Theoretical study of the structural, electronic and magnetic properties of equiatomic quaternary CoTcCrZ (Z = Si, Ge, P) Heusler alloys

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Theoretical study of the structural, electronic and magnetic properties of equiatomic quaternary CoTcCrZ (Z=Si, Ge, P) Heusler alloys M. Zafar , Halima Sadia , M. Rizwan , Hafsa Arshad , Shabbir Ahmad , S.S.A. Gillani , Islah-u-din , Cao Chuan Bao , Xiao-Ping Wei , M. Shakil PII: DOI: Reference:

S0577-9073(20)30009-5 https://doi.org/10.1016/j.cjph.2020.01.003 CJPH 1052

To appear in:

Chinese Journal of Physics

Received date: Revised date: Accepted date:

26 January 2019 28 December 2019 9 January 2020

Please cite this article as: M. Zafar , Halima Sadia , M. Rizwan , Hafsa Arshad , Shabbir Ahmad , S.S.A. Gillani , Islah-u-din , Cao Chuan Bao , Xiao-Ping Wei , M. Shakil , Theoretical study of the structural, electronic and magnetic properties of equiatomic quaternary CoTcCrZ (Z=Si, Ge, P) Heusler alloys, Chinese Journal of Physics (2020), doi: https://doi.org/10.1016/j.cjph.2020.01.003

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Highlights 

Investigation of structural, electronic and magnetic properties



Half-metallic character determination



Calculations of magnetic moments



Calculations of Curie temperature

Theoretical study of the structural, electronic and magnetic properties of equiatomic quaternary CoTcCrZ (Z = Si, Ge, P) Heusler alloys M. Zafara, Halima Sadiab, M. Rizwanb, Hafsa Arshadb, Shabbir Ahmadc, S. S. A. Gillanid, Islahu-dind, Cao Chuan Baoe, Xiao-Ping Weif, M. Shakil*,b a) Department of Physics, Govt. Rizwiya Islamia Post Graduate College Haroon Abad, Punjab, Pakistan b) Department of Physics, Hafiz Hayat Campus, University of Gujrat, Gujrat 50700, Pakistan c) Department of Physics, The Islamia University of Bahawalpur, Bahawalpur 63100, Pakistan d) Department of Physics, Govt. College University Lahore, Pakistan e) School of Materials Science and Engineering, Beijing Institute of Technology, Beijing 100081, PR China f) The School of Mathematics and Physics, Lanzhou Jiaotong University, Lanzhou 730070, PR China * Corresponding author: [email protected], [email protected] (M.Shakil) Abstract In this study, full potential linearized augmented plane wave (FP-LAPW) method has been used to calculate structural, electronic and magnetic characteristics of CoTcCrZ (Z = Si, Ge, P) Equiatomic Quaternary Heusler alloys (EQHAs). Furthermore, Generalized Gradient Approximation (GGA) and Hubbard potential (GGA+U) are adopted in the parametrization of Perdew-Burke-Ernzerhof (PBE). Structural stability calculations confirmed that the Type II is the most stable structure of all considered alloys. Further investigations were carried out for the

most stable Type (i.e. Type II). Comparison of electronic structure calculations performed by GGA and GGA+U methods concluded that considered alloys show half-metallic nature for GGA+U method. Magnetic moments for all these alloys are determined which are found in accordance to the Slater-Pauling rule. Half-metallicity has been verified in all these considered Heusler alloys (HAs) from the calculations of

spin-polarization at the Fermi level (EF).

Moreover, the Curie temperature (Tc) is estimated by employing mean field approximation (MFA). Keywords: Quaternary Heusler alloys; half-metallicity; Spin-polarization; Magnetic moments; Curie temperature; 1. Introduction Materials with complete spin-polarization have attracted much attention of researchers during the last few years due to their diverse properties and applications. An extensive research is going on to find out new series of HAs due to their promising applications in spintronic devices, such as non-volatile spin transfer torque magnetic random access memories (STT-MRAM), spin injectors and magnetic tunnel junctions [1-4]. The devices which are manufactured using HAs have high storage density, increased data processing speed and low power consumption [5]. Heusler alloys are of particular interest due to their Tc above room temperature [6-8] and tunable electronic properties. In half-metallic HAs, the electric current flows only due to spin up or spin down channel electrons [9-12]. Their band structure shows band gap in one spin channel and nonzero band gap at EF in other spin channel [9]. Furthermore, at EF, 100% spin-polarization occurs if either ρ↑(EF ) or ρ↓(EF) equals to zero. The literature survey showed that the first halfmetal (i.e. NiMnSb) was predicted by de Groot and collaborators in 1983 [9]. Also a number of HAs exhibiting half-metallic character have been reported in literature [13-17]. Heusler alloys

possessing high Curie temperature and 100% spin polarization have great importance for practical point of view. Co-based [18] QHAs have been reported with high spin-polarization and Tc which are necessary requirements for spintronics devives [19-21]. For instance, Co2FeSi has half- metallic nature with ~1100 K Tc and magnetic moment 6µB [22] and Co2MnSi with ~985 K Tc [23]. Enamullah et al. have reported in literature an alloy CoFeCeGe has been synthesized experimentally that has 866 K Tc [23]. Benkabou et al. have studied that CoRhMnZ (Z = Ge & Si) alloys are halfmetallic ferromagnets [24]. Bainsla et al. [25] synthesized CoRuFeSi and CoRuFeGe alloys and Tc values were reported as 867 and 833 K respectively. Xiong et al. predicted half-metallicity in CoFeTiZ (Z = Si & As) and CoFeVSb [17]. Gao et al. studied CoFeScZ (Z = P, As, Sb) and showed that CoFeScP is half-metallic with EHM = 0.6 eV. Alijani et al. [26] studied theoretically and experimentally other types of QHAs i.e. NiFeMnGa, NiCoMnGa and CuCoMnGa. Their calculations showed that NiFeMnGa and NiCoMnGa compounds are half-metallic ferromagnets on the other hand the CuCoMnGa is a conventional ferromagnet. Berri et al. predicted that CoFeTiSb [27] and ZrCoTiZ (Z = Si, Ge, Ga, Al) QHAs are half- metallic ferromagnets [28]. Therefore, it is imperative to explore new QHAs with half-metallic nature by changing composition of ternary HAs. To the best of authors knowledge, no work has been reported previously in literature on these CoTcCrZ (Z = Si, Ge, P) alloys. Therefore, in this work, these alloys are investigated theoretically to explore their structural, electronic and magnetic characteristics for the first time. Spin-polarization and Tc for these alloys have also been calculated and reported. In section 1, a brief literature survey has been presented. Section 2 contains the computational details. In section 3, the obtained results are discussed and explained in detail. Finally, in section 4, the conclusion is drawn.

2. Computational details The structural, electronic and magnetic properties of EQHAs CoTcCrZ (Z = Si, Ga, Ge) have been investigated using first-principles FP-LAPW method as implemented in WIEN2k provided by Blaha, Schwarz, and co-workers [29-31]. We adopted GGA and GGA+U in the parametrization of PBE for electronic exchange-correlation functional [32, 33]. The GGA+ U method used to esteem the on-site correlation of transition metals. Orbital dependence of Coulomb and exchange interaction was accounted in U parameter, which was absent in absolute GGA. The effective Hubbard potential Ueff (U-J) used for Co, Cr [34] and Tc transition elements was 1.92 eV, 1.59 eV and 2.0 eV respectively, where U and J are Hubbard and exchange parameters. In MFA, to obtain Tc we constructed a 4×4 matrix, and pair exchange coupling parameters adopted up to rmax = 4.0 a.u. To achieve self-consistent calculations, the charge and energy convergence were set to 10-3e and 10-6 Ry respectively. The threshold energy between core states and valence states was -6.0 Ry. Number of plane wave cut off parameter (RMT × kmax) have been set to 7.0 Ry, which determine convergence criteria, where RMT is the radius of smallest atomic sphere and appropriate value preferred so that spheres do not overlap. The Monkorst-Pack special k-points were performed using 1000

special k-points in the Brillouin

zone. 3. Results and discussion 3.1 Structural properties In this study, structural stability of EQHAs of Type LiMgPdSn has been studied. In order to investigate electronic and magnetic properties, firstly we optimized the crystal structures to obtain equilibrium lattice constants. The primitive cell contains four atoms at atomic sites provided in Table 1 and named as (Type I, Type II and Type III). The preferred occupation depends upon the electronegativity and sizes of individual atoms.

The sp elements and least electronegative element occupy positions on octahedral sites. While tetrahedral sites occupied by other two transition elements. By fixing position of sp element at 4a site, and exchange the occupancy of other three sitesthat originated three possible configurations [26, 35, 36] which are displayed in Table 1. For geometry optimization of EQHAs total energies as a function of unit cell volume have been calculated with respect to different atomic positions in ferromagnetic (FM) phase. Equilibrium state parameters are calculated using Murnaghan’s equation of state and listed in Table 2. It is observed that CoTcCrZ (Z = Si, Ge, P) alloys have minimum equilibrium energy for Type II atomic sites. Further investigations of properties have been carried out using Type II configuration. Crystal structures of CoTcCrZ (Z = Si, Ge, P) QHAs with Type II site occupations displayed in Fig. 1 (a)-(c). Optimization graphs (energy vs volume) for Type I, Type II and Type III of all considered alloys are shown in Fig. 2 (a).Table 1: Site occupations of CoTcCrZ (Z = Si, Ge, P) HAs in Type I, Type II and Type III structures. Type

4a (0, 0, 0)

4b (1/2, 1/2, 1/2)

4c (1/4, 1/4, 1/4)

4d (3/4, 3/4, 3/4)

I

Si/Ge/P

Tc

Co

Cr

II

Si/Ge/P

Cr

Co

Tc

III

Si/Ge/P

Co

Tc

Cr

(a) CoTcCrSi

(b) CoTcCrGe

(c) CoTcCrP

Furthermore magnetic stability was checked for CoTcCrZ (Z = Si, Ge, P) HAs with respect to stable Type II occupancy. Total energy as a function of unit cell volume was calculated for nonmagnetic (NM), anti-ferromagnetic (AFM) and FM phasesand it was observed that structures show minimum energy for FM phase shown in Fig. 2 (b). Hence, considered alloys are energetically stable in FM phase.

(b)

(a)

Fig. 2: Structure optimization curves of CoTcCrZ (Z = Si, Ge, P) HAs at equilibrium lattice constant (a) in Type I, Type II and Type III site occupations and (b) Comparison of AFM, NM and FM phase stability for Type II site occupancy.

The formation energy (Ef) gives the confirmation whether a material can be chemically synthesized or not. Therefore, it is obtained by subtracting total energy of CoTcCrZ material from total energy of atomic bulk states and can be computed using following relation [37]. =

where

-[

,

,

and

+

+

+

]

(1)

symbolize total energy per atom of Co, Tc, Cr and Z in

bulk respectively. Calculated Ef is -2.08 Ry, -1.93 Ry and -1.95 Ry for CoTcCrSi, CoTcCrGe and CoTcCrP alloys respectively. The negative values of formation energies revealed that compounds are thermodynamically stable [38-41]. Hence, these compounds can be synthesized experimentally. Table 2: Calculated lattice parameters (Å), total energy Etot, bulk modulus (B0), pressure derivative (B′) and formation energy (Ef). Alloys

Type Lattice constant (Å)

CoTcCrSi

CoTcCrGe

Etot (FM)

B0

B′

Ef

(Ry)

(GPa)

(GPa)

(Ry)

I

5.8125

-14042.148843 245.0171 3.6773

-

II

5.7964

-14042.206352 230.4842 9.4546

-2.08

III

5.8248

-14042.144940 231.7275 4.0459

-

I

5.9050

-17660.196307 221.6248 3.9322

-

II

5.8996

-17660.245626 221.6824 4.3269

-1.93

III

5.9159

-17660.197150 202.9219 6.4257

-

CoTcCrP

I

5.7937

-14146.215690 227.8491 4.2273

-

II

5.7950

-14146.282219 231.3219 4.4715

-1.95

III

5.7468

-14146.275597 250.6570 4.1014

-

3.2 Electronic properties In order to obtain electronic characteristics the calculations for considered alloys were carried out for both GGA and GGA+ U methods and results are discussed briefly. Spin-polarized total density of states (TDOS) and orbital resolved partial density of states (PDOS) of CoTcCrZ (Z = Si, Ge, P) HAs are patterned in Fig. (3) - (5). For GGA, HAs showed nearly half-metallic behaviour which can be seen from Fig. (3) - (5). However, half-metallic behaviour has been shown by CoTcCrZ (Z = Si, Ge, P) HAs by using GGA+U approximation, where CoTcCrSi and CoTcCrGe showed 100% spin-polarization and CoTcCrP showed 99.6% spin-polarization due to extremely small state present at EF. Valence states shifted towards lower energy level relative to EF as well as nature of alloys changed and energy band gap (Eg) appeared between the valence band maximum (VBM) and conduction band minimum (CBM). PDOS determine the atomic contribution in TDOS., Furthermore, the PDOS determine the hybridization caused by overlapping of energy states of the individual atoms. The DOS which are present at E F are of particular interest in order to understand the behaviour of alloys. Fig. 3 revealed that in CoTcCrSi alloy , GGA+U method showed the metallic nature of majority spin channel because states are overlapping at the EF and semiconducting nature in minority spin channel, as no states were occurred at the EF and 100% spin-polarization has been confirmed. Consequently, Eg is observed in minority channel which leads to half-metallic behaviour while nearly half-metallic nature was obtained by GGA approximation. The obtained value of Eg is 0.7 eV for CoTcCrSi

alloy by GGA+U method. It has been observed that in half-metallic compounds Eg keeps EF in region between edges of valence and conduction bands. Spin down channel in region from -1 to 3 eV below EF is mainly occupied by Co d-states and these states contribution in bonding region of majority spin channel consist from -1.5 to -3 eV. In minority spin channel above EF , energy region from 1 to 3 eV is

mainly populated by d-states of Cr. Co d-states have greater

contribution in bonding region of majority spin channel below the EF while in anti-bonding region of majority spin channel Cr d-states have noticeable contribution. In majority spin channel d-states of Cr have major role in metallic character due to compulsory states present at the EF. As far as degenerate states are concerned, Cr eg-states have more contribution in metallic nature of majority spin channel due to greater exchange splitting (Fig. 6). The p-states of atoms are located in deeper energy due to completely filled shell. Main contribution to the TDOS is from d-states of transition elements. Furthermore, p-states of transition elements have no role in influencing the half-metallic property. It can be observed from GGA+U calculation for CoTcCrGe (Fig. 4), the states of majority spin channel are crossing EF but in minority spin channel behaviour is non-conductive due to the absence of states at the EF. Metallic behaviour dominated by Cr eg-states at the EF and second contribution is of Co eg-states as shown in Fig. 6. This indicates that CoTcCrGe alloy is half-metallic which generates

spin-polarization and

corresponding Eg is 0.5 eV. Moreover, Co and Cr atoms have major contribution in TDOS. Fig. 5 shows spin-polarized TDOS and PDOS of CoTcCrP. After observing GGA+U findings, it is clear that majority spin channel behaviour is metallic because bonding and anti-bonding states are overlapping around EF. In minority spin channel behaviour is semiconducting. Hence the Eg is formed in minority spin channel between highest occupied states of valence band and the lowest occupied states of conduction band, which is the confirmation of half-metallic character

but extremely small state present at EF in minority spin channel gives 99.6% spin-polarization than ideal case [26, 42]. PDOS show that main contribution in majority spin channel contain the energy range -1.5 to -4 eV which is dominantly populated by Co d-states while the contribution in spin down channel below EF is from -1.8 to -3 eV. Anti-bonding states in spin down channel above EF populated by Cr d-states from energy range 0.5 to 2.8 eV. Furthermore, at EF in majority spin channel metallic behaviour is dominated by Cr d-states. Degenerate Cr t2g-states mainly contribute in metallic character. Secondly Co t2g-states have major role in metallic nature as shown in Fig. 6. Observed Eg in CoTcCrP is 0.3 eV. It ensured that CoTcCrZ (Z = Si, Ge, P) have half-metallic behaviour and conduction of electrons is only in majority spin channel.

CoTcCrSi-GGA

CoTcCrSi-GGA+U

Eg

CoTcCrGe-GGA

CoTcCrGe-GGA+U

Spin up channel

Spin down channel

Eg

GGA+U

Fig. 6: Spin resolved degenerate DOS of CoTcCrZ (Z = Si, Ge, P) HAs with GGA+U approximation.

At equilibrium lattice constants spin-polarized band structures have been computed at high symmetry points of Brillouin zone to further investigate electronic properties of CoTcCrZ (Z = Si, Ge, P) QHAs in GGA and GGA+U approximations as shown in Fig. 7 (a)-(l). By using GGA+U approximation, it is observed that band structures of all three alloys have conductive nature due to finite DOS at EF in majority spin channel and non-conductive nature in minority spin channel. Overlapping of bands in majority spin channel at EF are mainly due to Cr d-states. If we compare degenerate bands eg and t2g then prominent contribution of bands below the EF in

minority spin channel are Co-eg and Cr-t2g bands could be observed. For CoTcCrSi and CoTcCrGe band structure, appreciable Eg exists. VBM lies at high symmetry Γ point whereas CBM lies at high symmetry X point. Confirmed nature of existed Eg is indirect as it can be seen in Fig. 7 (d) and (h) respectively. Band structures for CoTcCrP are shown in Fig. 7 (i)-(l). It is clear that band diagram (Fig. 7 (k)) shows metallic behaviour, as majority bands crossed the EF. However, it might be seen that minority spin channel of DOS showed semiconducting behaviour and it is supposed that there is 100% spin-polarization as highlighted in Fig. 5(GGA+U). But close inspection of band structure in Fig. 7 (l) revealed that extremely small state present at E F in minority spin channel and as a result spin-polarization is less than ideal case. Inspection of minority band structure in Fig. 7 (l) revealed that VBM and CBM lie at same high symmetry point ℾ. Hence the nature of Eg is direct. From this observation it is concluded that the CoTcCrZ (Z = Si, Ge) HAs exhibit 100% spin-polarization while CoTcCrP possess 99.6% spinpolarization. Galankis et al. [43] described the origin of Eg around the EF due to d-d hybridization. Band gap arises due to the shifting of EF. This is due to the covalent hybridization between transition metals . The s and p-states of transition elements do not contribute much due to deep located states. Consider hybridization between Co and Tc atoms. Five d-states (dxy, dyz, dzx, dx2-y2, dz2) split into two degenerate states (I) doubly degenerate state dx2-y2, dz2 (2eg) (II) triply degenerate states dxy, dyz, dzx (3t2g). This splitting result in energy shift which causes the metallic nature in one of two spin channels. Two fold degenerate states of Co hybridize with two fold degenerate state of Tc and three fold degenerate states of Co hybridize with three fold degenerate states of Tc and form the five bonding (2eg and 3t2g with tetrahedral symmetry), five anti-bonding (2eu and 3t1u with octahedral symmetry) states. The transition metals possessing the degenerated

states hybridized with one another and with p-states of main group element to generate the Eg. Galankis et al. [43] also mentioned that Eg originates due to splitting of (2eu and 3t1u) states in full HAs and between eg-t2g in half HAs (here eu and t1u states do not exist). Due to symmetry tetrahedral bonding d-states which originate due to hybridization of Co and Tc further hybridize with tetrahedral symmetry of d-states of Cr. This forms five lower energy bonding states (eg, and t2g), five higher energy anti-bonding states (eg and t2g). Due to dissimilar symmetry eu and t1u (non-bonding) states do not hybridize with d-states of Cr atom. The EF is located between (2eu and 3t1u) states and occupied states are 12 below EF per unit cell. Eight states are with d character (2eg, 3t2g, and 3t1u), 1 × s and 3 × p character. Hybridization pattern is different in majority spin channel, which has metallic character. While in minority spin channel Eg produces due to hybridization, which has semiconducting character [44]. The Eg is obtained by subtracting energy of CBM and VBM while half-metallic energy gap (EHM ) is the lowest absolute value of CBM and VBM [45]. Calculated values of Eg, VBM, CBM, EHM and spin-polarization of studied HAs are listed in Table 3. To further examine the origin of Eg in minority spin channel, a schematic representation is shown Fig. 8. Table 3: Comparison of calculated values of VBM, CBM, EHM and spin-polarization of CoTcCrZ (Z = Si, Ge, P) HAs using GGA and GGA+U approximation methods. Heusler alloy

CoTcCrSi

CoTcCrGe

VBM

CBM

EHM

(eV)

(eV)

(eV)

GGA

0.3

0.3

-

85%

GGA+U

-0.3

0.4

0.3

100%

GGA

0.4

0.8

-

83%

Calculation method

Spin polarization

CoTcCrP

GGA+U

-0.1

0.4

0.1

100%

GGA

-0.1

-0.1

-

0%

GGA+U

-0.35

-0.05

-

99.6%

CoTcCrSi-GGA

a

Majority spin (↑

Minority spin (↓

b

CoTcCrSi-GGA+U

c

Majority spin (↑

Momentum (k)

Minority spin (↓

DOS (states/eV)

Momentum (k)

d

CoTcCrGe-GGA

e

Majority spin (↑

f

Minority spin (↓

eu t1u t2g eg

CoTcCrGe-GGA+U

g

Majority spin (↑

Momentum (k)

p Minority spin (↓

DOS (states/eV)

Momentum (k)

h

CoTcCrP-GGA

i

Majority spin (↑

eu

j

Minority spin (↓

t1u

t2g eg

p

CoTcCrP-GGA+U

k

Majority spin (↑

Momentum (k)

Minority spin (↓

DOS (states/eV)

Momentum (k)

l

Co

Fig. 7: Band structures calculated with GGA and GGA+U of CoTcCrZ (Z = Si, Ge, P) HAs

d4, d5 d1, d2, d3

Fig. 8: A schematic description of possible hybridization between transition metals of CoTcCrZ (Z = Si, Ge, P) HAs in minority spin channel.

3.3 Magnetic moment and Slater-Pauling rule Magnetic moments of CoTcCrZ (Z = Si, Ge, P) QHAs are calculated using GGA+U approximation and obtained results are presented in Table 4. Calculated magnetic moment are in accordance with Slater-Pauling rule [43, 46] as given below; Mt = (Zt – 24) µB

(2)

Where Mt represent the total magnetic moment and Zt represent the total number of valence electrons. Magnetic moment of integral value is the significant feature of half-metallic alloys

[47-50]. In order to find the reason that why considered compounds follow the particular SlaterPauling rule Mt = Zt – 24, determine the number of occupied states in spin channel where band gap exists. Minority band structures of CoTcCrSi, CoTcCrGe and CoTcCrP HAs occupied by 12 bands per unit cell below the EF as suggested from Fig. 7(d) (h) and (l) respectively. One band with s character, three bands with p character and eight with d character (3t2g, 2eg, 3t1u see the schematic description in Fig. 8). Minority band structure of CoTcCrP alloy (Fig. 7(l)) manifest that s-state is well separated from p and d-states and located at lower most energy below -13 eV. The p-states are laying from energy range -8.7 to -8.2 eV while d-states are higher laying states and located between energy ranges -6.6 to 0 eV. Considering minority band structure of CoTcCrGe alloy (Fig. 7(h)) it is found that s- state laying below -11eV while p-states are present in energy range -7.8 to -7.6 eV and d- states energy range is between -5.6 eV to 0 eV. By observing minority band structure of CoTcCrSi alloy (Fig. 7(d)) it is concluded that s-state is displayed below of energy -10 eV, however p-states reveal energy range from -7.8 eV to -7.6 eV and d-states are ranged between -5.7 eV to 0 eV. The s and p bands contribution is from sp elements while d character bands originated from transition elements. So, magnetic moment can be calculated by using relation Mt = (N↑ - N↓) µB = (Zt -2 N↓) µB= (Zt – 24) µB

(3)

Majority spin denoted by N↑ while minority spin denoted by N↓. Relation between valence electrons and unbalanced spin described by Galankis et al. by the Eg origin in HAs [51]. SlaterPauling rule deviates in case of any disorder in compound [52]. Direct relation exists between number of electrons in valence shell and magnetic moments. DOS calculated by GGA+U method described the magnetic characteristics. Greater exchange splitting leads to greater magnetic moment. From the results, it is evident that Cr has greater magnetic moment, so, the magnetic

behaviour is dominated by Cr atom. Magnetic moment contribution of Cr is 1.90, 2.06 and 2.63µB in CoTcCrSi, CoTcCrGe and CoTcCrP HAs respectively as presented in Table 4. Z and Tc elements have small magnetic moments and do not contribute much in overall magnetic moment. The negative sign with magnetic moments of constituent atoms indicate their antiparallel arrangement while positive sign indicate FM coupling between them. In CoTcCrP magnetic moment of P is anti-parallel to magnetic moment of Co, Tc and Cr atoms. Negative magnetic moment of P points out ferrimagnetic coupling with neighbour atoms. Magnetic moments of Si and Tc arranged anti-parallel to Co and Cr atoms of CoTcCrSi indicating ferrimagnetic arrangement while in CoTcCrGe alloy Ge, Tc and Co moments have small contribution and are anti-parallel to magnetic moment of Cr atom suggesting the

FM

arrangement. The Co, Tc and Cr have 9, 7 and 6 valence electrons respectively, while Si, Ge and P have 4, 4 and 5 valence electrons respectively in outer shell. So, valence electrons per unit cell of CoTcCrSi, CoTcCrGe and CoTcCrP are 26, 26 and 27 respectively. Table 4: Calculated values of magnetic moment of unit cell and individual atoms. Mt

MCo

MTc

MCr

MSi/Ge/P

(µB)

(µB)

(µB)

(µB)

(µB)

CoTcCrSi

2.00204

0.14441

-0.02438

1.90207

-0.04609

CoTcCrGe

2.00078

-0.04985

-0.01978

2.06504

-0.03722

CoTcCrP

3.00048

0.17343

0.01539

2.63938

-0.00342

Alloys

(a)

(b)

(c)

Exchange interactions Jij interatomic distance r are CoTcCrZ

as depicted

a

function

in

Fig.

Fig. 9: Exchange interactions Jij of CoTcCrZ (Z = Si, Ge, P) HAs

of

9

for

(Z

=

Si, Ge, P) HAs. All interactions are limited to r ≤ 2.5 distance. For CoTcCrSi only intrasublattice interactions Cr(C)-Cr(C) and inter-sublattice interactions Cr(C)-Co(B), Tc(D)-Co(B), Tc(D)-Cr(C) are displayed owing to weak effects of other interactions. It can be seen that dominant interaction comes from inter-sublattice Cr(C)-Co(B) exchange. By increasing

interatomic distance oscillatory behaviour is also observed, indicating RKKY (Ruderman-KittelKasuya-Yosida) exchange [32]. Intra-sublattice Cr(C)-Cr(C) exchange shows FM interaction for first nearest neighbour and

AFM interaction for second nearest neighbour. While inter-

sublattices Cr(C)-Co(B), Tc(D)-Co(B), Tc(D)-Cr(C) exchange depicts FM coupling for both first and second nearest neighbours. For CoTcCrGe inter-sublattices Cr(B)-Co(C), Tc(D)-Co(C), and Cr(B)-Tc(D) exchange indicated FM coupling for both first and second nearest neighbours, while intra-sublattice Cr(B)-Cr(B) exchange show FM coupling for first nearest neighbour and AFM coupling with second nearestneighbour. Major contribution comes from Cr(B)-Co(C) coupling. In case of CoTcCrP, intra-sublattices Co(B)-Co(B) and Cr(C)-Cr(C) exchange display FM coupling for first nearest neighbour and AFM coupling for second nearest neighbour. However, inter-sublattices Cr(C)-Co(B) and Cr(C)-Tc(D) coupling in

FM in both nearest

neighbours and major contribution comes from Cr(C)-Co(B) exchange. Hamiltonian of a spin system in classical Heisenberg model is given by [19]: ℋ=Where

(4)

is exchange coupling parameter and

are unit vectors pointing in the direction of

magnetic moment on atoms at different atomic sites. Exchange coupling parameter calculated within a real approach using an expression proposed by Liechtenstein et al. [20]. Calculations have been performed for various inter/intra sublattices. For exchange interactions confined for a single-lattice system, the Tc can be estimated within the MFA by =

(5)

While for multi-sublattice systems 〈

〉=

〈

〉

Average z component is denoted by 〈 field at site (ν, R) against the

〉 of unit vector

pointing in the direction of magnetic

is exchange coupling parameter of sublattices μ and ν. If interaction act

FM ordering on corresponding lattice then Tc reduced. The calculated Tc for

considered alloys in this study are listed in Table 5 which are comparable with other experimentally calculated Tc for Co based Heusler alloys [53]. Table 5: Comparison of Curie temperatures calculated by MFA with experimental values Heusler alloys

a

Tc (K) by MFA

Tc (K)

(Present work)

(Others Expe.)

CoTcCrSi

344

CoTcCrGe

485

CoTcCrP

449

CoRhMnGa

408a

CoRhMnSb

534a

Co0.5Rh1.5MnSb

424a

Ref. [53].

Spin-polarization at EF is defined by ↑

↓

↑

↓

(6)

ρ↑ (EF) represent majority DOS and ρ↓ (EF) represents minority DOS of two spin components at EF. Paramagnetic and FM materials give no spin-polarization even below the Tc [51]. Spinpolarization of considered HAs CoTcCrZ (Z = Si, Ge, P) at EF are presented in Table 3.

4. Conclusions Structural, electronic and magnetic properties of CoTcCrZ (Z = Si, Ge, P) EQHAs have been performed. We adopted FP-LAPW method within GGA and GGA+U in the parametrization of PBE for exchange correlation functional. Structural stability is confirmed for Type I, Type II and Type III atomic site occupations and Type II was found the most stable for all alloys. Magnetic stability is checked for Type II occupancy. Further investigations are carried out for the most stable Type II. Effect of correlation on electronic properties of Heusler compounds is discussed. The results established showed that owing to presence of U parameter, spin-polarization increased around the EF. DOS and band structures calculations performed by using GGA+U demonstrated that CoTcCrSi, CoTcCrGe and CoTcCrP HAs are half-metallic. Calculated magnetic moments for all these alloys obey Slater-Pauling rule. Total magnetic moments for CoTcCrSi, CoTcCrGe and CoTcCrP HAs are 2, 2 and 3µB respectively. Furthermore, the Tc is estimated by MFA which is above the room temperature. The robust half-metallic properties signify their usage in spintronic applications. Declaration of Interest It is stated that the work submitted have been performed by us and all authors have contributed. Furthermore, this work has not been submitted elsewhere partially or complete. Also, there is no ethical issue in any regard.

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