Tools Energy gap Eg (eV) Spin Polarization Emax(Γ) Emin(X) ΔE FE FE P% LSDA -0.25 0.00 0.25 2.50 0.00 100 GGA -0.25 0.20 0.45 2.40 0.00 100


224

The formation of gap for the half-metal compounds was discussed by Galanakis et. al. (2002) for Co2MnSi, is due to the strong hybridization between Co-d and Mn-d states, combined with large local magnetic moments and a sizeable separation of the d-like band centers.

Figure 3: Total DOS of Co2CrSb using LSDA

Figure 4: Band structure using LSDA


225

Figure 5: Total DOS using GGA

Figure 6: Band structure using GGA


226

Magnetic properties calculated in the LSDA Starting with the compound under investigation, all the information regarding the partial, total and the previously calculated magnetic moments are summarized in Table 3. It is shown in Table 3 the calculated total magnetic moment is almost an integer value in case of Co2CrSb as expected for the half-metallic systems. It is found that the partial as well as the total magnetic moment increases in GGA as shown in Table 3. We have found that the Co sites contribute much more to the magnetic moment in the Sb compound because of the indirect connection between the specific magnetic moment at Co and the hybridization arising from the interaction between the electrons at the Co sites with the neighboring electrons in the Co t-2g states. As shown in Table 3 the Sb atoms carry a negligible magnetic moment, which does not contribute much to the overall moment. We have also noticed that the partial moment of Sb atoms aligned anti-parallel to Co and Cr moments of the systems. It emerges from the hybridization with the transition metals and is caused by the overlap of the electron wave functions. The small moments found at the Z sites are mainly due to polarization of these atoms by the surroundings, magnetically active atoms as reported by Kandpal et al., (2006). Table 3: Total and partial magnetic moments Tools Magnetic Moment µB Previous Calculated Co Cr Z Total LSDA 5.023a 1.039 2.778 -0.010 4.939 GGA 1.058 2.853 -0.014 4.999 aRef : M. Gilleßen (2009) CONCLUSIONS We have performed the total-energy calculations to find the stable magnetic configuration and the optimized lattice constant. The DOS, magnetic moments and band structures of Co2CrSb were calculated using FP-LAPW method. The calculated results were in good agreement with the previously calculated results. The GGA gives wider gaps as well as the higher value of magnetic moment as compared to LSDA and the half-metallicity is more stable in Co2CrSb is the most stable HMF. For Ferromagnetic compounds the partial moment of Z being antiparallel to the Co and Cr atoms. We have investigated the possibility of appearance of half-metallicity in the case of the full Heusler compound Co2CrSb which shows 100% spin polarization at EF. The existence of energy gap in minority spin (DOS and band structure) of Co2CrSb is an indication of being a potential HMF. As well as the integral value of magnetic moment is also the evident of HMF. The calculated magnetic moment are in qualitative agreement with the integral value, supporting the HMF. ACKNOWLEDGEMENTS DPR acknowledges DST inspire research fellowship and RKT a research grant from UGC (New Delhi, India).


227

REFERENCES de Groot R. A., Mueller F. M., Van Engen P. G., and Buschow K. H. J. (1983) New Class of Materials: Half-Metallic Ferromagnets. Physical Review Letters, 50(25), 2024–2027. Katsnelson M. I., Irkhin V. Y., Chioncel L., Lichtenstein A. I., and de Groot R. A. (2008) Half-metallic ferromagnets: From band structure to many-body effects. Review of Modern Physics, 80(2), 315–378. Zutic I., Fabian J. and Das Sarma S. (2004) Spintronics: Fundamentals and applications Review of Modern Physics, 76(2), 323. Ohno H. (1998) Making Nonmagnetic Semiconductors Ferromagnetic. Science 281(5379), 951. Van Roy de Boeck W., Das J., Motsnyi V., Liu Z., Lagae L., Boeve H., Dessein K. and Borghs G. (2002) Technology and materials issues in semiconductor-based magnetoelectronics. Semiconductor Science Technology, 17(4), 342. Ishida S., Akazawa S., Kubo Y. and Ishida J. (1982) Band theory of Co2MnSn, Co2TiSn and Co2TiAl. Journal of Physics F: Metal Physics, 12(6), 1111. Webster P. J. and Ziebeck K. R. A. (1988) Alloys and Compounds of d-Elements with Main Group Elements. Part 2(Landolt Börnstein New Series, Group III, Vol. 19, Pt.c) ed H R J Wijn (Berlin: Springer) 75–184. Rai D. P., Hashemifar J., Jamal M., Lalmuanpuia, Ghimire M. P., Sandeep, Khathing D. T., Patra P. K., Sharma B. I., Rosangliana and Thapa R. K. (2010) Study of Co2MnAl Heusler alloy as half metallic Ferromagnet Indian Journal of Physics, 84(5), 593-595. Rai D. P., Sandeep, Ghimire M. P. and Thapa R. K. (2011) Study of energy bands and magnetic properties of Co2CrSi Heusler alloy Bulletin of Material Sciences, 34(6), 1219-1222. Manea A. S., Monnereau O., Notonier R., Guinneton F., Logofatu C., Tortet L., Garnier A., Mitrea M., Negrila C., Brandford W. and Grigorescu C. E. A. (2005) Journal of Crystal Growth 275(1-2), e1787-e1792. Rai D. P. and Thapa R. K. (2012) Electronic Structure and Magnetic Properties of X2YZ (X=Co, Y=Mn, Z=Ge, Sn) type Heusler Compounds by using A first Principle Study, Phase Transition: A Multinational Journal, iFirst, 1-11. Rai D. P., Sandeep, Ghimire M. P. and Thapa R. K. (2012) Electronic tructure and magnetic properties of Co2YZ (Y=Cr, V=Al, Ga) type Heusler compounds: A first Principle Study, International Journal of Modern Physics B 28(8), 1250071 Blaha P., Schwarz K., Madsen G. K. H., Kvasnicka D. and Luitz J. (2001) “WIEN2k, An augmented plane wave + local orbitals program for calculating crystal properties,” Karlheinz Schwarz, Techn. Universität, Wien, Austria, 3-9501031-1-2. Heusler F. (1903) Kristallstruktur und Ferromagnetismus der Mangan-Aluminium-Kupferlegierungen. Verh. Dtsch. Phys. Ges. 12, 219. Murnaghan F. D. (1944) The Compressibility of Media under Extreme Pressures. Proc. Natl. Acad. Sci. USA 30(9), 244. M. Gilleßen (2009) Von der Fakultät für Mathematik, Informatik und Naturwissenschaften der RWTH Aachen University zur Erlangung des akademischen Grades eines Doktors der Naturwissenschaften genehmigte Dissertation. Soulen Jr. R. J., Byers J. M., Osofsky M. S., Nadgorny B., Ambrose T., Cheng S. F., Broussard P. R., Tanaka C. T., Nowak J., Moodera J. S., Barry A. and Coey J. M. D. (1998) Measuring the Spin Polarization of a Metal with a Superconducting Point Contact. science 282(5386), 85.


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Ozdogan K., Aktas B., Galanakis I. and Sasioglu E. (2006) Influence of mixing the low-valent transition metal atoms (Y,Y∗=Cr,Mn,Fe) on the properties of the quaternary Co2[Y1−xY∗x]. arxiv: cond-mat/0612194v1 [cond-mat.mtrl-sci]. Kandpal H. C., Fecher G. H. and Felser C. (2006) Calculated electronic and magnetic properties of the half-metallic, transition metal based Heusler compounds. Journal of Physics D: Applied Physics, 40(6), 1507-1523. Galanakis I., Dederichs P. H. and Papanikolaou N. (2002) Slater-Pauling behavior and origin of the half-metallicity of the full-Heusler alloys. Physical Review B 66(17), 174429.

International Scholarly Research NetworkISRN Condensed Matter PhysicsVolume 2012, Article ID 410326, 5 pagesdoi:10.5402/2012/410326

Research Article

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

D. P. Rai and R. K. Thapa

Department of Physics, Mizoram University, Aizawl 796004, India

Correspondence should be addressed to D. P. Rai, [email protected]

Received 17 May 2012; Accepted 19 June 2012

Academic Editors: I. Galanakis, A. N. Kocharian, Y. Ohta, and A. D. Zaikin

Copyright © 2012 D. P. Rai and R. K. Thapa. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We performed the structure optimization of Co2TiAl based on the generalized gradient approximation (GGA) and linearizedaugmented plane wave (LAPW) method. The calculation of electronic structure was based on the full-potential linear augmentedplane wave (FP-LAPW) method and local spin density approximation exchange correlation LSDA+U . We also studied the impactof the Hubbard potential or onsite Coulomb repulsion (U) on electronic structure; the values are varied within reasonable limitsto 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.

1. Introduction

Semi-Heusler compound NiMnSb was the first found half-metal ferromagnets (HMFs) by using first principle calcula-tion based on density functional theory [1]. Co2TiAl is a fer-romagnetic half-metal with an integral magnetic moment of1 μB/atom [2]. It has been widely used in magnetic recordingtapes, spin valves, giant magnetoresistance (GMR), and soforth. In recent years, it attracts substantial interests becauseof the half-metallic property and the applicable potential forfuture spintronics. In half-metal, one spin channel is metallicand the other is insulating with 100% spin polarizationat the Fermi level EF [3, 4]. The electronic and magneticproperties of Co2MnAl [5] and Co2CrSi [6] using local spindensity approximation (LSDA) show the half-metallicity atthe ground state. Rai and Thapa have also investigated theelectronic structure and magnetic properties of X2YZ- (X= Co, Y = Mn, Z = Ge, Sn) type Heusler compounds byusing a first principle study and reported HMFs [7]. Rai et al.(2012) also studied the electronic and magnetic propertiesof Co2CrAl and Co2CrGa using both LSDA and LSDA+Uand 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 themajority spin, whereas in the minority spin, the Fermi energy

falls in an exchange-split gap between the occupied band andthe unoccupied 3d band. Since the magnetic properties arehighly spin polarized near the Fermi energy, it is thereforeinteresting to investigate the orbital contributions of theindividual atoms to the magnetic moment of Co2TiAl. TheLDA+U approach in which a Hubbard U repulsion term isadded to the LDA is functional for strong correlation of dor f electrons. Indeed, it provides a good description of theelectronic properties of a range of exotic magnetic materials,such as the Mott insulator KCuF3 [9] and the metallic oxideLaNiO2 [10]. Two main LDA+U schemes are in widespreaduse today: The Dudarev [11] approach in which an isotropicscreened on-site Coulomb interaction U is added and theLiechtenstein [9] approach in which the U and exchange (J)parameters are treated separately. Both the choice of LDA+Uschemes on the orbital occupation and subsequent properties[12], as well as the dependence of the magnetic propertieson the value of U [13], has recently been analyzed. It goeswithout saying that the Hubbard model [14] is of seminalimportance in the study of modern condensed mattertheory. It is believed that the Hubbard model can describemany properties of strongly correlated electronic systems.The discovery of high temperature superconductivity hasenhanced the interest in a set of Hubbard-like models that are

2 ISRN Condensed Matter Physics

Ti

Al

CoAlAl

Al

Al

Al

Al

Al

Co

Figure 1: Structure of Co2TiAl.

used to describe the strongly correlated electronic structureof transition metal oxides [15].

2. Crystal Structure and Calculation

2.1. Crystal Structure. X2YZ Heusler compounds crystallizein the cubic L21 structure (space group Fm3m) [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 fccsublattices, two of which are equally occupied by Co. The twoCo-site fcc sub-lattices combine to form a simple cubic sub-lattice as shown in Figure 1.

2.2. Method. In this work, we have performed the full-poten-tial linearized augmented plane wave (FP-LAPW) methodaccomplished by using the WIEN2K code [17] withinLSDA and LSDA+U [9] schemes. We have calculated onsiteCoulomb repulsion (U) based on Hubbard model. Thestandard Hubbard Hamiltonian [18] is of the form:

H = −t∑

〈i j〉,σ

c†iσ c jσ + U∑

ni↑ni↓, (1)

where niσ = c†iσ ciσ and c†iσ(ciσ) creates (annihilates) an elec-tron on site i with spin σ =↑ or ↓. A nearest neighbor isdenoted by 〈i j〉. U is the onsite Coulomb repulsion betweentwo electrons on the same site. The hybridization betweennearest neighbor orbitals is denoted by t, allowing the parti-cles to hop to adjacent sites. The on-site energies are takento be zero. Considering that the atoms are embedded in apolarizable surrounding, U is the energy required to movean electron from one atom to another, far away, in that case.U is equal to the difference of ionization potential (EI) andelectron affinity (EA) of the solid. Removing an electron froma site will polarize its surroundings thereby lowering theground state energy of the (N − 1) electron system [19, 20].Thus

EI = EN−1 − EN , EA = EN − EN+1

U =(EN−1 − EN

)−(EN − EN+1

),

(2)

where EN(±1) are the ground state energy of (N ± 1) electronsystem.

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 Ryfor 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) issituated close to the valence band; there exists a small gap of0.400 eV which is lower than the previously reported valueof energy gap 0.456 eV [2] which was calculated by usingGGA as given in Table 1. The robustness of half-metallicityin Co2TiAl can be explained by the impact of U on theDOS which is taken into consideration in LSDA+U . We haveplotted DOS for each value of U which is shown in Figure 2,and it is seen that the majority-spin bands shift towards lowenergy and the minority-spin bands shift toward high energyside. 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 minimumfor Co atoms. For a large U , the minority-spin band of Coextends across the Fermi level and gap disappears in EF . Asa result, the DOS is no longer half-metallic. The use of theLSDA+U method increases the width of the energy gap withincrease in U substantially up to some extent. The respectiveenergy 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 RyEg = 0.50 eV, and for UCo = 0.29 Ry and UTi = 0.053 RyEg = 0.32 eV. Kandpal et al. calculated the energy gap ofCo2TiAl, 1.12 eV, using LDA+U [2]. In LSDA, the transitionmetal d states are well separated from the sp states, whereasthe LSDA+U method increases the energetic overlap betweenthese states. In all cases, the gap is between the occupied andunoccupied transition metal d states [21]. It can be seen thatthe bandwidth of the d bands for the Co site is indeed smallerthan for the Ti site as shown in Figures 2(a) and 2(b). The dstates on the Co sites are more localized and one can expecta larger on-site Coulomb interaction than that on the Ti site,which is in agreement with UTi < UCo [22]. However, thehalf-metallicity is retained till some value of U as shown inFigure 2. Therefore, the dependency of DOS on U impliesthat the half-metallicity is robust sensitive to U .

3.1. Magnetic Properties. The calculated partial andtotal magnetic moments are summarized in Table 2. ForU = 0.10 Ry, LSDA+U gives the partial moment of1.0605 μB/atom for Co, −0.71433 μB/atom for Ti, and thetotal 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 totalsystem being in good agreement with the previouslycalculated orbital moment 1.00 μB/atom reported by Kandpal [2]. The opposite signs of spin moments between Co andTi indicate charge transfer from the Ti anion to the Co cation.With the increase of U , the total magnetic moment as well as

ISRN Condensed Matter Physics 3

Table 1: The calculated lattice parameters and magnetic moments are compared with the previous results.

CompoundsLattice constant ao (A) Magnetic moment μB (LSDA) Energy gap Eg (eV)

Previous Our calculation Previous Our calculation Mcal Previous Our result

Co2TiAl 5.828 [2] 6.210 1.00 [2] 0.999 0.456 [2] 0.400

86420−2−4−6−8−10

−10 −5 0 5 10 15

−10 −5 0 5 10 15

−10 −5 0 5 10

−10 −5 0 5 10

10

5

0

−5

−10

10

5

0

−5

−10

86420−2−4−6−8

10

−15

Energy (eV)

Energy (eV)

Energy (eV)

Energy (eV)

DO

S (s

tate

s/eV

)D

OS

(sta

tes/

eV)

DO

S (s

tate

s/eV

)D

OS

(sta

tes/

eV)

U = 0 Ry

U = 0.1 Ry

U = 0.2 Ry

UCo = 0.29 Ry

UTi = 0.053 Ry

Total DOS upCo tot. up

(a)

86420−2−4−6−8−10

−10 −5 0 5 10 15

−10 −5 0 5 10 15

−10 −5 0 5 10 15

10

5

0

−5

−10

10

5

0

−5

−10

86420−2−4−6−8

10

−10 −5 0 5 10−15

Energy (eV)

Energy (eV)

Energy (eV)

Energy (eV)

DO

S (s

tate

s/eV

)D

OS

(sta

tes/

eV)

DO

S (s

tate

s/eV

)D

OS

(sta

tes/

eV)

U = 0 Ry

U = 0.1 Ry

U = 0.2 Ry

UCo = 0.29 Ry

UTi = 0.053 Ry

Total DOS upTi tot. up

(b)

Figure 2: (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 andblue line denoted the partial DOS of Co atoms.

Table 2: The calculated partial and total magnetic moments versus U values.

Coulomb repulsion Magnetic moment μB of Co2TiAl Energy gap Eg (eV)

(U) Ry MCo MTi MAl Mtot

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

UCo = 0.290UTi = 0.053

1.579 −0.585 −0.057 2.258 0.32


−0.3

−0.4

−0.5

−0.6

−0.7

−0.8

−0.9

0 0.05 0.1 0.15 0.2

U (Ry)

Ti

Mag

net

ic m

omen

t (u

B)

−1

(a)

0 0.05 0.1 0.15 0.2

2.4

2.2

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.25 0.3

U (Ry)

CoTotal

Mag

net

ic m

omen

t (u

B)

(b)

Figure 3: (a) Plot of MTi versus U Ry. (b) Plot of (Mtot and MCo) versus U Ry.

the moment of Co increases and the moment of Ti decreasesas shown in Figures 3(a) and 3(b). The increase in magneticmoment is due the double occupancy which is a decreasingfunction of U reported by F. Mancini and F. P. Mancini [23].

4. Conclusion

In conclusion, we have performed FP-LAPW self-consistentcalculations for ferromagnetic half-metal Co2TiAl within theLSDA and the LSDA+U schemes. The spin-orbit couplingincluded in the self-consistent calculations; the orbitalmagnetic moments are obtained from both the LSDA andthe LSDA+U methods. It is found that the on-site Coulombinteraction U dramatically enhanced the orbital moments.For U = 0.00 Ry and U = 0.10 Ry, the calculated total orbitalmoments are 0.99999 μB/atom and 0.999 μB/atom, respec-tively, being in good agreement with the previously reportedresult 1.00 μB/atom [2]. The calculated energy gap was foundto be 0.84 eV forU = 0.10 Ry. It also appears that U decreasesdouble occupancy and hence increases local moments. Ourcalculated results ofU for Co and Ti are 0.29 Ry and 0.053 Ry;respectively, the corresponding magnetic moments is not theintegral value (HM) that is, 2.258 μB. Also Figure 2 showsthat EF does not lie at the middle of the gap at UCo =0.29 Ry and UTi = 0.053; Ry thus the half metallicity doesnot exist. By using LSDA+U , we have found that Co2TiAlis possible half-metal candidate having magnetic moment0.99994 uB at U = 0.10 Ry. This value of integral magneticmoment supports the condition of half-metallicity. Due tothese characteristics like integer value of magnetic moment,100% spin polarization at EF and the energy gap at the Fermilevel in spin-down channel make application of half-metallicferromagnets very important. The Co-based Heusler alloysCo2YZ (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 beyondroom temperature and the simple fabrication process such asdc-magnetron sputtering in Co2YZ.

Acknowledgments

D. P. RAI acknowledges DST inspire research fellowship andR. K. Thapa a research grant from UGC (New Delhi), India.

References

[1] R. A. de Groot, F. M. Mueller, P. G. van Engen, and K. H. J.Buschow, “New class of materials: half-metallic ferromagnets,”Physical Review Letters, vol. 50, no. 25, pp. 2024–2027, 1983.

[2] H. C. Kandpal, Computational studies on the structure andstabilities of magnetic inter-metallic compounds,dissertation zurErlanggung des Grades [Doktor der Naturwissenschaften], amFachbereich Chemie, Pharmazie und Geowissenschaften derJohannes Guttenberg-Universitat Mainz, 2006.

[3] I. Zutic, J. Fabian, and S. Das Sarma, “Spintronics: fundamen-tals and applications,” Reviews of Modern Physics, vol. 76, pp.323–410, 2004.

[4] J. de Boeck, W. Van Roy, J. Das et al., “Technology andmaterials issues in semiconductor-based magnetoelectronics,”Semiconductor Science and Technology, vol. 17, no. 4, pp. 342–354, 2002.

[5] D. P. Rai, J. Hashemifar, M. Jamal et al., “Study of Co2MnAlHeusler alloy as half metallic ferromagnet,” Indian Journal ofPhysics, vol. 84, no. 6, pp. 717–721, 2010.

[6] D. P. Rai, Sandeep, M. P. Ghimire, and R. K. Thapa, “Structuralstabilities, elastic and thermodynamic properties of ScandiumChalcogenides via first-principles calculations,” Bulletin desSciences Mathematiques, vol. 34, pp. 1219–1222, 2011.

[7] D. P. Rai and R. K. Thapa, “Electronic structure and magneticproperties of X2YZ (X = Co, Y = Mn, Z = Ge, Sn) type Heuslercompounds by using a first principle study,” Phase Transition:A Multinational Journal, pp. 1–11, 2012.


[8] D. P. Rai, Sandeep, M. P. Ghimire, and R. K. Thapa, “Elec-tronic tructure and magnetic properties of Co2YZ (Y = Cr,V = Al, Ga) type Heusler compounds: A first PrincipleStudy,” International Journal of Modern Physics B, vol. 26, pp.1250071–1250083, 2012.

[9] A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, “Density-functional theory and strong interactions: orbital ordering inMott-Hubbard insulators,” Physical Review B, vol. 52, no. 8,pp. R5467–R5470, 1995.

[10] K. W. Lee and W. E. Pickett, “Infinite-layer LaNiO2: Ni1+ is notCu2+ ,” Physical Review B, vol. 70, Article ID 165109, 2004.

[11] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys,and A. P. Sutton, “Electron-energy-loss spectra and thestructural stability of nickel oxide: an LSDA+U study,” PhysicalReview B, vol. 57, no. 3, pp. 1505–1509, 1998.

[12] E. R. Ylvisaker, W. E. Pickett, and K. Koepernik, “Anisotropyand magnetism in the LSDA+U method,” Physical Review B,vol. 79, Article ID 035103, 2009.

[13] S. Y. Savrasov, A. Toropova, M. I. Katsnelson, A. I. Lichten-stein, V. Antropovand, and G. Kotliar, “Electronic structureand magnetic properties of solids,” Zeitschrift Fur Kristallogra-phie, vol. 220, pp. 473–488, 2005.

[14] J. Hubbard, “Electron correlations in narrow energy bands,”Proceedings of the Royal Society A, vol. 276, pp. 238–257, 1963.

[15] P. W. Anderson, “The resonating valence bond state inLa2CuO4 and superconductivity,” Science, vol. 235, no. 4793,pp. 1196–1198, 1987.

[16] F. Heusler, “Uber magnetische Manganlegierungen,” Verhand-lungen der Deutschen Physikalischen Gesellschaft, vol. 12, p.219, 1903.

[17] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, J. Luitz,and K. Schwarz, “An augmented plane wave plus local orbitalsprogram for calculating crystal properties,” Wien2K User’sGuide, Technische Universitat Wien, Wien, Austria, 2008.

[18] L. M. Roth, “New method for linearizing many-body equa-tions of motion in statistical mechanics,” Physical ReviewLetters, vol. 20, pp. 1431–1434, 1968.

[19] J. van den Brink, M. B. J. Meinders, and G. A. Sawatzky, “Influ-ence of screening effects and inter-site Coulomb repulsion onthe insulating correlation gap,” Physica B, vol. 206-207, pp.682–684, 1995.

[20] K. H. G. Madsen and P. Novak, “Charge order in magnetite.An LDA+U study,” Europhysics Letters, vol. 69, p. 777, 2005.

[21] I. Galanakis, “Orbital magnetism in the half-metallic Heusleralloys,” Physical Review B, vol. 71, Article ID 012413, 2005.

[22] C. Ederer and M. Komelj, “Magnetic coupling in CoCr2O4

and MnCr2O4: an LSDA+U study,” Physical Review B, vol. 76,Article ID 064409, 9 pages, 2007.

[23] F. Mancini and F. P. Mancini, “One dimensional extendedHubbard model in the atomic limit,” Cond. Mat. Str-El, vol.77, pp. 061120–061121, 2008.

Ground state study of Electronic and Magnetic Properties of Co2MnZ (Z = Ge, Sn) type

Heusler Compounds : A first Principle Study

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Ground state study of Electronic and Magnetic Properties of Co2MnZ (Z = Ge, Sn) type Heusler Compounds : A first Principle Study.

D P Rai, Sandeep, M P Ghimire and R K Thapa

Department of Physics, Mizoram university, Aizawl, India 796004 E-mail: [email protected] Abstract. The structural optimization was performed based on generalized gradient approximation (GGA) followed by the calculation of electronic structure and magnetic properties on Co2MnGe and Co2MnSn. The calculation of electronic structure was based on local spin density approximation (LSDA) within full potential linear augmented plane wave (FPLAPW) method. We studied the structural, electronic and magnetic properties. The calculated density of states (DOS) and band structures shows the half-metallicity of Co2MnGe and Co2MnSn.

1. Introduction The term Heusler alloy is named after a German mining engineer and chemist Friedrich Heusler [1]. Half-metallicity attracted much attention [2], because of its prospective applications in spintronics [3]. Recently rapid development of magneto-electronics intensified the research on ferromagnetic materials that are suitable for spin injection into a semiconductor [4]. One of the promising classes of materials is the half-metallic ferrimagnets, i.e., compounds for which only one spin channel presents a gap at the Fermi level, while the other has a metallic character, leading to 100% carrier spin polarization at EF [5]. Ishida et. al. have also proposed that the full-Heusler alloy compounds of the type Co2MnZ, where Z stands for Si and Ge, are half-metals [6]. Heusler alloys have been particularly interesting systems because they exhibit much higher ferromagnetic Curie temperature than other half-metallic materials [7]. The electronic and magnetic properties of Co2MnAl [8] and Co2CrSi [9] using LSDA shows the half-metallicity at the ground state. Rai and Thapa investigated the Electronic Structure and Magnetic Properties of X2YZ (X=Co, Y=Cr, Z=Al, Ga) type Heusler Compounds by using A first Principle Study and reported HMFs [10]. In half-metals, the creation of a fully spin-polarized current should be possible that should maximize the efficiency of magnetoelectronic devices [11]. Materials with high spin polarization can be used for tunnel magnetoresistance (TMR) and giant magnetoresistance (GMR) [5]. The Co-based Heusler alloys Co2YZ (Y: transition metal, Z: sp atom) 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 [12] . In the present paper, we systematically study the electronic and magnetic structure of Co based full Heusler alloys Co2MnZ (Z = Ge and Sn) to search for new halfmetallic ferromagnetic candidates. Among the systems studied Co2MnZ (Z = Ge and Sn) are predicted to be nearly half-metallic.

2. Crystal structure and Computational Methods The considered full Heusler alloys Co2YZ adopt the ordered L21-type structure (space group Fm-3-m) which may be understood as the result of four interpenetrating face-centered-cubic (fcc) lattices. The Y and Z atoms occupy two fcc sublattices with origin at (0, 0, 0) and (½, ½, ½) respectively. The Co atoms are located in sublattices with origins at (¼, ¼, ¼,) and (¾, ¾, ¾,).

23rd International Conference on High Pressure Science and Technology (AIRAPT-23) IOP PublishingJournal of Physics: Conference Series 377 (2012) 012074 doi:10.1088/1742-6596/377/1/012074

Published under licence by IOP Publishing Ltd 1

Fig. 1: Structure of the Co2YZ Heusler alloys : Co (red), Mn(yellow) and Z (blue) atoms.

The first step, we optimized the parameters. The calculation was accomplished by using the

WIEN2K code [13]. In the next step, we calculated the electronic structure and magnetic properties using FP-LAPW method. The accuracy is up to 10-4 Ry. The exchange–correlation potential is chosen in the local spin density approximation (LSDA) [14]. The selfconsistent potentials were calculated on a 20 × 20 × 20 k-mesh in the Brillouin zone, which corresponds to 256 k points in the irreducible Brillouin zone. The sets of valence orbitals in the calculations were selected as 3s, 3p, 4s, 4p, 3d Co atoms, 3s, 3p, 4s, 4p, 3d for Mn atoms, 3d, 4s, 4p for Ge atoms and 4d, 5s, 5p for Sn atoms. All lower states were treated as core states.


3.1. The equilibrium energy and lattice parameter

The optimized lattice constant and isothermal bulk modulus are calculated by fitting the total energy to the Murnaghan’s equation of state [15] as shown in Fig. 2. It is clear IVB column sp element (Ge or Sn) with four valence electrons state is the most stable. In table 1 we have tabulated the equilibrium lattice constants and the equilibrium energy. We observed Co2MnGe has a higher energy compare to Co2MnSn.

5.2 5.4 5.6 5.8 6.0 6.2 6.4

-12089.05

-12089.00

-12088.95

-12088.90

-12088.85

-12088.80(a) Co2MnGe

Ener

gy [R

y]

Lattice Constant Ao

5.6 5.8 6.0 6.2 6.4 6.6 6.8

-20248.98

-20248.96

-20248.94

-20248.92

-20248.90

-20248.88

-20248.86

-20248.84

-20248.82

-20248.80

-20248.78 (b) Co2MnSn

Ener

gy [R

y]

Lattice Constant Ao

Fig. 2: Plot of Energy versus Lattice constant: (a) Co2MnGe and (b) Co2MnSn

3.2 Density of states and magnetic moments The total DOS plots of Co2MnGe and Co2MnSn are shown in Fig. 3 and Fig. 4 respictively. From the DOS plots of Co2MnGe and Co2MnSn, shown in Fig. 3 and Fig. 4, peaks are mostly due to d state electrons of Co atoms in the core, semi-core and the valence region below EF for both spin channels..


2

In spin up channel, the DOS intersects EF showing the metallic nature. For Co2MnGe we have observed from Fig. 3 that Mn-d electrons mainly contribute in the valence region with sharp peaks at -2.3 eV and -2.7 eV while, in case of Co2MnSn Mn-d electrons contribute with sharp peaks at -2.4 eV and -2.8 eV in spin-up configuration with a Fermi cut-off. In spin down channel, peaks are due to d states of Co atoms in the conduction region. In the conduction region of Co2MnGe peaks were observed at 1.0 eV, 2.0eV and for Co2MnSn, peaks are observed at 0.9 eV and 1.8 eV. The d electrons of Co are found to strongly hybridized with Mn-d electrons [16]. The partial magnetic moments of Co, Mn and Ge atoms are 0.975μB, 3.097μB and -0.044μB respectively for Co2MnGe. The effective magnetic moment is 5.004μB which is approximately an integer value 5.00 μB [16]. Similarly for Co2MnSn, the partial magnetic moments of Co, Mn and Sn atoms are 0.950µB, 3.253 µB and -0.059 µB respectively. The effective magnetic moment is thus 5.016 µB which is approximately an integer value 5.03 µB [16]. The effective magnetic moments, energy gaps, optimized lattice are tabulated and compared with the previous results in table 1. Table 1: The calculated equilibrium lattice parameters, spin magnetic moments in μB for the Co2MnZ ( Ge and Sn) compounds. Compound

Lattice constant ao Å Magnetic Moment µB Equilibrium Energy (Ry)

Previous Observed

Previous

Observed

Co2MnGe 5.737[17]

5.743[19]

5.749[16]

5.678 5.00[19]

5.00[16] 5.004

-12089.018

Co2MnSn 5.95[18]

5.984[16] 5.970 5.02[17]

5.04[18]

5.03[16]

5.016 -20248.956

Fig. 3: (a) total DOS plots of Co2MnGe, (b) partial DOS plots of Co (c) partial DOS plots of Mn (d) partial DOS plots of Ge. For Co2MnGe an indirect energy gap obtained between symmetry point Г and X is 0.60 eV which almost equals to 0.581 eV reported by Kandpal et. al. [16]. Similarly for Co2MnSn, an indirect energy gap obtained between Г and X is 0.40 eV which agrees with 0.411 eV reported by Kandpal et. al. [16]. The results of energy gap and bulk modulus are given in Table 2.


3

Fig. 4: (a) total DOS plots of Co2MnSn, (b) partial DOS plots of Co (c) partial DOS plots of Sn (d) partial DOS plots of Mn. Table 2: Energy gaps and bulk modulus.

Compounds

Energy gap Eg (eV) Bulk Modulus (GPa)

Previous

Observed

Co2MnGe 0.581[16] 0.600 409.33

Co2MnSn 0.411[16] 0.400 212.07

4. Conclusions We have studied the possibility of appearance of half-metallicity in the case of the full Heusler compounds Co2MnZ, where Z is an sp atom belonging to the IVB column of the periodic table. We found that compounds Co2MnZ (Z=Ge, Sn) the ferromagnetic is stable at the equilibrium lattice constant. Although all compounds are not half-metallic at their equilibrium lattice constant, small expansion of the lattice pushes the Fermi level within the gap. Thus these compounds follow the Slater–Pauling behaviour and the ‘rule of 24’ [20]. The lighter the transition elements and the smaller the number of valence electrons, the wider is the gaps and the more stable is the half-metallicity. The results were in support of the HMF nature for Co2MnGe and Co2MnSn . The existence of energy gap in DOS for spin down of both systems is an indication of being a potential HMF. The calculated magnetic moment for Co2MnGe is 5.004 µB and for Co2MnSn, it is 5.016 µB. : Acknowledgment DPR acknowledges MZU-UGC research fellowship, SD acknowledges a JRF and RKT a research grant from DAE (BRNS), Mumbai. References [1] Heusler F 1903 Verh. Dtsch. Phys. Ges. 12 219.


4

[2] Katsnelson M I, Irkhin V Y, Chioncel L, Lichtenstein A I and de Groot R A 2008 Rev. Mod. Phys. 80 315.

[3] Zutic I, Fabian J and Sarma S D 2004 Rev. Mod. Phys. 76 323. [4] Ohno H 1998 Science 281 951. [5] de Boeck, van Roy W, Das J, Motsnyi V, Liu Z, Lagae L, Boeve H, Dessein K and Borghs G

2002 Semicond. Sci. Technol. 17 342. [6] Ishida S, Akazawa S, Kubo Y and Ishida J 1982 J. Phys. F: Met. Phys. 12 1111. [7] Webster P J and Ziebeck K R A 1988 Alloys and Compounds of d-Elements with Main Group

Elements. Part 2(Landolt B¨ornstein New Series, Group III, Vol. 19, Pt.c) ed H R J Wijn (Berlin: Springer) pp 75–184.

[8] Rai D P, Hashemifar J, Jamal M, Lalmuanpuia, Ghimire M P, Sandeep, Khathing D T, Patra P K, Sharma B I, Rosangliana and Thapa R K 2010 Indian J. Phys. 84 (5) 593.

[9] Rai D P, Sandeep, Ghimire M P and Thapa R K 2011 Bull. Mat. Sc. 34 1219. [10] Rai D P, A. Shankar, Sandeep, Ghimire M P and Thapa R K 2012 Int. J. Mod. Phys. B 26 8

1250071. [11] Yakushi K, Saito K, Takanashi K, Takahashi Y K, Hondo K 2006 Appld. Phys. Lett. 88 222504l. [12] Miura Y, Nagao K and Shirai M 2004 Phys. Rev. B 69 144413. [13] Blaha P., Schwarz K., Madsen G. K. H., Kvasnicka D., Luitz J., Schwarz K., 2008. An

Augmented Plane Wave plus Local Orbitals Program for Calculating Crystal Properties:Wien2K User’s Guide, Techn. Universitat Wien, Wien.

[14] Perdew J P and Wang Y 1992 Phys. Rev. B 45 13244. [15] Murnaghan F D Proc. Natl. Acad. Sci. USA 30 244. [16] Kandpal H C, Fecher G H and Felser C 2006 J. Phys. D: Appl. Phys. 40 1507. [17] Piccozi S, Continenza A, Freeman A J 2002 Phys. Rev. B 66 094421 [18] Kubler J, Williams A R, Sommers C B 1983 Phys. Rev. B 28 4 1745. [19] Kurtulus Y and Dronskowski R 2005 Phys. Rev. B 71 014425. [20] Galanakis I, Dederichs P H and Papanikolaou N 2002 Phys. Rev. B 66 174429.


5

Asian Journal of Physical Sciences (2012), Vol.1, No.1, pp 10-25 Paper

*Correspondence: Condensed Matter Theory Group, Department of Physics, Mizoram University, Aizawl, India-796004 Published by Department of physics, Fatima Mata National College-691001, INDIA. All rights reserved.

For Permissions, please email: [email protected]

10

ELECTRONIC STRUCTURE AND MAGNETIC PROPERTIES OF Co2MnSi BY USING LSDA+U METHOD. D. P. Rai1, Sandeep1, A. Shankar1 , M. P. Ghimire2, R. K. Thapa* 1 and

M.P. Ghimire2

1Condensed Matter Theory Group,Department of Physics, Mizoram University, Aizawl, INDIA-796004. 2National Institute for Material Sciences, Tsukuba University, Tsukuba, JAPAN.

Abstract

The volume optimization of Co2MnSi was performed based on the

generalized gradient approximation (GGA) and linearized augmented plane

wave (LAPW) . The calculation of electronic structure was based on the full

potential linear augmented plane wave (FP-LAPW) method and we have

used local spin density approximation exchange correlation LSDA+U. We

also studied the variation of U on electronic structure, the values are

changed within reasonable limits to study the resulting effect on the physical

properties of Co2MnSi systems. The calculated density of states (DOS)

showed half-metallicity of Co2MnSi at a particular value of U.

Key words: GGA, LSDA+U, DOS, Half metallicity, Hubbard potential.

PACS No: 70, 71.5.-m, 71.15.Mb, 71.20.Be

Properties of CO2MnSi

11

INTRODUCTION

Full Heusler alloys are the ternary intermetallic compounds with composition

X2YZ, where X and Y are transition elements (Ni, Co, Fe, Mn, Cr, Ti, V etc.)

and Z is III, IV or V group elements (Al, Ga, Ge, AS, Sn, In etc.). Intermetallic

Heusler alloys are amongst the most attractive half-metallic systems due to the

high Curie temperatures and the structural similarity to the binary

semiconductors [1]. This opens the pathway for a new generation of devices

combining standard microelectronics with spin-dependent effects that arise from

the interaction between the carrier and the magnetic properties of the material.

However, this field is still in progress but the progress of these topics is very

impressive and they also branch into wider areas, which include strongly

correlated electron materials, multiferroic materials, and semiconductors,

including graphene. It is envisioned that the merging of electronics, photonics,

and magnetics will ultimately lead to new spin-based multifunctional devices

such as spin-FET (field effect transistor), spin-LED (light-emitting diode), spin

RTD (resonant tunnelling device), encoders, decoders, and quantum bits for

quantum computation and communication. The search for materials combining

properties of the ferromagnet and the semiconductor has been challenging

because of differences in crystal structure and chemical bonding [2, 3]. After 26

years, there have been great progresses on half-metal in both experiment and

theory. Ishida et al. [4] have proposed that the full-Heusler compounds like

Co2MnZ where Z stands for Si and Ge are half-metals. Marvorpoulos et al. [5]

studied the influence of the spin-orbit coupling on the spin polarization at the

Fermi level and found the effect to be very small, that is in agreement with a

small orbital moment calculated by Galanakis [6]. Recently rapid development of

magneto-electronics intensified the research on ferromagnetic materials which

are suitable for spin injection into a semiconductor [7]. One of the promising

Rai et al

12

classes of such materials are the half-metallic ferrimagnets, i.e., compounds for

which only one spin channel presents a gap at the Fermi level, while the other

has a metallic character, leading to 100% spin polarization at Fermi energy (EF)

[8]. Many of these systems have been predicted by means of electronic band

structure calculations [9] and some of them are in use already as elements in

multilayered magnetoelectronic devices such as magnetic tunnel junction [10]

and also in giant magnetoresistance spin valves [11]. Rai et al. [12, 13]

investigated the ground state study of Co2MnAl and Co2CrSi using LDA+U and

LSDA method respectively. Rai and Thapa 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 [14]. 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

[15]. In this paper we will present the electronic and magnetic properties of

Co2MnSi. For this we will use LSDA+U method.

CRYSTAL STRUCTURE AND COMPUTATIONAL METHODS Crystal Structure: The Heusler alloys [16] represent a class of ternary

intermetallic compounds of the form X2YZ in which X is the transition metal, Z

is a main groups III-V and Y is magnetically active transition metal. Co (blue),

Mn (red) and Si (yellow) atoms. The cubic L21 structure consists of four inter-

penetrating fcc sub-lattices, 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

Fig.1.


13

Fig. 1. Unit cell Structure of the Co2MnSi generated by xCrysden.

Computational method: The total energy is calculated by using GGA [17] and

linearized augmented wave (LAPW) method [18]. In the second step, we

calculated the electronic structure and magnetic properties using full potential

linear augmented plane wave (FP-LAPW) method [19], LSDA [20] and LSDA+

U [21]. This step is accomplished by using the WIEN2K code [22] with a

21x21x21 k-mesh in which the effective Coulomb-exchange interaction (Ueff = U

− J) is used for the LDSA+U calculations. The accuracy is up to 10-4 Ry. The

multipole exapansion of the crystal potential and the electron density within

muffin tin (MT) spheres was cut at l=10. Nonspherical contributions to the

charge density and potential within the MT spheres were considered up to lmax=6

. The cut-off parameter was RmtxKmax=7. In the interstitial region the charge

density and the potential were expands as a Fourier series with wave vectors up

to Gmax=12 a.u-1. The MT sphere radii(R) used were 2.34 a. u. for Co, 2.34 a.u.

for Mn and 2.21 a. u. for Si. The number of k-points used in the irreducible part

of the brillouin zone is 286. The interactions between the less localized s and p

electrons are treated within the standard local spin density approximation LSDA.

The LSDA+U method includes the Coulomb interaction between strongly

Rai et al

14

localized d or f electrons in the spirit of a mean-field Hubbard model. A Hubbard

interaction term U is added to the LSDA total energy [23]. The strongly

correlated systems are described by the Hubbard of Anderson_lattice type

models [24] which includes Hubbard potential U [21, 25] i.e the Coulomb energy

cost to two electrons at the same site. The additional onsite Coulomb repulsion

(U) from 0.00 Ry to 0.40 Ry is applied to the d orbits of the transition metal

atoms.

RESULTS AND DISCUSSIONS

Local Spin Density Approximation (LSDA): The optimized lattice constants of

Co2MnSi is found to be 5.678 Å. The total DOS of Co2MnSi is calculated by

using LSDA and LSDA+U which is depicted in Figs. (2, 3). For U = 0 eV,

which means normal LSDA. We applied the LSDA functional and FP-LAPW

method, we did not obtain the sharp peaks of DOS in the majority channel in

conduction region Fig 2.

Fig. 2: Total DOS using LSDA.


15

From the Figs.(2, 3) we have found the DOS at -1.4 eV below EF in spin down

and spin up regions are contributed by Mn-d atoms but Mn atoms has less

contribution in the spin down region. Similarly at -1.4eV, Co-d atoms also

contributed in both spin up and spin down regions. At 3.2eV and 4.2eV in spin

down region both Mn-d and Co-d atoms contributed to the total DOS. But

negligible contribution from Co and Mn atoms in the spin up channel above EF.

The partial magnetic moments of the atoms Co, Mn and Si are 1.029 µB, 3.058

µB and -0.055 µB respectively. Thus the total magnetic moment is 5.031 µB

which is approximately an integer value 5.00 µB [26].

Table 1. Comparison of lattice constants and magnetic moments with previous results

LSDA+U

The robustness of half-metallicity in Co2MnSi can be explained by the impact of

U on the DOS. When U is taken into consideration like in LSDA+U, the

majority-spin bands shift towards low energy and the minority-spin bands shift

toward high energy side shown in Fig. 3.

Compound Lattice constant ao

Å

Magnetic Moment µB

Energy gap Eg (eV)

Previous Our Result

Previous Our Result

Previous Our Result

Co2MnSi

5.64326 5.665 5.000 26 5.031 0.79826 0.720

Rai et al

16

Fig. 3. Total DOS of Co2MnSi with variation in U values.

In minority-spin of valence band, the maximum contribution is from the Co-d

atoms, the peaks will slightly shift towards high energy due to the hybridization.

The DOS in majority-spin of conduction band is minimum for both Co and Mn.

For a large U, the minority-spin band of Mn-d extends across the Fermi level and

the peaks disappear below EF. As a result, the system is no longer half-metallic.

The use of the LSDA+U method increases the width of the energy gap

substantially. The calculated energy gaps is 0.760 eV for Co2MnSi at U =

0.05Ry. While the LSDA gives smaller value of energy gap 0.720 eV for

Co2MnSi which is smaller than the previously reported value using GGA [26].

The bandwidth of the d bands for the Co site is indeed smaller than for the Mn

site. Thus, the d states on the Co sites are more localized and one can expect a

larger on-site Coulomb interaction than on the Mn site, which in agreement with

UMn<UCo. and the U values for transition metal lies between 0.147-0.368 Ry [27].

However, the half-metallicity is retained till some value of U as shown in Fig. 3.


17

Therefore, the dependency of DOS on U implies that the half-metallicity is

robust sensitive to U.

Fig. 4: Partial DOS of Co, Mn and Si (UCo = 0.29 Ry, UMn = 0.27 Ry)

We have used the calculated U values of Co and Mn i.e. 0.29 Ry and 0.27 Ry

respectively in LSDA. We performed the calculation supposing Hund's effect J =

0.00 Ry in which the effective Coulomb-exchange interaction (Ueff) becomes

simple U and used in the LSDA + U for further calculations. The application of

LSDA+U method increases the width of the energy gap as we increase the U

values form 0.00 to 0.40 Ry as shown in Table 2 as well as in Fig. 3. For a large

values of U i.e for U = 0.10 Ry and above, the total DOS extends across the

Fermi level EF in the spin down region as shown in Fig. 3. As a result, the system

under investigation is no longer half-metallic for U values higher than 0.10 Ry,

even though there exist a gap. The width of energy gap (Eg) is the difference in

energies of the highest occupied band at symmetry point Г in the valence region

Rai et al

18

and the lowest unoccupied band in the conduction region at symmetry point X

which is an indirect band gap. The origin of minority gap in Co2MnSi was

explained by Galanakis and Mavropoulos [28]. Based on the analysis of band

structures and DOS calculations it is seen that the 3d orbitals of Co atoms from

two different sub-lattices, Co1 (0, 0, 0) and Co2 (1/2, 1/2, 1/2) couple and form

bonding hybrids.

Fig. 5: Band structure (UCo = 0.29 Ry, UMn = 0.27 Ry)

In other words, the gap originates from the strong hybridization between the d

states of the higher valent and the lower valent transition metal atoms. As a result

the interaction of Mn with the Z-p states splits the Mn-3d states into a low lying

triplet of t2g states and a higher lying doublet of eg states. The splitting is partly

due to the different electrostatic repulsion, which is strongest for the eg states

which is directly point at the Z atoms. In the majority band the Mn-3d states are


19

shifted to lower energies and form a common 3d band with X(Co) 3d states,

while in the minority band the Mn-3d states are shifted to higher energies and are

unoccupied, so that a band gap at EF is formed, separating the occupied d

bonding states from the unoccupied d antibonding states.

Magnetic properties: We have also calculated the local magnetic moments of

Co and Mn atoms and their dependencies on U are shown in Table 2. As the

values of U increases, the partial magnetic moments of both Co and Mn

increases due to the increasing trend of localization. In the case of small

magnetic moment compounds, the Co atoms contribute mostly to the moment, as

compared with the compounds with large magnetic moments [26]. While going

from low to high U values, the Mn atoms contribute an increasing moment

shown in Fig.6. It should be noted that Mcal is the total magnetic moment but it is

not the sum of the moments of Co, Mn and Si sites but it also respects the

moment of interstitial between the sites. We have obtained the half metallicity at

U = 0.05 Ry for Co2MnSi and the value of magnetic moment is 5.000 µB. This

values of integral magnetic moment is in agreement with the DOS to support the

half metallicity. In LSDA method the calculated magnetic moments for Co2MnSi

is 5.031 µB as shown in Table 1, which shows a small deviation from integer

value. This proves that the systems treated with LSDA+U gives more accurate

half metallic ferromagnets than LSDA. While adding the calculated values i.e.

0.29 Ry for Co and 0.27 Ry for Mn in LSDA, we have observed that the system

does not shows a half-metallic character as the Fermi level EF does not lie at the

gap in the minority channel as shown in Figs.(4, 5). The total magnetic moment

is found to be 5.6118 µB [Table 2] which is more than the expected value 5.00 µB

[26]. The increase in the magnetic moment is due to the strong hybridization

between the Co-d and M-d states at a range of -2.0 eV to -2.600 eV as well as

Rai et al

20

Mn-d and Al-p states at around -5.20 eV in spin down region as shown in Fig. 4.

The partial magnetic moments of Co and Mn are 1.1039 and 3.7068 µB

respectively and the total magnetic moment is 5.6118 µB using LDSA+U. The

variation of magnetic moments and energy gaps are tabulated in Table 2.

Table 2. Variation of magnetic moments and Energy gaps with U values.

Coulomb

Repulsion

Co2MnSi Magnetic Moment µB Energy Gap (eV)

(U) Ry MCo MMn MSi Mtot Eg

0.05 0.9406 3.2118 -0.0598 5.0000 0.760

0.10 0.9136 3.3236 -0.0744 5.0116 0.840

0.20 0.9383 3.5372 -0.0904 5.1538 1.000

UCo=0.29

UMn=0.27

1.1039 3.7068 -0.1089 5.6118 1.200

0.40 1.7045 4.0696 -0.0699 7.3945 1.400

Fig. 6 depicted the plot of total and partial magnetic moments of Co2MnSi as

well as the change in energy gaps against the Coulomb repulsion (U). It is clearly

shown that the partial magnetic moments of Co and Mn sites are increasing with

U. In Fig. 6 the total and partial magnetic moments are represented as Mtot, MCo,

and MMn. The energy gap is denoted as Eg.


21

0.0 0.1 0.2 0.3 0.4

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

Ma

gn

etic m

om

en

t (u

B),

En

erg

y g

ap

(e

V)

U (Ry)

Mtot

MCo

MMn

Eg

Fig. 6. Evaluation of the magnetic moments of Co2MnSi and Energy gaps with

respect to the U values of Co atom.

CONCLUSION

Our result of U is UCo=0.27 Ry and UMn=0.29 Ry which is in agreement with the

reported values of U which lies between 0.147 Ry and 0.368 Ry for transition

metals Ederer et al. [27]. Our results of first-principle calculation based on

LSDA+U indicate that the half-metallicity of Co2MnSi is sensitive to U. The

Half-metallicity exist at the lower value of U. Also the increase of U value

increases the magnetic moments. For U = 0.05 Ry, the calculated total orbital

moments are 5.000 µB /atom, being in good agreement with the previously

Rai et al

22

reported result 5.00 µB/atom [26]. Our calculated results of U for Co and Mn are

0.29 Ry and 0.27 Ry respectively. Applying the calculated Coulomb repulsion

(U) on LSDA, the result thus obtained does not gives half-metallic character, the

corresponding magnetic moments is not an integer value (HM) i.e. 5.6118 µB

which is higher than the expected value. Also Figs.(4,5) shows that EF cuts

through the DOS as it does not lie at the middle of the gap at UCo=0.29 Ry and

UMn=0.27 Ry thus the system is no more half-metallic. Half-metallic

ferromagnetic materials, Co2YZ (Y: transition metal, Z: sp atom) compounds, are

much more desirable in magneto-electronic applications. We have studied the

electronic and magnetic structure of Co2MnSi a full Heusler alloy to search for

new half-metallic ferromagnetic candidates. The system studied is predicted to

be half-metallic at theoretical equilibrium lattice constants and U = 0.05 Ry. In

half-metals, the creation of a fully spin-polarized current should be possible that

should maximize the efficiency of magnetoelectronic devices. Materials with

high spin polarization can be used for tunnel magnetoresistance (TMR) and giant

magnetoresistance (GMR) [29]. Thus Co2MnSi has every chance to be used for

TMR and GMR. The Co-based Heusler alloys Co2YZ 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 [30]. Our study of Co2MnSi a type of

intermetallic compound can be an ideal spin injection devices to be used in

spintronics.

ACKNOWLEDGEMENTS

DPR acknowledges DST inspire research fellowship and RKT a research grant

from UGC (New Delhi, India).


23

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[14] Rai, D. P. and Thapa, R. K. Electronic Structure and Magnetic Properties of

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Bull. Mater. Sci., Vol. 34, No. 6, October 2011, pp. 1219–1222. c© Indian Academy of Sciences.

Study of energy bands and magnetic properties of Co2CrSi Heusler alloy

DIBYA PRAKASH RAI, SANDEEP, M P GHIMIRE and R K THAPA∗Department of Physics, Mizoram University, Aizawl 796 009, India

MS received 9 December 2010

Abstract. The electronic and magnetic properties of Co2CrSi is calculated by using full-potential linearized aug-mented plane wave (FP–LAPW) method based on density functional theory (DFT). Density of states (DOS), mag-netic moment and band structures of the system are presented. For the exchange and correlation energy, local spindensity approximation (LSDA+U) with the inclusion of Hubbard potential U is used. Our calculation shows indirectbandgap of 0·91 eV in the minority channel of DOS. This is supported by band structures and hence favoured thehalf metallic ferromagnetic (HMF) nature of the system. The effective magnetic moment of 4·006 μB also supportedour conclusion with a near integral value. The DOS of Co and Cr were found to hybridize and was also responsiblefor the ferromagnetic nature of the system.

Keywords. FP–LAPW; LDA+U; DOS; magnetic moments; band structures.

1. Introduction

The Heusler compounds, as named after their discovererare ternary intermetallics with a 2:1:1 stoichiometry andthe chemical formula, X2YZ. They usually consist of twotransition metals X and Y and a main group element Z.Half-metallic ferromagnetism was first predicted for the halfHeusler compound like NiMnSb by de Groot et al (1983).Half-metal ferromagnets (HMF) exhibit a real gap in minor-ity density of states (DOS). Due to ferromagnetic (FM)decoupling, spin up bands are metallic and spin-down bandsare semiconducting or vice-versa. In the present context ofscientific research of material science, HMFs (Ghimire et al2010) have become one of the most studied classes of mate-rials. Consequently only charge carriers of one spin directioncontributes to the conduction. The existence of a gap in theminority-spin band structure leads to 100% spin polarizationof the electron states at the Fermi level, which makes thesystems applicable for spintronic devices (Zutic et al 2004).In half-metals, the creation of a fully spin-polarized currentshould be possible that should maximize the efficiency ofmagnetoelectronics devices (de Boeck et al 2002). Materi-als with high spin polarization can be used for tunnel mag-netoresistance (TMR) and giant magnetoresistance (GMR)(Yakushi et al 2006). This is due to a high Curie tempera-ture beyond room temperature and a simple fabrication pro-cess such as d.c. magnetron sputtering in Co2YZ (Y=Mn, Cr,Fe and Z=Al, Si, Ge, Ga) (Miura et al 2004). We intend tocalculate the electronic and magnetic properties of Co2CrSi

∗Author for correspondence ([email protected])

by using the FP–LAPW method. DOS, magnetic momentsand band structures of these compounds are investigated andcompared.

2. Calculation details and crystal structuresof Heusler compounds

X2YZ Heusler compounds crystallize in the cubic L21 struc-ture with space group Fm3m as shown in figure 1. The cubicL21 structure consists of four inter-penetrating fcc sublatti-ces, two of which are equally occupied by Co. The twoCo-site fcc sublattices combine to form a simple cubicsublattice. First principles FP–LAPW method based onDFT (Kohn and Sham 1965) is used for Co2CrSi. For theexchange-correlation potential, LSDA+U method (Anisimovet al 1997) is used. In the LSDA+U calculations, the on-siteCoulomb energy, U and exchange parameter, J applied are0·52 and 0·00 Rydberg (Ry), respectively. Since the transi-tion d-state electron correlations are expected to be strong,LSDA+U method is used. The number of k-points usedfor optimization was 10000 and RMT × KMAX was takenas 7 for consistency. The muffin-tin sphere radii (RMT)used are 2·2, 2·0 and 1·89 a.u. for Co, Cr and Si, respec-tively. Self-consistency is achieved with energy convergenceof 10−3Ry. WIEN2k code (Blaha et al 2008) was employedfor computations which was also used by Rai et al (2010).

X2YZ Heusler compounds crystallize in the cubic L21 struc-ture (space group, Fm3m). The cubic L21 structure con-sists of four inter-penetrating fcc sublattices, two of whichare equally occupied by Co. The two Co-site fcc sublatticescombine to form a simple cubic sublattice.

1219

1220 Dibya Prakash Rai et al

Figure 1. Structure of Co2CrSi Heusler alloy: Co (red) atoms areat origin and (1/2, 1/2, 1/2), Cr (green) at (1/4, 1/4, 1/4) and Si (blue)atoms at (3/4, 3/4, 3/4).

3. Results and discussion

3.1 Density of states and magnetic moments

The partial DOS plots are shown in figure 2(a–d). Fromfigure 2(a–b), peaks due to Co atoms are found mostly inthe core, semi-core and the valence region below EF for bothspin channels due to its d state electrons. In spin up chan-nel, the DOS cuts EF showing the metallic nature shownin figure 2(a). In spin down channel, peaks are due to dstates of Co atoms of the conduction region (figure 2(b)).A bandgap of 0·91 eV is observed with the Fermi levellying midway between the gap (figure 2(b)) in spin downchannel showing semiconducting behaviour. The 3d elec-trons of Co atoms were decomposed into d-eg and d-t2gstates (Galanakis et al 2002). In the conduction region ofspin down, peaks were observed at 0·70 eV and 1·8 eVwhich is mainly due to d-eg and d-t2g states, respectively.We observe an exchange splitting between d-eg up and downwith a splitting energy of 1·4 eV. Also an exchange split-ting of 2·6 eV was observed for d-t2g up and down states(figure 2(a–b)). Similarly we observe that Cr atoms mainlycontribute in the valence region with two sharp peaks at

Figure 2. Partial DOS of Co and Cr atoms.

Study of energy bands and magnetic properties of Co2CrSi Heusler alloy 1221

Table 1. Experimental and theoretical values of lattice constants, magnetic moments, energy gaps and bulk modulus.

Lattice constant, a0 (Å) Magnetic moment, μB Energy gap, Eg (eV) Bulk modulus

Compound Experimental Theory Previous Observed Previous Observed k-value(GPa)

Co2CrSi 5·647b 5·699 4·000b 4·006 0·878b 0·910 10,000 405·556

bRaphael et al (2002)

Table 2. Partial and total magnetic moments.

Previous calculation Our calculation Energy gap, Eg

Magnetic moment, μB Magnetic moment, μB (eV)

Co Cr Total Co Cr Si Total Previous Our calculation

0·980 2·080 4·000b 0·980 2·102 −0·055 4·006 0·878b 0·910

bRaphael et al (2002)

Figure 3. (a) Energy band of Co2CrSi for spin up, (b) total DOS of Co2CrSi and (c) energy band of Co2CrSi for spin down.


−1·4 eV and −2·0 eV due to Cr-d state electrons in spin-upconfiguration with a Fermi cut-off (figure 2(c)). In spin-downconfiguration, a 3d state electron of Cr contributes both inthe valence and the conduction regions with an energy gapof 0·94 eV (figure 2(d)). Sharp peak at −0·8 eV is due toboth d-eg and d-t2g states of Cr-3d atoms in spin-up channel(figure 2(c)). In spin-down channel, the peaks in the valenceand conduction regions are due to both d-eg and d-t2g statesof Cr atoms (figure 2(d)). An exchange splitting of 2·4 eVis observed between spin-up and spin-down channel due tothe 3d states of Cr atoms. This splitting explains the highervalue of magnetic moment in Cr atoms (table 1). The partialmagnetic moments of the atoms Co, Cr and Si are 0·980 μB,2·102 μB and −0·055 μB, respectively. Thus the totalmagnetic moment is 4·006 μB which is approximately aninteger value of 4·00 μB (Kandpal et al 2006). The par-tial magnetic moments are tabulated and compared with theprevious results as shown in table 2.

3.2 Band structure of Co2CrSi

From figure 3 of the band structure the calculated energy gapalong -X symmetry is 0·91 eV which is almost similar to0·878 eV (Kandpal et al 2006). The Cr compounds give com-paratively high Eg compared to others as seen from DOS aswell as from the band structure. The band gap is determinedby the indirect bandgap.

4. Conclusions

We have calculated DOS, magnetic moments and band struc-tures of Co2CrSi using FP–LAPW method with LSDA+Uapproximation. The results were in support of the HMFnature of Co2CrSi. The existence of energy gap in DOS forminority spins is an indication of Co2CrSi being a poten-

tial HMF. This is also evident from the energy band resultscalculated. The calculated magnetic moment for Co2CrSi is4·006 μB which is in agreement with the integral value of4·00 μB (Kandpal et al 2006) supporting the HMF nature ofCo2CrSi.

Acknowledgements

One of the authors (SD) acknowledges a research fellow-ship (JRF) and (RKT), a research grant from DAE, BRNS,Mumbai.

References

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de Boeck, van Roy W, Das J, Motsnyi V, Liu Z, Lagae L, Boeve H,Dessein K and Borghs G 2002 Semicond. Sci. Technol. 17 342

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Galanakis I, Dederichs P H and Papanikolaou N 2002 Phys. Rev.B66 174429

Ghimire M P, Sandeep and Thapa R K 2010 Mod. Phys. Letts B242187

Kandpal H C, Fecher G H, Felser C and Schonhense G 2006 Phys.Rev. B73 094422

Kohn W and Sham L J 1965 Phys. Rev. A140 1133Miura Y, Nagao K and Shirai M 2004 Phys. Rev. B69 144413Rai D P et al 2010 Indian J. Phys. 84 593Raphael M P et al 2002 Phys. Rev. B66 104429Yakushi K, Saito K, Takanashi K, Takahashi Y K and Hondo K

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Electronic and magnetic properties of NdVSb3: A first principles study Sandeep, M. P. Ghimire, D. P. Rai, A. Shankar, A. K. Mohanty et al. Citation: AIP Conf. Proc. 1447, 1153 (2012); doi: 10.1063/1.4710417 View online: http://dx.doi.org/10.1063/1.4710417 View Table of Contents: http://proceedings.aip.org/dbt/dbt.jsp?KEY=APCPCS&Volume=1447&Issue=1 Published by the American Institute of Physics. Related ArticlesAnalysis of the Heyd-Scuseria-Ernzerhof density functional parameter space J. Chem. Phys. 136, 204117 (2012) Analytic model of energy spectrum and absorption spectra of bilayer graphene J. Appl. Phys. 111, 103714 (2012) First-principles study on the half-metallicity of full-Heusler alloy Co2VGa (111) surface J. Appl. Phys. 111, 093730 (2012) Calculation of electron-hole recombination probability using explicitly correlated Hartree-Fock method J. Chem. Phys. 136, 124105 (2012) Physical insights on comparable electron transport in (100) and (110) double-gate fin field-effect transistors Appl. Phys. Lett. 100, 123502 (2012) Additional information on AIP Conf. Proc.Journal Homepage: http://proceedings.aip.org/ Journal Information: http://proceedings.aip.org/about/about_the_proceedings Top downloads: http://proceedings.aip.org/dbt/most_downloaded.jsp?KEY=APCPCS Information for Authors: http://proceedings.aip.org/authors/information_for_authors

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Electronic And Magnetic Properties Of NdVSb3

Sandeep

: A First Principles Study

1, M. P. Ghimire2, D. P. Rai1, A. Shankar1, A. K. Mohanty3, Arthur Ernst4,D. Deka5, A. Rahman5 and R. K.Thapa1

1Department of Physics, Mizoram University, Tanhril, Aizawl-796009, Mizoram, India2Faculty of Science, Nepal Academy of Science and Technology, G.P.O.-3323, Kathmandu, Nepal

3Nuclear Physics Division, Bhabha Atomic Research Centre, Mumbai India-4000854Max-Plank-Institute fur Mikrostrukturphysik,Weinberg 2 D-06120 Halle, Germany

5

Email: [email protected] of Physics, Gauhati University, Guwahati 781 014, Assam, India

Abstract. The electronic density of states (DOS) and magnetic moments of rare-earth antimonides (NdVSb3) has been studied by the first principles full-potential linearized augmented plane wave (FP-LAPW) method based on density functional theory (DFT). For the exchange–correlation potential, the GGA+U method is used. The effective moments of NdVSb3 Bwas found to be 4.50 . The exchange-splittings of V-3d state electrons and 4f-states of Nd atoms were analyzed to explain the magnetic nature of these systems. The V atom plays a significant role on the magnetic properties due to the hybridization between V-3d and Sb-5p state orbitals. The results obtained are compared and found to be in qualitative agreement with the available results.

Keywords: Rare-earth antimonides; DFT; DOS; Magnetic momentPACS:

.71.15.Mb; 71.15.-m; 75.20.Hr

INTRODUCTION

The magnetic exchange mechanisms and spin-correlations mediated by itinerant electrons have attracted renewed interest in the last few years in the emerging field of spin electronics [1]. Due to the anisotropy in structure, and display of complex magnetic interaction RVSb3 is a material of interest which has major technological applications in spin electronics, Read Head, magnetic RAM, nano-systemsetc [2-3]. We calculate the DOS and magnetic moments of NdVSb3

COMPUATIONAL DETAIL

using first principles density functional theory using FP-LAPW method.

We have performed our calculations using the experimentally determined lattice parameters and the atomic positions [4] for NdVSb3. For rare earth elements, the 4f-electron correlations are expected to be strong. Consequently, the GGA+U calculations have been chosen to include the on-site Coulomb interaction. The onsite Coulomb energy (U) applied is

0.51 Rydberg (Ry.) for the Nd and 0.15 Rydberg (Ry.) for V respectively [5]. We have used 47 k points in the irreducible Brillouin zone, and the muffin-tin radii are 2.5 2.5, 2.27, 2.27 and 2.27 for Nd, V, Sb1, Sb2, Sb3The density plane cut-off RMT*KMAX is 7.0, where KMAX is the plane wave cut-off and RMT is the muffin-tin radii. The self-consistency is better than 0.001 e/a.u.3


for charge density and spin density and the stability is better than 0.01 mRy. for total energy per cell. For computations of DOS and magnetic moments, WIEN2k code [6] is used.

The total DOS for NdVSb3 is shown in Fig. 1 The regions are contributed by the Sb-5s and Sb-5p state electrons in both spin channels. In the valence regions DOS contributions were observed due to V-3d state electrons in spin-up channel. Sharp peak due to V-3dstate electrons were observed at 1.5 eV in the spin-up channel [Fig 1(c)]. At Fermi level contribution of Nd-4f state electrons were highest in the spin-up channel. The splitting of the DOS in the spin-up and spin-down

Solid State PhysicsAIP Conf. Proc. 1447, 1153-1154 (2012); doi: 10.1063/1.4710417© 2012 American Institute of Physics 978-0-7354-1044-2/$30.00

1153


channels were found to occur for both V-3d and Nd-4fstates which contribute to the magnetic nature of the compounds and were supported by the individual magnetic moments calculated [Table 1]. The magnetic moment calculated showed that the Nd-4f state is the main contributor towards the total magnetic moment of the system with moment equal to 3.37 B which is in qualitative agreement previous calculated moment[7] of Nd in NdVSb3. In addition we have presented the individual contributions of moments from V atom and Sb atoms. We have observed that the individual magnetic moment of V atom in NdVSb3 is higher by 28 % when compared to the experimental moment calculated by Hartjes et al. [7]. Thus the total moment of the system was found to be higher by 21% in the present study. The lower magnetic moment could be explained on the basis of hybridization between the states of Sb-5p and V-3d which leads to a less prominent splitting of the 3d states giving rise to lower magnetic moment for V atoms.

FIGURE 1. Total and partial DOS plots of NdVSb3 in spin-up and spin-down configurations.

TABLE 1. Total and partial magnetic moments of NdVSb3

Nd V Sb1 Sb2 Sb3 TotalOur Results 3.37 2.14 -0.06 -0.10 -0.03 4.50

Previous results

3.27[7] 1.53[7] 3.54[7]3.62[7]

ACKNOWLEDGMENTS

SD acknowledges a SRF and RKT a research grant from UGC. Delhi.

REFERENCES

1. D. D. Jackson, M. Torelli and Z. Fisk, Phys. Rev. B 65 0144211-0144217 (2001).

2. M. Inamdar, A. Thamizhavel and S. Ramakrishnan,J. Phys.: Condens. Matter 20 295226 (2008).

3. Sandeep, M. P.Ghimire, R. K.Thapa, J. Magn. Magn. Mater. 323 2883-2887 (2011)

4. M. Brylak and W. Jeitschko, Naturforsch. 50 899-904(1995).

5. S Hufner and G. K. Wertheim, Phys. Rev B 7 5086-5090(1973).

6. P. Blaha, K. Schwarz, G. H. K.. Madsen, D. Kvasnicka,J. Luitz WIEN2 k, An Augmented Plane wave + Local orbital program for calculating crystal properties (Karlheinz Schwarz, Techn. Universitat, Wien, Austria) 2008.

7. K. Hartjes, W. Jeitschko and M. Brylak, J. Magn. Magn. Mater. 173 109-116 (1997).

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International Journal of Computational Physical Sciences. ISSN 0976-5875 Volume 3, Number 1 (2012), pp. 21-27 © Research India Publications http://www.ripublication.com/ijcps.htm

A First Principles Study of Co2MnSi a Full-Heusler Compound: LSDA Method

D.P. Rai, A. Shankar, Sandeep and R.K. Thapa

Condensed Matter Theory Group (CMT), Department of Physics, Mizoram University, Aizawl-796004, India

E-mail: [email protected]

Abstract

We performed the structure optimization followed by the calculation of electronic structure and magnetic properties on Co2MnSi. The structure optimization was based on generalized gradient approximation (GGA) and linearized augmented plane wave (LAPW) method. The calculation of electronic structure was based on full potential linear augmented plane wave (FP-LAPW) method and exchange correlation. We studied the electronic structure and magnetic properties. Results of density of states (DOS) and band structures shows the half-metallicity of Co2MnSi with an integral value of magnetic moment 5.031 µB and the energy band gap of 0.760 eV. Keywords: GGA, LSDA, half-metallicity, DOS and band structure. PACS No.: --01.30.-y

Introduction Half-metallic ferromagnets (HMFs), where the majority spin band is metallic and the minority–spin band is semiconducting with an energy gap at the Fermi level. In present scientific research of material science, the half-metal ferromagnets have become one of the most studied classes of materials. The existence of a gap in the minority-spin band structure leads to 100% spin polarization of the electron states at the Fermi level which is makes the systems applicable for the developing field of spintronics [1]. In half-metals, the creation of a fully spin-polarized current should be possible that should maximize the efficiency of magnetoelectronics devices [2]. Materials have high spin polarization can be used for tunnel magnetoresistance(TMR) and the giant magnetoresistance (GMR) [3]. The Co-based Heusler alloys Co2YZ (Y: transition metal, Z: sp atom) are the most prospective candidates for the application in

22 D.P. Rai et al

spintronics. This is due to a high Curie temperature beyond room temperature and the simple fabrication process suc as dc magnetron sputtering in Co2YZ [4]. The electronic and magnetic properties of a series of heusler alloys have been investigated by using first principle method by Galanakis et al. [5] and [Block et al.[6]. It was found that many of the Co2YZ are half metallic and follow Slater-Pauling behoviour where the total spin magnetic moment per unit cell in µB (Mt) scales with the total number of valence electrons (Zt) following the rule: Mt=Zt-24. Miura et al. [4] found some Co-based Heusler alloys shows more than 70% spin polarizations are Co2CrAl(99.9%), Co2CrSi(100%), Co2CrGa(93.2%), Co2CrGe(99.8%), Co2MnSi(100%), Co2FeAl(86.5%) etc. The half-metallicity was first predicted by de Groot et al.[7] in 1983 when studying the band structure of a Heusler alloy. They found that the spin down channel is semiconducting. In 2006, Galanakis et al.[5] have shown that the gap arises from the interaction between the d orbitals of X and Y in X2YZ compounds creating bonding and antibonding states separated by a gap. Ishida et al.[8] have proposed Co2MnZ compound where Z stand for Si is half metal. Rai et al. [9, 10] studied the band structure and the density of state for Co2MnAl and Co2CrSi which also tells the similar facts. Crystal structure and Calculation Details Crystal structure: X2Y Z Heusler compounds crystallize in the cubic L21 structure (space group Fm3m) [11]. Co (red) atoms are at the origin and ( 1/2, 1/2, 1/2 ), Mn (green) at (1/4, 1/4, 1/4) and Si (blue) atoms at (3/4, 3/4, 3/4 ).The cubic L21 structure consists of four inter-penetrating fcc sub-lattices, 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: Unit Cell Structure of the Co2MnSi Heusler alloy.

Calculation Details: For calculation experimental lattice parameter is used to find the optimized lattice constant for which the optimized value of wave vector k was used

A First Principles Study of Co2MnSi a Full-Heusler Compound: LSDA Method 23

resulting in minimum energy. The theoretical lattice constant was found out from volume optimization. With these values of lattice constants, WIEN2k code Blaha et al. [12] was used to calculate DOS, energy bands, magnetic moments etc in these kinds of Heusler intermetallic systems. For calculation the various sets of muffin tin radii were taken to ensure almost nearly touching sphere, RMT.Kmax=7 was used for the number of plane waves, and the expansion wave functions was set up as l=14 inside the muffin tins. LSDA[13] was used and the convergence criterion for self–consistence calculations was set up to charge convergence equal to 10-4. The experimental and the theoretical values of lattice constants, magnetic moments and energy gaps are tabulated in Table 2. and the respective bulk modulus is given in Table 1. For overall band structure calculation of Co2MnSi compounds there exist an energy gap Eg in the spin down region. The width of the energy gap Eg is the difference of the energies of the highest occupied band in the valence region at the Г-point and the lowest unoccupied band in the conduction region at the X-point. The smaller value is found between Г and X, thus it is an indirect gap. With the help of the DOS, it is clear that the energy region lower than -3eV consists mainly of s and p electrons of the Si atoms which is called the core and the energy region between -3eV and 2eV consists mainly of the d-electrons of Co and Mn atoms. Results and Discussions

Table 1: The previous lattice constant and calculated bulk modulus is given below

Compound Experimental lattice constant ao Å

Calculated Bulk Modulous

RMT.Kmax k-value

Co2MnSi 5.645a 866.499 7 10000 a: Ref [14, 15, 16]

Table 2: The previous and the theoretical lattice constants, magnetic moment and energy gap are given below.

Compounds Lattice constant ao Å Magnetic Moment µB Energy gap Eg (eV) k-value

Experimental Theory PreviousOur CalculationPreviousOur Calculation Co2MnSi 5.645a 5.665 4.900a

5.000 b5.031 0.798b 0.760 10000

b: Ref [17] Co2MnSi From the Figs.(2, 3) we have found the DOS at -1.4 eV below EF in spin down and spin up regions are contributed by Mn-d atoms but Mn atoms has less contribution in the spin down region. Similarly at -1.4eV, Co-d atoms also contributed in both spin up and spin down regions. At 3.2eV and 4.2eV in spin down region both Mn-d and Co-d atoms contributed to the total DOS. But negligible contribution from Co and Mn atoms in the spin up channel above EF.

24 D.P. Rai et al

Table 3: The partial and total magnetic moments are tabulated and compared with the previous results.

Previous Calculation Magnetic Moment µB

Our Calculation Magnetic Moment µB

Energy gap Eg eV

Co Mn Total Co Mn Si Total Previous Our Calculation 1.00 3.00 5.000b

4.900a 1.029 3.058 -0.055 5.031 0.798b 0.760

a: Ref [14, 15, 16] b: Ref [17]

Figure 2: Total DOS plot of Co2MnSi.


Figure 3: Partial DOS of Co-d and Mn-d

Band Structure of Co2MnSi The origin of minority gap in Co2MnSi was explained by Galanakis et al. [5]. Based on the analysis of the band-structure calculation it was shown that the 3d orbitals of Co atoms from two different sub-lattices, Co1(0, 0, 0) and Co2(1/2, 1/2, 1/2), couple and form bonding hybrids Co1(t2g/eg) -Co2(t2g/eg). In other words, the t2g/eg orbitals of one of the Co atoms can couple only with the t2g/eg orbitals the other Co atom. Furthermore, the Co-Co hybrid bonding orbitals hybridize with the Mn(d)-t2g, eg manifold, while the Co-Co hybrid antibonding orbitals remain uncoupled owing to their symmetry. The Co-Co hybrid antibonding t2g is situated below the Fermi energy EF and the Co-Co hybrid antibonding eg is unoccupied and lies just above the Fermi level. Thus due to the missing Mn(d)-t2g, eg and Co-Co hybrid antibonding hybridization, the Fermi energy is situated within the minority gap formed by the triply degenerate Co-Co antibonding t2g and the doubly degenerate Co-Co antibonding eg. From Figs.4(b, c) it is clearly shown that the EF lies exactly at the middle of the gap between the valence and the conduction bands in spin down region where as there is no such gap occur in the spin up region. Thus there exist a half metallicity in Co2MnSi. The value of energy gap between Г and X is 0.760eV which is an indirect band gap. Almost similar results of 0.798 had been reported by Kandpal et al. (2006) [17].

26 D.P. Rai et al

Figure 4: Energy band of Co2MnSi (a) Spin up (b) Total DOS and (c) Spin down.

Conclusion It can be said that due to the existence of gap in DOS for the minority spins Co2MnSi is a potenntial half-metallic ferromagnet. This is also evident from the energy band results as discussed. The calculated magnetic moment for Co2MnSi is 5.031 µB which is almost equal to an integral value of 5.00 µB [17] and 4.90 µB [15]. The integral value of magnetic moment is also one of the evidences for the half metallicity. Co2MnSi together with a Mn interface layer is a possible candidate for spin injection. Acknowledgements DPR acknowledges DST INSPIRE research fellowship and RKT a research grant from UGC, New Delhi. References

[1] I Zutic, J Fabian and S D Sharma, Rev. Mod. Phys. 76, 323 (2004) [2] de Boeck, W van Roy, J Das, V Motsnyi, Z Liu, L Lagae, H Boeve, K Dessein

and G Borghs, Semicond. Sci. Technol. 17, 342 (2002) [3] K Yakushi, K Saito, K Takanashi, Y K Takahashi and K Hondo, Appld. Phys.

Lett., 88, 222504l (2006). [4] Y Miura, K Nagao and M Shirai, Phys. Rev. B 69, 144413 (2004). [5] I Galanakis, Mavropoulos Ph., J. Phys. Conens. Matter, 0610827:1-17 (2006).


[6] T Block, C Felser, G Jakob, J Ensling, B Muhling, P Gutlich, V Beaumont, F Studer and R J Cava, J. Solid Solid State Chem., 176, 646(2003).

[7] R A de Groot, F M Mueller, P G Van Engen and K H J Buschow, Phys. Rev. Lett. 50, 2024–2027 (1983).

[8] S Ishida, S Akazawa, Y Kubo and J Ishida, J. Phys. F: Met. Phys. 12, 1111 (1982).

[9] D P Rai, J Hashemifar, M Jamal, Lalmuanpuia, M P Ghimire, Sandeep, D T Khathing, P K Patra, B I Sharma, Rosangliana and R K Thapa, Indian J. Phys. 84 (5), 593-595 (2010).

[10] D P Rai, Sandeep, M P Ghimire and R K Thapa, Bull. Mat. Sc. (accepted). [11] F Heusler, Verh. Dtsch. Phys. Ges., 12: 219 (1903). [12] P Blaha, K Schwarz, G K H Madsen, D Kvasnicka, J Luitz, K Schwarz, 2008.

An Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal Properties: Wien2K User’s Guide, Techn. Universitat Wien, Wien., 1-108.

[13] U Von Barth and L Hedin J. Phys C:Solid State Phys. 5 1629 (1972). [14] Ido H, J. Magn. Magn. Mater. 54-57, 937 (1986). [15] M P Raphael, B Ravel, Q Huang, M A Willard, S F Cheng, B N Das, R M

Stroud, K M Bussmann, J H Claassen, and V G Harris, Phys. Rev. B 66, 104429 (2002).

[16] L Ritchien, G Xiao, Y Ji, T Y Chen, C L Chien, M Zhang, J Chen, Z Liu, G Wu and X X Zhang, Phys. Rev. B 68, 104330 (2003).

[17] H C Kandpal, G H Fecher, C Felser and G Schonhense, Phys. Rev. B 73: 094422-094433 (2006).

March 26, 2012 10:17 WSPC/Guidelines-IJMPB S0217979212500713

International Journal of Modern Physics BVol. 26, No. 8 (2012) 1250071 (12 pages)c© World Scientific Publishing Company

DOI: 10.1142/S0217979212500713

ELECTRONIC STRUCTURE AND MAGNETIC PROPERTIES

OF Co2YZ (Y = Cr, Z =Al, Ga) TYPE HEUSLER

COMPOUNDS: A FIRST PRINCIPLE STUDY

D. P. RAI∗,‡, A. SHANKAR∗, SANDEEP∗, M. P. GHIMIRE† and R. K. THAPA∗

∗Condensed Matter Theory Group, Department of Physics,

Mizoram University, Aizawl 796004, India†Nepal Academy of Science and Technology, Kathmandu, Nepal

‡[email protected]

Received 22 November 2011Revised 20 February 2012Published 26 March 2012

The structural optimization was followed by the calculation of electronic structure andmagnetic properties on Co2CrAl and Co2CrGa. The structure optimization was based ongeneralized gradient approximation (GGA). The calculation of electronic structure wasbased on full potential linear augmented plane wave (FPLAPW) method within localspin density approximation (LSDA). We studied the electronic structure and magneticproperties. Results of density of states (DOS) and band structures shows that Co2CrAland Co2CrGa are half-metallic ferromagnets (HMFS). The calculated magnetic momentsof Co2CrAl and Co2CrGa are 2.915 and 3.075 µB, respectively. We have calculated theonsite d–d coulomb and exchange interaction (U) For 3d elements like Co and Cr. Thestrongly localized d states were treated with LSDA+U method.

Keywords: GGA; FP-LAPW; LSDA; LSDA+U; half-metallicity; DOS; band structure.

1. Introduction

In 1983, de Groot et al.1 discovered half-metallic ferromagnetism in semi-Heusler

compound NiMnSb by using first-principle calculation based on density functional

theory. After that half-metallicity attracted much attention2 because of its prospec-

tive application in spintronics.3 Recently rapid development of magneto-electronics

intensified the research on ferromagnetic materials that are suitable for spin injec-

tion into a semiconductor.4 One of the promising classes of materials is the half-

metallic ferrimagnets, i.e., compounds for which only one spin channel presents a

gap at the Fermi level, while the other has a metallic character, leading to 100% car-

rier spin polarization at EF.5,6 Ishida et al.7 have also proposed that the full-Heusler

alloy compounds of the type Co2MnZ, where Z stands for Si and Ge, are half-metals.

Jiang et al.8 examined the magnetic structure of Mn2VAl by X-ray diffraction and

magnetization measurements. Zhang et al.9 studied the electronic band structure

1250071-1

http://dx.doi.org/10.1142/S0217979212500713

mailto:[email protected]


D. P. Rai et al.

and transport properties and reported the half-metallicity of Co2CrAl. Umetsu

et al.10 studied the magnetic properties of L21 type Co2CrGa alloy and reported

the saturated magnetic moment at 4.2 K is 3.01 µB consistent with the generalized

Slater–Pauling rule Zt = Zt−24. Miura et al.11 have proposed that stable Co-based

full-Heusler alloys with excellent prospects for half-metallic ferromagnets (HMFs).

Rai et al.12,13 investigated the ground state study of Co2MnAl and Co2CrSi using

LDA+U and local spin density approximation (LSDA) method respectively. The

electronic and magnetic properties of Co2(Cr1−2xFex)Al with the ordered L21 struc-

ture have been investigated using the first-principles calculation.14 In this paper, we

have studied the various ground state properties of Co2CrAl and Co2CrGa using

LSDA and LSDA+U.

2. Crystal Structure and Computational Methods

Heusler alloy with chemical formula Co2CrZ (Z = Al, Ga). The full Heusler struc-

ture consists of four penetrating fcc sublattices with atoms at X1(1/4, 1/4, 1/4),

X2(3/4, 3/4, 3/4), Y(1/2, 1/2, 1/2) and Z(0, 0, 0) positions which results in L21

crystal structure having space group Fm-3-m as shown in Fig. 1.

We performed the structural optimization as shown in Fig. 2, followed by the

calculation of electronic structure and magnetic properties. In Figs. 3–6, we have

projected density of states (DOS) of Co2YZ. The total energy is calculated by us-

ing parameterization of generalized gradient approximation (GGA) and linearized

augmented plane wave (LAPW) method. The calculation is accomplished by using

the WIEN2K code.15 In the next step, we calculated the electronic structure and

magnetic properties using full potential linear augmented plane wave (FPLAPW)

method and GGA exchange correlation. The accuracy is up to 10−4 Ry. The

exchange–correlation potential is chosen in the LSDA.16 The self consistent po-

tentials were calculated on a 20 × 20 × 20 k-mesh in the Brillouin zone, which

corresponds to 256 k points in the irreducible Brillouin zone. The sets of valence

orbitals in the calculations were selected as 3s, 3p, 4s, 3d Co atoms, 3s, 3p, 4s,

Fig. 1. Unit cell Structure of Co2YZ Heusler alloy: Co (yellow), Y (blue) and Z (red/green)atoms.

1250071-2


Electronic Structure and Magnetic Properties

4.5 5.0 5.5 6.0 6.5 7.0 7.5 8.0

-8161.6

-8161.4

-8161.2

-8161.0

-8160.8

-8160.6

-8160.4

En

erg

y [

Ry]

Lattice Constant Ao

(a) Co2CrAl

5.6 5.8 6.0 6.2 6.4 6.6 6.8 7.0

-11564.0

-11563.9

-11563.8

-11563.7

-11563.6

-11563.5

-11563.4

En

erg

y [

Ry]

Lattice Constant Ao

(b) Co2CrGa

Fig. 2. Plot of energy versus lattice constant: (a) Co2CrAl and (b) Co2CrGa.

3d for Cr atoms, 3s, 3p for Al and 3d, 4s, 4p for Ga atoms. All lower states were

treated as core states.


3.1. Local spin density approximation (LSDA)

In Table 1 we have tabulated the equilibrium lattice constants and the equilibrium

energy. It is shown that Co2CrAl has a higher energy compared to Co2CrGa in the

calculated equilibrium lattice constant. Galanakis et al. [17] have shown that in the

case of half-metallic full-Heusler alloys the total spin moment follows the relation

Zt− 24, where Zt is the total number of valence electrons.

The total and partial DOS plots of Co2CrAl and Co2CrGa are shown in Figs. 3–

6. From the DOS plots of Co2CrAl and Co2CrGa, the peaks are mostly due to d

state electrons of Co atoms in the core, semi-core and the valence region below EF

for both spin channels (Figs. 4–6).

In spin up channel, the DOS intersects EF showing the metallic nature. For

Co2CrAl we have observed from Fig. 4 that Co-d electrons mainly contribute in the

valence region with sharp peaks at −1.4 and −2.4 eV while, in case of Co2CrGa

Cr-d electrons contribute with sharp peak at −1.8 eV in spin-up configuration with

a Fermi cut-off. In spin down channel, peaks are due to d states of Co and Cr atoms

in the conduction region. Energy band gap of 0.750 eV and 0.380 eV is found out

with EF lying midway between the gap in spin down channel for both Co2CrAl and

Table 1. Lattice parameters and bulk modulus.

Lattice constant ao (A)

Compounds Previous Our calculation Bulk modulus Electronic charge

Co2CrAl 5.80519

5.7420 5.756 840.4398 91

Co2CrGa 5.80519 5.8794 1191.2628 109

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D. P. Rai et al.

Fig. 3. (a) Total DOS plot of Co2CrAl, (b) partial DOS of Co atoms, (c) partial DOS of Cratoms and (d) partial DOS of Al atoms.

Fig. 4. DOS plots of Co2CrAl: (a) Co (total, d, d, deg) states in spin-up, (b) Co (total, d, d, deg)states in spin-down, (c) Cr (total, d, d, deg) states in spin-up and (d) Cr (total, d, d, deg) statesin spin down.

Co2CrGa respectively, showing the semi-conducting behavior. In the conduction

region of Co2CrAl peaks were observed at 1.0 and 1.6 eV and for Co2CrGa, peaks

are observed at 1.6 and 2.0 eV. These peaks are mainly due to d − eg and d− t2g

states of Co and Cr atoms respectively in spin down region as shown in Figs. 3–5.

The d electrons of Co are found to strongly hybridize with Cr-d electrons.18

1250071-4



Fig. 5. (a) total DOS of Co2CrGa, (b) partial DOS of Co atoms and (c) partial DOS of Cratoms.

Fig. 6. DOS plots of Co2CrGa: (a) Co (total, d, d, deg) states in spin-up, (b) Co (total, d, d, deg)states in spin-down, (c) Cr (total, d, d, deg) states in spin-up and (d) Cr (total, d, d, deg) statesin spin down.

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D. P. Rai et al.

Table 2. Partial magnetic moments and energy gap of Co2CrAl.

Magnetic moment (µB)

Previous Our calculation Energy gap Eg (eV)

Co Cr Al Total Co Cr Al Total Previous Our calculation

0.83 1.47 — 3.0019 0.746 1.558 −0.0568 2.915 0.74819 0.750.65 1.745 −0.045 3.0019

Table 3. Partial magnetic moments and energy gap of Co2CrGa.

Magnetic moment (µB)

Previous Our calculation Energy gap Eg (eV)

Co Cr Al Total Co Cr Al Total Previous Our calculation

0.823 1.437 −0.058 3.1019 0.736 1.693 −0.0485 3.075 0.42520 0.380.901 1.283 −0.074 3.01110

The partial magnetic moments of Co, Cr and Al atoms are 0.746, 1.558 and

−0.0568 µB, respectively for Co2CrAl. The effective magnetic moment is 2.915 µB

which is approximately an integer value which is equal to 3.00 µB.19 Similarly for

Co2CrGa, the partial magnetic moments of Co, Cr and Ga atoms are 0.736, 1.693

and −0.0485 µB, respectively. The effective magnetic moment is thus 3.075 µB

which is approximately an integer value 3.010 µB.20 The effective magnetic mo-

ments, energy gaps, optimized lattice constants and bulk modulus are tabulated

and compared with the previous results in Tables 1–3. Figures 7 and 8 shows the

band structure plots of Co2CrAl and Co2CrGa in both spin channels. In the valence

region of the spin up and down channels, more number of bands were seen which

are due to the 3d states of Co and Cr atoms of Co2CrAl and Co2CrGa. Spin-down

channel comprises of thick energy bands in the conduction region above EF due to

3d electrons of Cr atoms.

The width of energy gap (Eg) is the difference in energies of the highest occupied

band at symmetry point Γ in the valence region and the lowest unoccupied band in

the conduction region at symmetry point X which is an indirect band gap. With the

help of DOS, it is clear that the energy region lower than −3 eV consists mainly of

s and p electrons (not shown) of the Al atoms in the valence region and the energy

region between −3 and 2 eV consists mainly of the d-electrons of Co and Cr atoms.

From Figs. 7 and 8 it is seen that EF lies almost at the middle of the gap between

the valence and the conduction region in spin down channel. The origin of minority

gap in Co2YZ explained by Kandpal et al.19 and Galanakis and Mavropoulos.18

Based on the analysis of band structures and DOS calculations it is seen that the

3d orbitals of Co atoms from two different sub-lattices, Co1 (0, 0, 0) and Co2 (1/2,

1/2, 1/2) couple and form bonding hybrids. In other words, the gap originates from

the strong hybridization between the d states of the higher valent and the lower

1250071-6



Fig. 7. (a) Energy band of Co2CrAl for spin up case and (b) energy band of Co2CrAl for spindown.

Fig. 8. (a) Energy band of Co2CrGa for spin up and (b) energy band of Co2CrGa for spin down.

valent transition metal atoms. As a result the interaction of Y(Cr) atoms with the

Z–p states splits the Cr-3d states into a low lying triplet of t2g states and a higher

lying doublet of eg states. The splitting is partly due to the different electrostatic

repulsion, which is strongest for the eg states which is directly pointed at the Z(Al,

1250071-7


D. P. Rai et al.

Ga) atoms. In the majority band the Cr-3d states are shifted to lower energies and

form a common 3d band with X(Co) 3d states, while in the minority band the Cr-3d

states are shifted to higher energies and are unoccupied, so that a band gap at EF

is formed, separating the occupied d bonding states from the unoccupied d anti

bonding states. Thus X2YZ is a half-metal with gap at EF in minority band and a

metallic DOS at the Fermi level in majority band. This explains half metallicity in

Co2CrAl and Co2CrGa. Less number of bands were found in the conduction regions

of the spin-up regions indicating the absence of DOS contributions.

For Co2CrAl an indirect energy gap obtained between symmetry point Γ and

X is 0.75 eV which almost equals 0.748 eV reported by Kandpal et al.19 Similarly

for Co2CrGa, an indirect energy gap obtained between Γ and X is 0.38 eV which

agrees with 0.425 eV reported by Kandpal et al.19

3.2. LSDA+U

The Cr-d states are strongly localized at EF in spin up region as shown in Figs. 4

and 6 so that these states have to be treated in a more suitable way in terms of

LDA+U approach as described previously for the study of Co2FeSi.19 In our case

Fig. 9. Partial DOS (Co-d, Cr-d, Al-p).

1250071-8



Fig. 10. Partial DOS (Co-d, Cr-d, Ga-p).

we have applied the approach of Dudarev et al.21 which needs only the difference

Ω = U − J of the on site Coulomb and exchange parameters. We have calculated

the onsite d–d Coulomb and exchange interaction (U) for 3d elements like Co and

Cr by using the technique proposed by Novak et al.22 In our LSDA+U calculation

the screening effect of electrons (J = 0) was neglected. The calculated values of

Coulomb repulsion (U) for Co and Cr are 0.293 and 0.257 Ry, respectively. The

d states on the Co sites are more localized and one can expect a larger on-site

Coulomb interaction than on the Cr site, which in agreement with UCr < UCo. and

the U values for transition metal lies between 0.147–0.368 Ry.23 Figures 9 and 10

summarizes the results of the DOS which were calculated using LSDA+U . With

the inclusion of U the EF has shifted towards the lower energy side as shown in

Figs. 11 and 12. More number of peaks are seen in conduction region due to Cr-d

states. There occurs an exchange splitting in Cr-d sites between d–eg and d–t2g,

which lies at −1.3 and 3.8 eV respectively for Co2CrAl (Fig. 9). As a result of

this exchange splitting Co2CrAl shows larger gap in minority channel (Fig. 11).

1250071-9


D. P. Rai et al.

Fig. 11. DOS and band structure of Co2CrAl (LSDA+U).

Fig. 12. DOS and band structure of Co2CrGa (LSDA+U).

The same explanation may follow for Co2CrGa as well (Figs. 10 and 12). The total

magnetic moment of Co2CrGa is higher because there lies a strong hybridization

between Co-d and Ga-p states at −4 eV to −6 eV in spin down region and also a

small d−d hybridization between Co and Cr at −1.8 and 2.2 eV as shown in Fig. 10.

Where as in case of Co2CrAl the hybridization of Co-d and Al-p states lies below

−4.1 eV in the spin down region and d–d hybridization between Co and Cr lies at

4.5–7 eV in majority channel as given in Fig. 9. However, U = 0.293 Ry for Co and

U = 0.257 Ry for Cr, the Fermi energy cuts through the minority spin states and

1250071-10



Table 4. Partial magnetic moments and energy gaps from LSDA+U .

CalculatedCalculated magnetic moment (µB) energy gap Eg (eV)

Co2CrAl Co2CrGa Co2CrAl Co2CrGa

Co Cr Al Total Co Cr Al Total

1.576 2.959 −0.962 6.0299 1.674 3.020 −0.0952 6.375 2.000 1.700

the half metal character is destroyed in both the systems. The main effect of the

localization enforced by LSDA+U treatment are an increased gap width and the

occupation of more majority spin states than for the simple LSDA approach. The

calculated magnetic moments and energy gaps are given in Table 4.

4. Conclusion

We have studied the possibility of appearance of half-metallicity in the case of

the full Heusler compounds Co2CrZ, where Z is an sp atom belonging to the IVB

column of the periodic table. These compounds show ferromagnetism with the Cr

and Z spin moments being antiparallel to the Co ones. We found that compounds

Co2CrZ (Z = Al, Ga) the ferromagnetic is stable at the equilibrium lattice constant.

Thus these compounds follow the Slater–Pauling behavior and the “rule of 24”.17

The lighter the transition elements and the smaller the number of valence electrons,

the wider is the gaps and the more stable is the half-metallicity. We have calcu-

lated the DOS, magnetic moments and band structures of Co2CrAl and Co2CrGa

using FP-LAPW method using LSDA approximation. The existence of energy gap

in the DOS for spin down of both systems is an indication of being a potential

HMF. This is also evident from the energy band results calculated. The calculated

magnetic moment for Co2CrAl is 2.915 µB and for Co2CrGa, it is 3.075 µB. The

calculated results are in qualitative agreement with the integral value, supporting

the HMF nature of Co2CrAl and Co2CrGa. The Cr-d states are strongly localized

at EF in so that these states have to be treated in a more suitable way in terms of

LDA+U approach. The calculated values of Coulomb repulsion (U) for Co and Cr

are 0.293 and 0.257 Ry, respectively. These calculated values of U were added to

conventional LSDA to study the electronic and magnetic properties. The hybridiza-

tion takes place between d–d states and d–p states of Co–Cr and Co–Z (Z = Al,

Ga) respectively which is responsible for creating the high magnetic moments. The

calculated magnetic moments of Co2CrAl and Co2CrGa using LSDA+U are 6.0299

and 6.375 µB, respectively. The LSDA+U gives the larger energy gap compared to

LSDA but the EF is not located exactly at the gap in spin down channel which will

destroy the HMF. The calculated energy gap for Co2CrAl is 2.00 eV and that of

Co2CrGa is 1.7 eV.

1250071-11


D. P. Rai et al.

Acknowledgment

DPR acknowledges DST inspire research fellowship (India), AS a fellowship and

RKT a research grant from UGC (New Delhi).

References

1. R. A. de Groot et al., Phys. Rev. Lett. 50, 2024 (1983).2. M. I. Katsnelson et al., Rev. Mod. Phys. 80, 315 (2008).3. I. Zutic, J. Fabian and S. D. Sarma, Rev. Mod. Phys. 76, 323 (2004).4. H. Ohno, Science 281, 951 (1998).5. K. Inomata, S. Okmura and N. Tezuka, J. Magn. Magn. Mat. 282, 269 (2004).6. de Boeck et al., Semicond. Sci. Technol. 17, 342 (2002).7. S. Ishida et al., J. Phys. F: Met. Phys. 12, 1111 (1982).8. C. Jiang, M. Venkatesan and J. M. D. Coey, Solid State Commun. 118, 513 (2001).9. M. Zhang et al., J. Magn. Magn. Mater. 277, 130 (2004).

10. R. Y. Umetsu et al., Appl. Phys. Lett. 85, 2011 (2004).11. Y. Miura, K. Nagao and M. Shirai, Phys. Rev. B 69, 144413 (2004).12. D. P. Rai et al., Indian J. Phys. 84(5), 593 (2010).13. D. P. Rai et al., Bull. Mat. Sc. 34, 1219 (2011).14. T. Block et al., J. Solid Solid State Chem. 176, 646 (2003).15. P. Blaha et al., An augmented plane wave plus local orbitals program for calculating

crystal properties: Wien2K User’s Guide, Techn. Universitat Wien (Wien, 2008).16. U. Von Barth and L. Hedin, J. Phys. C: Solid State Phys. 5, 1629 (1972).17. I. Galanakis, P. H. Dederichs and N. Papanikolaou, Phys. Rev. B 66, 174429 (2002).18. I. Galanakis and Ph. Mavropoulos, J. Phys. Condens. Matter 19, 315213 (2007).19. H. C. Kandpal, G. H. Fecher and C. Felser, J. Phys. D: Appl. Phys. 40, 1507 (2006).20. E. Sasioglu, L. M. Sandratskii and P. Bruno, Phys. Rev. B 72, 184415 (2005).21. S. L. Dudarev et al., Phys. Rev. B 57, 1505 (1998).22. K. H. G. Madsen and Novak, Europhys. Lett. 69, 777 (2005).23. C. Ederer and M. Komelj, Phys. Rev. B 76, 064409 (2007).

1250071-12

Indian J. Phys. 84 (6), 717-721 (2010)

© 2010 IACS

Study of Co2MnAl Heusler alloy as half metallicferromagnet

Dibya Prakash Rai1, Javad Hashemifar2, Morteeza Jamal2, Lalmuanpuia1,M P Ghimire1, Sandeep1, D T Khathing3, P K Patra4, B Indrajit Sharma5,

Rosangliana6 and R K Thapa1*

1Department of Physics, Mizoram University, Tanhril, Aizawl-796 009, India2Department of Physics, Isfahan University of Technology, Isfahan 84156 Iran

3Department of Physics, Jharkhand University, Ranchi-835 205, Jharkhand, India4Science Centre, North-Eastern Hill University, Shillong-793 022, India

5Department of Physics, Assam University, Silchar-788 011, Assam, India 6Department of Physics, Govt. Zirtiri Residential Science College,

Aizawl-796 001 Mizoram, India

E-mail : [email protected]

Abstract : We present the study of half metallacity of Co2MnAl as half-Heusler alloy in L21 structure whichconsists of four inter-penetrating fcc sub-lattices. Density functional theory based electronic structure calculationswill be performed by using the full-potential linear augmented plane wave (FP-LAPW). We will use the generalgradient approximation method (GGA) and local density approximation method called LDA+U. Density of statesand band structure results will be presented in this paper.

Keywords: Half metal, Heusler alloy, Density functional theory, GGA, LDA+U.

PACS Nos.: 71.15.-m, 71.15.Ap, 71.15.Dx, 71.15.Mb, 71.15.Ne, 71.15. Rf, 71.20.Be, 71.20.Ip

1. Introduction

Half-metallic ferromagnets (HMF) were predicted to exhibit 100% spin polarization at theFermi energy (EF), and have been now intensively investigated in the field of spintronics.Several kinds of Co2-based full Heusler alloys in L21 and B2 phase structures, have beenreported from the theoretical investigations to be HMF or exhibit high spin polarization. Inthe applicable viewpoints, it is required that the ferromagnetic materials used as an electrodeof magnetic tunnel junctions (MTJ) for tunneling magnetoresistance (TMR) have high

*Corresponding Author


Curie temperature TC as well as the high spin polarization. Co2-based Heusler alloyshave high TC and are more favorable, display large TMR ratios even at room temperature.

We present a preliminary study of the half metallacity of Co2MnAl half-Heusler alloy.For this purpose, density of states (DOS), energy bands for spin up and spin downcases, will be studied along with the calculations of magnetic moments. The theoreticalvalue of lattice constant is determined by using the volume optimization method. Densityfunctional theory based electronic structure calculations will be performed by using thefull-potential linear augmented plane wave (FP-LAPW). We will use the general gradientapproximation method (GGA) and local density approximation method called LDA+U, wherethe total Coulomb and orbital potentials will be taken into consideration.

2. Crystal structure and calculation details

Co2MnAl is a type of Heusler compound which crystallizes in the cubic L21 structure(space group Fm m2). In general Co (red) and Mn (green) atoms are transition metalsand Al (blue) is a main group element. The Co atoms are placed on 8c (1/4, 1/4,1/4)and Mn and Al atoms on 4a (0,0,0) and 4b (1/2, 1/2, ½) Wycloff positions, respectively.The crystal structure of Co2MnAl Heusler compounds is illustrated in Figure 1.

Figure 1. Structure of the Co2MnAl Heusler compounds.

We have used the experimental [1] lattice constant ao = 5.749 Å for the initialcalculations for which the optimized value of wave vector k = 3000 was used. Theoptimum value of k was calculated by plotting the converged values of energies againstk. The theoretical value of lattice constant ao = 5.7261 Å was found out from volumeoptimization. With this value of lattice constant, WIEN2k code [2] was used to calculateDOS, energy bands, magnetic moments etc in this Heusler intermetallic system. Theenergy threshold between the core and the valence states was set to 108.80 eV. Varioussets of muffin-tin radii were taken to ensure almost nearly touching sphere. RMT . Kmax =7 was used for the number of plane waves, and the expansion wave functions was set up

Study of Co2MnAl Heusler alloy as half metallic ferromagnet 719

as l = 14 inside the muffin tins. The self-consistent calculations employed a grid of 104k points in the irreducible Brillouine zone. LDA+U method [2] was used in which the J= 0, and hence Ueff = U = 6.8 eV. The convergence criterion for self-consistencecalculations was set up to charge convergence equal to 10-5.

3. Results and discussions

In the half-metallic ferromagnetic compound discussed here, that is Co2MnAl, we havefound that the energy gap stays in the minority spin channel (spin down), whereas EFcuts through bands in the majority spin channel, that is spin up case (Figure 2). Thismeans that majority-spin states have metallic character while the minority-spin bandcontains an energy gap at the Fermi level (EF ), which is a semiconducting behaviour. Forthe majority spin channel, the position of EF is in the region of the d derived bands.These states are shifted to lower energies with respect to the corresponding minority spinstates by the exchange splitting. Thus we find that for spin up case, Co2MnAl behaveslike a metal with DOS concentrated at EF whereas for spin down case, energy gapexists around EF. The origin of such gaps has been attributed to be arisen from thehybridization of Co and Mn d orbitals. The overall gap is determined by the Co-Cointeraction while the effective gap in Mn partial density of states (Figure 3), is muchlarger as a result of Co-Co-Mn hybridization. This also indicates that the contribution ofMn-atom is more effective than Co or Co2MnAl atom as a whole to create the band gapin spin down. Further there is not much difference between the behaviour in atomic DOSfor Co (Figure 4) and total DOS for Co2MnAl (Figure 2). The reason for this may bethat only the Co-Co interaction contributes to the DOS at the EF. Similar reports havebeen also given by Telling et al. [3].

We have also plotted the energy bands for Co2MnAl for both the spin up and spindown cases. The value of Fermi level calculated is EF = 8.501224 eV. It is found thatfor the spin up the Co2MnAl alloy behaves simply like a metal in which the majority

Figure 3. Plot of d-states DOS for spin up andspin down cases in Mn.

Figure 2. Total DOS for spin up and down case inCo2MnAl.

DO

S (S

tate

s/eV

)

DO

S (S

tate

s/eV

)

spin up

spin down

spin up

spin down

Mn d

–15 –10 –5 0 5 10Energy (eV)

–15 –10 –5 0 5 10Energy (eV)

8

6

4

2

0

–2

–4

–6

–8

total DOS 4

3

2

1

0

–1

–2

–3

–4


bands (spin up) crosses or touch the Fermi-level (EF ) in rather all the directions of thehigh symmetry (Figure 5). On the other hand, the minority bands (spin down) exhibit aclear band gap (Figure 5), the width of which is given by the energies of the highestoccupied band at and the lowest unoccupied band at the X. The value of the energy

gap between and X along - direction is ~ 0.7 eV. This is an indirect band gap.Similar results had been reported by Kandpal et al. [4]. The - direction plays animportant role in understanding the half-metallic ferromagnets which has been pointed outby others [5-6].

Figure 5. Energy bands plot for spin up and spin down cases in Co2MnAl.

Figure 4. Plot of total DOS for spin up and spin down cases in Co-atom.

DO

S (S

tate

s/eV

)

spin up

spin down

Co tot

–15 –10 –5 0 5 10Energy (eV)

4

2

0

–2

–4

–6


4. Conclusions

In conclusion, it can be said that due to existence of gap in DOS for the minority spins,Co2MnAl is a potential half-metallic ferromagnet. This is also evident from the energyband results as discussed. The calculated magnetic moment for Co2MnAl is equal to4.02986 in the unit cell, which is same as experimental value ~ 4.28 [4].

Acknowledgment

RKT is grateful to DAE(BRNS), Government of India, Mumbai, for the sanction of a researchgrant. (No. 2008/37/39/BRNS/2482, Dt. 27/01/09). RKT also thanks Prof. Hadi Akbarzadehfor hospitality at ICTP Centre, Physics Department, Isfahan University of Technology, Isfahan,Iran, during his visit in January 2009.

References

[1] J M D Coey and M Venkatesan J. Appl. Phys. 91 8345 (2002)

[2] Peter Blaha, Karlheinz Schwartz, George K H Madsen, Dieter Kvasnicka and Joacim Luitz AnAugmented Plane Wave+Local Orbitals Program for Calculating Crystal Properties, revised ed.WIEN2k_08.3 (Release 18/9/08)

[3] N D Telling, P S Keatly, G van der Laan, R J Hicken, E Arenholz, Y Sakurba, M Oogane, Y Ando,K Takanashi, A Sakuma and T Miyazaki Phys. Rev. B78 184438 (2008)

[4] Hem Chandra Kandpal PhD Thesis (Gutenberg-University of Johannes, Mainz, Germany) (2007)

[5] S Ogut and K M Rabe Phys. Rev. B51 51 (1995)

[6] I Galanakis, P H Dederichs and N Papanikolaou Phys. Rev. 66 174429 (2002)

Indian J. Phys. 84 (6), 717-721 (2010)

© 2010 IACS

Study of Co2MnAl Heusler alloy as half metallicferromagnet

Dibya Prakash Rai1, Javad Hashemifar2, Morteeza Jamal2, Lalmuanpuia1,M P Ghimire1, Sandeep1, D T Khathing3, P K Patra4, B Indrajit Sharma5,

Rosangliana6 and R K Thapa1*

1Department of Physics, Mizoram University, Tanhril, Aizawl-796 009, India2Department of Physics, Isfahan University of Technology, Isfahan 84156 Iran

3Department of Physics, Jharkhand University, Ranchi-835 205, Jharkhand, India4Science Centre, North-Eastern Hill University, Shillong-793 022, India

5Department of Physics, Assam University, Silchar-788 011, Assam, India 6Department of Physics, Govt. Zirtiri Residential Science College,

Aizawl-796 001 Mizoram, India

E-mail : [email protected]

Abstract : We present the study of half metallacity of Co2MnAl as half-Heusler alloy in L21 structure whichconsists of four inter-penetrating fcc sub-lattices. Density functional theory based electronic structure calculationswill be performed by using the full-potential linear augmented plane wave (FP-LAPW). We will use the generalgradient approximation method (GGA) and local density approximation method called LDA+U. Density of statesand band structure results will be presented in this paper.

Keywords: Half metal, Heusler alloy, Density functional theory, GGA, LDA+U.

PACS Nos.: 71.15.-m, 71.15.Ap, 71.15.Dx, 71.15.Mb, 71.15.Ne, 71.15. Rf, 71.20.Be, 71.20.Ip

1. Introduction

Half-metallic ferromagnets (HMF) were predicted to exhibit 100% spin polarization at theFermi energy (EF), and have been now intensively investigated in the field of spintronics.Several kinds of Co2-based full Heusler alloys in L21 and B2 phase structures, have beenreported from the theoretical investigations to be HMF or exhibit high spin polarization. Inthe applicable viewpoints, it is required that the ferromagnetic materials used as an electrodeof magnetic tunnel junctions (MTJ) for tunneling magnetoresistance (TMR) have high

*Corresponding Author


Curie temperature TC as well as the high spin polarization. Co2-based Heusler alloyshave high TC and are more favorable, display large TMR ratios even at room temperature.

We present a preliminary study of the half metallacity of Co2MnAl half-Heusler alloy.For this purpose, density of states (DOS), energy bands for spin up and spin downcases, will be studied along with the calculations of magnetic moments. The theoreticalvalue of lattice constant is determined by using the volume optimization method. Densityfunctional theory based electronic structure calculations will be performed by using thefull-potential linear augmented plane wave (FP-LAPW). We will use the general gradientapproximation method (GGA) and local density approximation method called LDA+U, wherethe total Coulomb and orbital potentials will be taken into consideration.

2. Crystal structure and calculation details

Co2MnAl is a type of Heusler compound which crystallizes in the cubic L21 structure(space group Fm m2). In general Co (red) and Mn (green) atoms are transition metalsand Al (blue) is a main group element. The Co atoms are placed on 8c (1/4, 1/4,1/4)and Mn and Al atoms on 4a (0,0,0) and 4b (1/2, 1/2, ½) Wycloff positions, respectively.The crystal structure of Co2MnAl Heusler compounds is illustrated in Figure 1.

Figure 1. Structure of the Co2MnAl Heusler compounds.

We have used the experimental [1] lattice constant ao = 5.749 Å for the initialcalculations for which the optimized value of wave vector k = 3000 was used. Theoptimum value of k was calculated by plotting the converged values of energies againstk. The theoretical value of lattice constant ao = 5.7261 Å was found out from volumeoptimization. With this value of lattice constant, WIEN2k code [2] was used to calculateDOS, energy bands, magnetic moments etc in this Heusler intermetallic system. Theenergy threshold between the core and the valence states was set to 108.80 eV. Varioussets of muffin-tin radii were taken to ensure almost nearly touching sphere. RMT . Kmax =7 was used for the number of plane waves, and the expansion wave functions was set up


as l = 14 inside the muffin tins. The self-consistent calculations employed a grid of 104k points in the irreducible Brillouine zone. LDA+U method [2] was used in which the J= 0, and hence Ueff = U = 6.8 eV. The convergence criterion for self-consistencecalculations was set up to charge convergence equal to 10-5.


In the half-metallic ferromagnetic compound discussed here, that is Co2MnAl, we havefound that the energy gap stays in the minority spin channel (spin down), whereas EFcuts through bands in the majority spin channel, that is spin up case (Figure 2). Thismeans that majority-spin states have metallic character while the minority-spin bandcontains an energy gap at the Fermi level (EF ), which is a semiconducting behaviour. Forthe majority spin channel, the position of EF is in the region of the d derived bands.These states are shifted to lower energies with respect to the corresponding minority spinstates by the exchange splitting. Thus we find that for spin up case, Co2MnAl behaveslike a metal with DOS concentrated at EF whereas for spin down case, energy gapexists around EF. The origin of such gaps has been attributed to be arisen from thehybridization of Co and Mn d orbitals. The overall gap is determined by the Co-Cointeraction while the effective gap in Mn partial density of states (Figure 3), is muchlarger as a result of Co-Co-Mn hybridization. This also indicates that the contribution ofMn-atom is more effective than Co or Co2MnAl atom as a whole to create the band gapin spin down. Further there is not much difference between the behaviour in atomic DOSfor Co (Figure 4) and total DOS for Co2MnAl (Figure 2). The reason for this may bethat only the Co-Co interaction contributes to the DOS at the EF. Similar reports havebeen also given by Telling et al. [3].

We have also plotted the energy bands for Co2MnAl for both the spin up and spindown cases. The value of Fermi level calculated is EF = 8.501224 eV. It is found thatfor the spin up the Co2MnAl alloy behaves simply like a metal in which the majority

Figure 3. Plot of d-states DOS for spin up andspin down cases in Mn.

Figure 2. Total DOS for spin up and down case inCo2MnAl.

DO

S (S

tate

s/eV

)

DO

S (S

tate

s/eV

)

spin up

spin down

spin up

spin down

Mn d

–15 –10 –5 0 5 10Energy (eV)

–15 –10 –5 0 5 10Energy (eV)

8

6

4

2

0

–2

–4

–6

–8

total DOS 4

3

2

1

0

–1

–2

–3

–4


bands (spin up) crosses or touch the Fermi-level (EF ) in rather all the directions of thehigh symmetry (Figure 5). On the other hand, the minority bands (spin down) exhibit aclear band gap (Figure 5), the width of which is given by the energies of the highestoccupied band at and the lowest unoccupied band at the X. The value of the energy

gap between and X along - direction is ~ 0.7 eV. This is an indirect band gap.Similar results had been reported by Kandpal et al. [4]. The - direction plays animportant role in understanding the half-metallic ferromagnets which has been pointed outby others [5-6].

Figure 5. Energy bands plot for spin up and spin down cases in Co2MnAl.

Figure 4. Plot of total DOS for spin up and spin down cases in Co-atom.

DO

S (S

tate

s/eV

)

spin up

spin down

Co tot

–15 –10 –5 0 5 10Energy (eV)

4

2

0

–2

–4

–6


4. Conclusions

In conclusion, it can be said that due to existence of gap in DOS for the minority spins,Co2MnAl is a potential half-metallic ferromagnet. This is also evident from the energyband results as discussed. The calculated magnetic moment for Co2MnAl is equal to4.02986 in the unit cell, which is same as experimental value ~ 4.28 [4].

Acknowledgment

RKT is grateful to DAE(BRNS), Government of India, Mumbai, for the sanction of a researchgrant. (No. 2008/37/39/BRNS/2482, Dt. 27/01/09). RKT also thanks Prof. Hadi Akbarzadehfor hospitality at ICTP Centre, Physics Department, Isfahan University of Technology, Isfahan,Iran, during his visit in January 2009.

References

[1] J M D Coey and M Venkatesan J. Appl. Phys. 91 8345 (2002)

[2] Peter Blaha, Karlheinz Schwartz, George K H Madsen, Dieter Kvasnicka and Joacim Luitz AnAugmented Plane Wave+Local Orbitals Program for Calculating Crystal Properties, revised ed.WIEN2k_08.3 (Release 18/9/08)

[3] N D Telling, P S Keatly, G van der Laan, R J Hicken, E Arenholz, Y Sakurba, M Oogane, Y Ando,K Takanashi, A Sakuma and T Miyazaki Phys. Rev. B78 184438 (2008)

[4] Hem Chandra Kandpal PhD Thesis (Gutenberg-University of Johannes, Mainz, Germany) (2007)

[5] S Ogut and K M Rabe Phys. Rev. B51 51 (1995)

[6] I Galanakis, P H Dederichs and N Papanikolaou Phys. Rev. 66 174429 (2002)

International Journal of Physics and Mathematical Sciences ISSN: 2277-2111 (Online)

An Online International Journal Available at http://www.cibtech.org/jpms.htm

2012 Vol. 2 (1) January-March, pp.46-49/ Rai et al.

Research Article

______________________________________________________________________________

STUDY OF HALF-METALLIC PROPERTIES IN CO2CrSb USING GGA AND LSDA

D. P. Rai, A. Shankar, Sandeep, M.P. Ghimire and R. K. Thapa*

Department of Physics, Mizoram University, Aizawl, India-796004

Author for correspondence: Email:[email protected]

ABSTRACT

Based on the density functional theory (DFT) calculations, the electronic and magnetic

properties of Heusler compound Co2CrSb were investigated. The system Co2CrSb were treated

with generalized gradient approximation (GGA) as well as local spin density approximation

(LSDA). Co2CrSb gives the 100% spin polarization at EF. Co2CrSb is the most stable Half-

Metallic Ferromagnets (HMF). But the energy gap when treated with GGA is much larger as

compared with the system treated with LSDA. We have also found that the increase in the total

magnetic moments in GGA. Based on the calculated results we have predicted the half-metal

ferromagnetic character for Co2CrSb.

Key Words: GGA, LSDA, DOS, band structure, HMF, spin polarization.

PACS No: 70, 71.5.-m, 71.15.Mb, 71.20.Be

INTRODUCTION

In 1983, de Groot discovered half-metallic ferromagnetism in semi-Heusler compound NiMnSb

[de Groot et al. (1983)] by using first-principle calculation based on density functional theory.

After that, half-metallicity attracted much attention [Katsnelson et al. (2008)] because of its

prospective applications in spintronics [Zutic et al. (2004)]. Recently rapid development of

magneto-electronics intensified the research on ferromagnetic materials that are suitable for spin

injection into a semiconductor [Ohno (1998)]. One of the promising classes of materials is the

half-metallic ferrimagnets (HMF), i.e., compounds for which only one spin channel presents a

gap at the Fermi level, while the other has a metallic character, leading to 100% carrier spin

polarization at EF [Zutic et al. (2004), de Boeck et al. (2002)]. Ishida et al. [Ishida et al.

(1982)] have also proposed that the full-Heusler alloy compounds of the type Co2MnZ, (Z=Ge,

Sn), are half-metals. Heusler alloys have been particularly interesting systems because they

exhibit much higher ferromagnetic Curie temperature than other half-metallic materials

[Webster et al. (1988)]. Rai et al. [Rai et al. (2010), Rai et al. (2011)] investigated the ground

state study of Co2MnAl and Co2CrSi using LDA+U and LSDA method respectively. The

preparation and characterization of bulk Co2MnZ (Z=Si, Ge, Ga and Sn) to be used as targets for




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pulsed laser deposition (PLD) of magnetic contacts for spintronic devices [Manea et al. (2005)].

Rai and Thapa 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 [Rai and Thapa (2012)]. 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. In

our present work, we have studied the structural, electronic and magnetic properties of Co2CrSb

using full potential linearized augmented plane wave (FP-LAPW) method.

COMPUTATION DETAIL

A computational code (WIEN2K) [Blaha et al. (2001)] based on FP-LAPW method was applied

for structure calculations of Co2CrSb. GGA was used for the exchange correlation potential. The

multipole exapansion of the crystal potential and the electron density within muffin tin (MT)

spheres was cut at l=10. Nonspherical contributions to the charge density and potential within the

MT spheres were considered up yo lmax=6 . The cut-off parameter was RKmax=7. In the

interstitial region the charge density and the potential were expands as a Fourier series with wave

vectors up to Gmax=12 a.u-1

. The number of k-points used in the irreducible part of the brillouin

zone is 286. The Muffin Tin sphere radii (RMT) for each atom are 2.45 a.u. for Co, 2.45 a.u. for

Cr and 2.31 a.u. for Sb.

CRYSTAL STRUCTURE

Heusler alloy [Heusler (1903)] with chemical formula Co2CrZ (Z = Sb). The full Heusler

structure consists of four penetrating fcc sublattices with atoms at Co1(1/4,1/4,1/4),

Co2(3/4,3/4,3/4), Cr(1/2,1/2,1/2) and Z(0,0,0) positions which results in L21 crystal structure

having space group Fm-3-m as shown in Fig. 1.




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.

Fig. 1 Unit cell Structure of Co2CrZ: Co(green), Cr(red) and Z(blue) generated by

xCrysden package.


Structural optimization for Co2CrSb

Systematic calculations of the electronic and magnetic properties of the Heusler compounds

Co2CrSb were carried out in this work. The results of the electronic properties calculations are

compared to study the effect of the different kinds of atoms and valence electron concentration

on the magnetic properties and in particular the band gap in the minority states. The electronic

properties were calculated using GGA and LSDA respectively. The optimized lattice constant,

isothermal bulk moduli, its pressure derivative are calculated by fitting the total energy to the

Murnaghan’s equation of state [Murnaghan (1944)]. The optimized lattice parameters were

slightly higher than the experimental lattice parameters, the change in lattice parameters are

given by Δ(ao). It is confirmed that the ferromagnetic configuration is lower in energy in case of

the systems Co2CrSb [Table 1]. The results of the structural optimization are shown in Fig. 1.

The detail values of the optimized Lattice parameters and bulk moduli are given in Table 1.




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Fig.2: Volume optimization of Co2CrSb

Table 1: Lattice parameters, Bulk modulus and Equilibrium energy.

Compounds Lattice Constants ao (Å) Bulk Modulus Equilibrium

Previous Calculated Δ(ao) B(GPa) Energy (Ry)

Co2CrSb 6.011a 6.034 0.023 170.763 -20642.8466

aRef : M. Gilleßen (2009)

Spin Polarization and half-metallic ferromagnets.

The electron spin polarization (P) at Fermi energy (EF) of a material is defined by equation (1)

[Soulen et al. (1998)].

F F

F F

E EP

E E

(1)

Where FE and FE are the spin dependent density of states at the EF. The ↑ and ↓

assigns the majority and the minority states respectively. P vanishes for paramagnetic or anti-

ferromagnetic materials even below the magnetic transition temperature. It has a finite value in

ferromagnetic materials below Curie temperature [Ozdogan et al. (2006)]. The electrons at EF

are fully spin polarized (P=100%) when FE or FE equals to zero. In present work, we

have studied the system Co2CrSb shows 100% spin polarization at EF [Table 2]. According to

our results, the compound Co2CrSb is interesting as it shows large DOS at the EF of FE =2.5

states/eV within both GGA as well as LSDA.

Fig.3 summarizes the results of the DOS which were calculated using LSDA. As Compared to

LSDA, GGA increases the exchange splitting between the occupied majority and the unoccupied




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minority states and thus to larger gap for Co2CrSb [Fig. 5]. According to Figs. (4, 6) the indirect

band gap along the Γ-X symmetry for Co2CrSb are 0.25 eV and 0.45 eV using LSDA and GGA

respectively. Incase of both LSDA and GGA the Fermi energy (EF) lies in the middle of the gap

Table 2: Energy gap and Spin polarization

Tools Energy gap Eg (eV) Spin Polarization

Emax(Γ) Emin(X) ΔE FE FE P%

LSDA -0.25 0.00 0.25 2.50 0.00 100

GGA -0.25 0.20 0.45 2.40 0.00 100

of the minority-spin states, determining the half-metal character [Figs. (5, 6)]. The formation of

gap for the half-metal compounds was discussed by Galanakis et. al. (2002) for Co2MnSi, is due

to the strong hybridization between Co-d and Mn-d states, combined with large local magnetic

moments and a sizeable separation of the d-like band centers.

Figure 3: Total DOS of Co2CrSb using LSDA




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Figure.4: Band structure using LSDA

Figure.5: Total DOS using GGA




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Figure 6: Band structure using GGA

Magnetic properties calculated in the LSDA

Starting with the compound under investigation, all the information regarding the partial, total

and the previously calculated magnetic moments are summarized in Table 3. It is shown in Table

3 the calculated total magnetic moment is almost an integer value in case of Co2CrSb as expected

for the half-metallic systems. It is found that the partial as well as the total magnetic moment

increases in GGA as shown in Table 3. We have found that the Co sites contribute much more to

the magnetic moment in the Sb compound because of the indirect connection between the

specific magnetic moment at Co and the hybridization arising from the interaction between the

electrons at the Co sites with the neighboring electrons in the Co t-2g states. As shown in Table 3

the Sb atoms carry a negligible magnetic moment, which does not contribute much to the overall

moment. We have also noticed that the partial moment of Sb atoms aligned anti-parallel to Co

and Cr moments of the systems. It emerges from the hybridization with the transition metals and

is caused by the overlap of the electron wave functions. The small moments found at the Z sites

are mainly due to polarization of these atoms by the surroundings, magnetically active atoms as

reported by Kandpal et al.(2006).




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Table 3: Total and partial magnetic moments

Tools Magnetic Moment µB

Previous Calculated

Co Cr Z Total

LSDA 5.023a 1.039 2.778 -0.010 4.939

GGA 1.058 2.853 -0.014 4.999

aRef : M. Gilleßen (2009)

Conclusions

We have performed the total-energy calculations to find the stable magnetic configuration and

the optimized lattice constant. The DOS, magnetic moments and band structures of Co2CrSb

were calculated using FP-LAPW method. The calculated results were in good agreement with

the previously calculated results. The GGA gives wider gaps as well as the higher value of

magnetic moment as compared to LSDA and the half-metallicity is more stable in Co2CrSb is the

most stable HMF. For Ferromagnetic compounds the partial moment of Z being antiparallel to

the Co and Cr atoms. We have investigated the possibility of appearance of half-metallicity in

the case of the full Heusler compound Co2CrSb which shows 100% spin polarization at EF. The

existence of energy gap in minority spin (DOS and band structure) of Co2CrSb is an indication

of being a potential HMF. As well as the integral value of magnetic moment is also the evident

of HMF. The calculated magnetic moment are in qualitative agreement with the integral value,

supporting the HMF.

ACKNOWLEDGEMENT

DPR acknowledges DST inspire research fellowship and RKT a research grant from UGC (New

Delhi, India).

REFERENCES

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International Scholarly Research NetworkISRN Condensed Matter PhysicsVolume 2012, Article ID 410326, 5 pagesdoi:10.5402/2012/410326

Research Article

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

D. P. Rai and R. K. Thapa

Department of Physics, Mizoram University, Aizawl 796004, India

Correspondence should be addressed to D. P. Rai, [email protected]

Received 17 May 2012; Accepted 19 June 2012

Academic Editors: I. Galanakis, A. N. Kocharian, Y. Ohta, and A. D. Zaikin

Copyright © 2012 D. P. Rai and R. K. Thapa. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

We performed the structure optimization of Co2TiAl based on the generalized gradient approximation (GGA) and linearizedaugmented plane wave (LAPW) method. The calculation of electronic structure was based on the full-potential linear augmentedplane wave (FP-LAPW) method and local spin density approximation exchange correlation LSDA+U . We also studied the impactof the Hubbard potential or onsite Coulomb repulsion (U) on electronic structure; the values are varied within reasonable limitsto 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.

1. Introduction

Semi-Heusler compound NiMnSb was the first found half-metal ferromagnets (HMFs) by using first principle calcula-tion based on density functional theory [1]. Co2TiAl is a fer-romagnetic half-metal with an integral magnetic moment of1 μB/atom [2]. It has been widely used in magnetic recordingtapes, spin valves, giant magnetoresistance (GMR), and soforth. In recent years, it attracts substantial interests becauseof the half-metallic property and the applicable potential forfuture spintronics. In half-metal, one spin channel is metallicand the other is insulating with 100% spin polarizationat the Fermi level EF [3, 4]. The electronic and magneticproperties of Co2MnAl [5] and Co2CrSi [6] using local spindensity approximation (LSDA) show the half-metallicity atthe ground state. Rai and Thapa have also investigated theelectronic structure and magnetic properties of X2YZ- (X= Co, Y = Mn, Z = Ge, Sn) type Heusler compounds byusing a first principle study and reported HMFs [7]. Rai et al.(2012) also studied the electronic and magnetic propertiesof Co2CrAl and Co2CrGa using both LSDA and LSDA+Uand 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 themajority spin, whereas in the minority spin, the Fermi energy

falls in an exchange-split gap between the occupied band andthe unoccupied 3d band. Since the magnetic properties arehighly spin polarized near the Fermi energy, it is thereforeinteresting to investigate the orbital contributions of theindividual atoms to the magnetic moment of Co2TiAl. TheLDA+U approach in which a Hubbard U repulsion term isadded to the LDA is functional for strong correlation of dor f electrons. Indeed, it provides a good description of theelectronic properties of a range of exotic magnetic materials,such as the Mott insulator KCuF3 [9] and the metallic oxideLaNiO2 [10]. Two main LDA+U schemes are in widespreaduse today: The Dudarev [11] approach in which an isotropicscreened on-site Coulomb interaction U is added and theLiechtenstein [9] approach in which the U and exchange (J)parameters are treated separately. Both the choice of LDA+Uschemes on the orbital occupation and subsequent properties[12], as well as the dependence of the magnetic propertieson the value of U [13], has recently been analyzed. It goeswithout saying that the Hubbard model [14] is of seminalimportance in the study of modern condensed mattertheory. It is believed that the Hubbard model can describemany properties of strongly correlated electronic systems.The discovery of high temperature superconductivity hasenhanced the interest in a set of Hubbard-like models that are


Ti

Al

CoAlAl

Al

Al

Al

Al

Al

Co

Figure 1: Structure of Co2TiAl.

used to describe the strongly correlated electronic structureof transition metal oxides [15].

2. Crystal Structure and Calculation

2.1. Crystal Structure. X2YZ Heusler compounds crystallizein the cubic L21 structure (space group Fm3m) [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 fccsublattices, two of which are equally occupied by Co. The twoCo-site fcc sub-lattices combine to form a simple cubic sub-lattice as shown in Figure 1.

2.2. Method. In this work, we have performed the full-poten-tial linearized augmented plane wave (FP-LAPW) methodaccomplished by using the WIEN2K code [17] withinLSDA and LSDA+U [9] schemes. We have calculated onsiteCoulomb repulsion (U) based on Hubbard model. Thestandard Hubbard Hamiltonian [18] is of the form:

H = −t∑

〈i j〉,σ

c†iσ c jσ + U∑

ni↑ni↓, (1)

where niσ = c†iσ ciσ and c†iσ(ciσ) creates (annihilates) an elec-tron on site i with spin σ =↑ or ↓. A nearest neighbor isdenoted by 〈i j〉. U is the onsite Coulomb repulsion betweentwo electrons on the same site. The hybridization betweennearest neighbor orbitals is denoted by t, allowing the parti-cles to hop to adjacent sites. The on-site energies are takento be zero. Considering that the atoms are embedded in apolarizable surrounding, U is the energy required to movean electron from one atom to another, far away, in that case.U is equal to the difference of ionization potential (EI) andelectron affinity (EA) of the solid. Removing an electron froma site will polarize its surroundings thereby lowering theground state energy of the (N − 1) electron system [19, 20].Thus

EI = EN−1 − EN , EA = EN − EN+1

U =(EN−1 − EN

)−(EN − EN+1

),

(2)

where EN(±1) are the ground state energy of (N ± 1) electronsystem.

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 Ryfor Ti were used in the LSDA+U calculations.


We have studied Co2TiAl using simple LSDA; that is, U =0.00 Ry as shown in Figure 2. The Fermi energy (EF) issituated close to the valence band; there exists a small gap of0.400 eV which is lower than the previously reported valueof energy gap 0.456 eV [2] which was calculated by usingGGA as given in Table 1. The robustness of half-metallicityin Co2TiAl can be explained by the impact of U on theDOS which is taken into consideration in LSDA+U . We haveplotted DOS for each value of U which is shown in Figure 2,and it is seen that the majority-spin bands shift towards lowenergy and the minority-spin bands shift toward high energyside. 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 minimumfor Co atoms. For a large U , the minority-spin band of Coextends across the Fermi level and gap disappears in EF . Asa result, the DOS is no longer half-metallic. The use of theLSDA+U method increases the width of the energy gap withincrease in U substantially up to some extent. The respectiveenergy 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 RyEg = 0.50 eV, and for UCo = 0.29 Ry and UTi = 0.053 RyEg = 0.32 eV. Kandpal et al. calculated the energy gap ofCo2TiAl, 1.12 eV, using LDA+U [2]. In LSDA, the transitionmetal d states are well separated from the sp states, whereasthe LSDA+U method increases the energetic overlap betweenthese states. In all cases, the gap is between the occupied andunoccupied transition metal d states [21]. It can be seen thatthe bandwidth of the d bands for the Co site is indeed smallerthan for the Ti site as shown in Figures 2(a) and 2(b). The dstates on the Co sites are more localized and one can expecta larger on-site Coulomb interaction than that on the Ti site,which is in agreement with UTi < UCo [22]. However, thehalf-metallicity is retained till some value of U as shown inFigure 2. Therefore, the dependency of DOS on U impliesthat the half-metallicity is robust sensitive to U .

3.1. Magnetic Properties. The calculated partial andtotal magnetic moments are summarized in Table 2. ForU = 0.10 Ry, LSDA+U gives the partial moment of1.0605 μB/atom for Co, −0.71433 μB/atom for Ti, and thetotal 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 totalsystem being in good agreement with the previouslycalculated orbital moment 1.00 μB/atom reported by Kandpal [2]. The opposite signs of spin moments between Co andTi indicate charge transfer from the Ti anion to the Co cation.With the increase of U , the total magnetic moment as well as


Table 1: The calculated lattice parameters and magnetic moments are compared with the previous results.

CompoundsLattice constant ao (A) Magnetic moment μB (LSDA) Energy gap Eg (eV)

Previous Our calculation Previous Our calculation Mcal Previous Our result

Co2TiAl 5.828 [2] 6.210 1.00 [2] 0.999 0.456 [2] 0.400

86420−2−4−6−8−10

−10 −5 0 5 10 15

−10 −5 0 5 10 15

−10 −5 0 5 10

−10 −5 0 5 10

10

5

0

−5

−10

10

5

0

−5

−10

86420−2−4−6−8

10

−15

Energy (eV)

Energy (eV)

Energy (eV)

Energy (eV)

DO

S (s

tate

s/eV

)D

OS

(sta

tes/

eV)

DO

S (s

tate

s/eV

)D

OS

(sta

tes/

eV)

U = 0 Ry

U = 0.1 Ry

U = 0.2 Ry

UCo = 0.29 Ry

UTi = 0.053 Ry

Total DOS upCo tot. up

(a)

86420−2−4−6−8−10

−10 −5 0 5 10 15

−10 −5 0 5 10 15

−10 −5 0 5 10 15

10

5

0

−5

−10

10

5

0

−5

−10

86420−2−4−6−8

10

−10 −5 0 5 10−15

Energy (eV)

Energy (eV)

Energy (eV)

Energy (eV)

DO

S (s

tate

s/eV

)D

OS

(sta

tes/

eV)

DO

S (s

tate

s/eV

)D

OS

(sta

tes/

eV)

U = 0 Ry

U = 0.1 Ry

U = 0.2 Ry

UCo = 0.29 Ry

UTi = 0.053 Ry

Total DOS upTi tot. up

(b)

Figure 2: (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 andblue line denoted the partial DOS of Co atoms.

Table 2: The calculated partial and total magnetic moments versus U values.

Coulomb repulsion Magnetic moment μB of Co2TiAl Energy gap Eg (eV)

(U) Ry MCo MTi MAl Mtot

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

UCo = 0.290UTi = 0.053

1.579 −0.585 −0.057 2.258 0.32


−0.3

−0.4

−0.5

−0.6

−0.7

−0.8

−0.9

0 0.05 0.1 0.15 0.2

U (Ry)

Ti

Mag

net

ic m

omen

t (u

B)

−1

(a)

0 0.05 0.1 0.15 0.2

2.4

2.2

2

1.8

1.6

1.4

1.2

1

0.8

0.6

0.25 0.3

U (Ry)

CoTotal

Mag

net

ic m

omen

t (u

B)

(b)

Figure 3: (a) Plot of MTi versus U Ry. (b) Plot of (Mtot and MCo) versus U Ry.

the moment of Co increases and the moment of Ti decreasesas shown in Figures 3(a) and 3(b). The increase in magneticmoment is due the double occupancy which is a decreasingfunction of U reported by F. Mancini and F. P. Mancini [23].

4. Conclusion

In conclusion, we have performed FP-LAPW self-consistentcalculations for ferromagnetic half-metal Co2TiAl within theLSDA and the LSDA+U schemes. The spin-orbit couplingincluded in the self-consistent calculations; the orbitalmagnetic moments are obtained from both the LSDA andthe LSDA+U methods. It is found that the on-site Coulombinteraction U dramatically enhanced the orbital moments.For U = 0.00 Ry and U = 0.10 Ry, the calculated total orbitalmoments are 0.99999 μB/atom and 0.999 μB/atom, respec-tively, being in good agreement with the previously reportedresult 1.00 μB/atom [2]. The calculated energy gap was foundto be 0.84 eV forU = 0.10 Ry. It also appears that U decreasesdouble occupancy and hence increases local moments. Ourcalculated results ofU for Co and Ti are 0.29 Ry and 0.053 Ry;respectively, the corresponding magnetic moments is not theintegral value (HM) that is, 2.258 μB. Also Figure 2 showsthat EF does not lie at the middle of the gap at UCo =0.29 Ry and UTi = 0.053; Ry thus the half metallicity doesnot exist. By using LSDA+U , we have found that Co2TiAlis possible half-metal candidate having magnetic moment0.99994 uB at U = 0.10 Ry. This value of integral magneticmoment supports the condition of half-metallicity. Due tothese characteristics like integer value of magnetic moment,100% spin polarization at EF and the energy gap at the Fermilevel in spin-down channel make application of half-metallicferromagnets very important. The Co-based Heusler alloysCo2YZ (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 beyondroom temperature and the simple fabrication process such asdc-magnetron sputtering in Co2YZ.

Acknowledgments

D. P. RAI acknowledges DST inspire research fellowship andR. K. Thapa a research grant from UGC (New Delhi), India.

References

[1] R. A. de Groot, F. M. Mueller, P. G. van Engen, and K. H. J.Buschow, “New class of materials: half-metallic ferromagnets,”Physical Review Letters, vol. 50, no. 25, pp. 2024–2027, 1983.

[2] H. C. Kandpal, Computational studies on the structure andstabilities of magnetic inter-metallic compounds,dissertation zurErlanggung des Grades [Doktor der Naturwissenschaften], amFachbereich Chemie, Pharmazie und Geowissenschaften derJohannes Guttenberg-Universitat Mainz, 2006.

[3] I. Zutic, J. Fabian, and S. Das Sarma, “Spintronics: fundamen-tals and applications,” Reviews of Modern Physics, vol. 76, pp.323–410, 2004.

[4] J. de Boeck, W. Van Roy, J. Das et al., “Technology andmaterials issues in semiconductor-based magnetoelectronics,”Semiconductor Science and Technology, vol. 17, no. 4, pp. 342–354, 2002.

[5] D. P. Rai, J. Hashemifar, M. Jamal et al., “Study of Co2MnAlHeusler alloy as half metallic ferromagnet,” Indian Journal ofPhysics, vol. 84, no. 6, pp. 717–721, 2010.

[6] D. P. Rai, Sandeep, M. P. Ghimire, and R. K. Thapa, “Structuralstabilities, elastic and thermodynamic properties of ScandiumChalcogenides via first-principles calculations,” Bulletin desSciences Mathematiques, vol. 34, pp. 1219–1222, 2011.

[7] D. P. Rai and R. K. Thapa, “Electronic structure and magneticproperties of X2YZ (X = Co, Y = Mn, Z = Ge, Sn) type Heuslercompounds by using a first principle study,” Phase Transition:A Multinational Journal, pp. 1–11, 2012.


[8] D. P. Rai, Sandeep, M. P. Ghimire, and R. K. Thapa, “Elec-tronic tructure and magnetic properties of Co2YZ (Y = Cr,V = Al, Ga) type Heusler compounds: A first PrincipleStudy,” International Journal of Modern Physics B, vol. 26, pp.1250071–1250083, 2012.

[9] A. I. Liechtenstein, V. I. Anisimov, and J. Zaanen, “Density-functional theory and strong interactions: orbital ordering inMott-Hubbard insulators,” Physical Review B, vol. 52, no. 8,pp. R5467–R5470, 1995.

[10] K. W. Lee and W. E. Pickett, “Infinite-layer LaNiO2: Ni1+ is notCu2+ ,” Physical Review B, vol. 70, Article ID 165109, 2004.

[11] S. L. Dudarev, G. A. Botton, S. Y. Savrasov, C. J. Humphreys,and A. P. Sutton, “Electron-energy-loss spectra and thestructural stability of nickel oxide: an LSDA+U study,” PhysicalReview B, vol. 57, no. 3, pp. 1505–1509, 1998.

[12] E. R. Ylvisaker, W. E. Pickett, and K. Koepernik, “Anisotropyand magnetism in the LSDA+U method,” Physical Review B,vol. 79, Article ID 035103, 2009.

[13] S. Y. Savrasov, A. Toropova, M. I. Katsnelson, A. I. Lichten-stein, V. Antropovand, and G. Kotliar, “Electronic structureand magnetic properties of solids,” Zeitschrift Fur Kristallogra-phie, vol. 220, pp. 473–488, 2005.

[14] J. Hubbard, “Electron correlations in narrow energy bands,”Proceedings of the Royal Society A, vol. 276, pp. 238–257, 1963.

[15] P. W. Anderson, “The resonating valence bond state inLa2CuO4 and superconductivity,” Science, vol. 235, no. 4793,pp. 1196–1198, 1987.

[16] F. Heusler, “Uber magnetische Manganlegierungen,” Verhand-lungen der Deutschen Physikalischen Gesellschaft, vol. 12, p.219, 1903.

[17] P. Blaha, K. Schwarz, G. K. H. Madsen, D. Kvasnicka, J. Luitz,and K. Schwarz, “An augmented plane wave plus local orbitalsprogram for calculating crystal properties,” Wien2K User’sGuide, Technische Universitat Wien, Wien, Austria, 2008.

[18] L. M. Roth, “New method for linearizing many-body equa-tions of motion in statistical mechanics,” Physical ReviewLetters, vol. 20, pp. 1431–1434, 1968.

[19] J. van den Brink, M. B. J. Meinders, and G. A. Sawatzky, “Influ-ence of screening effects and inter-site Coulomb repulsion onthe insulating correlation gap,” Physica B, vol. 206-207, pp.682–684, 1995.

[20] K. H. G. Madsen and P. Novak, “Charge order in magnetite.An LDA+U study,” Europhysics Letters, vol. 69, p. 777, 2005.

[21] I. Galanakis, “Orbital magnetism in the half-metallic Heusleralloys,” Physical Review B, vol. 71, Article ID 012413, 2005.

[22] C. Ederer and M. Komelj, “Magnetic coupling in CoCr2O4

and MnCr2O4: an LSDA+U study,” Physical Review B, vol. 76,Article ID 064409, 9 pages, 2007.

[23] F. Mancini and F. P. Mancini, “One dimensional extendedHubbard model in the atomic limit,” Cond. Mat. Str-El, vol.77, pp. 061120–061121, 2008.

Vol. 33, No. 8 Journal of Semiconductors August 2012

Structural, electronic, magnetic and optical properties of neodymium chalcogenidesusing LSDACU method

A Shankar, D P Rai, Sandeep, and R K Thapa

Condensed Matter Theory Group, Department of Physics, Mizoram University, Aizawl, Mizoram -796 004, India

Abstract: We have studied the electronic, magnetic and optical properties of neodymium chalcogenides by per-forming LSDACU and full potential linearized augmented plane wave (FP-LAPW) method. The electronic struc-ture calculation shows that the electronic states in Nd-chalcogenides were mainly contributed by Nd-4f electronsnear Fermi energy and 3p, 4p and 5p state electrons of X (S, Se and Te), respectively. We have also studied theabsorption of light via the imaginary parts of the dielectric function of Nd-chalcogenides.

Key words: DFT; DOS; magnetic moment; UDOI: 10.1088/1674-4926/33/8/082001 EEACC: 2570

1. Introduction

In the last few years Nd-chalcogenides have been widelyinvestigated due to their trivalent electronic properties at roomtemperature. They have many applications in spintronics andfiltering devicesŒ1, non-linear optics, electro-optical compo-nents and electronicsŒ2. All three Nd-chalcogenides are sta-ble in the rock salt structure at ambient temperature and pres-sure with lattice parameters of 5.743 A, 5.975 A and 6.387 Afor NdS, NdSe and NdTe, respectively. These are the typicalmembers of binary rare-earth chalcogenides with space groupFm3m. Several studies have been made on Nd-chalcogenides.Papamantellos et al.Œ3 have studied the magnetic structuresand ordered moments of these chalcogenides at 4.2 K by usingthe neutron diffraction method. Antonov et al.Œ4 have studiedthe electronic, optical and magneto optical properties of thesematerials using the LSDACU method. Furrer et al.Œ5 mea-sured the width of the crystal field by using the neutron inelas-tic scattering method. VermaŒ6 studied the electronic and opti-cal properties of rare-earth chalcogenides and pnictides. Singhet al.Œ7 have studied the structural, elastic and electronic prop-erties of Nd-chalcogenides using the GGA method, respec-tively.

In this paper we have studied the electronic structure, mag-netic properties and optical properties of Nd-chalcogenides inthe rock salt structure using an FP-LAPW based LSDACU

method within the formalism of the density functional the-ory (DFT)Œ8. The local spin density approximation (LSDA)method is used for the calculation of Coulomb repulsion (U).

2. Computational details

The full potential linearized augmented plane wave (FP-LAPW) methodŒ9 within the DFT, implemented in theWIEN2k codeŒ10 has been applied for the study of structural,electronic, magnetic and optical properties of NaCl type struc-tured neodymium chalcogenides, viz. NdS, NdSe and NdTe.The total energy is calculated by using the parametrization ofLSDACU Œ11 and the FP-LAPWmethod. The sets of 5p, 6s, 4f

orbitals for Nd atoms, 3s, 3p orbitals for S atoms, 3s, 3p, 4s, 3d,4p orbitals for Se atoms and 4p, 5s, 4d, 5p for Te atomswere se-lected as valence states whereas other lower states were treatedas core states in the calculations. The plane-wave cut-off forthe basis functions was set to RMTKmax D 7 where Kmax is themaximum value of the wave vector K D k C G . The poten-tial and charge density were expanded up to a cut-off Gmax D

12 a.u.1. The muffin-tin radii is set to RMT D 2.5 a.u. The ex-pansion of wave functions, as well as the density along with thepotentials inside the muffin-tin spheres, were upto lmax D 10.LSDACU calculations were carried out using the parametersU D 0.47 Ry and J D 0 eV for Nd atoms. We have calculatedU for Nd in NdX, based on the Hubbard modelŒ12. The on-site energies are taken to be zero. Considering that the atomsare embedded in a polarizable surrounding, U is the energy re-quired to move an electron from one atom to another far awayone. In that case U is equal to the difference of the ionizationpotential (EI/ and the electron affinity (EA/ of the solid. Re-moving an electron from a site will polarize its surroundings,thereby lowering the ground state energy of the N – 1 electronsystemŒ13; 14. Thus8<

:EI D EN 1 EN ;

EA D EN EN C1;

U D EI EA;

(1)

where EN.˙1/ are the ground state energy of the N ˙ 1 elec-tron system. A mesh of 12000 k-points for NdS and NdSeand 10000 k-points for NdTe were used to obtain 111 specialk-points in the irreducible wedge of the Brillouin zone. Foreach calculation, the energy convergence criteria were set tobe 0.0001 Ry and charge 0.0001e. The optical calculation wasperformed by using the dipole approximationŒ15.


We observed how the total energy of the system changeswith the variation of volume, which we have summarized inFig. 1. The curve is obtained by fitting the calculated values of

Corresponding author. Email: [email protected] 20 February 2012, revised manuscript received 23 March 2012 c 2012 Chinese Institute of Electronics

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J. Semicond. 2012, 33(8) A Shankar et al.

Fig. 1. Plots of energy versus volume of (a) NdS, (b) NdSe and (c)NdTe.

energies to the Murnaghan equation of stateŒ16. The theoreti-cal equilibrium lattice parameters and bulk modulus were de-duced from the volume corresponding to the minimum energyvalue. We have tabulated the equilibrium lattice parameters,bulk modulus and the equilibrium energy for NdX (X D S,Se and Te) in Table 1. From Table 1 it is clear that the cal-culated lattice parameters are in between the experimental andprevious theoretical values. We can conclude that they are inqualitative agreement with the previous results. We have alsocompared the calculated value of the bulk modulus with thepreviously presented value. The optimized lattice parametersso obtained were used to calculate the electronic, magnetic andoptical properties, the results of which are given in the next sec-tions.

3.1. Density of states

The total density of states (DOS) and partial DOS for NdX(X D S, Se, and Te) are shown in Fig. 2 for both spin channels.

As seen from the figure, the total and partial DOS are quitesimilar for all three compounds. From the total DOS plot inFigs. 2(a), 2(e) and 2(i), we observed a sharp peak for the spinup channel centred at around 0 eV, which is taken as the Fermilevel (EF/. Now if we examine the core region, a small peakis observed almost at –12.86 eV for both spin channels, whichare mainly due to the contribution of the 3s, 4s and 5s stateelectrons of S, Se and Te, respectively, as can be seen fromthe partial DOS plot in Figs. 2(b), 2(f), and 2(j). Similarly, inthe valence region, a small sharp peak is observed almost at–3.6 eV for both channels, which are mainly due to the contri-bution of the 3p, 4p and 5p state electrons of S, Se and Te, re-spectively, and the 4d state electrons of Nd, as can be seen fromFigs. 2(b), 2(f), 2(j) and 2(c), 2(g), 2(k). In the conduction re-gion a sharp peak is observed almost at 2.14 eV for spin-downchannels, which is mainly due to the contribution of the 4f stateelectrons of Nd. In addition, the 4f state of Nd also dominatesat EF for the spin-up channel as can be seen from the partialDOS plot in Figs. 2(d), 2(h), 2(l). From the total and partialDOS plot we can say that the contribution due to chalcogen ishigher in the core and valence region and, due to Nd electrons,is higher in the conduction and in the Fermi level. As can beseen from the total DOS plot, there is an exchange splitting ofthe order of 2.14 eV in the two spin channels. From the par-tial DOS plot in Figs. 2(b, c, f, g, j, k) we can say that thereis small hybridization between the p-state electrons of chalco-genides and the d and d-eg state electrons of Nd, and that theydegenerate over large part of their extension. This gives rise tothe covalent bond in between the Nd and chalcogenides. Butthe relative amount of chalcogenides and Nd is different aboveand below the Fermi level. At and above theEF, Nd DOS dom-inates the chalcogenides DOS and below theEF vice versa. So,from this argument we can say that an electrovalent bond alsoexists in between the chalcogenides andNd. According to Paul-ing, Nd has an electronegativity of 1.1 and S, Se and Te haveelectronegativities of 2.5, 2.4 and 2.1, respectively. From theseelectronegativity values, we simply expect that NdS to be themost ionic and NdTe the most covalent of the three systems.In the present calculation we have taken equal sized muffin-tinspheres for Nd and its chalcogenides, which are touching eachother. The lattice parameters are also increasing for S, Se andTe, respectively. Thus absolute size of the sphere in NdTe islarger than NdS since the lattice parameter is larger in the caseof NdTe, and the sphere size also does not correspond to theatomic radii of the element.

3.2. Band structure

The electronic band structure plots for NdX are shown inFigs. 3, 4 and 5. We find that the observed core level bandswere due to X-s state electrons followed by X-p state electronsin the valence region in both the spin channels, as shown inFigs. 2(b, f, j). At the EF, Nd-4f state, electrons were foundto be contributing, which is observed in terms of flat bands inthe spin-up channels for NdX, which are shown in Figs. 2(d,h, l). In the spin-down channels, bands due to Nd-4f state elec-trons were observed in the conduction region at about 2.14 eV,as shown in Figs. 3, 4 and 5. From the plot of the total DOSwe find that exchange splitting is observed, which is due to theinclusion of U . This contradicts the result obtained by Singh

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Fig. 2. Total and partial DOS for (a, b, c, d) NdS, (e, f, g, h) NdSe and (i, j, k, l) NdTe.

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Table 1. The calculated equilibrium lattice parameter, bulk modulus and the energy E at the equilibrium lattice constant.

CompoundLattice parameter (A) Bulk modulus (GPa)

Equilibrium energy (Ry)Experimental Theoretical(previous)

Our result Previous Present

NdS 5.69Œ17 5.59Œ7 5.629 94.88Œ7 65.266 20042:56

NdSe 5.789Œ18 5.87Œ7 5.83 64.81Œ7 83.195 24100:686

NdTe 6.249Œ18 6.28Œ7 6.273 52.95Œ7 109.347 32829:011

Fig. 3. Plot of band structure of NdS.

et al.Œ7 where GGA was employed. The magnetic moment ofNd chalcogenides are given Table 2 along with previous theo-retical data. The contribution of the Nd atom to the magneticmoment is 3.34B, 3.38B and 3.35B, for NdS, NdSe andNdTe respectively, which is evident from the exchange split-ting in the DOS of Nd-f states [Figs. 2(d, h, l)].

Fig. 4. Plot of band structure of NdSe.

3.3. Optical properties

The absorptive parts of the dielectric function "2 are shownin Fig. 6, which is plotted as a function of energy. Optical spec-tra have been analyzed for the energy range 0–14 eV. The mainfeature is a sharp peak with maximum around the Fermi level.Thus the critical points are embedded at the EF, unlike the op-

082001-4


Fig. 5. Plot of the band structure of NdTe.

Table 2. The magnetic moments for Nd chalcogenides.

CompoundMagnetic moment B

PreviousŒ3 Our resultNdS 3.62 3.65NdSe 3.52 3.69NdTe 3.54 3.78

tical critical points for BeXŒ19. These were followed by smallstructures 1.5 eV (Fig. 6), which were observed for all threechalcogenides. The peaks reproduced in our calculation, arethe general form of the spectra. The trends in "2 may be linkedto the trends observed in the DOS and band structures. Ana-lysis of Fig. 6 shows that the 0–0.5 eV photon-energy range ischaracterized by small absorption and appreciable reflectivity.The 0.6–1.4 eV photon energy range is characterized by high

Fig. 6. Absorptive part of the optical constants for (a) NdS, (b) NdSeand (c) NdTe.

transparency, no absorption and small reflectivity. The photonenergy ranging from 2.2 to 12.0 eV is characterized by maxi-mum reflectivity with small absorption.

4. Conclusion

From the first principles study of the structure, DOS, bandstructure and optical properties of Nd-chalcogenides (S, Se andTe), we find that the DOS contributions were mainly due toNd-4f states in the spin up configuration at EF. The DOS andband structures suggested weak hybridization of Nd-5d and X-p state electrons. The magnetic moments were explained on

082001-5

J. Semicond. 2012, 33(8) A Shankar et al.the basis of exchange splittings of Nd-4f state electrons. Thebehavior of the imaginary part of the dielectric constant sug-gested that the system exhibits small absorptionŒ13 and appre-ciable reflectivity just above EF, which is followed by hightransparency and small reflectivity in the conduction region.

Acknowledgments

AS acknowledges fellowship and RKT a research grantfrom UGC (New Delhi). DPR acknowledges the INSPIRE fel-lowship from INSA (New Delhi).

References[1] Horne M, Strange P, Temmerman W M, et al. The electronic

structure of europium chalcogenides and pnictides. J Phys Con-dens Matter, 2004, 16: 5061

[2] Singh O P, Gupta V P. Electronic properties of europium chalco-genides (EuO, EuS, EuSe, EuTe). Phys Status Solidi B, 1985,129: K153

[3] Papamantellos P S, Fischer P, Niggli A, et al. Magnetic orderingof rare earth monochalcogenides: I. neutron diffraction investiga-tion of CeS, NdS, NdSe, NdTe and TbSe. J Phys C: Solid StatePhys, 1974, 7: 2023

[4] AntonovVN, HarmonBN, Perlor A, et al. Fully relativistic spin-polarized LMTO calculations of the magneto-optical Kerr effectof d and f ferromagnetic materials. II. neodymium chalcogenides.Phys Rev B, 1999, 59: 14561

[5] Furrer A, Warming E. Crystal-field splittings of NdS and NdSe.J Phys C: Solid State Phys, 1974, 7: 3365

[6] VermaA S. Electronic and optical properties of rare-earth chalco-genides and pnictides. African Physical Review, 2009, 3: 11

[7] Pal S R, Kumar S R, Rajagopalan M. Structural, elastic and elec-tronic properties of neodymium chalcogenides (NdX, X D S, Se,Te): first principles study. Chalcogenide Lett, 2011, 8: 325

[8] Kohn C W, Sham L J. Self-consistent equations including ex-change and correlation effects. Phys Rev A, 1965, 140: 1133

[9] Thapa R K, Sandeep, Ghimire M P, et al. Study of DOS and en-ergy band structures in beryllium chalcogenides. Indian J Phys,2011, 85: 727

[10] Blaha P, Schwarz K, Madsen G, et al. An augmented planewave plus local orbitals program for calculating crystal proper-ties. Tech University, Vienna-Austria, 2010

[11] Anisimov V I. Aryasetiawan F, Lichtenstein A I. First-principlescalculations of the electronic structure and spectra of stronglycorrelated systems: the LDA+U method. J Phys Condens Mat-ter, 1997, 9: 767

[12] Hubbard J. Electron correlations in narrow energy bands. ProcRoy Soc London, 1963, A27: 238

[13] Van den Brink, Meinders M B J, Sawatzk G A. Influence ofscreening effects and inter-site Coulomb repulsion on the insu-lating correlation gap. Physica B, 1995, 206/207: 682

[14] Madsen G K H, Nova P. Charge order in magnetite. An LDA+Ustudy. Europhys Lett, 2005, 69: 777

[15] Ghimire M P, Sandeep, Sinha T P, et al. First principles study ofthe electronic and optical properties of SbTaO4. Physica B, 2011,406: 3454

[16] Murnaghan F D. The compressibility of media under extremepressures. Proc Natl Acad Sci USA, 1944, 30: 5390

[17] Modukuru Y, Thachery J, Cahay M, et al. Growth and character-ization of LaS and NdS thin films on Si, GaAs and InP substrates.Electrochemical Society Proceedings, 2002, 18: 367

[18] Wyckoff RWG. Crystal structures. NewYork: Interscience Pub-lishers, 1963, 1: 85

[19] Okoye C M I. Structural, electronic, and optical properties ofberyllium monochalcogenides. Eur Phys J, 2004, B39: 5

082001-6

Lat. Am. J. Phys. Educ. Vol. 6, No. 2, June 2012 317 http://www.lajpe.org

A density functional theory (DFT) study of Co2CrGe: LSDA method

D. P. Rai, and R. K. Thapa

*

Department of Physics, Mizoram Universty, Aizawl, India 796004.


(Received 11 February 2012, accepted 18 May 2012)

Abstract The structural, electronic and magnetic properties of Co2CrGe, a Heusler alloy, have been evaluated by first principles

density functional theory and compared with the known experimental and theoretical results. Generalized gradient

approximation (GGA) is used for structural study where as Local spin density approximation (LSDA) for electronic

calculation. First principles structure optimizations were done through total energy calculations at 0K by the full

potential linearized augmented plane wave (FP-LAPW) method as implemented in WIEN2K code.

Keywords: GGA, half-metallicity, DOS and band structure.

Resumen Las propiedades estructurales, electrónicas y magnéticas del Co2CrGe, una aleación Heusler, han sido evaluadas por la

teoría de densidad funcional de primeros principios y son comparados con los experimentos conocidos y resultados

teóricos. La aproximación generalizad de gradiente (GGA) es usada para el estudio estructural con la aproximación de

la densidad spin local (LSDA) para el cálculo electrónico. Las optimizaciones de estructura de los primeros principios

fueron realizados a través de los cálculos de energía total a 0K por el potencial completo linealizado de onda plana (FP-

LAPW) como método implementado en el código WIEN2K.

Palabras clave: GGA, media-metalicidad, DOS y estructura de banda.

PACS: 70, 71.5.-m, 71.15.Mb, 71.20.Be ISSN 1870-9095

I. INTRODUCTION

In 1983, de Groot discovered half-metallic ferromagnetism

in semi-Heusler compound NiMnSb [1] by using first-

principle calculation based on density functional theory.

Heusler alloys are the ternary intermetallich compounds

with compostion X2YZ, where X and Y are transition

elements (Ni, Co, Fe, Mn, Cr, Ti, V etc.) and Z is III, IV or

V group elements (Al,Ga, Ge, AS, Sn, In etc.). One of the

promising classes of materials is the half-metallic

ferrimagnets, i.e., compounds for which only one spin

channel presents a gap at the Fermi level, while the other

has a metallic character, leading to 100% carrier spin

polarization at EF [4]. After that, half-metallicity attracted

much attention [2], because of its prospective applications

in spintronics [3]. The electronic and magnetic properties of

Co2MnAl [5] and Co2CrSi [6] using LSDA shows the half-

metallicity at the ground state. In this present work, we

report the result of GGA and LSDA of bulk electronic

structure and magnetic properties of Co2CrGe.

II. COMPUTATIONAL DETAILS AND CRYSTAL STRUCTURE

The FP-LAPW method (WIEN2K) [7] was applied to band

structure calculations of Co2CrGe. GGA [8] and LSDA

were used for the exchange correlation potential. The

multipole exapansion of the crystal potential and the

electron density within muffin tin (MT) spheres was cut at

l=10. Nonspherical contributions to the charge density and

potential within the MT spheres were considered up yo

lmax=6. The cut-off parameter was RKmax=7. In the

interstitial region the charge density and the potential were

expands as a Fourier series with wave vectors up to

Gmax=12a.u-1

. The MT sphere radii(R) used were 2.35a.u.

for Co, 2.35a.u. for Cr and 2.21a.u. for Ge. The number of

k-points used in the irreducible part of the brillouin zone is

286.

Crystal structure: Heusler alloy [9] with chemical

formula X2YZ (X = Co, Y = Cr and Z = Ge). The full

Heusler structure consists of four penetrating fcc sublattices

with atoms at X1(1/4, 1/4, 1/4), X2(3/4, 3/4, 3/4), Y(1/2,

D. P. Rai, A. Shankar, Sandeep, M. P. Ghimire, R. K. Thapa


1/2, 1/2) and Z(0, 0, 0) positions which results in L21 crystal

structure having space group Fm-3-m as shown in Fig. 1.

FIGURE 1. Unit Cell Structure of Co2CrGe.

III. RESULTS AND DISCUSSIONS

The volume optimization was performed using the lattice

constant by taking the experimental one. The calculated

total energies within GGA as function of the volume were

used for determination of theoretical lattice constant and

bulk modulus. The bulk modulus was calculated using the

Murnaghan’s equation of state [10]. The calculated values

of lattice constant and bulk modulus are presented in Table

I.

FIGURE 2. Volume optimization of Co2CrGe.

TABLE I. Lattice constant and Bulk modulus.

Lattice Constants ao (Å) Bulk

Modulus

B(GPa)

Equilibrium

Energy (Ry) Previous Calculated Δ(ao)

5.740[11] 5.770 0.03 250.437 -11873.836

-35 -30 -25 -20 -15 -10 -5 0 5 10-28000

-26000

-24000

-22000

-20000

-18000

-16000

-14000

-12000

-10000

-8000

E+P

V

P (GPa)

FIGURE 4. Enthalpy versus Pressure derivative.

The calculated bulk modus is 250.4376GPa and its pressure

derivative is found to be 7.4730 for Co2CrGe. The

optimized lattice constants for Co2CrGe is 5.770Å and the

change in the lattice constant of Co2CrGe with that of

experimental one is 0.030. The optimized lattice parameters

were slightly higher than the experimental lattice

parameters, the change in lattice parameter is given by

Δ(ao). As shown in above Figs. (2, 3) the volume derivative

decreases with the increase in pressure on the other hand

the enthalpy (H) is increasing and finally reaches -

9452.0364Ry at stable volume 324.0732a.u-3

and pressure

derivative of 7.4730. The values of total and local moments

are given in Table III in comparison with the earlier results.

In order to understand the formation of magnetic properties,

it is necessary to consider their density of states (DOS) and

band structure.

-25 -20 -15 -10 -5 0

1.00

1.05

1.10

1.15

1.20

V/Vo

P (GPa)

FIGURE 3. Change in the volume versus pressure derivative.

A first principle study of Co2CrGe: LSDA method


A. Spin Polarization and half-metallic ferromagnets

The electron spin polarization (P) at Fermi energy (EF) of a

material is defined by Eq. (1) [12].

F F

F F

E EP

E E

. (1)

Where FE and FE are the spin dependent density

of states at the EF. The ↑ and ↓ assigns the majority and the

minority states respectively. P vanishes for paramagnetic or

anti-ferromagnetic materials even below the magnetic

transition temperature. It has a finite value in ferromagnetic

materials below Curie temperature [13]. The electrons at EF

are fully spin polarized (P=100%) when FE or

FE equals to zero. In present work, we have studied the

Co2CrGe system which shows 100% spin polarization at EF

[TABLE II]. According to our result, the compound

Co2CrGe is interesting as it shows large DOS at the EF of

FE =3.00 states/eV [TABLE II]. The reason for this

large value is that EF cuts through strongly localized states

of Cr-d whereas the contribution of Co-d states to FE

is very small as illustrated in Fig. 4(b). On the other hand

FE =0.00 states/eV for both Co and Cr atoms according

to this Co2CrGe is a half-metal which gives 100% spin

polarization at EF.

TABLE II. Energy gap and Spin polarization.

Energy gap Eg (eV) Spin Polarization


0.24 0.00 0.24 3.00 0.00 100

FIGURE 5. Total DOS of Co2CrGe.

Figs. (5, 6) summarizes the results of the DOS which were

calculated using LSDA. It is shown in Figs. 6(a, c) the

majority contribution of DOS is from the d states of Co and

Cr atoms. The Cr-d gives almost an exchange splitting type

pattern as shown in Fig. 6(c). The sharp peaks appear at the

fermi level in spin up region for Cr-d atoms. The

contribution of Co-d atoms are very small at the conduction

region. The hybridization between the Co-d and Cr-d atoms

are appear between 3.5eV and 5.5eV, which responsible for

the creation of magnetic moment. According to Fig. 7 the

indirect band gap along the Γ-X symmetry for Co2CrGe is

0.24eV. For Co2CrGe the Fermi energy (EF) lies in the

middle of the gap of the minority-spin states, determining

the half-metal character [Fig. 5(a)]. The formation of gap

for the half-metal compounds was discussed by Galanakis

et al. [14] for Co2MnSi, is due to the strong hybridization

between Co-d and Mn-d states, combined with large local

magnetic moments and a sizeable separation of the d-like

band centers.

FIGURE 6. Partial DOS of Co and Cr atoms.

FIGURE 7. Band Structure.



B. Magnetic properties calculated in the LSDA

Starting with the compound under investigation, all the

information regarding the partial, total and the previously

calculated magnetic moments are summarized in Table III.

It is shown in Table III the calculated total magnetic

moment is almost an integer value in case of Co2CrGe as

expected for the half-metallic systems. In most cases the

calculated magnetic moments are in good agreement with

the previous results. We have found that the Co sites

contribute much less compared with the Cr sites this is

because Cr-d states shows exchange splitting. The same

observation was also reported and explained by Kandpal et

al. [15] because of the indirect connection between the

specific magnetic moment at Co and the hybridization

arising from the interaction between the electrons at the Co

sites with the neighboring electrons in the Co t-2g states. As

shown in Table III the Ge atom carry a negligible magnetic

moment, which does not contribute much to the overall

moment. We have also noticed that the partial moment of

Ge atoms aligned anti-parallel to Co and Cr moments of the

systems. It emerges from the hybridization with the

transition metals and is caused by the overlap of the

electron wave functions. The small moments found at the

Ge sites are mainly due to polarization of these atoms by

the surroundings, magnetically active atoms as reported by

Kandpal et al. [15].

TABLE III. Total and partial magnetic moments.

Magnetic Moment µB of Co2CrGe

Previous Calculated

Co Cr Ge Total

4.00[11] 0.932 2.122 -0.029 3.999

IV. CONCLUSIONS

We have performed the total-energy calculations to find the

stable magnetic configuration and the optimized lattice

constant. The DOS, magnetic moments and band structure

of Co2CrGe were calculated using FP-LAPW method. The

calculated results were in good agreement with the

previously calculated results. For Ferromagnetic

compounds the partial moment of Ge being very small and

the contribution is very less in the total magnetic moment.

We have investigated the possibility of appearance of half-

metallicity in the case of the full Heusler compound

Co2CrGe which shows 100% spin polarization at EF. The

existence of energy gap in minority spin (DOS and band

structure) of Co2CrGe is an indication of being a potential

HMF. This is also evident from the calculated magnetic

moment for Co2CrGe is 3.9999µB. The calculated result is

in qualitative agreement with the integral value, supporting

the HMF.

ACKNOWLEDGEMENT

DPR acknowledges DST inspires research fellowship and

RKT a research grant from UGC (New Delhi, India).

REFERENCES

[1] de Groot, R. A., Mueller, F. M., Van Engen, P. G. and

Buschow, K. H. J., New Class of Materials: Half-Metallic

Ferromagnets, Phys. Rev. Lett. 50, 2024–2027 (1983).

[2] Katsnelson, M. I., Irkhin, V. Y., Chioncel, L.,

Lichtenstein, A. I. and de Groot, R. A., Half-metallic

ferromagnets: From band structure to many-body effects,

Rev. Mod. Phys. 80, 315–378 (2008).

[3] Zutic, I., Fabian, J. and Sarma, S. D., Spintronics:

Fundamentals and applications, Rev. Mod. Phys. 76, 323–

410 (2004).

[4] de Boeck, W., van Roy, W., Das, J., Motsnyi, V., Liu,

Z., Lagae, L., Boeve, H., Dessein, K. and Borghs, G.,

Technology and materials issues in semiconductor-based

magnetoelectronics, Semicond. Sci. Technol. 17, 342-354

(2002).

[5] Rai, D. P., Hashemifar, J., Jamal, M., Lalmuanpuia.,

Ghimire, M. P., Sandeep., Khathing, D. T., Patra, P. K.,

Sharma, B. I., Rosangliana. and Thapa, R. K., Study of

Co2MnAl Heusler alloy as half metallic Ferromagnet,

Indian J. Phys. 84, 717-721 (2010).

[6] Rai, D. P., Sandeep, Ghimire, M. P. and Thapa, R. K.,

Study of energy bands and magnetic properties of Co2CrSi

Heusler alloy, Bull. Mat. Sc. 34, 1219-1222 (2011).

[7] Blaha, P., Schwarz, K., Madsen, G. K. H., Kvasnicka,

D., Luitz, J., Schwarz, K., Wien2K User’s Guide, (Vienna

University of Technology, Viena, 2008).

[8] Perdew, J. P., Burke, S., Ernzerhof, M., Generalized

Gradient Approximation Made Simple, Phys. Rev. Lett. 77,

3865-3868 (1996).

[9] Heusler, O., Kristallstruktur und Ferromagnetismus der

Mangan-Aluminium-Kupferlegierungen, Ann.Phys. 19,

155-201 (1934).

[10] Murnaghan, F. D., The Compressibility of Media under

Extreme Pressures, Proc. Natl. Acad. Sci. USA 30, 244-247

(1944).

[11] Gilleßen, M., Von der Fakult¨at f¨ur Mathematik,

Informatik und Naturwissenschaften der RWTH Aachen

University zur Erlangung des akademischen Grades eines

Doktors der Naturwissenschaften genehmigte Dissertation,

Thesis (2009).

[12] Soulen Jr., R. J., Byers, J. M., Osofsky, M. S.,

Nadgorny, B., Ambrose, T., Cheng, S. F., Broussard, P. R.,

Tanaka, C. T., Nowak, J., Moodera, J. S., Barry, A., and

Coey, J. M. D., Measuring the Spin Polarization of a Metal

with a Superconducting Point Contact, Science 282, 85-88

(1998).

[13] Ozdogan, K., Aktas, B., Galanakis, I. and Sasioglu, E.,

Influence of mixing the low-valent transition metal atoms

(Y, Y∗ = Cr, Mn, Fe) on the properties of the quaternary


















Co2[Y1−xY∗x], arxiv: cond-mat/0612194v1 [cond-

mat.mtrl-sci] (2006).

[14] Galanakis, I., Dederichs, P. H. and Papanikolaou, N.,

Slater-Pauling behavior and origin of the half-metallicity of

the full-Heusler alloys, Phys. Rev. B 66, 1744291-1744299

(2002).

[15] Kandpal, H. C., Fecher, G. H. and Felser, C.,

Calculated electronic and magnetic properties of the half-

metallic, transition metal based Heusler compounds, arXiv:

cond-mat/0611179v1 [cond-matm-matrl-sci] (2006).


A density functional theory (DFT) study of Co2CrGe: LSDA method

D. P. Rai, and R. K. Thapa

*

Department of Physics, Mizoram Universty, Aizawl, India 796004.


(Received 11 February 2012, accepted 18 May 2012)

Abstract The structural, electronic and magnetic properties of Co2CrGe, a Heusler alloy, have been evaluated by first principles

density functional theory and compared with the known experimental and theoretical results. Generalized gradient

approximation (GGA) is used for structural study where as Local spin density approximation (LSDA) for electronic

calculation. First principles structure optimizations were done through total energy calculations at 0K by the full

potential linearized augmented plane wave (FP-LAPW) method as implemented in WIEN2K code.

Keywords: GGA, half-metallicity, DOS and band structure.

Resumen Las propiedades estructurales, electrónicas y magnéticas del Co2CrGe, una aleación Heusler, han sido evaluadas por la

teoría de densidad funcional de primeros principios y son comparados con los experimentos conocidos y resultados

teóricos. La aproximación generalizad de gradiente (GGA) es usada para el estudio estructural con la aproximación de

la densidad spin local (LSDA) para el cálculo electrónico. Las optimizaciones de estructura de los primeros principios

fueron realizados a través de los cálculos de energía total a 0K por el potencial completo linealizado de onda plana (FP-

LAPW) como método implementado en el código WIEN2K.

Palabras clave: GGA, media-metalicidad, DOS y estructura de banda.

PACS: 70, 71.5.-m, 71.15.Mb, 71.20.Be ISSN 1870-9095

I. INTRODUCTION

In 1983, de Groot discovered half-metallic ferromagnetism

in semi-Heusler compound NiMnSb [1] by using first-

principle calculation based on density functional theory.

Heusler alloys are the ternary intermetallich compounds

with compostion X2YZ, where X and Y are transition

elements (Ni, Co, Fe, Mn, Cr, Ti, V etc.) and Z is III, IV or

V group elements (Al,Ga, Ge, AS, Sn, In etc.). One of the

promising classes of materials is the half-metallic

ferrimagnets, i.e., compounds for which only one spin

channel presents a gap at the Fermi level, while the other

has a metallic character, leading to 100% carrier spin

polarization at EF [4]. After that, half-metallicity attracted

much attention [2], because of its prospective applications

in spintronics [3]. The electronic and magnetic properties of

Co2MnAl [5] and Co2CrSi [6] using LSDA shows the half-

metallicity at the ground state. In this present work, we

report the result of GGA and LSDA of bulk electronic

structure and magnetic properties of Co2CrGe.

II. COMPUTATIONAL DETAILS AND CRYSTAL STRUCTURE

The FP-LAPW method (WIEN2K) [7] was applied to band

structure calculations of Co2CrGe. GGA [8] and LSDA

were used for the exchange correlation potential. The

multipole exapansion of the crystal potential and the

electron density within muffin tin (MT) spheres was cut at

l=10. Nonspherical contributions to the charge density and

potential within the MT spheres were considered up yo

lmax=6. The cut-off parameter was RKmax=7. In the



Gmax=12a.u-1

. The MT sphere radii(R) used were 2.35a.u.

for Co, 2.35a.u. for Cr and 2.21a.u. for Ge. The number of

k-points used in the irreducible part of the brillouin zone is

286.

Crystal structure: Heusler alloy [9] with chemical

formula X2YZ (X = Co, Y = Cr and Z = Ge). The full

Heusler structure consists of four penetrating fcc sublattices

with atoms at X1(1/4, 1/4, 1/4), X2(3/4, 3/4, 3/4), Y(1/2,



1/2, 1/2) and Z(0, 0, 0) positions which results in L21 crystal

structure having space group Fm-3-m as shown in Fig. 1.

FIGURE 1. Unit Cell Structure of Co2CrGe.

III. RESULTS AND DISCUSSIONS







of lattice constant and bulk modulus are presented in Table

I.

FIGURE 2. Volume optimization of Co2CrGe.

TABLE I. Lattice constant and Bulk modulus.

Lattice Constants ao (Å) Bulk

Modulus

B(GPa)

Equilibrium


5.740[11] 5.770 0.03 250.437 -11873.836

-35 -30 -25 -20 -15 -10 -5 0 5 10-28000

-26000

-24000

-22000

-20000

-18000

-16000

-14000

-12000

-10000

-8000

E+P

V

P (GPa)

FIGURE 4. Enthalpy versus Pressure derivative.

The calculated bulk modus is 250.4376GPa and its pressure

derivative is found to be 7.4730 for Co2CrGe. The

optimized lattice constants for Co2CrGe is 5.770Å and the

change in the lattice constant of Co2CrGe with that of

experimental one is 0.030. The optimized lattice parameters

were slightly higher than the experimental lattice

parameters, the change in lattice parameter is given by

Δ(ao). As shown in above Figs. (2, 3) the volume derivative

decreases with the increase in pressure on the other hand

the enthalpy (H) is increasing and finally reaches -

9452.0364Ry at stable volume 324.0732a.u-3

and pressure

derivative of 7.4730. The values of total and local moments

are given in Table III in comparison with the earlier results.

In order to understand the formation of magnetic properties,

it is necessary to consider their density of states (DOS) and

band structure.

-25 -20 -15 -10 -5 0

1.00

1.05

1.10

1.15

1.20

V/Vo

P (GPa)

FIGURE 3. Change in the volume versus pressure derivative.



A. Spin Polarization and half-metallic ferromagnets

The electron spin polarization (P) at Fermi energy (EF) of a

material is defined by Eq. (1) [12].

F F

F F

E EP

E E

. (1)

Where FE and FE are the spin dependent density

of states at the EF. The ↑ and ↓ assigns the majority and the

minority states respectively. P vanishes for paramagnetic or

anti-ferromagnetic materials even below the magnetic

transition temperature. It has a finite value in ferromagnetic

materials below Curie temperature [13]. The electrons at EF

are fully spin polarized (P=100%) when FE or

FE equals to zero. In present work, we have studied the

Co2CrGe system which shows 100% spin polarization at EF

[TABLE II]. According to our result, the compound

Co2CrGe is interesting as it shows large DOS at the EF of

FE =3.00 states/eV [TABLE II]. The reason for this

large value is that EF cuts through strongly localized states

of Cr-d whereas the contribution of Co-d states to FE

is very small as illustrated in Fig. 4(b). On the other hand

FE =0.00 states/eV for both Co and Cr atoms according

to this Co2CrGe is a half-metal which gives 100% spin

polarization at EF.

TABLE II. Energy gap and Spin polarization.

Energy gap Eg (eV) Spin Polarization


0.24 0.00 0.24 3.00 0.00 100

FIGURE 5. Total DOS of Co2CrGe.

Figs. (5, 6) summarizes the results of the DOS which were

calculated using LSDA. It is shown in Figs. 6(a, c) the

majority contribution of DOS is from the d states of Co and

Cr atoms. The Cr-d gives almost an exchange splitting type

pattern as shown in Fig. 6(c). The sharp peaks appear at the

fermi level in spin up region for Cr-d atoms. The

contribution of Co-d atoms are very small at the conduction

region. The hybridization between the Co-d and Cr-d atoms

are appear between 3.5eV and 5.5eV, which responsible for

the creation of magnetic moment. According to Fig. 7 the

indirect band gap along the Γ-X symmetry for Co2CrGe is

0.24eV. For Co2CrGe the Fermi energy (EF) lies in the

middle of the gap of the minority-spin states, determining

the half-metal character [Fig. 5(a)]. The formation of gap

for the half-metal compounds was discussed by Galanakis

et al. [14] for Co2MnSi, is due to the strong hybridization

between Co-d and Mn-d states, combined with large local

magnetic moments and a sizeable separation of the d-like

band centers.

FIGURE 6. Partial DOS of Co and Cr atoms.

FIGURE 7. Band Structure.



B. Magnetic properties calculated in the LSDA

Starting with the compound under investigation, all the

information regarding the partial, total and the previously

calculated magnetic moments are summarized in Table III.

It is shown in Table III the calculated total magnetic

moment is almost an integer value in case of Co2CrGe as

expected for the half-metallic systems. In most cases the

calculated magnetic moments are in good agreement with

the previous results. We have found that the Co sites

contribute much less compared with the Cr sites this is

because Cr-d states shows exchange splitting. The same

observation was also reported and explained by Kandpal et

al. [15] because of the indirect connection between the

specific magnetic moment at Co and the hybridization

arising from the interaction between the electrons at the Co

sites with the neighboring electrons in the Co t-2g states. As

shown in Table III the Ge atom carry a negligible magnetic

moment, which does not contribute much to the overall

moment. We have also noticed that the partial moment of

Ge atoms aligned anti-parallel to Co and Cr moments of the

systems. It emerges from the hybridization with the

transition metals and is caused by the overlap of the

electron wave functions. The small moments found at the

Ge sites are mainly due to polarization of these atoms by

the surroundings, magnetically active atoms as reported by

Kandpal et al. [15].

TABLE III. Total and partial magnetic moments.

Magnetic Moment µB of Co2CrGe

Previous Calculated

Co Cr Ge Total

4.00[11] 0.932 2.122 -0.029 3.999

IV. CONCLUSIONS

We have performed the total-energy calculations to find the

stable magnetic configuration and the optimized lattice

constant. The DOS, magnetic moments and band structure

of Co2CrGe were calculated using FP-LAPW method. The

calculated results were in good agreement with the

previously calculated results. For Ferromagnetic

compounds the partial moment of Ge being very small and

the contribution is very less in the total magnetic moment.

We have investigated the possibility of appearance of half-

metallicity in the case of the full Heusler compound

Co2CrGe which shows 100% spin polarization at EF. The

existence of energy gap in minority spin (DOS and band

structure) of Co2CrGe is an indication of being a potential

HMF. This is also evident from the calculated magnetic

moment for Co2CrGe is 3.9999µB. The calculated result is

in qualitative agreement with the integral value, supporting

the HMF.

ACKNOWLEDGEMENT

DPR acknowledges DST inspires research fellowship and

RKT a research grant from UGC (New Delhi, India).

REFERENCES

[1] de Groot, R. A., Mueller, F. M., Van Engen, P. G. and

Buschow, K. H. J., New Class of Materials: Half-Metallic

Ferromagnets, Phys. Rev. Lett. 50, 2024–2027 (1983).

[2] Katsnelson, M. I., Irkhin, V. Y., Chioncel, L.,

Lichtenstein, A. I. and de Groot, R. A., Half-metallic

ferromagnets: From band structure to many-body effects,

Rev. Mod. Phys. 80, 315–378 (2008).

[3] Zutic, I., Fabian, J. and Sarma, S. D., Spintronics:

Fundamentals and applications, Rev. Mod. Phys. 76, 323–

410 (2004).

[4] de Boeck, W., van Roy, W., Das, J., Motsnyi, V., Liu,

Z., Lagae, L., Boeve, H., Dessein, K. and Borghs, G.,

Technology and materials issues in semiconductor-based

magnetoelectronics, Semicond. Sci. Technol. 17, 342-354

(2002).

[5] Rai, D. P., Hashemifar, J., Jamal, M., Lalmuanpuia.,

Ghimire, M. P., Sandeep., Khathing, D. T., Patra, P. K.,

Sharma, B. I., Rosangliana. and Thapa, R. K., Study of

Co2MnAl Heusler alloy as half metallic Ferromagnet,

Indian J. Phys. 84, 717-721 (2010).

[6] Rai, D. P., Sandeep, Ghimire, M. P. and Thapa, R. K.,


Heusler alloy, Bull. Mat. Sc. 34, 1219-1222 (2011).

[7] Blaha, P., Schwarz, K., Madsen, G. K. H., Kvasnicka,

D., Luitz, J., Schwarz, K., Wien2K User’s Guide, (Vienna

University of Technology, Viena, 2008).

[8] Perdew, J. P., Burke, S., Ernzerhof, M., Generalized

Gradient Approximation Made Simple, Phys. Rev. Lett. 77,

3865-3868 (1996).

[9] Heusler, O., Kristallstruktur und Ferromagnetismus der

Mangan-Aluminium-Kupferlegierungen, Ann.Phys. 19,

155-201 (1934).

[10] Murnaghan, F. D., The Compressibility of Media under

Extreme Pressures, Proc. Natl. Acad. Sci. USA 30, 244-247

(1944).

[11] Gilleßen, M., Von der Fakult¨at f¨ur Mathematik,

Informatik und Naturwissenschaften der RWTH Aachen

University zur Erlangung des akademischen Grades eines

Doktors der Naturwissenschaften genehmigte Dissertation,

Thesis (2009).

[12] Soulen Jr., R. J., Byers, J. M., Osofsky, M. S.,

Nadgorny, B., Ambrose, T., Cheng, S. F., Broussard, P. R.,

Tanaka, C. T., Nowak, J., Moodera, J. S., Barry, A., and

Coey, J. M. D., Measuring the Spin Polarization of a Metal

with a Superconducting Point Contact, Science 282, 85-88

(1998).

[13] Ozdogan, K., Aktas, B., Galanakis, I. and Sasioglu, E.,

Influence of mixing the low-valent transition metal atoms

(Y, Y∗ = Cr, Mn, Fe) on the properties of the quaternary


















Co2[Y1−xY∗x], arxiv: cond-mat/0612194v1 [cond-

mat.mtrl-sci] (2006).

[14] Galanakis, I., Dederichs, P. H. and Papanikolaou, N.,

Slater-Pauling behavior and origin of the half-metallicity of

the full-Heusler alloys, Phys. Rev. B 66, 1744291-1744299

(2002).

[15] Kandpal, H. C., Fecher, G. H. and Felser, C.,


metallic, transition metal based Heusler compounds, arXiv:

cond-mat/0611179v1 [cond-matm-matrl-sci] (2006).




10

ELECTRONIC STRUCTURE AND MAGNETIC PROPERTIES OF Co2MnSi BY USING LSDA+U METHOD. D. P. Rai1, Sandeep1, A. Shankar1 , R. K. Thapa* 1 and M.P. Ghimire2

1Condensed Matter Theory Group,Department of Physics, Mizoram University, Aizawl, INDIA-796004. 2MANA, National Institute for Material Sciences (NIMS), Tsukuba, JAPAN.

Abstract











PACS No: 70, 71.5.-m, 71.15.Mb, 71.20.Be


11

INTRODUCTION



























Rai et al

12























Fig.1.


13



















Rai et al

14







atoms.










15











LSDA+U






Å

Magnetic Moment µB

Energy gap Eg (eV)

Previous Our Result

Previous Our Result

Previous Our Result

Co2MnSi

5.64326 5.665 5.000 26 5.031 0.79826 0.720

Rai et al

16

















17















Rai et al

18






bonding hybrids.









19



























Rai et al

20






Coulomb

Repulsion



0.05 0.9406 3.2118 -0.0598 5.0000 0.760

0.10 0.9136 3.3236 -0.0744 5.0116 0.840

0.20 0.9383 3.5372 -0.0904 5.1538 1.000

UCo=0.29

UMn=0.27

1.1039 3.7068 -0.1089 5.6118 1.200

0.40 1.7045 4.0696 -0.0699 7.3945 1.400







21

0.0 0.1 0.2 0.3 0.4

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

Ma

gn

etic m

om

en

t (u

B),

En

erg

y g

ap

(e

V)

U (Ry)

Mtot

MCo

MMn

Eg



CONCLUSION








Rai et al

22






















spintronics.

ACKNOWLEDGEMENTS




23






















Rai et al

24


























25







Phys. Rev. B 69, 144413 (2004).




10

ELECTRONIC STRUCTURE AND MAGNETIC PROPERTIES OF Co2MnSi BY USING LSDA+U METHOD. D. P. Rai1, Sandeep1, A. Shankar1 , R. K. Thapa* 1 and M.P. Ghimire2

1Condensed Matter Theory Group,Department of Physics, Mizoram University, Aizawl, INDIA-796004. 2MANA, National Institute for Material Sciences (NIMS), Tsukuba, JAPAN.

Abstract











PACS No: 70, 71.5.-m, 71.15.Mb, 71.20.Be


11

INTRODUCTION



























Rai et al

12























Fig.1.


13



















Rai et al

14







atoms.










15











LSDA+U






Å

Magnetic Moment µB

Energy gap Eg (eV)

Previous Our Result

Previous Our Result

Previous Our Result

Co2MnSi

5.64326 5.665 5.000 26 5.031 0.79826 0.720

Rai et al

16

















17















Rai et al

18






bonding hybrids.









19



























Rai et al

20






Coulomb

Repulsion



0.05 0.9406 3.2118 -0.0598 5.0000 0.760

0.10 0.9136 3.3236 -0.0744 5.0116 0.840

0.20 0.9383 3.5372 -0.0904 5.1538 1.000

UCo=0.29

UMn=0.27

1.1039 3.7068 -0.1089 5.6118 1.200

0.40 1.7045 4.0696 -0.0699 7.3945 1.400







21

0.0 0.1 0.2 0.3 0.4

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

6.5

7.0

7.5

Ma

gn

etic m

om

en

t (u

B),

En

erg

y g

ap

(e

V)

U (Ry)

Mtot

MCo

MMn

Eg



CONCLUSION








Rai et al

22






















spintronics.

ACKNOWLEDGEMENTS




23






















Rai et al

24


























25







Phys. Rev. B 69, 144413 (2004).

M. J. CONDENSED MATTER VOLUME 14, NUMBER 1 MARCH 2012

MJCM, VOLUME 14, NUMBER 1 © 2010 The Moroccan Statistical Physical and Condensed Matter Society

Study of Heusler compounds Co2YSi (Y = Mn, Cr) using a full potential linearized

augmented plane wave (FP-LAPW) method

D. P. Rai1, Sandeep

1, A. Shankar

1, M. P. Ghimire

2, and R. K. Thapa

1*

1. Condensed Matter Theory Group, Department of Physics, Mizoram University,

Aizawl-796004, India

2. Nepal Academy of Science and Technology, Kathmandu, Nepal

Email:[email protected]

Tel: +918014613900, Fax: 0389 - 2330522

Abstract: We have performed the volume optimization followed by the calculation of electronic structure and magnetic properties on

Co2MnSi and Co2CrSi. The structure optimization was based on generalized gradient approximation (GGA) method. The calculation of

electronic structure was based on full potential linear augmented plane wave (FP-LAPW) method and exchange correlation. Results of

density of states (DOS) and band structures shows the half-metallicity of Co2MnSi and Co2CrSi with an integer value of magnetic moments

5.031 µB and 4.006 µB respectively which follows the Slater-Pauling rule.

Key words: GGA, LSDA, half-metallicity, DOS and band structure.

PACS numbers: --71.15.Mb, 71.15.m, 71.20.-b, 71.15.Ap

I. Introduction

Half-metallic ferromagnets (HMFs), is a magnetic

material where the majority spin band is metallic and the

minority–spin band is semiconducting with an energy gap

at the Fermi level. In present scientific research of material

science, the half-metal ferromagnets have become one of

the most studied classes of materials. The existence of a

gap in the minority-spin band structure leads to 100% spin

polarization of the electron states at the Fermi level which

makes the systems applicable for the developing field of

spintronics [1]. In half-metals, the creation of a fully spin-

polarized current should be possible, that should maximize

the efficiency of magnetoelectronics devices [2]. Materials

having high spin polarization can be used for tunnel

magnetoresistance(TMR) and the giant magnetoresistance

(GMR) [3]. The Co-based Heusler alloys Co2YZ (Y:

transition metal, Z: sp atom) 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 [4]. Sandeep et al. [5, 6] studied the

electronic properties of compounds like NdCrSb3,

SmCrSb3 and GdCrSb3 by first principles method. Ghimire

et al. [7, 8] studied the electronic properties of CrO2 using

density functional theory and also studied the electronic

and optical properties of SbTaO4. Rai et al. [9, 10]

investigated the ground state of Co2MnAl and Co2CrSi

using LDA+U and LSDA method respectively and

reported the half-metallicity. 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 [11]. 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 [12]. In this

paper we have studied the full-Heusler compounds

Co2MnSi and Co2CrSi within FP-LAPW method. Our

main aim is to investigate the half-metallic behaviour of

these compounds, which are a better prospective for the

spintronic devices.

II. Crystal structure and Calculation Details

2.1 Crystal structure: Heusler alloy [13] with chemical

formula Co2YSb (Y = Sc, Ti), has full Heusler structure

with four penetrating fcc sublattices with atoms at

X1(1/4,1/4,1/4), X2(3/4,3/4,3/4), Y(1/2,1/2,1/2) and

14 1

D. P. Rai, Sandeep, A. Shankar, M. P. Ghimire, and R. K. Thapa


Z(0,0,0) positions which results in L21 crystal structure

having space group Fm-3-m as shown in Fig-1.

Fig. 1: Unit cell Structure of Co2YSi: Co (red), Y (yellow)

and Si (blue) atoms.

2.2 Calculation Details: The FP-LAPW method

(WIEN2K) [14] was applied to band structure calculations

of Co2CrGe. In this method the space is divided into non-

overlapping muffin-tin (MT) spheres separated by an

interstitial region. The basis functions are expanded into

spherical harmonic functions inside the muffin-tin sphere

and the Fourier series in the interstitial region. The

convergence of basis set was controlled by a cutoff

parameter RMT x Kmax = 7 where RMT is the smallest of the

MT sphere radii and Kmax is the largest reciprocal lattice

vector used in the plane wave expansion and made the

expansion up to lmax = 6 in the muffin tins, where lmax is

the maximum value of angular momentum. The magnitude

of the largest vector in charge density Fourier expansion

(Gmax) was 12 a.u-1

. The cutoff energy which defines the

separation of valence and core states was chosen as -6.0

Ry. For k-point sampling, a 21x21x21 k-point mesh in the

first Brillouin zone was used. LSDA [15] was used and the

convergence criterion for self–consistence calculations

was set up to charge convergence equal to 10-4

. In the



Gmax=12 a.u-1

. The MT sphere radii(R) used were 2.35 a.

u. for Co, 2.35 a.u. for Cr and 2.21 a. u. for Si and in case

of Co2MnSi, 2.34 a. u. for Co, 2.34 a.u. for Mn and 2.20

a. u. for Si. The number of k-points used in the irreducible

part of the brillouin zone is 286.

III. Results and Discussions







of lattice constant and bulk modulus are presented in

Table 1.

Table 1: The previous lattice constant, calculated lattice constant and bulk modulus

Compound Lattice constant ao (Å) Bulk modulus (GPa)

Previous Our Calculation

Co2MnSi 5.645[18]

5.665 866.499

Co2CrSi 5.647[17]

5.699 405.556

Fig-2 shows the DOS of Co2MnSi, at -1.4 eV below EF in

spin down and spin up regions are contributed by Mn-d

atoms but Mn atoms has less contribution in the spin down

region. Similarly at -1.4eV, Co-d atoms also contributed in

both spin up and spin down regions. At 3.2eV and 4.2eV

in spin down region both Mn-d and Co-d atoms

contributed to the total DOS. But negligible contribution

from Co and Mn atoms in the spin up channel above EF. It

is shown in Fig-3 that the hybridization takes place

between the Co-d-eg states and Mn-d-eg states below EF

in the spin up channel, which is responsible for

contribution of magnetic moments. The exchange splitting

occurs between the Mn-d electrons at -1.8 eV and 1.8 eV

is also responsible for the creation of gap in the spin down

channel shown in Fig-2(b). From Fig-4(b), in Co2CrSi

peaks due to Co atoms are found in the valence region for

both the spins. In the conduction region sharp peaks were

observed at 0.70 eV and 1.8 eV which were contributed by

Co-d states. An exchange splitting also occurs between

Co-d up and down with a splitting energy of 1.4 eV. In

spin down channel, Cr-d contributes both in the valence

and conduction regions. We have observed two sharp

2 14

3




peaks -1.4 eV and -2.0 eV in valence region due to Cr-d

states. An exchange splitting of 2.4 eV is observed

between spin down and spin up channel due to Cr-d states.

This exchane splitting is responsible for the creation of

energy gap as well as magnetic moment. The total

magnetic moments and energy gaps of Co2MnSi and

Co2CrSi are given in Table 2.

Table 2: The magnetic moments and energy gap are given below.

Compounds Magnetic Moment µB Energy gap Eg (eV)

Previuos Our Calculation Previous Our Calculation

Co2MnSi 4.900[17]

5.031 0.798[18]

0.760

5.000[18]

Co2CrSi 4.000[17]

4.006 0.878[17]

0.910

Fig. 2: (a) DOS plot for Mn atoms in spin up (b) DOS plot for Mn atoms in spin down (c) DOS plot for Co atoms in spin up

and (d) DOS plot for Co atoms in spin down.

The partial magnetic moments of the atoms Co, Mn and Si

are 1.029 µB, 3.058 µB and -0.055 µB respectively. Thus

the total magnetic moment is 5.031 µB which is

approximately an integer value 5.00 µB [18]. There are 9

electrons in the valence shell of Co atoms, 7 electrons for

Mn atom, 6 electrons for Cr atom and 4 electrons for Si

atom. According to Slater-Pauling rule [19] the magnetic

moment of Co2MnSi is 2x9+7+4-24= 5 µB, similarly 4 µB

for Co2CrSi which are consistent with our results. The

total and partial magnetic moments of Co2MnSi and

Co2CrSi are tabulated in Table 3.

Table 3: The partial and total magnetic moments are tabulated and compared with the previous results.

Compounds Previous Magnetic Moment µB Calculated Magnetic Moment µB

Co Y Total Co Y Si Total

Co2MnSi 1.06 2.99 5.00[18]

1.03 3.06 -0.06 5.031

Co2CrSi 0.98 2.08 4.00[17]

0.98 2.10 -0.505 4.006

1.00 2.03 4.00[18]

14 3



3.1. Band Structure

For overall band structure calculation of Co2MnSi

compounds there exist an energy gap Eg in the spin down

region. The width of the energy gap Eg is the difference of

the energies of the highest occupied band in the valence

region at the Г-point and the lowest unoccupied band in

the conduction region at the X-point, thus it is an indirect

gap. With the help of the DOS, it is clear that the energy

region lower than -3eV consists mainly of s and p

electrons of the Si atoms which is called the core and the

energy region between -3eV and 2eV consists mainly of

the d-electrons of Co and Mn atoms. Based on the analysis

of the band-structure calculation it was shown that the 3d

orbitals of Co atoms from two different sub-lattices,

Co1(0, 0, 0) and Co

2(1/2, 1/2, 1/2), couple and form

bonding hybrids Co1(t2g/eg) -Co

2(t2g/eg). In other words,

the t2g/eg orbitals of one of the Co atoms can couple only

with the t2g/eg orbitals the other Co atom. Furthermore, the

Co-Co hybrid bonding orbitals hybridize with the Mn(d)-

t2g,eg manifold, while the Co-Co hybrid antibonding

orbitals remain uncoupled owing to their symmetry. The

Co-Co hybrid antibonding t2g is situated below the Fermi

energy EF and the Co-Co hybrid antibonding eg is

unoccupied and lies just above the Fermi level.

Fig. 3: (a) Energy band of Co2MnSi for spin up (b) Total DOS of Co2MnSi and

(c) Energy band of Co2MnSi for spin down.

Fig. 4: (a) Energy band of Co2CrSi for spin up (b) Total DOS of Co2CrSi and

(c) Energy band of Co2CrSi for spin down.

4 14




Thus due to the missing Mn(d)-t2g,eg and Co-Co hybrid

antibonding hybridization, the Fermi energy is situated

within the minority gap formed by the triply degenerate

Co-Co antibonding t2g and the doubly degenerate Co-Co

antibonding eg. From Figs-3 (a-c) it is clearly shown that

the EF lies exactly at the middle of the gap between the

valence and the conduction bands in spin down region

where as there is no such gap occur in the spin up region.

Thus there exist a half metallicity in Co2MnSi. The value

of energy gap between Г and X is 0.760 eV which is an

indirect band gap. Almost similar results of 0.798 eV had

been reported by Kandpal et al. (2006) [18]. From Figs-

4(a-c) for Co2CrSi, we have observed an indirect band gap

in spin down channel. The calculated energy gap along Г-

X symmetry is 0.91 eV which is almost close to 0.878 eV

[17].

IV. Conclusions

It can be said that due to the existence of gap in DOS for

the minority spins, Co2MnSi and Co2CrSi is a potential

half-metallic ferromagnet. This is also evident from the

energy band results as discussed. The calculated magnetic

moment for Co2MnSi is 5.031 µB and 4.006 µB for

Co2CrSi which is equal to an integer value and follows the

Slater-Pauling rule. The integral value of magnetic

moment is also one of the evidences for the 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 makes

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.

Acknowledgements

DPR acknowledges DST Inspire fellowship; AS and RKT

a research grant from UGC, New Delhi.

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Slater-Pauling behavior and origin of the half-metallicity

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14 6

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Phase Transitions: A MultinationalJournalPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/gpht20

Electronic structure and magneticproperties of X2YZ (X = Co, Y = Mn,Z = Ge, Sn) type Heusler compounds: afirst principle studyD.P. Rai a & R.K. Thapa aa Condensed Matter Theory Group, Department of Physics,Mizoram University, Aizawl, Mizoram 796004, India

Available online: 13 Mar 2012

To cite this article: D.P. Rai & R.K. Thapa (2012): Electronic structure and magnetic properties ofX2YZ (X = Co, Y = Mn, Z = Ge, Sn) type Heusler compounds: a first principle study, Phase Transitions: AMultinational Journal, DOI:10.1080/01411594.2012.661860

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Phase Transitions2012, 1–11, iFirst

Electronic structure and magnetic properties of X2YZ (X^Co,

Y^Mn, Z^Ge, Sn) type Heusler compounds: a first principle study

D.P. Rai and R.K. Thapa*

Condensed Matter Theory Group, Department of Physics, Mizoram University, Aizawl,Mizoram 796004, India

(Received 4 September 2011; final version received 24 January 2012)

We performed the structure optimization followed by the calculation of electronicstructure and magnetic properties on Co2MnGe and Co2MnSn. The structureoptimization was based on generalized gradient approximation exchange corre-lation and full potential linearized augmented plane wave (FP-LAPW) method.The calculation of electronic structure was based on FP-LAPW method usinglocal spin density approximation. We have studied the electronic structure andmagnetic properties. The calculated density of states and band structures showsthe half-metallic ferromagnets character of Co2MnGe and Co2MnSn.

Keywords: GGA; half-metallicity; DOS and band structure

1. Introduction

In 1983, de Groot discovered half-metallic ferromagnetism (HMF) in semi-Heuslercompound NiMnSb [1] by using first-principle calculation based on density functionaltheory. After that, half-metallicity attracted much attention [2], because of its prospectiveapplications in spintronics [3]. Recently rapid development of magneto-electronicsintensified the research on ferromagnetic materials that are suitable for spin injectioninto a semiconductor [4]. One of the promising classes of materials is the HMF, i.e.,compounds for which only one spin channel presents a gap at the Fermi level, while theother has a metallic character, leading to 100% carrier spin polarization at EF [3,5]. Ishidaet al. [6] have also proposed that the full-Heusler alloy compounds of the type Co2MnZ,(Z¼Ge, Sn), are half-metals. Heusler alloys have been particularly interesting systemsbecause they exhibit much higher ferromagnetic Curie temperature than other half-metallic materials [7]. Among the other properties useful for the applications are thecrystal structure and lattice matching compatible with zinc-blende semiconductors usedindustrially [8,9]. Jiang et al. [10] examined the magnetic structure of Mn2VAl by X-raydiffraction and magnetization measurements. Rai et al. [11,12] investigated the groundstate study of Co2MnAl and Co2CrSi using LDAþU and local spin density approximation(LSDA) method, respectively. The preparation and characterization of bulk Co2Mn(Si, Ge, Ga, and Sn) to be used as targets for pulsed laser deposition of magnetic contactsfor spintronic devices [13]. Weht and Pickett [14] performed the theoretical studies of

*Corresponding author. Email: [email protected]

ISSN 0141–1594 print/ISSN 1029–0338 online

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Mn2VAl using generalized gradient approximation (GGA) [15] which shows the half-metallic character with a gap at the spin-up band instead of the spin-down band as for theother half-metallic Heusler alloys.

2. Motivation

In this article, we systematically study the electronic and magnetic structure of Co-basedfull Heusler alloys Co2MnZ (Z¼Ge and Sn) to search for new HMF candidates. Amongthe systems studied Co2MnZ (Z¼Ge and Sn) are predicted to be nearly half-metallic attheoretical equilibrium lattice constants. Materials with high spin polarization can be usedfor tunnel magnetoresistance and giant magnetoresistance [16]. The Co-based Heusleralloys Co2YZ (Y: transition metal, Z: sp atom) are the most prospective candidates for theapplication in spintronics. This is due to a high Curie temperature beyond roomtemperature and the simple fabrication process such as DC magnetron sputtering inCo2YZ [17].

3. Crystal structure

The considered full-Heusler alloys Co2YZ adopt the ordered L21-type structure (spacegroup Fm-3-m) which may be understood as the result of four interpenetrating face-centered-cubic (fcc) lattices. The Y and Z atoms occupy two fcc sublattices with origin at(0, 0, 0) and (½,½,½), respectively. The Co atoms are located in sublattices with originsat (¼,¼,¼,) and (g,g,g,) (Figure 1).

3.1. Computational methods

We performed the structural optimization followed by the calculation of electronicstructure and magnetic properties. In Figures 2 and 3 we have projected density of states(DOS) of Co2YZ. The majority-spin states are shown in the upper portion and theminority-spin states are in the lower portion of every half. The first step, we optimized theparameters (lattice constant). The total energy is calculated by using parameterization ofGGA method. The calculation is accomplished by using the WIEN2K code [18]. In thenext step, we calculated the electronic structure and magnetic properties using fullpotential linear augmented plane wave (FP-LAPW) method. The accuracy is up to 104

Ry. The exchange–correlation potential is chosen in the LSDA [19]. The self-consistentpotentials were calculated on a 20 20 20 k-mesh in the Brillouin zone, which

Figure 1. (Color online). Structure of the Co2YZ Heusler alloy: Co (red) atoms are at the origin and(1/2, 1/2, 1/2), Y (yellow) at (1/4, 1/4, 1/4), and Z (blue) atoms at (3/4, 3/4, 3/4). The cubic L21

structure consists of four inter-penetrating fcc sub-lattices, two of which are equally occupied by Co.The two Co-site fcc sub-lattices combine to form a simple cubic sub-lattice.

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corresponds to 256 k points in the irreducible Brillouin zone. The sets of valence orbitals in

the calculations were selected as 3s, 3p, 4s, 4p, 3d Co atoms, 3s, 3p, 4s, 4p, 3d for Mn

atoms, 3d, 4s, 4p for Ge atoms, and 4d, 5s, 5p for Sn atoms. All lower states were treated

as core states.


4.1. The equilibrium energy and lattice parameter

The first task which we undertook was to calculate how the total energy changes with the

lattice constant and thus find the theoretical equilibrium lattice parameter. The optimized

lattice constant, isothermal bulk modulus, its pressure derivative are calculated by fitting

the total energy to Murnaghan’s equation of state [20]. We summarize our results

in Figure 4. The optimized lattice parameters were slightly higher than the experimental

lattice parameters. It is confirmed that the HMF configuration is lower in energy for the

system Co2MnSn (Table 1). The results of the structural optimization are shown in

Figure 4. The detail values of the optimized lattice parameters and bulk modulus are given

in Table 1.

Figure 2. DOS and band structure of Co2MnGe. (a) Total DOS plots of Co2MnGe, (b) partial DOSplots of Co, (c) partial DOS plots of Mn, and (d) partial DOS plots of Ge.

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4.1.1. DOS and magnetic moments

The total and partial DOS plots of Co2MnGe and Co2MnSn are shown in Figures 2 and 3and Figures 5 and 6, respectively. From the DOS plots of Co2MnGe and Co2MnSn,shown in Figures 2(b) and 5(b), peaks are mostly due to d state electrons of Co atoms inthe semi-core and the valence region below EF for both spin channels. Similarly, the DOSplots of Co2MnGe and Co2MnSn, as shown in Figures 2(c) and 5(d), sharp peaks arefound in spin up and spin down channels above and below EF, respectively, which are dueto d electrons of Mn atoms. In spin up channel, the DOS intersects EF showing the metallicnature. For Co2MnGe we have found in Figures 2(c) and 3 that the contribution is due toMn-d electrons in the valence region with sharp peaks at 2.3 eV and 2.7 eV while, incase of Co2MnSn, Mn-d electrons contribute at 2.4 eV and 2.8 eV. Figures 5(d) and 6show the spin-up configuration with a Fermi cut-off. In spin down channel, peaks are dueto d states of Co atoms in the conduction region. It is shown in Figures 2 and 5, EF lyingmidway between the gap in spin down channel for both Co2MnGe and Co2MnSn,respectively, showing the semi-conducting behavior. In the conduction region ofCo2MnGe, peaks were observed at 1.0 and 2.0 eV and for Co2MnSn, peaks are observedat 0.9 and 1.8 eV. These peaks are mainly due to d-eg and d-t2g states of Co and Mn atoms,respectively, in spin down region (Figures 3(b and d) and Figures 6(b and d)). The delectrons of Co are found to strongly hybridized with Mn-d electrons [21]. The partialmagnetic moments of Co, Mn, and Ge atoms are 0.975 mB, 3.097 mB, and 0.044 mB,respectively, for Co2MnGe. The effective magnetic moment is 5.004 mB which is

Figure 3. DOS plots of Co2MnGe, (a) Co (d, deg) and Mn (d, deg) states in spin-up; (b) Co (d, deg)and Mn (d, deg) states in spin-down; (c) Co (d, dt2g) and Mn (d, dt2g) states in spin-up; and (d) Co (d,dt2g) and Mn (d, dt2g) states in spin-down.


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approximately an integer value 5.00 mB [21]. Similarly for Co2MnSn, the partial magneticmoments of Co, Mn, and Sn atoms are 0.950 mB, 3.253 mB, and 0.059 mB, respectively. Theeffective magnetic moment is thus 5.016 mB which is approximately an integer value 5.03mB[21] (Table 2).

4.1.2. Band structures

Figures 7(a and b) and 8(a and b) show the band structure plots of Co2MnGe andCo2MnSn in both spin channels. In the valence region of the spin up and down channels,more number of bands were seen which are due to the 3d states of Mn atoms of Co2MnGe

Figure 4. Volume optimization of (a) Co2MnGe and (b) Co2MnSn.

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and Co2MnSn. Spin-down channel comprises of dense or large number of energy bands inthe conduction region above EF due to 3d electrons of Mn atoms. The width of energy gap(Eg) is the difference in energies of the highest occupied band at symmetry point D in thevalence region and the lowest unoccupied band in the conduction region at symmetrypoint X, which is an indirect band gap. With the help of DOS, it is clear that the energyregion lower than 3 eV consists mainly of s and p electrons (not shown) of the Ge atomsin the valence region and the energy region between 3 eV and 2 eV consists mainly of the

Figure 5. DOS and band structure of Co2MnSn. (a) Total DOS plots of Co2MnSn, (b) partial DOSplots of Co, (c) partial DOS plots of Sn, and (d) partial DOS plots of Mn.

Table 1. The calculated lattice parameters, the equilibrium energy, and magnetic moments in mB arecompared with the previous results.

Compound

Lattice constant ao (A)

k-value Equilibrium energy (Ry) Electronic chargeExperimental Theoretical

Co2MnGe 5.737 [22]5.743 [26]5.749 [21]

5.678 10,000 12089.018 111

Co2MnSn 6.00 [24]5.997 [23]5.95 [25]5.984 [21]

5.97 10,000 20248.956 129


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d-electrons of Co and Mn atoms. From Figures 7(b) and 8(b) it is seen that EF lies almostat the middle of the gap between the valence and the conduction region in spin downchannel. The origin of minority gap in Co2MnGe and Co2MnSn was explained byKandpal et al. [21] and Galanakis and Mavropoulos [27]. Based on the analysis of bandstructures and DOS calculations it is seen that the 3d orbitals of Co atoms from twodifferent sub-lattices, Co1 (0, 0, 0) and Co2 (1/2, 1/2, 1/2) couple and form bonding hybrids.In other words, the gap originates from the strong hybridization between the d states of

Figure 6. DOS plots of Co2MnSn, (a) Co (d, deg) and Mn (d, deg) states in spin-up; (b) Co (d, deg)and Mn (d, deg) states in spin-down; (c) Co (d, dt2g) and Mn (d, dt2g) states in spin-up; and (d) Co (d,dt2g) and Mn (d, dt2g) states in spin-down.

Table 2. The experimental, the theoretical lattice constants, magnetic moments, energy gaps, andbulk modulus.

Compound

Magnetic moment (mB) Energy gap Eg (eV)Bulk modulus

(GPa)Previous Observed Previous Observed

Co2MnGe 5.00 [26]5.00 [21]

5.004 0.581 [21] 0.600 409.33

Co2MnSn 5.02 [25]5.04 [26]5.03 [21]

5.016 0.411 [21] 0.400 212.07

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the higher valent and the lower valent transition metal atoms. As a result, the interactionof Mn with the Z-p states splits the Mn-3d states into a low-lying triplet of t2g states and ahigher-lying doublet of eg states. The splitting is partly due to the different electrostaticrepulsion, which is strongest for the eg states which is directly points at the Z atoms. In themajority band the Mn-3d states are shifted to lower energies and form a common 3d bandwith X(Co) 3d states, while in the minority band the Mn-3d states are shifted to higherenergies and are unoccupied, so that a band gap at EF is formed, separating the occupied dbonding states from the unoccupied d antibonding states. Thus, X2YZ is a half-metal withgap at EF in minority band and a metallic DOS at the Fermi level in majority band. Thisexplains half metallicity in Co2MnGe and Co2MnSn. Less number of bands was found inthe conduction region of the spin-up channels indicating the absence of sharp peaks in theDOS. For Co2MnGe an indirect energy gap obtained between symmetry point D and X is0.60 eV which almost equals 0.581 eV reported by Kandpal et al. [21]. Similarly forCo2MnSn, an indirect energy gap obtained between D and X is 0.40 eV which agrees with0.411 eV reported by Kandpal et al. [22].

5. Conclusions

We have studied the possibility of appearance of half-metallicity in the case of the full-Heusler compounds Co2MnZ, where Z is an sp atom belonging to the IVB column of theperiodic table. These compounds show ferromagnetism with the Mn- and Z-spin moments

Figure 7. (a) Energy bands of Co2MnGe and (b) energy bands of Co2MnGe for spin down.


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being antiparallel to the Co ones. Firstly, we performed total-energy calculations to findthe stable magnetic configuration and the equilibrium lattice constant. We have found thatthe compounds Co2MnZ (Z¼Ge, Sn) are stable HMF at their equilibrium latticeconstants. Thus these compounds follow the Slater-Pauling behavior and the ‘rule of 24’[28]. We have calculated the DOS, magnetic moments, and band structures of Co2MnGeand Co2MnSn using FP-LAPW. The results were in support of the HMF nature forCo2MnGe and Co2MnSn. The existence of energy gap in DOS for spin down of bothsystems is an indication of being a potential HMF. This is also evident from the energyband results calculated. The calculated magnetic moment for Co2MnGe is 5.004mB and forCo2MnSn, it is 5.016 mB. The observed results are in qualitative agreement with the integralvalue, supporting the HMF nature of Co2MnGe and Co2MnSn.

Acknowledgments

DPR acknowledges DST INSPIRE research fellowship and RKT, a research grant from UGC,New Delhi, India.

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Figure 8. (a) Energy bands of Co2MnSn for spin up and (b) energy bands of Co2MnSn forspin down.

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A comparative study of a Heusler alloy Co2FeGe using LSDA and LSDAþU

D.P. Rai a,n, A. Shankar a, Sandeep a, M.P. Ghimire b, R.K. Thapa a

a Department of Physics, Mizoram University, Aizawl 796004, Indiab Nepal Academy of Science and Technology, Kathmandu, Nepal

a r t i c l e i n f o

Article history:

Received 6 January 2012

Received in revised form

19 April 2012

Accepted 21 April 2012Available online 19 May 2012

Keywords:

GGA

LSDA

DOS

Band structure

HMF

Hubbard potential (U)

a b s t r a c t

We have calculated the on-site Coulomb repulsion (U) for the transition elements Co and Fe. To study

the impact of Hubbard potential or on-site Coulomb repulsion (U) on structural and electronic

properties the calculated values of U were added on GGA and LSDA. We performed the structure

optimization of Co2FeGe based on the generalized gradient approximation (GGA and GGAþU).

The calculation of electronic structure was based on the full potential linear augmented plane wave

(FP-LAPW) method and local spin density approximation (LSDA) as well as exchange correlation

LSDAþU. The Heusler alloy Co2FeGe fails to give the half-metallic ferromagnetism (HMF) when treated

with LSDA. The LSDAþU gives a good result to prove that Co2FeGe is a HMF with a large gap of 1.10 eV

and the Fermi energy (EF) lies at the middle of the gap of minority spin. The calculated density of states

(DOS) and band structure show that Co2FeGe is a HMF when treated with LSDAþU.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

In 1983, de Groot discovered half-metallic ferromagnetism inthe semi-Heusler compound NiMnSb [1] by using first-principlecalculation based on the density functional theory. In recent years,it attracts substantial interest because of the half-metallic propertyand the applicable potential for future spintronics. In half-metals,one spin channel is metallic and the other is insulating with 100%spin polarization at the Fermi level EF [2]. The Fermi level lies in thepartially filled 3d band of the majority spin, whereas in theminority spin, the Fermi energy falls in an exchange-split gapbetween the occupied band and the unoccupied 3d band. Rai et al.[3,4] have studied the electronic and magnetic properties ofCo2MnAl and Co2CrSi using LSDA which shows the half-metallicityat the ground state. Blake et al. studied the compounds Co2FeZ(Z¼Al, Si, Ga and Ge) using the x-ray diffraction (XRD) andextended x-ray absorption fine structure (EXAFS) techniques. UsingEXAFS they found that the compounds Co2FeGa and Co2FeGecrystallize in the L21 structure [5]. In the LDAþU approach aHubbard repulsion (U) term is added to the LDA functional forstrong correlation of d or f electrons. Indeed, it provides a gooddescription of the electronic properties of a range of exoticmagnetic materials, such as the Mott insulator KCuF3 [6] and themetallic oxide LaNiO2 [7]. Two main LDAþU schemes are inwidespread use today: the Dudarev et al. [8] approach in which

an isotropic screened on-site Coulomb interaction U is added, andthe Liechtenstein [6] approach in which the U and exchange (J)parameters are treated separately. Both the choice of LDAþU

schemes on the orbital occupation and subsequent properties [9]and the dependence of the magnetic properties on the value of U[10] have recently been analyzed. It goes without saying that theHubbard model [11] is of seminal importance in the study ofmodern condensed matter theory. It is believed that the Hubbardmodel can describe many properties of strongly correlated electro-nic systems. The discovery of high temperature superconductivityhas enhanced the interest in a set of Hubbard-like models that areused to describe the strongly correlated electronic structure oftransition metal oxides [12]. Kim et al. studied the magneto-optical(MO) properties of ferromagnetic Co2YGe full-Heusler alloys (Y¼Feand Mn) using GGAþU. They have studied the correlation level forthe Co2YGe using MO spectra [13]. In this paper we have investi-gated the HMF character of Co2FeGe using LSDAþU.

2. Crystal structure and computational method

The Heusler alloy [14] is with the chemical formula X2YZ(X¼Co, Y¼Sc, Ti, V, Cr, Mn, Fe and Z¼Ge). The full Heuslerstructure consists of four penetrating fcc sublattices with atomsat X1(1/4,1/4,1/4), X2(3/4,3/4,3/4), Y(1/2,1/2,1/2) and Z(0,0,0)positions which results in the L21 crystal structure having spacegroup Fm-3-m.

In this work, we have used the full-potential linearizedaugmented plane wave (FP-LAPW) method accomplished by

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Physica B

0921-4526/$ - see front matter & 2012 Elsevier B.V. All rights reserved.

http://dx.doi.org/10.1016/j.physb.2012.04.055

n Corresponding author. Tel.: þ91 8014613900; fax: þ91 389 2330552.

E-mail address: [email protected] (D.P. Rai).

Physica B 407 (2012) 3689–3693


using the WIEN2K code [15] within LSDA and LSDAþU [6]schemes. We have calculated on-site Coulomb repulsion(U) based on the Hubbard model. The standard Hubbard Hamil-tonian [16] is of the form

H¼tX

/ijS,scyiscjsþU

Xnimnik ð1Þ

where nis ¼ cyiscis, and cyisðcisÞ creates (annihilates) an electron onsite i with spin s¼m or k. A nearest neighbor is denoted by /ijS.For the calculation of the parameter U we used the methodproposed by Gunnarsson et al. [17]. We used a 8-unit supercelland set the hopping integrals to the d shell of the centraltransition-metal ion equal to zero. The d occupancies of the other3d ions were kept fixed at integral values by removing thehopping. The values of U barely depend on how one constrainsthe other 3d shells as long as the systems are rather localized. U isthe on-site Coulomb repulsion between two electrons on thesame site. The hybridization between nearest neighbor orbitals,denoted by t, allows the particles to hop to adjacent sites. Theon-site energies are taken to be zero. Considering that the atomsare embedded in a polarizable surrounding, U is the energyrequired to move an electron from one atom to another, far away.In this case U is equal to the difference of ionization potential (EI)and electron affinity (EA) of the solid. Removing an electron from asite will polarize its surroundings, thereby lowering the groundstate energy of the (N1) electron system [18,19]. Thus

EI ¼ EN1EN and EA ¼ EN

ENþ1

U ¼ ðEN1ENÞðEN

ENþ1Þ ð2Þ

where EN(71) are the ground state energies of (N71) electronsystem.


We have calculated the on-site Coulomb repulsion (U) usingEq. (2) [18,19] for Co and Fe atoms. The calculated values of U forCo and Fe are 0.29 Ry and 0.28 Ry respectively, which are in goodagreement with the previously reported values, 0.31 Ry for Co and0.32 Ry for Fe in the case of Co2FeSi [20]. GGA or LSDA is widelyused to describe the properties of a wide variety of materials.However there exists a class of materials which are poorlydescribed by GGA or LSDA. They are strongly correlated materialscontaining atoms with d or f shells. It was reported that LSDA andGGA schemes are not sufficient to describe the electronic struc-ture of an Fe based compound like Co2FeSi [20]. In order to studythese types of compounds a modified version of LSDA is required.The LSDAþU approach was introduced by Anisimov et al. [21], totreat correlated materials that add an intra-atomic Hubbardrepulsion (U) in the energy functional. Thus the calculated valuesof Coulomb repulsion (U), UCo¼0.29 Ry and UFe¼0.28 Ry wereadded to LSDA to make it LSDAþU in our present work to studythe structural and electronic properties of Co2FeGe.

3.1. Structural optimization for Co2FeGe

A GGA and GGAþU method were used to study the structuralproperties of a Heusler compound Co2FeGe. The optimized latticeconstant, isothermal bulk modulus, and its pressure derivative arecalculated by fitting the total energy to the Murnaghan’s equationof state [22]. The optimized lattice parameters were slightlyhigher than the experimental lattice parameters. It is confirmedthat the ferromagnetic configuration is stable for GGA andGGAþU (Table 1) with the same minimum energy. The resultsof the structural optimization are shown in Fig. 1. The detailed

values of the optimized Lattice parameters and bulk moduli aregiven in Table 1 (Fig. 2).

3.2. Spin polarization and half-metallic ferromagnets

The electron spin polarization (P) at Fermi energy (EF) of amaterial is defined by [25]

P¼rmðEF ÞrkðEF Þ

rmðEF ÞþrkðEF Þð3Þ

where rm(EF) and rk(EF) are the spin dependent densities ofstates at EF. The m and k assigns the majority and the minoritystates respectively. P vanishes for paramagnetic or anti-ferromag-netic materials even below the magnetic transition temperature.It has a finite value in ferromagnetic materials below Curietemperature [24]. The electrons at EF are fully spin polarized(P¼100%) when rm(EF) or rk(EF) equals zero. In the present work,we have studied the electronic properties of Co2FeGe using LSDAand LSDAþU methods. It is confirmed that Co2FeGe does notshow 100% spin polarization at EF when treated with LSDAwhereas LSDAþU gives 100% spin polarization (Table 2). Accord-ing to our results, the compound Co2FeGe when treated withLSDAþU is interesting as it shows considerable value of DOS atthe EF of rm(EF)¼0.85 states/eV (Table 2). The reason for thisvalue of DOS is that EF cuts through strongly localized states ofFe-d. On the other hand rk(EF)¼0.00 states/eV according to whichCo2FeGe is a half-metal which gives 100% spin polarization at EF.

Fig. 3 summarizes the density of states (DOS) calculated usingLSDA. According to Fig. 3 it is clearly shown that the Heusler alloyCo2FeGe is not a half-metal because EF falls into an uprising peakof the minority-spin states. The Fermi energy stays close to theconducton band. The DOS below EF is dominated by d states beinglocated at Co and Fe sites in both the spin channels. Thecontribution of Fe-d states is maximum above EF in minority-spin. The reason behind the disappearance of the gap wasexplained by Ozdagan et al. [26]; in case of an Fe containingcompound like Co2FeGe the extra electrons with respect to Cr orMn containing compounds lead to an overlap of the Co bonding

Table 1Lattice parameters, bulk moduli and equilibrium energies.

Co2FeGe Lattice constants ao (A) Bulk modulus Energy (Ry)

Previous 5.739 [23]

5.738 [24]

5.743 [5]

5.737 [13]

GGA 5.758 162.677 12,317.674

GGAþU 5.762 144.684 12,317.674

Fig. 1. Unit cell Structure of Co2FeGe: Co (green), Fe (red) and Ge (purple) atoms.

(For interpretation of the references to color in this figure legend, the reader is

referred to the web version of this article.)

D.P. Rai et al. / Physica B 407 (2012) 3689–36933690


and antibonding minority d-hybrids and the gap is destroyed forthe Co atoms. Moreover the unoccupied Fe states move lower inenergy since the majority occupied Fe states are very deep inenergy. It is clear from Figs. 3 and 4 that the system Co2FeGe isnot a HMF as EF is not located in the gap of the minority spinwhen LSDA is applied. In order to show that Co2FeGe is a HMFsystem, it is treated with the LSDAþU method.

The impact of onsite Coulomb repulsion (U) on the electronicstates is an increase of the splitting between the bands of

different symmetries. This causes a shift of the Fermi energy withrespect to the gap in the minority states as well as increases thegap. The spin resolved DOS and band structure of Co2FeGe arecalculated using LSDAþU. The Fermi energy is located at themiddle of the gap in the minority-spin as shown in Figs. 5 and 6.

The system treated with LSDA does not give any gap in theminority spin. On the other hand LSDAþU gives a large gap of1.10 eV in the minority spin as shown in Fig. 6 which is in goodagreement with the previous result 1.00 eV [13]. The crucial gapfor the HMF compounds, as discussed by Galanakis et al. [27] forCo2MnSi, is due to the strong hybridization between Co-d andMn-d states, combined with large local magnetic moments and asizeable separation of the d-like band centers. For Co2FeGe, thewidth of the gap DE is given by the difference of energies of thehighest occupied band at the G-point and the lowest unoccupiedband at the X-point. The energy gap was measured between Gand X; thus it is an indirect gap. The conclusion drawn from thedisplayed electronic structure is that the states around the fermienergy are strongly polarized and the system according toLSDAþU calculations is indeed a HMF.

Fig. 2. Volume optimization of Co2FeGe (GGA and GGAþU).

Table 2Energy gap and spin polarization.

Compounds Energy gap Eg (eV) Spin polarization

Co2FeGe Emax(C) Emin(X) DE rm(EF) rk(EF) P (%)

Previous [24] 0.517 0.43 0.087 2.3 0.74 51.32

LSDA – – – 1.25 3.50 47.4

LSDAþU 0.2 0.9 1.10 0.85 0.00 100

GGAþU [13] – – 1.00 – – –

D.P. Rai et al. / Physica B 407 (2012) 3689–3693 3691


3.3. Magnetic properties

The main focus was put on the magnetic moment as thisshows a strong discrepancy on comparing the experimental andcalculated values reported for Co2FeGe. Kandpal et al. [24]reported a magnetic moment of 5.70mB using GGA. The presentLSDA calculation revealed a total magnetic moment of 5.391mB.

This value is too low and not an integer. Such a large discrepancypoints out that the actual electronic structure of this compound isdifferent from the calculated one. Now treating the d-states as

strongly localized states by means of LSDAþU the compoundCo2FeGe becomes HMF with a magnetic moment of 5.999mBE6.00mB which is an integer value in agreement with the previousresult of 5.90mB [24]. The information regarding the partial andtotal magnetic moments calculated using LSDA and LSDAþU aresummarized in Table 3. It is shown that the calculated totalmagnetic moment is exactly an integer value in the case ofCo2FeGe as expected for half-metallic systems when treated withLSDAþU. The Ge atoms carry a negligible magnetic moment,which does not contribute much to the overall moment (Table 3).

Fig. 3. DOS using LSDA.

Fig. 4. Band structure using LSDA.

Fig. 5. DOS using LSDAþU.

D.P. Rai et al. / Physica B 407 (2012) 3689–36933692


We have noticed that the partial moment of Ge atoms alignedanti-parallel to Co and Fe moments of a HMF system. The partialmagnetic moment of Ge atoms for the HMF compound Co2FeGeusing LSDAþU is 0.057mB. It emerges from hybridization withthe transition metals and is caused by the overlap of the electronwave functions as reported by Kandpal et al. [24].

4. Conclusion

The on-site Coulomb repulsion (U) was calculated for thetransition elements Co and Fe. Structural parameters, magneticmoments and electronic properties of the Heusler compoundCo2FeGe were presented. The obtained, optimized lattice para-meter for Co2FeGe leads to a small value of magnetic momentusing LSDA calculations. The inclusion of Coulomb repulsion(U) term, i.e. UCo¼0.29 Ry and UFe¼0.28 Ry in LSDA, makes itLSDAþU. It was found that the LSDAþU scheme reproducessatisfactorily good results which are in qualitative agreementwith the experimental values. At the same time the compoundCo2FeGe was predicted to be a HMF when treated with LSDAþU.

Acknowledgment

DPR acknowledges DST Inspire research fellowship and RKT aresearch grant from UGC (New Delhi, India).

References

[1] R.A. De Groot, F.M. Mueller, P.G. Van Engen, K.H.J. Buschow, Phys. Rev. Lett.50 (1983) 2024.

[2] W. De Boeck, J. Van Roy, V. Das, Z. Motsnyi, L. Liu, H. Lagae, K. Boeve, Dessein,G. Borghs, Semicond. Sci. Technol. 17 (2002) 342.

[3] D.P.Please provide the initial(s) of Lalmuanpuia and Sandeep in Ref. [3], ifany. Rai, J. Hashemifar, M. Jamal, Lalmuanpuia, M.P Ghimire, Sandeep,D.T. Khathing, P.K. Patra, B.I. Sharma, Rosangliana, R.K. Thapa, Indian J. Phys84 (2010) 593.

[4] D.P. Rai, M.P. Sandeep, Ghimire, R.K. Thapa, Bull. Mater. Sci. 34 (2011) 1219.[5] B. Blake, S. Wurmehl, G.H. Fecher, C. Felser, Appl. Phys. Lett. 90 (2007)

172501.[6] A.I. Liechtenstein, V.I. Anisimov, J. Zaanen, Phys. Rev. B 52 (1995) R5467.[7] K.W. Lee, W.E. Pickett, Phys. Rev. B 70 (2004) 165109.[8] S.L. Dudarev, G.A. Botton, S.Y. Savrasov, C.J. Humphreys, A.P. Sutton, Phys.

Rev. B 57 (1998) 1505.[9] E.R. Ylvisaker, W.E. Pickett, K. Koepernik, Phys. Rev. B 79 (2009) 035103.

[10] S.Y. Savrasov, A. Toropova, Kat, K.M.I.L.A.I.,A.V, G. Kotliar, Z. Kristallogr. 220(2005) 473.

[11] J. Hubbard, Proc. R. Soc. London A 276 (1963) 238.[12] P.W. Anderson, Science 235 (1987) 1196.[13] M. Kim, H. Lim, J.I. Lee, Thin Solid Films 519 (2011) 8419.[14] F. Heusler, Verh. Dtsch, Mangan-Aluminium-Kupferlegierungen, Phys. Ges 12

(1903) 219.[15] P. Blaha, K. Schwarz, G.K.H. Madsen, D. Kvasnicka, J. Luitz, K. Schwarz, An

Augmented Plane Wave Plus Local Orbitals Program for Calculating CrystalProperties: Wien2K User’s Guide, Techn. Universitat Wien, Wien, 2008.

[16] L.M. Roth, J. Appl. Phys 40 (1969) 1103.[17] O. Gunnarsson, O.K. Andersen, O. Jepsen and J. Zaanen, Phys. Rev. B 39, (1989)

1708.[18] Van den Brink, M.B.J. Meinders, G.A. Sawatzky, Physica B 206–207 (1995)

682.[19] G.K.H. Madsen, Novak, Europhys. Lett. 69 (2005) 777.[20] H.C. Kandpal, G.H. Fecher, C. Felser, Phys. Rev. B 73 (2006) 094422.[21] V.I. Anisimov, F. Aryasetiawan, A.I. Lichtenstein, J. Phys. Condens. Matter. 9

(1997) 767.[22] F.D. Murnaghan, Proc. Natl. Acad. Sci. USA 30 (1944) 244.[23] M. Gilleßen, Von der Fakult€at f€ur Mathematik, Informatik und Naturwis-

senschaften der RWTH Aachen University zur Erlangung des akademischenGrades eines Doktors der Naturwissenschaften genehmigte Dissertation,2009.

[24] H.C. Kandpal, G.H. Fecher, C. Felser, arXiv:cond-mat/0611179v1 [cond-matm-matrl-sci] (2006).

[25] R.J. Soulen Jr., et al., Science 282 (1998) 85.[26] K. Ozdogan, B. Aktas, I. Galanakis, E. Sasioglu, arXiv:cond-mat/0612194v1

[cond-mat.mtrl-sci] (2006).[27] I. Galanakis, P.H. Dederichs, N. Papanikolaou, Phys. Rev B 66 (2002) 174429.

Fig. 6. Band structure using LSDAþU.

Table 3Total and partial magnetic moments.

Compound Magnetic moment lB (LSDA)

Co2FeGe Co Fe Ge Total

Previous 1.42 [24] 2.92 [24] – 5.70 [24]

LSDA 1.320 2.777 0.004 5.391

LSDAþU 1.5335 3.1916 0.057 5.999

D.P. Rai et al. / Physica B 407 (2012) 3689–3693 3693

Study of Bulk modulus, Volume, Energy, lattice parameters and magnetic moments in rare

earth hexaborides using density functional theory

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Study of Bulk modulus, Volume, Energy, lattice parameters and magnetic moments in rare earth hexaborides using density functional theory

Sandeep1, M. P.Ghimire2, 5, D. P. Rai1, P. K. Patra3, A. K. Mohanty4 and R. K. Thapa1

1Condensed Matter Theory Group, Department of Physics, Mizoram University, Aizawl, Mizoram, India 796009 2Faculty of Science, Nepal Academy of Science and Technology, Kathmandu, Nepal. 3Science Centre, North-Eastern Hill University, Shillong. Meghalaya, India. 4Nuclear Physics Division, Bhaba Atomic Research Centre, Mumbai, India. 5Condensed Matter Physics Research Center, Butwal, Rupendehi, Nepal. E-mail: [email protected] Abstract. We have calculated the theoretical lattice parameters, Bulk modulus, volume, energy, lattice parameters and magnetic moments for RB6 (R=La, Ce, Pr and Sm) of CaB6 type crystal structure with space group Pm3m using full potential linearized augmented plane wave (FP-LAPW) method. The bulk modulus was found to be 9.56 % higher for LaB6 and 2.4% lower for CeB6 compared to the experimental results Gupta et al. [4] and Ogita et al [5]. Magnetic moments for LaB6, CeB6 were found in qualitative agreement with the earlier reported results. The results based on generalized gradient approximation (GGA) were found and compared with local spin density approximation (LSDA) results for CeB6 and SmB6 as well.

1. Introduction Rare earth hexaborides have unusual combination of properties with metallic conductivity and low work function as well as low volatility at temperatures providing technologically useful thermionic electron current density, micro-beam applications [1]. Hexaborides of the rare-earth elements are considered for applications as wear- and corrosion-resistant hard coatings for decoration of consumer products such as eye-glass frames and wristwatch casings etc [2]. We have calculated the theoretical lattice parameters, Bulk modulus, volume, energy, lattice parameters and magnetic moments for RB6 (R=La, Ce, Pr and Sm) of CaB6 type crystal structure using full potential linearized augmented plane wave (FP-LAPW) method.

2. Computational Details We have performed our calculations using the theoretically optimized lattice parameters and the atomic positions Wyckoff (1965) [3] for RB6. RB6 have the CaB6 type crystal structure and may be viewed as a CsCl-type lattice with the Cs replaced by a R ion and the chlorine by a B6 octahedron. The atomic arrangement is described in terms of the space group Pm3m. Since R 4f-orbitals are rather localized, the 4f-electron correlations are expected to be strong. Consequently, the LSDA+U


Published under licence by IOP Publishing Ltd 1

calculations [4] have been chosen to include the on-site Coulomb interaction. The values of on-site Coulomb energies (U) used are 5.00 eV for LaB6 and CeB6 [5], 6.00 eV for PrB6 and NdB6, 7.00 eV for SmB6 [6] and the exchange parameter (J) 1.00 eV was chosen fixed as the usual value for R respectively. The density plane cut-off RMT*K MAX is chosen to be 7.0, where KMAX is the plane wave cut-off and RMT is the muffin-tin radii. This amounts 419, 407, 408, 413 and 416 plane waves for LaB6, CeB6, PrB6, NdB6 and SmB6 respectively. To provide a reliable Brillouin zone integration, a set of 120, 84, 165, 120 and 120 k points for LaB6, CeB6, PrB6, NdB6 and SmB6 respectively in the irreducible wedge of the Brillouin zone (IBZ) were used. The self-consistency was better than 0.0001 e/a.u.3 for charge density. The stability was better than 0.01 mRy for total energy per cell. For computations, WIEN2k code [7] is used.

3. Results and Discussions The calculated lattice parameters are compared with the previous results in table 1.

Table 1. Lattice constant and total magnetic moment of rare earth hexaborides.

Lattice Constant (Ǻ) Our Results Experimental Results Previous results

LaB6 4.176 4.156 4.138[24]

4.185[24] CeB6 4.144 4.141 4.084[24] PrB6 4.151 4.121 NdB6 4.157 4.128 SmB6 4.172 4.133 4.111[24]

4.156[24]

The theoretically calculated Bulk modulus, volume and energy with earlier results are tabulated for RB6 in table 2.

Table 2 Bulk modulus, volume, Pressure derivative and energy of RB6 Bulk Modulus Bulk Modulus Bulk Modulus Bulk Modulus (Present) Theoretical Theoretical Expt. Vol. Energy

LDA GGA (GPa) (GPa) (GPa) (GPa) (a.u.3) (Ryd) LaB6 189.5336 178.8[11] 166.2 [11] 164 [9] 491.2956 -17293.8 173 [9] CeB6 162.1642 173 [11] 166 [10] 480.2547 -18029.2 PrB6 165.8493 482.5787 -18784.0 SmB6 146.6851 179.6[11] 167.2 [11] 489.0972 -21168.0

The volume versus energy curve was obtained by fitting the calculated values with the

Murnaghan equation [8] to calculate the ground state lattice parameters for these compounds [Table 1]. The bulk modulus for LaB6 was calculated to be 189.5336 GPa and was 9.56% higher than the experimental value of 173 GPa as determined by Ogita et al. [9] in the study of Raman scattering spectra for RB6. We have observed a decrease of 2.4 % in the calculated Bulk modulus for CeB6 (162.1642 GPa) when compared to the experimentally derived value by Leger et al. [10]. We have observed that the bulk modulus for LaB6 was higher compared to the earlier experimental [9] as well as theoretical LDA and GGA results [11]. On the other hand we noticed found higher values of bulk modulus for CeB6 [10, 11] and SmB6 [11] compared to the earlier theoretical and experimental results.


2

Fig. 1 (a-d) shows the volume optimization curve for LaB6, CeB6, PrB6 and SmB6 respectively. The high value of the LaB6 may be an indication of charge transfer from La to B atoms. It seems that the form of the GGA approach does not affect the results and they give closer values compared to those obtained by other. Therefore, we may conclude that the LSDA do not improve the quality of the structural properties, but rather it gives worse, especially for the bulk modulus. Thus the inclusion of the LSDA in the calculations of the structural properties of RB6 is not recommended, even if it is known that the LSDA may play an important role in the accuracy of the electronic properties of heavy compounds. We have observed zero magnetic moment for LaB6 as was reported by earlier studies of Gupta et al. [11] and Singh et al. [12]. The magnetic moment for CeB6 was calculated to be 1.03 µB which is also in agreement with the results of Singh et al. [12]. This was found slightly on the higher side when compared with the value obtained by Kubo et al. [13], Biasini et al. [14] and Gupta et al [11]. The moment for PrB6 was calculated to be 0.09% less than the value reported by Singh et al 12]. The NdB6 moment was calculated to be 0.15% higher than the value reported by Singh et al [12]. The SmB6 was found to have a moment of 5.87 µB which agrees well with the result obtained by Gupta et al. [11] and is higher than the moment calculated by Singh et al [12] by 4.2%.

Figure 1 (a-d): Volume optimization curves for LaB6, CeB6, PrB6 and SmB6.

4. Conclusions The spin polarized full potential linerized augmented plane wave method was employed for calculations of the bulk modulus, volume and total energy of RB6. The bulk modulus was found to be 9.56 % higher for LaB6 and 2.4% lower for CeB6 compared to the experimental results Gupta et al.[11] and Ogita et al [9]. Magnetic moments for RB6 were also found in qualitative agreement with the earlier reported results.


3

5. References [1] Davis P R, Swanson L W, Hutta J J and Jones D L 1986 J. Mater. Science 21 825. [2] Mitterer C, Waldhauser W, Beck U and Reiners G 1996 Surface and Coatings Technology, 86-87 715. [3] Wyckoff R W G (ed) (1965) Crystal Structures vol 2 (Tucson, AZ: University of Arizona) [4] Anisimov V I, Aryasetiawan F and Lichtenstein A I 1997 J. Phys.: Condens. Matter 9 767 [5] Gschneidner K A Jr and Eyring L (ed) (1995) Handbook on the Physics and Chemistry of Rare Earths vol 20 (Amsterdam: North-Holland) [6]Antonov V N, Harmon B N and Yaresko A N 2002 Phys. Rev. B 66 165209 [7]Blaha P, Schwartz K, Madsen G K H, Kvasnicka D, Luitz J 2008: An Augmented Plane Wave+Local Orbitals Program for Calculating Crystal Properties. (Vienna University of Technology: Inst. of Physical and Theoretical Chemistry, Getreidemarkt 9/156, A-1060 2008, Vienna / Austria) [8] Murnaghan F D 1944 Proc. Natl. Acad. Sci. USA, 30 244. [9] Ogita N, Nagai S, Okamoto N and Udagawa M 2003 Phys. Rev. B 68 224 305. [10] Leger J M, Rossat-Mignod J, Kunii S and Kasuya T 1985 solid state communication 54 995 [11]Gupta H C and S. Kumar 2010 AIP Conf. Proc. Xxii International Conference On Raman Spectroscopy Vol 1267 p. 856-857 [12] Singh N, Saini S M, Nautiyal T and Auluck S 2007 J. Phys.: condens. Matter 19 346226 [13] Kubo Y, Asano S, Harima H and Yanase A 1993 J. Phys. Soc. Japan 62 205 [14] Biasini M, Fretwell H M, Dugdale S B, Alam M A, Kubo Y, Harima H and Sato N 1997 Phys Rev. B 56 10192.


4

74 Science Vision © 2012 MIPOGRASS. All rights reserved

Original Research

Study of the structural properties of Co2YGe (Y=Sc, Ti, V, Cr, Mn, Fe)

by GGA method

D. P. Rai1*, A. Shankar1, Sandeep1, Rosangliana2 and R. K. Thapa1

1 Condensed Matter Theory Group, Department of Physics, Mizoram University, Aizawl 796 004, India

2 Department of Physics, Govt. Zirtiri Residential Science College, Aizawl 796 001, India

Received 2 February 2012 | Accepted 2 April 2012

ABSTRACT The structural properties of Co2YGe, a Heusler alloy have been evaluated by first principles density functional theory through total energy calculations at 0 K by the full potential linearized augmented plane wave (FP-LAPW) method as implemented in WIEN2K code. The calculated results were com-pared with the previously reported results. Generalized gradient approximation (GGA) was used to study the structural properties of Co2YGe. The calculated values of lattice parameters were in quali-tative agreement with the previously reported results. Key words: GGA; FP-LAPW; structural properties; Wien2k.

Corresponding author: D.P. Rai Phone: E-mail: [email protected]

INTRODUCTION Heusler alloys are the ternary intermetallic

compounds with the composition X2YZ, where X and Y are transition elements (Ni, Co, Fe, Mn, Cr, Ti, V etc.) and Z is III, IV or V group elements (Al, Ga, Ge, AS, Sn, In, etc.). One of the promising classes of materi-als is the half-metallic ferrimagnets, i.e. com-pounds for which only one spin channel pre-sents a gap at the Fermi level, while the other has a metallic character, leading to 100% car-rier spin polarization1 at EF. After that, half-

metallicity attracted much attention,2 because of its prospective applications in spintronics.3

The electronic and magnetic properties of Co2MnAl and Co2CrSi,4,5 using LSDA shows the half-metallicity at the ground state.

Rai and Thapa have also investigated the electronic structure and magnetic properties as well as structural properties of Like Co2M-nGe, Co2MnSn, Co2CrAl and Co2CrGa Heusler compounds by using a first principle study and reported HMFs.6,7 High pressure research on structural or electronic phase transformations and behaviour of materials under compression based on their calculations or measurements have become quite interest-ing in the recent few years; as it provides in-sight into the nature of the solid state theo-ries, and determine the values of fundamental parameters8. In this paper, we have studied the various ground state and structural prop-erties of Co2YGe using full potential lin-

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April-June 2012 ISSN (print) 0975-6175

ISSN (online) 2229-6026

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earized augmented plane wave (FP-LAPW) method within GGA.

Heusler alloys9 are with chemical formula X2YZ (X = Co, Y and Z = Ge) and the full Heusler alloys crystallize in the cubic L21 structure (space group Fm-3-m). The X atoms are placed on 8a X (1/4,1/4,1/4), 4a Y

(1/2,1/2,1/2) and 4b Z (0,0,0) positions. The

cubic L21 crystal structure consists of four inter-penetrating fcc sub-lattices, two of which

are equally occupied by X. The two X fcc sub-

lattices combine to form a simple cubic sub-lattice. The Y and Z atoms occupy alterna-tively the center of the simple cubic X2 sub-lattice resulting in a CsCl type super struc-ture. The crystal structure of Co2YGe Heusler

compounds is illustrated in Figure 1.

THEORY AND COMPUTATIONAL DETAILS The calculated total energies (E) within

GGA as function of the volume (V) were used for determination of theoretical lattice con-stant and bulk modulus. Equilibrium lattice constant, isothermal bulk modulus, its pres-sure derivative are calculated by fitting the calculated total energy to the Murnaghan’s equation of state.10 A series of total energy calculations as a function of volume can be

fitted to an equation of states according to Murnaghan.

(1) where E0 is the minimum energy at T = 0K,

B0 is the bulk modulus at the equilibrium vol-

ume and B'0 is pressure derivative of the bulk

modulus at the equilibrium volume. The equi-librium volume is given by the corresponding total energy minimum as shown in Figure 2.11

Pressure, Bulk modulus, A computational code (WIEN2K)12 based

on FP-LAPW method was applied for struc-ture calculations of Co2YGe. GGA13 was used for the exchange correlation potential. The multipole expansion of the crystal potential and the electron density within muffin tin (MT) spheres was cut at l =10. Nonspherical

contributions to the charge density and poten-tial within the MT spheres were considered up to lmax=6. The cut-off parameter was

RKmax=7. In the interstitial region the charge density and the potential were expands as a Fourier series with wave vectors up to

Figure 1. Co (red), Y (yellow) and Ge (blue).

Figure 2. Total energy of TiC as a function of vol-

ume.10

'0

0 0 00 ' '

0 0

( ) 11 1

BV V B V

E V EB B

2

0 2

dP d EB V V

dV dV

dEP

dV

Rai et al.


RMT Compounds

(a. u) Co2ScGe Co2TiGe Co2VGe Co2CrGe Co2MnGe Co2FeGe

Co 2.43 2.39 2.36 2.35 2.34 2.34

Y 2.43 2.39 2.36 2.35 2.34 2.34

Ge 2.29 2.24 2.21 2.21 2.20 2.20

Table 1. Muffin Tin Radius (RMT).

Compounds Lattice Constants ao (Å) Bulk Modulus

B(GPa)

Equilibrium


Co2ScGe 5.953[14] 5.978 0.025 109.969 -11300.629

Co2TiGe 5.842[14] 5.867 0.025 200.378 -11479.814

Co2VGe 5.766[14] 5.792 0.026 202.158 -11670.736

Co2CrGe 5.740[14] 5.770 0.030 250.438 -11873.835

Co2MnGe 5.738[14] 5.749 0.011 219.479 -12089.405

Co2FeGe 5.739[14] 5.758 0.019 162.677 12317.674

Table 2. Lattice parameters and Bulk modulus.

Figure 3. Energy as a function of volume.

Study of the structural properties of Co2YGe (Y=Sc, Ti, V, Cr, Mn, Fe) by GGA method


Gmax=12 a.u-1. The Muffin Tin sphere radii (RMT) are given in Table 1. The number of k-points used in the irreducible part of the bril-louin zone is 286.


The calculated total energies within GGA

as function of the volume were used for deter-mination of theoretical lattice constant and bulk modulus. Equilibrium lattice constant, isothermal bulk modulus, its pressure deriva-tive are calculated by fitting the calculated total energy to the Murnaghan’s equation of state given in Eq. (1). The plot of energy ver-sus volume is shown in Figure 3. The volume corresponds to the lowest energy is used to determination of equilibrium lattice constant. The calculated values of lattice constant and bulk modulus are presented in Table 2.

The change in the optimized lattice con-stant of Co2YGe with that of previous result

is given as Δ(ao). The enthalpy (H) of the sys-

tem was calculated by using the Eq. (2), (2)

where P is the bulk pressure, Vo is bulk vol-

ume corresponds to minimum energy (Eo) of

the system. The calculated enthalpy (H), pres-

sure (P) and volume (Vo) is depicted in Table

3. Kandpal et al. has reported the ferromag-

netic configuration is lower in energy than the non-spinpolarized case.15 We have also found the same trend where the energy is minimum for the highest ferromagnetic compound, i.e. Co2FeGe and the energy is maximum for low-est ferromagntic compound Co2ScGe as shown in Figure 4. The minimum of the total energies of the phases with E(Co2FeGe) < E(Co2MnGe) < E(Co2CrGe) < (Co2VGe) < E(Co2TiGe) < E(Co2ScGe) rank indicates that Co2FeGe is the most stable structure as it was reported in the case of ScN.16 According to Figure 4, the lowest enthalpy is observed at the low ferromagnetic compound Co2ScGe as well as in strong ferromagnetic compound

like Co2FeGe. It is also seen in Figure 4 that there is a slow variation of enthalpy as we move from Co2TiGe to Co2MnGe which pre-dicts the stability of bulk modulus shown in Table 2.

CONCLUSION The ground state structural optimization

was performed to obtain the equilibrium en-ergy. The optimize Lattice parameters of the full Heusler compounds Co2YGe, where Ge is a sp atom belonging to the IVB column of the

periodic table were compared with the previ-ously available results. The calculated values of lattice parameters were in qualitative agreement with the previously reported results.14 We have also calculated the enthalpy (H) of the system. The sharp

variation of enthalpy was noticed for high and low ferromagnetic compounds like Co2FeGe and Co2ScGe respectively. The minimum of the total energies of the phases with E(Co2FeGe) < E(Co2MnGe) < E(Co2CrGe) < E(Co2VGe) < E(Co2TiGe) < E(Co2ScGe) rank indicates that Co2FeGe is the most stable structure as it was reported in the case of ScN.16

Figure 4. Plot of energy and enthalpy against valence

electron (Z).

0 0 H E PV

Rai et al.


ACKNOWLEDGEMENT

DPR acknowledges DST Inspire research fellowship, AS and RKT a research grant from UGC, (New Delhi, India).

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