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Structural, Elastic, Electronic, and Magnetic Properties of Full-Heusler Alloys Sc_{2}TiAl and Sc_{2}TiSi Using the FP-LAPW Method

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## Abstract

**:**

_{2}TiAl and Sc

_{2}TiSi, were investigated using a full-potential, linearized augmented plane-wave (FP-LAPW) method, within the density functional theory. The optimized structural parameters were determined from the minimization of the total energy versus the volume of the unit cell. The band structure and DOS calculations were performed within the generalized gradient approximation (GGA) and modified Becke–Johnson approaches (mBJ-GGA), employed in the Wien2K code. The density of states (DOS) and band structure (BS) indicate the metallic nature of the regular structure of the two compounds. The total spin magnetic moments for the two compounds were consistent with the previous theoretical results. We calculated the elastic properties: bulk moduli, $B$, Poisson’s ratio, $\nu $, shear modulus, S, Young’s modulus (Y), and the B/s ratio. Additionally, we used Blackman’s diagram and Every’s diagram to compare the elastic properties of the studied compounds, whereas Pugh’s and Poisson’s ratios were used in the analysis of the relationship between interatomic bonding type and physical properties. Mechanically, we found that the regular and inverse full-Heusler compounds Sc

_{2}TiAl and Sc

_{2}TiSi were stable. The results agree with previous studies, providing a road map for possible uses in electronic devices.

## 1. Introduction

_{2}YZ, crystallize in the regular Heusler (L2

_{1})(225) crystal structure with the prototype AlCu

_{2}Mn, when the valence number of Y is less than that of X. (ii) The inverse Heusler (Xa)(216), which has the same chemical formula as full Heusler, but it crystallizes in the C1

_{b}type with prototype CuHg

_{2}Ti, where the valence number of Y is greater than that of X. (iii) Half-Heusler alloys (HHAs) have the chemical formula XYZ, crystalline in the C1

_{b}structure, as an inverse Heusler with one X missing. (iv) Quaternary Heusler alloys (QHAs) have the chemical formula XX′YZ, crystalline in the LiMgPdSn (Y-type). In all categories, X, X′, and Y are transition metals, while Z is the main-group element.

_{1}-type structure (regular structure) is more stable than the Xa structure (inverse structure) since the energy difference between the two structures (E = E

_{XA}− E

_{L21}) is greater than zero. This means that the L2

_{1}-type structure is more likely to be synthesized. They also calculated the band structure of Sc

_{2}TiAl and Sc

_{2}TiSi alloys using the first-principle calculations. The calculated electronic structures of the L2

_{1}-type structure and the Xa structure of Sc

_{2}TiAl and Sc

_{2}TiSi are ferromagnetic.

_{2}PtGa. These studies demonstrate the growing interest in metallic full Heusler compounds and the potential they hold for various applications [15,16,17].

_{2}-based Heusler compounds by calculating their elastic constants using a full-potential all-electron method [19].

_{2}CrAl and Cr

_{2}MnSb Heusler alloys. The results revealed that both compounds have half-metallic characters [20]. In 2020, Abu Baker et al. [21] used the FP-LAPW approach to explore the structural, electronic, magnetic, and elastic properties of the regular Co

_{2}TiSn and inverse Co

_{2}TiSn full Heusler alloys. The inverse Zr

_{2}RhGa and regular Co

_{2}TiSn have lattice parameters of 6.619 Å and 6.094 Å, respectively. Additionally, it was discovered that these compounds have total magnetic moments of 1.9786 ${\mu}_{B}$ and 1.99 ${\mu}_{B}$, respectively. Furthermore, the indirect energy gaps for these compounds are 0.482 eV and 0.573 eV, respectively. They are also mechanically stable. On the other hand, Alrahamneh et al. calculated the thermoelectric properties for Zr

_{2}RhGa and Zr

_{2}RhIn compounds, and they found that both compounds exhibit half-metallic behavior and are unsuitable for thermoelectric applications [5].

_{2}Y (Al/Si) full Heusler alloys based on first-principle calculations. There are 26 stable alloys, out of which 24 are magnetic and 10 are half-metallic. For alloys with magnetism, DOS were calculated by applying the GGA. The (Ag/Cu/Mg)

_{2}Y(Al/Si) alloys were found to be unstable, while (Mn/Co)2Y(Al/Si) alloys are always stable. X

_{2}Cr(Al/Si) alloys are stable, while X

_{2}Nb(Al/Si) alloys are always unstable. The (Co/Mn)

_{2}YZ alloys always exhibit magnetism and half-metallicity. Moreover, they predicted that (Mn/Co)

_{2}Y(Al/Si) alloys can be considered as candidates for spintronic devices and serve as a valuable reference for future experimental work.

_{2}TiAl and Sc

_{2}TiSi in the cubic phase for possible spintronic applications. Spintronics reveals a possible high capacity for next-generation information technology [23]. To determine if our compounds are suitable for spintronics, we have studied various properties using the FP-LAPW method.

_{2}TiAl and Sc

_{2}TiSi specifically, and Heusler alloys generally, in addition to outlining the main purpose of this work. The second section describes the computational method used. The third section presents the results obtained by referring to other results available in the literature review; therefore, all the results obtained are systematically compared. The last section is a conclusion, that summarizes the physical properties of the regular and inverse Sc

_{2}TiAl and Sc

_{2}TiSi compounds.

## 2. Computational Method

_{2}TiAl, the muffin-tin (MT) radii of Sc, Ti, and Al atoms are 2.50, 2.50, and 2.39 a.u., respectively, whereas, for Sc2TiSi, Sc, Ti, and Si atoms are 2.45, 2.45, and 2.14 a.u., respectively. To obtain a self-consistency for Sc

_{2}TiAl and Sc

_{2}TiSi, 1240 special k-points in the irreducible Brillion zone (IBZ) were used with a grid equivalent to 50,000 k-points in the full BZ [28]. Furthermore, the expansions of the wave functions were set to $l=10$ inside the MT spheres and the number of plane waves was restricted by ${K}_{max}\times {R}_{MT}=8$. The self-consistent calculations were considered to converge only when the calculated total energy of the crystal converged to less than 0.01 mRy. Finally, using the second-order derivative within the IRelast formalism [29], the elastic constants were calculated.

## 3. Results

#### 3.1. Structural Properties

^{3}) graphs were fitted using Murnaghan’s equation of state (EOS), which is given by [30,31]:

_{2}TiAl and Sc

_{2}TiSi compounds have the space group Fm-3m L2

_{1}(225), while the inverse Heusler Sc

_{2}TiAl and Sc

_{2}TiSi compounds have the space group F-43m Xa (216) [1]. Figure 1 shows the crystal structures of the full Heusler Sc

_{2}TiAl and Sc

_{2}TiSi compounds.

_{2}TiAl and Sc

_{2}TiSi alloys is shown in Figure 2 and Figure 3. Figure 2 indicates that the regular Sc

_{2}TiAl has a minimum energy, ${E}_{0}$, lower than the inverse Sc

_{2}TiAl, indicating that the regular structure of Sc

_{2}TiAl is more mechanically stable than the inverse structure. Similarly, Figure 3 demonstrates that the regular Sc

_{2}TiSi has a minimum energy, ${E}_{0}$, lower than the inverse Sc

_{2}TiSi implying that the regular structure of Sc

_{2}TiSi is more mechanically stable than the inverse structure.

_{2}TiAl and Sc

_{2}TiSi are listed in Table 1. The table demonstrates that the present calculated lattice constants are compatible with the previous theoretical results for both the regular and inverse Heusler structures of Sc

_{2}TiAl and Sc

_{2}TiSi [9].

#### 3.2. Formation Energy

_{2}TiX (X = Si and Al) compounds in both regular and inverse structures. The formation energy (E

_{f}) is defined as the difference between the total energy of a compound and the energy of its components, and is given by the following formula [32]:

_{2}TiX compound in the structure being studied, while ${E}_{Sc}^{hex}$ and ${E}_{Ti}^{hex}$ represent the energy of scandium and titanium in hexagonal structures, respectively. Additionally, ${E}_{X}$ represents the energy for aluminum (face-centered cubic) and silicon (diamond structure). The calculated values of E

_{f}are listed in Table 1.

_{2}TiAl and Sc

_{2}TiSi compounds are negative in both regular and inverse structures, with the regular structure being the most stable for both systems. This is because the formation energy is less than that of the inverse structure. As the formation energy decreases, the stability of the compound increases. Therefore, we can conclude that the Sc

_{2}TiX compounds exist in nature in the regular structure.

#### 3.3. Magnetic Properties

_{2}TiAl and Sc

_{2}TiSi were calculated, and the results were compared with those of other theoretical works, as shown in Table 2. As indicated in Table 2, we concluded that the computed total spin magnetic moment (MM

^{tot}) of the regular and inverse Heusler compounds Sc

_{2}TiAl and Sc

_{2}TiSi is reasonably comparable to the findings of prior theoretical studies [9].

^{tot}for the regular Sc

_{2}TiAl was 2.72 ${\mu}_{B}$, while it was 2.07 ${\mu}_{B}$ for the inverse Sc

_{2}TiAl. The MM

^{tot}in the inverse structure is lower than that in the normal structure due to the low contribution of the Ti atom in the inverse case, whereas its contribution was higher in the normal case. Therefore, it can be noted from the results produced here that the regular and inverse Sc

_{2}TiAl compound’s computed MM

^{tot}values are close to the prior theoretical findings [9].

_{2}TiSi compounds. The main contribution in the MM

^{tot}was due to the high contribution of the Ti atom in the regular case, while its contribution was lower by $~$34% in the inverse case, which caused the MM

^{tot}to be lower (MM

^{tot}= 2.204 ${\mu}_{B}$) in the inverse case compared to the case of regular Heusler Sc

_{2}TiSi (MM

^{tot}= 2.808 ${\mu}_{B}$). It was noticed that the atoms that occupied the Z position in X

_{2}YZ or XYXZ structures for both molecules Sc

_{2}TiAl and Sc

_{2}TiSi did not significantly contribute to the MM

^{tot}. The non-integral total magnetic moment of both compounds confirmed their metallic nature and was consistent with the results of their band structures and DOS. Based on the magnetic results, our studied compounds are unlikely to be half-metallic.

#### 3.4. Electronic Structure

_{2}TiAl and Sc

_{2}TiSi alloys. Since the regular L2

_{1}cubic structure is the ground state, we will focus on the regular L2

_{1}electronic structure. The analysis of the BS for regular Heusler structures of Sc

_{2}TiAl and Sc

_{2}TiSi showed that they exhibited metallic behavior for both spin-up and spin-down within the GGA-PBE and mBJ-GGA methods, with zero-energy gaps, as shown in Figure 4.

_{2}TiAl and Sc

_{2}TiSi Heusler alloys. These figures also show the metallic behavior for both alloys.

_{2}TiAl (Figure 5), the main contributions to the valence band (VB) and conduction band come from the Sc-d state, the Ti-d state, and both the s- and p-states of Al. The main contributions come from the Ti atom, then the Sc atom, and then a small contribution from the Al atom in this compound. In addition, the main contributions to the conduction band come from the Sc-d state, the Ti-d state, and the s- and p-states of Al, with the main contributions coming from the Sc atom, then the Ti atom, and then a small contribution from the Al atom in this compound. In spin-down of regular Sc

_{2}TiAl (Figure 5), the main contributions to the valence band come from the Sc-d state, the Ti-d state, and the s- and p-states of Al, with the main contributions coming from the Sc atom, then a small contribution from the Al atom, and then the Ti atom in this compound. The main contributions to the conduction band come from the Sc-d state, the Ti-d state, and the s- and p-states of Al, with the main contributions coming from the Sc atom, then the Ti atom, and then a small contribution from the Al atom in this compound.

_{2}TiSi (Figure 6), the Sc-d and Ti-d states, as well as the s- and p-states of Si, make the main contributions to the valence and conduction bands.

#### 3.5. Elastic Properties

_{ij}), Bugh ratio (B/S), Poisson’s ratio (ν), Young’s modulus (Y), and the anisotropic factor (A) for the regular and inverse Heusler alloys Sc

_{2}TiAl and Sc

_{2}TiSi. To investigate the stability, we computed three symmetry elements, ${C}_{11}$,${C}_{12}$, and ${C}_{44}$, for cubic bulk alloys. The Born–Huang criteria of mechanical stability are as follows [33,34,35]:

_{44}-shear, and the tetragonal shear moduli, ${C}^{\prime}$, must be positive [19]. The third criterion is known as the spinodal pressure, ${p}_{s}={C}_{11}+2{C}_{12}$, while the last criterion is used to define an additional elastic constant, called the tetragonal shear modulus:

_{2}TiAl and Sc

_{2}TiSi were stable (see Table 3).

_{2}TiAl and Sc

_{2}TiSi are ductile, and they have dominant ionic bonds.

_{2}TiAl and Sc

_{2}TiSi compounds exhibit elastic anisotropy, as shown in Table 3.

_{2}TiAl and Sc

_{2}TiSi, in both regular and inverse structures, satisfy the stability conditions and are considered mechanically stable with elastic anisotropy. Table 4 shows that the regular and inverse structures of Sc

_{2}TiSi are ductile, as $B/S>1.75$ and $\nu >0.26$. However, the regular structure of Sc

_{2}TiAl is brittle ($B/S<1.75$ and $v<0.26$), while the inverse structure is ductile ($B/S>1.75$ and $\nu >0.26$).

_{3}= 0 are isotropic, and the two quantities, s

_{2}and s

_{3}, are related to the elastic waves. Since c

_{44}> 0, then F

_{44}> 0, whereas F

_{12}is restricted to −0.5 < F

_{12}< 1. These are the restrictions for Blackman’s diagram, and all values within these ranges are allowed. In Every’s diagram, the values for stability are restricted to fall into the triangle with (s

_{2}, s

_{3}), equal to (−0.5, −1)–(1, 0)–(1, 1.5), as shown in Figure 7. It is noticed that the studied compounds in this study were below the isotropy line, where s

_{3}= 0 [19].

_{s}= 0. Additionally, all the studied compounds were in the region where the anisotropy index is positive [19].

_{2}TiAl (k = 1.7), which lay slightly to the left of the vertical line between brittle and ductile. Therefore, it was considered a brittle compound.

## 4. Conclusions

_{2}TiAl and Sc

_{2}TiSi. The results were as follows: According to the electronic properties, both regular Heusler Sc

_{2}TiAl and Sc

_{2}TiSi compounds are metals with zero-energy bandgaps in both GGA-PBE and mBJ methods. As for the magnetic properties, both Sc

_{2}TiAl and Sc

_{2}TiSi alloys are ferromagnetic compounds with a total magnetic moment of ${\mathrm{MM}}^{\mathrm{tot}}=$2.86, 2.09, 2.97, and 2.54 ${\mu}_{B}$, respectively. Additionally, the elastic properties of these compounds emphasized that both regular and inverse Sc

_{2}TiAl and regular Sc

_{2}TiSi are mechanically stable. The B/S ratios showed that both regular and inverse Heusler Sc

_{2}TiAl and Sc

_{2}TiSi compounds have ductile natures. Moreover, the Poisson’s ratio values indicated that both Sc

_{2}TiAl and Sc

_{2}TiSi alloys have ionic bonds. Finally, both Sc

_{2}TiAl and Sc

_{2}TiSi compounds exhibited elastic anisotropy.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Paudel, R.; Zhu, J. Investigation of half-metallicity and magnetism of (Ni/Pd/Ru) ZrTiAl quaternary Heusler alloys for spintronic applications. Phys. Condens. Matter
**2019**, 557, 45–51. [Google Scholar] [CrossRef] - Azar, S.; Mousa, A.; Khalifeh, J. Structural, electronic and magnetic properties of Ti1+xFeSb Heusler alloys. Intermetallics
**2017**, 85, 197–205. [Google Scholar] [CrossRef] - Berrahal, M.; Bentouuaf, A.; Rached, H.; Mebsout, R.; Aissa, B. Investigation of Ruthenium based Full-Heusler compound for thermic, spintronic and thermoelectric applications: DFT computation. Mater. Sci. Semicond. Process.
**2021**, 134, 106047. [Google Scholar] [CrossRef] - Patel, P.; Pandaya, J.; Shinde, S.; Gupta, S.; Narayan, S. Investigation of Full-Heusler compound Mn2MgGe for magnetism, spintronics and thermoelectric applications: DFT study. Comput. Condens. Matter
**2020**, 23, e00472. [Google Scholar] [CrossRef] - Alrahamneh, M.; Mousa, A.; Khalifeh, J. First principles study of the structural, electronic, magnetic and thermoelectric properties of Zr
_{2}RhAl. Phys. Condens. Matter**2019**, 552, 227–235. [Google Scholar] [CrossRef] - Remil, G.; Zitouni, A.; Bouadjemi, B.; Houari, M.; Abbad, A.; Benstaali, W.; Cherid, S.; Matougui, M.; Lantri, T.; Bentata, S. A potential full Heusler thermoelectric material CO
_{2}ZrZ (Z=Al, Si, Ga and Sn) in low temperature: An Ab-initio investigation. Solid State Commun.**2021**, 336, 114422. [Google Scholar] [CrossRef] - Srivastava, V.; Kaur, N.; Khenata, R.; Dar, S. Investigation of the electronic, magnetic, elastic, thermodynamic and thermoelectric properties of Mn
_{2}CoCr Heusler compound A DFT-based simulation. J. Magn. Magn. Mater.**2020**, 513, 167107. [Google Scholar] [CrossRef] - Mushtaq, M.; Khalid, S.; Atif Sattar, M.; Khenata, R.; Seddik, T.; Ahmad Dar, S.; Muhammad, I.; Bin Omran, S. Electronic band structure, phase stability, magnetic and thermoelectric characteristics of the quaternary Heusler alloys CoCuZrAs and CoRhMoAl: Insights from DFT computations. Inorg. Chem. Commun.
**2021**, 124, 108384. [Google Scholar] [CrossRef] - Han, Y.; Chen, Z.; Kuang, M.; Liu, Z.; Wang, X.; Wang, X. 171 Scandium-based full Heusler compounds: A comprehensive study of competition between XA and L21 atomic ordering. Results Phys.
**2019**, 12, 435–446. [Google Scholar] [CrossRef] - Wang, C.; Casper, F.; Gasi, T.; Ksenofontov, V.; Balke, B.; Fecher, G.H.; Felser, C.; Hwu, Y.; Lee, J. Structural and magnetic properties of Fe2CoGa Heusler nanoparticles. J. Phys. Appl. Phys.
**2012**, 45, 295001. [Google Scholar] [CrossRef] - Shan, R.; Ouardi, S.; Fecher, G.H.; Gao, L.; Kellock, A.; Gloskovskii, A.; ViolBarbosa, C.E.; Ikenaga, E.; Felser, C.; Parkin, S.S.P. A p-type Heusler compound: Growth, structure, and properties of epitaxial thin NiYBi films on MgO(100). Appl. Phys. Lett.
**2012**, 101, 212102. [Google Scholar] [CrossRef] - Galanakis, I.; Şaşıoğlu, E. High T
_{C}half-metallic fully-compensated ferrimagnetic Heusler compounds. Appl. Phys. Lett.**2011**, 99, 052509. [Google Scholar] [CrossRef] - Nayak, A.K.; Shekhar, C.; Winterlik, J.; Gupta, A.; Felser, C. Mn2PtIn: A tetragonal Heusler compound with exchange bias behavior. Appl. Phys. Lett.
**2012**, 100, 152404. [Google Scholar] [CrossRef] - Wei, X.P.; Hu, X.R.; Liu, B.; Lei, Y.; Deng, H.; Yang, M.K.; Deng, J.B. Electronic structure and magnetism in full-Heusler compound Mn2ZnGe. J. Magn. Magn. Mater.
**2011**, 323, 1606. [Google Scholar] [CrossRef] - Graf, T.; Felser, C.; Parkin, S.S. Simple rules for the understanding of Heusler compounds. Prog. Solid State Chem.
**2011**, 39, 1–50. [Google Scholar] [CrossRef] - Felser, C.; Hirohata, A. Heusler Alloys Properties, Growth, Applications. In Springer Series in Materials Science; Springer: Berlin/Heidelberg, Germany, 2016; ISBN 978-3-319-21449-8. [Google Scholar]
- Nayak, A.K.; Nicklas, M.; Chadov, S.; Shekhar, C.; Skourski, Y.; Winterlik, J.; Felser, C. Large zero-field cooled exchange-bias in bulk Mn2PtGa. Phys. Rev. Lett.
**2013**, 110, 127204. [Google Scholar] [CrossRef] - Elphick, K.; Frost, W.; Samiepour, M.; Kubota, T.; Takanashi, K.; Sukegawa, H.; Mitani, S.; Hirohata, A. Heusler alloys for spintronic devices: Review on recent development and future perspectives. Sci. Technol. Adv. Mater.
**2021**, 22, 235. [Google Scholar] [CrossRef] - Wu, S.-C.; Fecher, G.H.; Naghavi, S.S.; Felser, C. Elastic properties and stability of Heusler compounds: Cubic Co
_{2}YZ compounds with L21 structure. J. Appl. Phys.**2019**, 125, 082523. [Google Scholar] [CrossRef] - Yahya, S.J.; Abu-Jafar, M.S.; Al Azar, S.; Mousa, A.A.; Khenata, R.; Abu-Baker, D.; Farout, M. The Structural, Electronic, Magnetic and Elastic Properties of Full-Heusler Co
_{2}CrAl and Cr_{2}MnSb: An Ab Initio Study. Crystals**2022**, 12, 1580. [Google Scholar] [CrossRef] - Abu Baker, D.N.; Abu-Jafar, M.S.; Mousa, A.; Jaradat, R.; Ilaiwi, K.; Khenata, R. Structural, magnetic, electronic and elastic properties of half-metallic ferromagnetism full-Heusler alloys: Regular- Co
_{2}TiSn and inverse- Zr_{2}RhGa using FP-LAPW method. Mater. Chem. Phys.**2020**, 240, 122122. [Google Scholar] [CrossRef] - Li, Y.; Zhu, J.; Paudel, R.; Huang, J.; Zhou, F. Screen the half metallic X
_{2}Y (Al/Si) full Heusler alloys based on the first principle calculations. Comput. Mater. Sci.**2021**, 193, 110391. [Google Scholar] [CrossRef] - Oudrane, D.; Bourachid, I.; Bouafia, H.; Abidri, B.; Rached, D. Computational insights in predicting structural, mechanical, electronic, magnetic and optical properties of EuAlO3 cubic Perovskite using FP-LAPW. Comput. Condens. Matter
**2021**, 26, e00537. [Google Scholar] [CrossRef] - Blaha, P.; Schwarz, K.; Tran, F.; Laskowski, R.; Madsen, G.K.H.; Marks, L.D. WIEN2k: An APW+lo program for calculating the properties of solids. J. Chem. Phys.
**2020**, 152, 074101. [Google Scholar] [CrossRef] - Perdew, J.; Burke, K.; Ernzerhof, M. Generalized gradient approximation made simple. Phys. Rev. Lett.
**1996**, 77, 3865–3868. [Google Scholar] [CrossRef] - Becke, A.; Johnson, E. A simple effective potential for exchange. J. Chem. Phys.
**2006**, 124, 221101. [Google Scholar] [CrossRef] [PubMed] - Tran, F.; Blaha, P. Accurate band gaps of semiconductors and insulators with a semilocal exchange-correlation potential. Phys. Rev. Lett.
**2009**, 102, 226401. [Google Scholar] [CrossRef] [PubMed] - Blaha, P.; Schwarz, K.; Medsen, G.K.H.; Kvasnicka, D.; Luitz, J. WIEN2k, An Augmented Plane Wave Plus Local Orbitals Program for Calculating Crystal Properties; Vienna University Technology: Vienna, Austria, 2001; Available online: https://www.scirp.org/(S(i43dyn45teexjx455qlt3d2q))/reference/ReferencesPapers.aspx?ReferenceID=1880359 (accessed on 5 January 2023).
- IRelast Package is Provided by M. Jamal as Part of the Commercial Code WIEN2K. 2014. Available online: http://www.wien2k.at/ (accessed on 5 January 2023).
- Murnaghan, F.D. The compressibility of media under extreme pressures. Proc. Natl. Acad. Sci. U.S.A.
**1944**, 30, 244. [Google Scholar] [CrossRef] - Tyuterev, V.G.; Vast, N. Murnaghan’s equation of state for the electronic ground state energy. Comput. Mater. Sci.
**2006**, 38, 350. [Google Scholar] [CrossRef] - Mousa, A.; Hamad, B.; Khalifeh, J. Structure, electronic and elastic properties of the NbRu shape memory alloys. Eur. Phys. J.
**2009**, 72, 575–581. [Google Scholar] [CrossRef] - Born, M.; Huang, K. Dynamical Theory of Crystal Lattices. Am. J. Phys.
**1956**, 23, 474. [Google Scholar] [CrossRef] - Gupta, Y.; Sinha, M.; Verma, S. Exploring the structural, elastic, lattice dynamical stability and thermoelectric properties of semiconducting novel quaternary Heusler alloy LiScPdPb. J. Solid State Chem.
**2021**, 304, 122601. [Google Scholar] [CrossRef] - Abu-Jafar, M.; Dayton-Oxland, R.; Jaradat, R.; Mousa, A.; Khenata, R. Structural, electronic, mechanical and elastic properties of Scandium Chalcogenides by first-principles calculations. Phase
**2020**, 93, 773. [Google Scholar] [CrossRef] - Pugh, S.F. Relations between the elastic moduli and the plastic properties of polycrystalline pure metals. Philos. Mag.
**1954**, 45, 823. [Google Scholar] [CrossRef] - Abu-Jafar, M.; Leonhardi, v.; Jaradat, R.; Mousa, A.; Al-Qaisi, S.; Mahmoud, N.; Bassalat, A.; Khenata, R.; Bouhemadou, A. Structural, electronic, mechanical, and dynamical properties of scandium carbide. Results Phys.
**2021**, 21, 103804. [Google Scholar] [CrossRef] - Cheriet, A.; Khenchoul, S.; Aissani, L.; Lagoun, B.; Zaabat, M.; Alhussein, A. First-principles calculations to investigate structural, magnetic, electronic and elastic properties of full-Heusler alloys Co2MB (M=V, Mn). Solid State Commun.
**2021**, 337, 114426. [Google Scholar] [CrossRef] - Abada, A.; Marbouh, N.; Bentayeb, A. First-principles calculations to investigate structural, elastic, electronic and magnetic properties of novel d half metallic half Heusler alloys XSrB (X=Be, Mg). Intermetallics
**2022**, 140, 107392. [Google Scholar] [CrossRef] - Voigt, W. Ueber die Beziehung Zwischen den Beiden Elasticitätsconstanten Isotroper Körper; Wiley Online Library: San Marcos, CA, USA, 1889; p. 38. [Google Scholar] [CrossRef]
- Reuss, A. Berechnung der Fließgrenze von Mischkristallen auf Grund der Plastizitätsbedingung für Einkristalle. Math Phys.
**1929**, 9, 49. [Google Scholar] [CrossRef] - Hill, R. The Elastic Behaviour of a Crystalline Aggregate. Proc. Phys. Soc.
**1952**, 65, 349. [Google Scholar] [CrossRef] - Teter, D.M. Computational alchemy: The search for new superhard materials. MRS Bull.
**1998**, 23, 22. [Google Scholar] [CrossRef] - Zener, C. Elasticity and Anelasticity of Metals; University of Chicago Press: Chicago, IL, USA, 1948. [Google Scholar] [CrossRef]
- Ravindran, P.; Fast, L.; Korzhavyi, P.A.; Johansson, B. Density functional theory for calculation of elastic properties of orthorhombic crystals: Application to TiSi
_{2}. J. Appl. Phys.**1998**, 84, 4891. [Google Scholar] [CrossRef]

**Figure 1.**Crystal structures of Sc

_{2}TiAl and Sc

_{2}TiSi in (

**a**) regular Heusler (L2

_{1}) and (

**b**) inverse Heusler (Xa) (Red: Sc, Green: Ti, Blue: Al/Si).

**Figure 2.**The total energy (Ry) versus volume (a.u

^{3}) of non-magnetic, regular, and inverse Sc

_{2}TiAl Heusler.

**Figure 3.**The total energy (Ry) versus volume (a.u

^{3}) of non-magnetic, regular, and inverse Sc

_{2}TiSi Heusler.

**Figure 4.**Calculated band structure for regular Sc

_{2}TiAl and Sc

_{2}TiSi using GGA and mBJ (blue lines for spin-up, red lines for spin-down).

**Figure 5.**(

**a**) TDOS for regular Sc

_{2}TiAl, and PDOS for the (

**b**) Sc atom, (

**c**) Ti atom, and the (

**d**) Al atom.

**Figure 6.**(

**a**) TDOS for regular Sc

_{2}TiSi, and PDOS for the (

**b**) Sc atom, (

**c**) Ti atom, and the (

**d**) Si atom.

**Figure 7.**(

**a**) Blackman’s and (

**b**) Every’s diagrams. The full green line refers to the isotropic case, where A

_{e}= 1, and the dashed green line refers to the case where Cauchy pressure = 0. In Every’s diagram, the stability triangle is marked by dashed brown lines.

**Figure 8.**Ductile–brittle diagram for the studied compounds. The horizontal dashed line corresponds to the metallic-covalent criteria of Pettiford, and the vertical dashed line corresponds to the brittleness–ductile criteria of Pugh.

**Table 1.**The obtained lattice constant (a), bulk modulus (B), pressure derivative of the bulk modulus (B’), minimum energy (E

_{0}), and formation energy (E

_{f}) for the regular and inverse Heusler of both Sc

_{2}TiAl and Sc

_{2}TiSi.

Structure | Space Group | Reference | a (A ^{0}) | B (GPa) | B^{′}(GPa) | E_{0}(eV) | E_{f}(meV) |
---|---|---|---|---|---|---|---|

Sc_{2}TiAl | Fm-3m (225) | Present Theoretical | 6.88 6.87 [9] | 75.9 | 2.59 | −5250.241 | −391.1 |

F-43m (216) | Present Theoretical | 6.84 6.83 [9] | 73.9 | 3.30 | −5250.219 | −109.8 | |

Sc_{2}TiSi | Fm-3m (225) | Present Theoretical | 6.69 6.69 [9] | 85.28 | 4.35 | −5344.693 | −913.3 |

F-43m (216) | Present Theoretical | 6.64 6.64 [9] | 82.79 | 5.26 | −5344.669 | −584.7 |

**Table 2.**Total, atom-resolved, and interstitial spin magnetic moments for inverse and regular Sc

_{2}TiAl and Sc

_{2}TiSi.

Structure | Reference | $\mathbf{Magnetic}\mathbf{Moment}\mathbf{in}{\mathit{\mu}}_{\mathit{B}}$ | |||||
---|---|---|---|---|---|---|---|

Sc | Sc | Ti | Al | Interstitial | MM^{tot} | ||

Regular Sc _{2}TiAl | Present Theoretical | 0.38 | 0.38 | 1.71 | −0.04 | 0.43 | 2.86 2.92 [9] |

Inverse Sc _{2}TiAl | Present Theoretical | 0.18 | 0.42 | 0.91 | −0.02 | 0.60 | 2.09 2.24 [9] |

Regular Sc _{2}TiSi | Present Theoretical | 0.33 | 0.33 | 1.60 | −0.05 | 0.76 | 2.97 2.96 [9] |

Inverse Sc _{2}TiSi | Present Theoretical | 0.20 | 0.63 | 1.05 | −0.04 | 0.70 | 2.54 2.51 [9] |

**Table 3.**Elastic constants (${C}_{ii}$), bulk modulus (B), and anisotropic factor (A) of Sc

_{2}TiAl and Sc

_{2}TiSi.

Compound | C_{11} (GPa) | C_{12} (GPa) | C_{44} (GPa) | B (GPa) | A |
---|---|---|---|---|---|

Regular Sc _{2}TiAl | 96.914 | 67.463 | 63.778 | 77.280 | 4.331 |

Inverse Sc _{2}TiAl | 75.932 | 73.512 | 63.281 | 74.318 | 52.298 |

Regular Sc _{2}TiSi | 106.781 | 81.347 | 64.582 | 89.825 | 5.078 |

Inverse Sc _{2}TiSi | 95.248 | 94.762 | 63.792 | 94.924 | 262.519 |

**Table 4.**Shear modulus ($S$), $B/S$ ratio, Voigt Poisson’s ratio ($\nu $), and Young’s modulus ($Y$) of Sc

_{2}TiAl and Sc

_{2}TiSi.

Compound | $\mathit{S}\left(\mathrm{GPa}\right)$ | $\mathit{B}/\mathit{S}\left(\mathrm{GPa}\right)$ | $\mathit{Y}\left(\mathrm{GPa}\right)$ | $\mathit{\nu}$ |
---|---|---|---|---|

Regular Sc _{2}TiAl | 45.41 | 1.70 | 113.30 | 0.247 |

Inverse Sc _{2}TiAl | 38.45 | 1.93 | 98.39 | 0.279 |

Regular Sc _{2}TiSi | 43.84 | 2.05 | 113.11 | 0.290 |

Inverse Sc _{2}TiSi | 38.37 | 2.74 | 101.45 | 0.321 |

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## Share and Cite

**MDPI and ACS Style**

Al-Masri, K.M.; Abu-Jafar, M.S.; Farout, M.; Dahliah, D.; Mousa, A.A.; Azar, S.M.; Khenata, R.
Structural, Elastic, Electronic, and Magnetic Properties of Full-Heusler Alloys Sc_{2}TiAl and Sc_{2}TiSi Using the FP-LAPW Method. *Magnetochemistry* **2023**, *9*, 108.
https://doi.org/10.3390/magnetochemistry9040108

**AMA Style**

Al-Masri KM, Abu-Jafar MS, Farout M, Dahliah D, Mousa AA, Azar SM, Khenata R.
Structural, Elastic, Electronic, and Magnetic Properties of Full-Heusler Alloys Sc_{2}TiAl and Sc_{2}TiSi Using the FP-LAPW Method. *Magnetochemistry*. 2023; 9(4):108.
https://doi.org/10.3390/magnetochemistry9040108

**Chicago/Turabian Style**

Al-Masri, Khadejah M., Mohammed S. Abu-Jafar, Mahmoud Farout, Diana Dahliah, Ahmad A. Mousa, Said M. Azar, and Rabah Khenata.
2023. "Structural, Elastic, Electronic, and Magnetic Properties of Full-Heusler Alloys Sc_{2}TiAl and Sc_{2}TiSi Using the FP-LAPW Method" *Magnetochemistry* 9, no. 4: 108.
https://doi.org/10.3390/magnetochemistry9040108