An-Najah National University Faculty of Graduate studies Structural, electronic, magnetic & elastic properties of Full-Heusler alloys: normal and inverse Zr2RhGa, Co2TiSn using FP-LAPW method By Doha Naser Abu Baker Supervisor Prof. Mohammed Abu-Jafar This Thesis is Submitted in Partial Ful fillment of the Requirements for the Degree of Master of Physics, Faculty of Graduated Studies, An-Najah National University, Nablus - Palestine. 2018 II Structural, electronic, magnetic & elastic properties of Full-Heusler alloys: normal and inverse Zr2RhGa, Co2TiSn using FP-LAPW method By Doha Naser Abu Baker This thesis was defended successfully on 24 /10/2018 and approved by: Defense committee members: Signature  Prof. Mohammed Abu-Jafar / Supervisor .…….…….  Prof. Jihad Asad / External Examiner …………..  Prof. Mohammed El-Saeed / Internal Examiner ………….. III Dedication To my kind parents, who are impossible to be thanked adequately for everything they have done for me and my future and learn me to be ambitious person. They are really the best model for perfect parents and they are the main cause of success in my life. ALLAH bless them. To my beloved husband (Hasan), who shares me all moments and helps me to overcome difficulties to continue moving forward. He also provides me encouragement, motivation and support to achieve my ambitions successfully. To my lovely sister (Saja) and my lovely brothers (Thaer, Fadi & Rami),who always encourage me and give me a powerful support. They are always present to help and motivate me .To my cute babies (Sara & Saeed). IV Acknowledgments All thanks to Allah, who gives me health, patience and knowledge to complete my thesis.I would like to express sincere thanks to my advisor and instructor Dr. Mohammed Abu-Jafar for his guidance, assistance, supervision and contribution of valuable suggestions, I would like to thank Ms. Nisreen Hamadneh who helps me to use some programs. Never forget my faculty members of physics department for their help and encouragement. Finally, special thanks to my friends for their moral support and cares. V االقرار الرسالة التي تحمل عنوان: ةانا الموقعة أدناه, مقدم Dyslipidemia in young patients with type I diabetes mellitus in Nablus city: a cross-sectional study أقر بأن ما اشتممت عميو ىذه الرسالة ىي من نتاج جيدي الخاص, باستثناء ما تمت االشارة اليو ء منيا لم يقدم من قبل لنيل درجة أو لقب عممي أو حيثما ورد, وأن ىذه الرسالة ككل أو أي جز بحثي لدى أي مؤسسة تعميمية أو بحثية أخرى. Declaration The work provided in this thesis unless otherwise referenced, is the researcher's own work, and has not been submitted elsewhere for any other degree or qualification. Student's name: :اسم الطالبة Signature: :التوقيع Date: :التاريخ VI Table of contents No. Content Page Dedication Iii Acknowledgment Iv Declaration V List of Tables Vii List of Figures Viii Abstract X 1 Chapter One: Introduction 1 2 Chapter Two: Methodology 5 2.1 The Open-Heimer Approximation 5 2.2 Hartree and Hartree-Fock Approximation 6 2.3 Density Functional Theory 6 2.4 Single Particle Kohn-Sham Equation 7 2.5 The Exchange-Correlation Functional 9 2.6 Local Density Approximation(LDA) 10 2.7 Generalized Gradient Approximation(GGA) 11 2.8 Agumented Plane Wave (APW) Method. 12 2.9 The Linearized Augmented Plane Wave (LAPW) Method. 13 2.10 The Augmented Plane Wave+Local Orbits (LAPW+Lo) Method 14 3 Chapter Three: Results and Discussion 16 3.1 Computational Method 16 3.2 Structural Properties 17 3.3 Magnetic Properties 22 3.4 Electronic Properties 23 3.5 Elastic Properties 38 4 Chapter Four: Conclusion 43 References 45 b الملخص VII List of Tables No. Table captions Page 1 Calculated lattice parameter, bulk modulus, pressure derivative for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds. 21 2 Total magnetic moment for normal and inverse Heusler Co2TiSn Compound. 22 3 Total magnetic moment for normal and inverse Heusler Zr2RhGa Compound. 22 4 Energy band gaps for normal Heusler Co2TiSn and normal Heusler Zr2RhGa Compounds. 28 5 Energy band gaps for inverse Heusler Co2TiSn and inverse Heusler Zr2RhGa Compounds. 29 6 Elastic constants for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa Compounds. 40 VIII List of Figures No. Figure captions Page 1 Flowchart for the iteration in the self-consistent procedure to solve Hartree-Fock or Kohn-Sham equations. 9 2 Scheme of Augmented Plane Wave. 12 3 The crystal structure of (a) normal Heusler Co2TiSn (b) inverse Heusler Co2TiSn (c) normal Heusler Zr2RhGa (d) inverse Heusler Zr2RhGa compounds. 18 4 Total energy as function of the volume for (a)normal Co2TiSn (b) normal Zr2RhGa and (c) inverse Zr2RhGa (d) inverse Co2TiSn compounds on unit cell of volume. 19 5 Band structure spin up by using PBE-GGA method for (a) normal Co2TiSn (b) inverse Co2TiSn (c) inverse Zr2RhGa compounds. 25 6 Band structure spin down by using PBE-GGA method for (a) normal Co2TiSn (b) inverse Co2TiSn (c) inverse Zr2RhGa compounds 26 7 Band structure by using PBE-GGA method for normal Zr2RhGa compound. 27 8 Band structure spin up by using mBJ-GGA method (a) normal Co2TiSn (b) inverse Zr2RhGa compounds . 27 9 Band structure spin down by using mBJ-GGA method (a) normal Co2TiSn (b) inverse Zr2RhGa compounds . 28 10 (a)Total density of states of spin up for normal Co2TiSn and partial density of states of spin up for (b) Co atom (c) Ti atom (d) Sn atom of normal Co2TiSn compound. 31 11 (a)Total density of states of spin down for normal Co2TiSn and partial density of states of spin up for (b) Co atom (c) Ti atom (d) Sn atom of normal Co2TiSn compound. 32 12 (a)Total density of states for normal Zr2RhGa and partial density of states for (b) Zr atom (c) Rh atom (d) Ga atom of normal Zr2RhGa compound. 33 13 (a)Total density of states of spin up for inverse Co2Ti Sn and partial density of states of spin down for (b) Co atom (c) Ti atom (d) Sn atom of inverse Co2TiSn compound. 34 14 (a)Total density of states of spin down for inverse Co2Ti Sn and partial density of states of spin down for (b) Co atom (c) Ti atom (d) Sn atom of inverse Co2TiSn compound. 35 IX 15 (a)Total density of states of spin up for inverse Zr2RhGa and partial density of states of spin up for (b) Zr atom (c) Rh atom (d) Ga atom of inverse Zr2RhGa compound. 36 16 (a)Total density of states of spin down for inverse Zr2RhGa and partial density of states of spin down for (b) Zr atom (c) Rh atom (d) Ga atom of inverse Zr2RhGa compound. 37 X Structural, electronic, magnetic & elastic properties of Full-Heusler alloys: normal and inverse Zr2RhGa, Co2TiSn using FP-LAPW method By Doha Naser Abu Baker Supervisor Prof. Mohammed Abu-Jafar Abstract The equilibrium structural parameters, electronic, magnetic and elastic properties of the normal and inverse Zr2RhGa and Co2TiSn Full-Heusler compounds have been studied using density functional theory (DFT) and full–potential linearized augmented plane wave (FP-LAPW) method as implemented in the WIEN2k package. The Generalized Gradient Approximation (GGA) has been used for the exchange-correlation potential (VXC) to compute the equilibrium structural parameters; lattice constant (a), bulk modulus (B), bulk modulus pressure derivative (B). In addition to GGA approach, the modified Becke Johnson (mBJ) scheme has been used to calculate the band gap energies. The normal Heusler Co2TiSn compound and inverse Heusler Zr2RhGa compound within GGA and mBJ approaches are found to have a half-metallic behavior, with an indirect energy gap in the spin down configuration. The total magnetic moment for normal Co2TiSn and inverse Zr2RhGa Full-Heusler compounds are to some extent compatible with the experimental and theoretical results. The normal Co2TiSn and inverse Zr2RhGa Full-Heusler compounds are mechanically stable; are satisfy the Born mechanical stability criteria. B/S ratio shows that the normal Co2TiSn has a brittle nature. While the inverse has a ductile XI nature. Poisson’s ratio (ν) values show that the normal Co2TiSn and inverse Zr2RhGa Full-Heusler compounds have an ionic bond nature. 1 Chapter One Introduction In recent years, Heusler alloys received growing attention due to their interesting physical properties[1-4], especially the half- metallic (HM) character, half-metallic (HM) materials exhibiting a 100% spin polarization around the Fermi surface [1-3] (a half metal is a ferromagnetic with a gap in one of the spin directions at the Fermi energy εf). Half metals can be used as spin injectors for magnetic random access memories and other spin dependent devices [2]. A lot of alloys were predicted to be half-metallic (HM) materials. In fact, investigating and searching for new (HM) materials are mainly focusing on the Heusler alloys [3-17]. From structural point of view, Heusler family can be described by two variants: Full Heusler X2YZ phases, which typically crystallize in Cu2MnAl(L21)-type structure and the Half-Heusler XYZ which typically crystallize in NiMnSb (C1b) type structure, X and Y transition elements are 3d,4d or 5d elements and Z is s-p elements. The Full-Heusler compounds are divided into two types: normal Heusler and inverse Heusler. The atoms in normal Heusler compounds are lined up in X2: (1/4,1/4,1/4), (3/4,3/4,3/4), Y(1/2,1/2,1/2), and Z(0,0,0), and the atoms in inverse 2 Heusler compounds are lined up in X2:(1/4,1/4,1/4), (1/2,1/2,1/2), Y(3/4,3/4,3/4), and Z(0,0,0) [18] . There are a lot of previous studies that have been done on Heusler compounds and other compounds (half-metallic compounds) by using different methods [19-24]. In 2006, Kandpal et al. [19] measured the lattice parameter and the total magnetic moment for inverse Heusler Co2TiSn compound and found to be 7.072 A o , 2 µB, respectively. Also, the magnetic moment was calculated using LMTO-ASA code [20], SPRKKR code [21] , FP-LAPW method [22] and FPLMTO method [23] ,it is found to be 1.40 µB, 1.55 µB, 2 µB, 2 µB, respectively. It is clear that the LMTO-ASA and SPRKKR codes fail to get the correct measured total magnetic moment, while Wien2k and FPLMTO codes do correctly obtain minority gap in this compound and also measured correctly the magnetic moment 2 µB per formula unit . In 2014, Birsan and Kuncser [24] studied the electronic, structural and magnetic properties of Zr2CoSn Full-Heusler compound by using FP- LAPW method. They calculated the energy band gap (Eg= 0.543 eV), total magnetic moment (Mtot = 3 µB) and the lattice parameter (a=6.76 A o ). In 2014, A. Birsan [25] investigated structural and magnetic properties of the Full-Heusler compound, Zr2CoAl using FP-LAPW method. He calculated the lattice parameter (a= 6.54 A 0 ), energy band gap (Eg = 0.48 eV), and total magnetic moment (Mtot = 2 µB ). 3 In 2015, Wang et al. [26] studied the half-metallic state and magnetic properties versus the lattice constant in Zr2RhZ (Z = Al, Ga, In) Heusler alloys by using CASTEP code. The CASTEP code is based on the density functional theory (DFT) pseudo-potential method [27,28] .The calculated lattice parameters were found to be 6.66, 6.64 and 6.81 A O ,respectively and magnetic moment for all alloys was found to be 2 µB [26] . In 2017, Jain et al. [29] studied electronic structure, magnetic and optical properties of Co2TiZ (Z= B,AL,Ga,In ) Heusler alloys by using FP-LAPW code. They found the lattice parameters to be 5.494, 5.842, 5.845 and 6.087 A 0 , respectively. Bulk modulus is found to be 233, 182, 184 and 161 GPa, respectively. Energy band gap is found to be 0.1, 0.44, 0.16 and 0.06 eV, respectively, and total magnetic moment is found to be 1, 0.99, 1 and 1.03 µB, respectively. In 2017, Weia et al. [30] estimated the electronic, Fermi surface, Curie temperature and optical properties of Zr2CoAl compound by using FPLO code [31,32]. They calculated the lattice parameter (a = 6.629 A 0 ), Bulk modulus (B = 106.8 GPa, ), and the total magnetic moment( Mtot = 2 µB ) for the normal Full-Heusler structure (Cu2MnAl). In present work, the motivation is to investigate electronic, structural, magnetic and elastic properties of normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds by using the full potential-linearized augmented plane wave method [FP-LAPW method] within the Perdew- 4 Burke-Ernzerhof generalized gradient approximation [PBE-GGA] [33] integrated in Wien2k code [22] . In chapter two, presents methodology details. In chapter three, we report and discuss our obtained results for the system of interest, Finally Conclusions are summarized in chapter four. 5 Chapter Two Methodology Schrodinger equation; given in Equation 1 below; is very hard to be solved for a N-body system. It has become evident that we must involve some approximations to face a lot of problems of the body [22]. The first term is the kinetic energy operator for the nuclei (Tn), the second term is for the kinetic energy of electrons (Te) and the last three terms describe the coulomb interaction between the electron and the nuclei ( Ven), between the electron and other electrons (Vee), and between the nuclei and other nuclei (Vnn) . There isn’t an exact solution to this Schrodinger equation. However, to solve this problem, there are some approximations. The approximations are as follows: 2.1 The Open-Heimer Approximation This approximation assumes that the nuclei are more massive than electrons, and then Tn =zero and Vnn =constant then H= Te +Vee + Vext (2) Where the External Potential is given by Vext =(Vnn+Ven) 6 2.2 Hartree and Hartree-Fock Approximation Hartree approximation depends on the principle of pauli exclusion. This means that no two electrons can have the same set of quantum numbers. In Hartree-Fock Approximation, many system particles have been solved by assuming that the electrons are independent and separated from each other. Thus, the wave function of electrons can be written as: Ψ (r1,r2,r3….rN) =Ψ1(r1 )Ψ2(r2)Ψ3(r3)….ΨN(rN) (3) The Ψn (rn) is the wave function for the electrons, and so the total Hamiltonian can be written as: (Ts+ Vext + VH) Ψ(r) = E Ψ(r) (4) The Ts is the kinetic energy and Vext is the external potential whereas the VH is the Hartree potential for non-interacting electrons, and it can be written as: (5) 2.3 Density Functional Theory (DFT) DFT provides a way to solve many body problem into a single body problem, its successful minimizing of the energy functional will produce the ground state density ρ0. So E (ρ) is rewritten as the Hartree total energy plus another smaller unknown functional called exchange-correlation functional, Exc (ρ). 7 E (ρ) = Ts (ρ) + Ec (ρ) + EH (ρ) + Eii (ρ) + Exc (ρ) (6) Whereas Ts is single kinetic energy, Ec is coloumb energy between nuclei and electrons. Eii(ρ) represents the interaction between the nuclei. Exc is found to be the exchange correlation energy and it is unknown part, while EH is Hartree potential. (7) Based on the variational principle, a set of effective one-particle Schrodinger equation (Kohn-Sham Equation) is given as: [Ts +Vext(r) +VH(ρ(r)) +VXC(ρ(r))]Фi(r) = εiФi(r) (8) Where εi is the single particle energy, Фi is the electron wave function, VH is the Hartree potential, Vext is the coulomb potential and VXC is the exchange-correlation potential. 2.4 Single Particle Kohn-Sham Equation. The LAPW method is a procedure to solve the Kohn-Sham equations of the ground state density, total energy and (Kohn-Sham) eigenvalues (energy bands) of a many electrons system. In such method, we can interpret Equation 7 as the energy function of a non-interacting classical electron gas, subject to two external potentials: one is due to the nuclei , while the other is due to the exchange correlation effects . The corresponding Hamiltonian is called the Kohn-Sham Hamiltonian. The exact ground-state density of an N-electron system is given as: 8 9) Where the single-particle wave functions are the N lowest-energy solutions of the Kohn- Sham equation ) In certain occasions, Kohn-Sham equation is written as: Where is known to be the Hamiltonian operator, the effective potential is the sum of the external, Hartree (electrostatic), and the exchange correlation potential. is stated as: (12) Where It is shown from eqn. (12) that VH and Vxc depend on the charge density , which in turn depends on the . This means that a self-consistency problem occurs and therefore needs to be dealt with the solution . The solution determines the original equation (VH and Vxc in Hks), and the equation cannot be written down and solved before its solution is being recognized. Some starting density ρo is guessed, and a Hamiltonian HKS1 can be constructed with it. The eigenvalue problem is also solved, and Φ1 can be determined by ρ1. And now ρ1 can be used to construct HKS2 which will yield ρ2, etc. The procedure can be subsequently used as long as the series convergence and ρf get out. 9 Figure 1: Flowchart for the iteration in the self-consistent procedure to solve Hartree- Fock or Kohn-Sham equations. 2.5 The Exchange-Correlation Functional. The Kohn-Sham scheme described above was accurate: apart from the preceding Born-Oppenheimer approximation, no other approximations were made. But the fact that we do not know the exchange-correlation functional was neglected so far and is still made unknown. 10 Consequently, the introduction of an approximation is needed. The two often - used approximations are LDA (Local Density Approximation) and GGA (Generalized Gradient Approximation). 2.6 Local Density Approximation (LDA). A widely used approximation-called (LDA) – is to postulate that the exchange-correlation functional has the following form: Where is the exchange-correlation energy per particle of a uniform (a homogeneous) electron gas, it only depends on the electron density , thus the name given is the “local density approximation". The exchange-correlation energy due to a particular density could be found by dividing the material into infinitesimally small volumes with a constant density [17]. Each volume contributes to the total exchange correlation energy by an amount equal to the exchange correlation energy of an identical volume and filled with a homogeneous electron gas. The exchange-correlation energy is decomposed into exchange and correlation terms linearly as shown below: 11 The first contribution is the exchange energy that comes from the Pauli Exclusion Principle. The second contribution, called the correlation energy , originates from the interaction of electrons having the same spin [18]. The next logical step to improve the LDA is to make the exchange- correlation contribute to every infinitesimal volume not only to be dependent on the local density in that volume, but also to contribute to the density in the neighboring volumes. To sum up, the gradient of the density will play a role. This approximation is called the Generalized Gradient Approximation (GGA). 2.7 Generalized Gradient Approximation (GGA). Where as LDA uses the exchange energy density of the uniform electron gas, regardless of the homogeneity of the real charge density, the GGA is concerned with such inhomogeneities (non-uniform charge density) by including the gradient of the electron density in the functional. The GGA uses the gradient of the charge density . The GGA can be conveniently written as follows 15) GGA seeks to improve the accuracy of the local density approximation 12 2.8 Augmented Plane Wave (APW) Method. Augmented plane wave method is introduced by Slater as a basis of functions for solving the one-electron equation. APW method is a procedure for solving the Kohn-Sham equation. In the APW scheme, the unit cell is divided into two types of regions: (I) atomic centered muffin-tin (MT) spheres with radius , and (II) the remaining interstitial region as shown in figure 2. Figure 2: Scheme of Augmented Plane Wave. In both types of regions, different basis sets are used. In the region far away from the nuclei, electrons are almost free, so plane waves are employed. However, close to the nuclei, the electrons behave almost as if they were in a free atom and therefore they can be described as atomic like- wave functions (radial solution of Schrödinger’s equation) The introduction of such a basis is due to the fact that close to the nuclei the potential and wave functions are very similar to those in an atom, while 13 they appear to be smoother between the atoms. In other words, the further the region from the nuclei, the more or less “free” the electrons would become. The APWs consists of: 16) Where is the reciprocal lattice vectors and is the wave vector inside the Brillion zone, V is the volume of the unit cell, is the poison vector inside the sphere , is the numerical solution to the radial Schrodinger equation at the energy . 2.9 The linearized Augmented Plane Wave (LAPW) Method. The linearized augmented plane wave method (LAPW) scheme was introduced by Andersen [ref.] who suggested the expansion of the energy dependence on radial wave functions u(r′) inside the atomic spheres with its energy derivative . In this scheme, a linear combination of radial function times spherical harmonics are used. Inside the atomic sphere of radius Rt a linear combination of the radial functions times spherical harmonics Ylm(r) is used, where ul(r,El) - at the origin - is the regular solution of the radial Schrödinger equation for energy El and the spherical part of the potential inside the atomic sphere is the energy derivative of ul taken at the same energy El 14 )17) In the interstitial region, a plane wave expansion is used. 2.10 The Augmented Plane Wave + Local Orbits (LAPW+Lo) Method. This alternative approach was proposed by Sjöstedt et al [18], namely the APW+lo (local orbital) method. They have shown that the standard LAPW method with the additional constraint on the PWs of matching in value and slope to the solution inside the sphere is not the most efficient way to linearize Slater's APW method. It can be made much more efficient when one uses the standard APW basis, but, in fact, with ul(r,El) at a fixed energy El in order to keep the linear eigen value problem. One then adds a new local orbital (lo) to have enough variation flexibility in the radial basis functions [19]: (19) 20) 15 The coefficient , are decided by necessities that should be regularized and has a zero value with a slope at the sphere border. In its general form, the LAPW method expands the potential in the following form: 16 Chapter Three Results and Discussion 3.1 Computational Method In this work, First-principles full-potential linearized augmented plane wave [FP-LAPW] computations have been performed as implemented in Wien2k package [22] within the generalized gradient approximation [PBE- GGA] [33] . For the compound Co2TiSn,the muffin-tin radii (RMT) of Co, Ti and Sn atoms are 2.22, 2.17 and 2.22 a.u. , respectively and for the compound Zr2RhGa ,RMT of Zr, Rh and Ga atoms are 2.37, 2.49 and 2.37 a.u., respectively . The number of plane waves was restricted by KMAX RMT =8 and the expansions of the wave functions was set to l=10 inside the muffin thin spheres. 35k points in the irreducible Brillion zone (BZ) with grid 10 x 10 x10 Monkhorst-Pack (MP) [34] meshes (equivalent to 1000k points in the full Brillion zone (BZ)) are used to obtain self-consistency for Co2TiSn and Zr2RhGa compounds. The self-consistent calculations are considered to converge only when the calculated total energy of the crystal converges to less than 10 -5 Ry. The elastic component C11, C12 and C44 are calculated by using the method developed by Morteza Jamal [35] and integrated in Wien2k code as the IRelast package. The elastic constants of the cubic phase are calculated by using second-order derivative within formalism of the Wien2k code. 17 3.2 Structural Properties The optimized lattice constant (a), bulk modulus (B), and its pressure derivative (Bʹ) were calculated by fitting the total energy to Murnaghan’s equation of state (EOS) [36]. Murnaghan’s equation of state (EOS) is given by: (22) Where E0 is the minimum energy, B is the bulk modulus at the equilibrium volume and Bʹ is the pressure derivative of the bulk modulus at the equilibrium volume. Pressure, , Bulk modulus , Normal Heusler Co2TiSn and Heusler Zr2RhGa compounds have space group Fm-3m (225),and inverse Heusler Co2TiSn and Heusler Zr2RhGa compounds have space group F-43m (216) [19]. We have calculated the structural properties for Full-Heusler Co2TiSn and Zr2RhGa compounds. The crystal structure of Full-Heusler Co2TiSn and Zr2RhGa compounds are shown in Figure 3. The total energy as function of the volume for normal and inverse Heusler Co2TiSn, Zr2RhGa compounds are shown in Figure 4 (a-d).The optimized structural parameters are calculated from the equation of state (EOS) and tabulated in Table 1 18 (a) (b) (c) (d) Figure 3: The crystal structure of (a) normal Heusler Co2TiSn (b) inverse Heusler Co2TiSn (c) normal Heusler Zr2RhGa (d) inverse Heusler Zr2RhGa compounds. 19 (a) (b) 20 (c) (d) Figure 4: Total energy as function of the volume for (a) normal Co2TiSn (b) normal Heusler Zr2RhGa and (c) inverse Zr2RhGa (d) inverse Co2 TiSn compounds. 21 The estimated lattice parameter (a), bulk modulus (B) and pressure derivative (Bʹ) at zero pressure are tabulated in Table 1 : Table 1: Calculated lattice parameter(a), bulk modulus(B), pressure derivative (Bʹ) for normal and inverse Heusler Co2TiSn andn Zr2RhGa compounds. Compounds Reference Lattice parameter(a) Ǻ B (GPa) Bʹ (GPa) Normal Co2TiSn Present 6.094 166.932 4.627 Experimental 6.072 [19] - - Normal Zr2RhGa Present 6.679 134.1 4.062 Inverse Co2TiSn Present 6.151 139.179 3.807 Inverse Zr2RhGa Present 6.619 129.319 5.073 Theoretical 6.64 [26] - - Table 1 shows that our calculated lattice parameter is in good agreement with the experimental lattice parameter for normal Heusler Co2TiSn compound. The calculated lattice parameter of normal Heusler Co2TiSn compound is slightly overestimated the experimental lattice parameter with 0.36% larger [19] . Our calculated lattice parameter of inverse Heusler Zr2RhGa compound is found to be closer to the other theoretical result [26]. 22 3.3 Magnetic Properties In this part, the total and partial magnetic moments for normal Heusler Co2TiSn and Heusler Zr2RhGa ,and inverse Heusler Co2TiSn and Heusler Zr2RhGa Compounds were calculated, and compared with the experimental and other theoretical results as shown in Tables 2 and 3. Table 2: Total magnetic moment for normal and inverse Heusler Co2TiSn Compounds. Compounds Magnetic Moment (µB) Co Co Ti Sn Interstitial Total magnetic moment (µB) Normal Co2TiSn Present 1.32398 1.32398 0.19866 0.00709 ˉ0.47779 1.9786 Experimental Result - - - - - 2 [19] Inverse iiiiiiii Co2TiSn Present 0.46092 1.54481 0.25961 0.01579 ˉ0.12165 1.64026 Table 3: Total magnetic moment for normal and inverse Heusler Zr2RhGa Compound. Compounds Magnetic Moment (µB) Zr Zr Rh Ga Interstitial Total magnetic moment (µB) Normal Zr2RhGa Present 0 0 0 0 0 0 Inverse Zr2RhGa Present 0.88019 0.44447 0.15415 0.00905 0.50214 1.99 Theoretical Result - - - - - 2 [26] The integer value of the total magnetic moment (µB ) is characteristic of half-metallic materials. 23 We noticed that the total magnetic moment are found to be in the range from 1.64026 to 2 µB. We found that the normal and inverse Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds are ferromagnetic compounds .The total magnetic moment for normal Zr2RhGa compound is 0 µB which means it does not have magnetic behavior. Present results show that the calculated total magnetic moment of normal Heusler Co2TiSn compound is underestimated the experimental value with 1.07% less[19] . Our calculated total magnetic moment of inverse Heusler Zr2RhGa compound is found to be closer to the other theoretical result [26]. Present total magnetic moment results are to some extent compatible with experimental and theoretical results. The calculated total spin magnetic moments are clearly integral values and are in agreement with the slater-Pauling rule [37]. 3.4 Electronic Properties In this part, the band structure, the total and partial density of states for normal Heusler Co2TiSn and Heusler Zr2RhGa, and inverse Heusler Co2TiSn and Heusler Zr2RhGa compounds were calculated. It is clear from the band structure and density of states that the normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds both have a half -metallic behavior , which means at spin up the materials behave as metallic nature while at spin down the materials behave as semiconducting nature, and inverse Heusler Co2TiSn and normal Heusler Zr2RhGa compounds both have a metallic behavior. 24 Figure 5 (a, b and c) shows that the band structure spin up of normal and inverse Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds have metallic nature by using PBE-GGA method. Figure 6(a and c) shows that the band structure spin down of normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds have an indirect energy band gap using PBE-GGA method. The values of the energy band gaps of spin down are calculated for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds using PBE-GGA method . The energy band gaps of normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds are found to be 0.482 eV and 0.573 eV, respectively as shown in Tables 4 and 5. Figure 6b shows that the spin up of inverse Heusler Co2TiSn has metallic behavior using PBE-GGA method. The band structure of normal Heusler Co2TiSn and inverse Heusler Zr2RhGa are also calculated by using mBJ- GGA. Figure 7 shows that the normal Heusler Zr2RhGa compound has metallic behavior. Figure 8 (a and b) also shows that the spin up of normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds both have metallic behavior using mBJ-GGA method. Figure 9 (a and b) shows the energy band gap within mBJ-GGA method for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds is still indirect band gap and the energy gap increases for normal Co2TiSn to 1.430 eV and for inverse Zr2RhGa to 0.641 eV at spin down. 25 (a) (b) (c) Figure 5: Band structure spin up by using PBE-GGA method for (a) normal Co2TiSn (b) inverse Co2TiSn (c) inverse Zr2RhGa compounds 26 (a) (b) (c) Figure 6: Band structure spin down by using PBE-GGA method for (a) normal Co2TiSn (b) inverse Co2TiSn (c) inverse Zr2RhGa compounds 27 Figure 7: Band structure by using PBE-GGA method for normal Zr2RhGa compound . )a) )b) Figure 8: Band structure spin up by using mBJ-GGA method for (a) normal Co2TiSn (b) inverse Zr2RhGa compounds 28 . (a) (b) Figure 9: Band structure spin down by using mBJ-GGA method for (a) normal Co2TiSn (b) inverse Zr2RhGa compounds . The band gap types and high symmetry lines are presented in Tables4and5: Table 4: Energy band gaps for normal Heusler Co2TiSn and normal Heusler Zr2RhGa Compounds using PBE-GGA and mBJ-GGA methods Compounds. Compounds Band gap type High Symmetry Lines Eg-PBE-GGA (eV) Eg-mBJ-GGA (eV) Co2TiSn Indirect Γ-X 0.482 1.430 Zr2RhGa Metallic - - - 29 Table 5: Energy band gaps for inverse Heusler Co2TiSn and inverse Heusler Zr2RhGa Compounds using PBE-GGA and mBJ-GGA methods. Compounds Band gap type High Symmetry Lines Eg-PBE-GGA (eV) Eg-mBJ-GGA (eV) Co2TiSn Metallic - - - Zr2RhGa Indirect L-Δ 0.573 0.641 Total and partial density of states of spin up and spin down for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds are shown in Figures (10 – 16). Density of state figures (10-16) show also half metallic property for normal Co2TiSn and inverse Zr2RhGa compounds with existing small energy band gap in the spin down direction, and this means that these compounds have half metallic property. In the spin up of normal Co2TiSn (Figure 10), the valence band is due to d- state of Co, s-state and p-state of Sn and small contribution from Ti d-state, while the conduction band is due to d-state of Ti and small contribution from Sn p-state. In spin down of normal Co2TiSn (Figure 11), the valence band is due to Co d-state, Ti d-state and Sn s-state and p-state, while the conduction band is due to Co d-state and Ti d-state and small contribution from s-state and p-state. In normal Zr2RhGa (Figure 12), the valence band is due to Rh d-state and Ga s-state, and small contribution from Zr near to fermi level due to Zr d-state, and the conduction band is due to Zr d-state, and small contribution of Rh due to d-state. 30 In spin up of inverse Co2TiSn (Figure 13), the valance band is due to Co d-state, Sn s-state and p-state, and small contribution from Ti d-state, and the conduction band is due to Ti d-state and small contribution from s-state and p-state. In spin down of inverse Co2TiSn(Figure 14), the valence band is due to Co d-state near to fermi level and Sn s-state and p-state, and small contribution from Ti d-state, and the conduction band is due to Co d-state near to femi energy and Ti d-state, and small contribution from Sn s-state and p-state. In spin up of inverse Zr2RhGa (Figure 15), the valence band is due to Rh d-state, Ga d-state, and small contribution from Zr d-state near to fermi energy level, and the conduction band is due to Zr d-state near to fermi energy level. In spin down of inverse Zr2RhGa (Figure 16), the valence band is due to Rh d-state , Ga d-state and small contribution from Zr d-state, and the conduction band is due to Zr d-state near to fermi energy level, and small contribution from Rh and Ga d-states . 31 (a) (b) (c) (d) Figure 10 : (a) Total density of states of spin up for normal Co2TiSn and partial density of states of spin up for (b) Co atom (c) Ti atom (d) Sn atom of normal Co2TiSn compound. 32 (a) (b) (c) (d) Figure 11: (a) Total density of states of spin down for normal Co2TiSn compound and partial density of states of spin down for (b) Co atom (c) Ti atom (d) Sn atom of normal Co2TiSn compound . 33 (a) (b) (c) (d) Figure 12: (a) Total density of states for normal Zr2RhGa and partial density of states for (b) Zr atom (c) Rh atom (d) Ga atom of normal Zr2RhGa compound. 34 (a) (b) (c) (d) Figure 13: (a)Total density of states of spin up for inverse Co2Ti Sn and partial density of states of spin up for (b) Co atom (c) Ti atom (d) Sn atom of inverse Co2TiSn compound. 35 (a) (b) (c) (b) Figure 14: (a) Total density of states of spin down for inverse Co2Ti Sn and partial density of states of spin down for (b) Co atom (c) Ti atom (d) Sn atom of inverse Co2TiSn compound. 36 (a) (b) (c) (d) Figure 15: (a)Total density of states of spin up for inverse Zr2RhGa and partial density of states of spin up for (b) Zr atom (c) Rh atom (d) Ga atom of inverse Zr2RhGa compound. 37 (a) (b) (c) (d) Figure 16: (a) Total density of states of spin down for inverse Zr2RhGa and partial density of states of spin down for (b) Zr atom (c) Rh atom (d) Ga atom of inverse Zr2RhGa compound. 38 3.5 Elastic Properties In this part, elastic constants(Cii), bulk modulus (B), shear modulus (S), B/S ratio, Young’s modulus (Y), Poisson’s ratio (v)and anisotropic factor (A) of the normal Co2TiSn and inverse Zr2RhGa compounds were calculated. For a cubic crystal, the standard mechanical stability is C11 > 0, C11 –C12 > 0, C11 +2 C12 > 0 and C44 > 0 [38]. Present calculations satisfied all the above conditions. We noticed that the normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds are found to be mechanically stable, while inverse Heusler Co2TiSn and normal Heusler Zr2RhGa compounds are mechanically unstable. In our calculation, we focused on normal Heusler Co2TiSn and inverse Heusler Zr2RhGa Compounds. For the face center cubic crystal, the bulk modulus and shear modulus were calculated using Voigt and Reuss approximations, Bulk modulus for cubic structure can be calculated from the following equations [39,40]: (23) Voigt Shear modulus and Reuss shear modulus are given by the following two equations: (24) (25) 39 The average value of Voigt shear modulus and Reuses shear modulus is called Hill shear modulus ,and can be estimated from the following equation [39,40] : (26) Young's modulus (Y) is defined as the ratio of the stress to strain, and given by: (27) Poisson's ratio and anisotropic factor can be computed by using bulk and shear moduli, Poisson's ratio and anisotropic factor can be given by: (29) Elastic constants, Voight bulk modulus (B), Voight shear modulus (S), B/S ratio, Voight Young’s modulus (Y), Voight Poisson’s ratio (v) and anisotropic factor (A) are presented in Table 6: 40 Table 6: Elastic constants for normal Co2TiSn and inverse Zr2RhGa Full Heusler Compounds. Materials C11 (GPa) C12 (GPa) C44 (GPa) B (GPa) S (GPa) B/S Y (GPa) V A Normal Co2TiSn 246.976 136.962 109.226 173.633 87.53 0.841 224.83 0.284 1.985 Normal Zr2RhGa 118.2706 141.5202 2.1653 133.77 ˉ3.350 ˉ39.931 ˉ10.13 0.512 ˉ0.186 Inverse Co2TiSn 120.027 151.389 104.594 140.934 56.482 2.495 149.48 0.323 ˉ6.67 Inverse Zr2RhGa 145.279 116.5186 70.4579 126.105 48.026 2.6257 127.85 0.331 4.9 41 The Bulk (B) or shear modulus (S) measures the hardness of materials [37]. The ratio B/S measures the ductility and brittleness of the materials. When B/S > 1.75, the materials behave in a ductile nature, otherwise it behaves in a brittle nature [38]. In the present calculations, the B/S ratio of normal Heusler Co2TiSn, normal Heusler Zr2RhGa, inverse Heusler Co2TiSn, and inverse Heusler Zr2RhGa compounds are 0.841, -39.931, 2.495 and 2.6257, respectively, The normal Heusler Co2TiSn and normal Heusler Zr2RhGa both have brittle nature, while inverse Heusler Co2TiSn and inverse Heusler Zr2RhGa both have ductile nature. Young modulus (Y) measures the stiffness of materials. The highest the value of Young modulus (Y), the stiffest is the material and the solids will have covalent bonds. The Poisson ratio (ν) measures the stability of the material and provides useful information about the nature of the bonding [39]. When Poisson ration (ν) is greater than 1/3, the materials behave in a ductile nature. Otherwise, it behaves in a brittle nature [40], and if the value of Poison's ratio is greater than 0.25, the material will have ionic bond; otherwise, the material has covalent bond. In the present calculations, the ν of normal Heusler Co2TiSn, normal Heusler Zr2RhGa, inverse Heusler Co2TiSn, and inverse Heusle Zr2RhGa compounds are found to be 0.284, 0.512, 0.323 , and 0.331, respectively, The normal Heusler Co2TiSn, normal Heusler Zr2RhGa, inverse Heusler Co2TiSn, inverse Heusler Zr2RhGa compounds have ionic bonds. Likewise, the elastic anisotropy is an important parameter to measure the degree of anisotropy of materials [41]. For an isotropic material, the value of A is 42 unity. Otherwise, the material has an elastic anisotropy [42]. In the present calculations, the A of normal Heusler Co2TiSn, normal Heusler Zr2RhGa, inverse Heusler Co2TiSn, and inverse Heusler Zr2RhGa compounds are found to be 1.985, -0.186, -6.67, 4.899, respectively and the normal Heusler Co2TiSn, normal Heusler Zr2RhGa ,inverse Heusler Co2TiSn, and inverse Heusler Zr2RhGa Compounds are elastic anisotropy [43]. 43 Chapter Four Conclusion In this work, the structural, electronic, magnetic and elastic properties for normal Heusler Co2TiSn and inverse Heusler Zr2RhGa Full Heusler compounds have been studied. We found that normal Heusler Co2TiSn compound and the inverse Heusler Zr2RhGa compound have half-metallic behavior. The normal Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds have an indirect energy gap of 0.482 eV and 0.573 eV using PBE-GGA method. It was shown that the energy band gap within mBJ-GGA for normal Heusler Co2TiSn compound and for inverse Heusler Zr2RhGa compound are still indirect band gap and the energy gap increases for normal Heusler Co2TiSn to be 1.430 eV and for inverse Heusler Zr2RhGa to be 0.641 eV. The calculated total magnetic moment for these compounds are in the range from 1.64 to 2 µB,which means that present results are to some extent compatible with the experimental and theoretical results . The elastic properties indicate that the normal Heusler Co2TiSn compound and the inverse Heusler Zr2RhGa compound are mechanically stable. B/S results show that the normal Heusler Co2TiSn and normal Heusler Zr2RhGa compounds both have brittle nature, while inverse Heusler Co2TiSn and inverse Heusler Zr2RhGa compounds both have ductile nature. The Poisson's ratio (ν) of normal Heusler Co2TiSn, normal Heusler Zr2RhGa, inverse Heusler Co2TiSn, and inverse Heusler Zr2RhGa compounds are 44 found to be 0.284, 0.512, 0.323, and 0.331, respectively. The normal Heusler Co2TiSn, normal Heusler Zr2RhGa, inverse Heusler Co2TiSn, and inverse Heusler Zr2RhGa compounds have ionic bonds. The A of normal Heusler Co2TiSn, normal Heusler Zr2RhGa, inverse Heusler Co2TiSn, and inverse Heusler Zr2RhGa compounds are found to be 1.985, -0.186, -6.67, 4.899, respectively, and the normal Heusler Co2TiSn and normal Heusler Zr2RhGa, inverse Heusler Co2TiSn, and inverse Heusler Zr2RhGa compounds are elastic anisotropy. 45 References 1. R.A. de Groot, F.M. Mueller, P.G. van Engen, K.H.J. Buschow, Phys. Rev. Lett. 50 2024 (1983). 2. G. A. Prinz, Science 282 1660 (1998). 3. G.D. Liu, X.F. Dai, S.Y. Yu, Z.Y. Zhu, J.L. Chen, G.H. Wu, H. Zhu, and J.Q. Xiao, Phys. Rev. B 74 054435 (2006). 4. X.F. Dai, G.D. Liu, L.J. Chen, J.L. Chen, and G.H. Wu, Solid State Commun.140 533 (2006). 5. G.D. Liu, X.F. Dai, H.Y. Liu, J.L. Chen, Y.X. Li, Phys. Rev. B 77 014424 (2008). 6. E. Bayar, N. Kervan, S. Kervan, J. Magn. Magn. Mater. 323 2945 (2011). 7. Q.L. Fang, J.M. Zhang, K.W. Xu, V. Ji, J. Magn. Magn. Mater. 345 171 (2013). 8. N. Kervan, S. Kervan, J. Magn. Magn. Mater. 324 645 (2012). 9. H.Y. Jia, X.F. Dai, L.Y. Wang, R. Liu, X.T. Wang, P.P. Li, Y.T. Cui, G.D.Liu, J. Magn. Magn. Mater. 367 33(2014). 10. A. Birsan, P. Palade, V. Kuncser, J. Magn. Magn. Mater. 331 109 (2013). 46 11. Yamamoto M, Marukame T, Ishikawa T, Matsuda K, Uemura T and Arita M J. Phys. D:Appl. Phys. 39 824 (2006) . 12. K¨ammerer S, Thomas A, H¨utten A and Reiss G Appl. Phys. Lett. 85 79 (2004) . 13. Okamura S, Miyazaki A, Sugimoto S, Tezuka N and Inomata K Appl. Phys. Lett. 86 232503 (2005). 14. Sakuraba Y, Hattori M, Oogane M, Ando Y, Kato H, Sakuma A, Miyazaki T and Kubota H Appl. Phys. Lett. 88 192508 (2006). 15.Sakuraba Y, Miyakoshi T, Oogane M, Ando Y, Sakuma A, Miyazaki T and Kubota H Appl. Phys. Lett. 89 0528508 (2006). 16. Oogane M, Sakuraba Y, Nakata J, Kubota H, Ando Y, Sakuma A and Miyazaki T J. Phys. D: Appl. Phys. 39 834 (2006). 17. Tezuka N, Ikeda N, Miyazaki A, Sugimoto S, Kikuchi M and Inomata K Appl. Phys. Lett. 89 112514 (2006). 18.Dunlap,R.,Effects of composition on the properties of magnetic shape memory alloys, Halifax, Nova Scotia, March (2007). 19. Hem Chandra Kandpal, Vadim Ksenofontov ,Marek Wojcik , Ram Seshadri ,and Claudia Felser, Electronic structure, magnetism, and disorder in the Heusler compound Co2TiSn, Journal of Physics D: Applied Physics(2006). http://iopscience.iop.org/journal/0022-3727 http://iopscience.iop.org/journal/0022-3727 47 20. Jepsen O and Andersen O K Stuttgart tb-lmto-asa program version 47 http://www.fkf.mpg.de/andersen/, (2000). 21. Ebert H The Munich spr-kkr package, version 3.6, http://olymp.cup.uni-muenchen.de/ak/ebert/sprkkr ,(2005) . 22. Blaha, P. Schwarz, k., Madsen,G., Kvasnicka ,D.,& Luitz,J.2016. WIEN2k An Augmented PlaneWavePlus Local Orbitals Program for Calculating Crystal Properties. In User’s Guide. Place Published: Vienna University of Technology (Release 12/12/2016). 23. Savrasov S Y and Savrasov D Y Phys. Rev. B 46 12181(1992). 24. A. Birsan, V. Kuncser, Theoretical investigations of electronic structure and magnetism in Zr2CoSn full-Heusler compound,arXiv:14117154v1[cond-mat.mtrl-sci]26Nov (2014). 25. A. Birsan, Magnetism in the new full-Heusler compound, Zr2CoAl: A first-principles study, Current Applied Physics 14 1434e1436 (2014). 26. X.T. Wang, J.W. Lu, H. Rozale, X.F. Liu, Y.T. Gui, G.D. Liu , Half-metallic state and magnetic properties versus the lattice constant in Zr2RhZ (Z = Al, Ga, In) Heusler alloys, School of Material Sciences and Engineering, Hebei University of Technology, 8 DingZiGu 1st Road, Tianjin, PR China (2015). 27. M.C. Payne, M.P. Teter, D.C. Allan, T.A. Arias, J.D. Joannopoolous, Reviews of Modern Physics 64 1065 (1992). http://www.fkf.mpg.de/andersen/ http://olymp.cup.uni-muenchen.de/ak/ebert/sprkkr%20,(2005 48 28. M.D. Segall, P.L.D. Lindan, M.J. Probert, C.J. Pickard, P.J. Hasnip, S.J. Clark, M.C. Payne, Journal of Physics: Condensed Matter 14 2717(2002). 29.Rakesh Jain, N. Lakshmi,Vivek Kumar Jain, Vishal Jain, Aarti R Chandra and K. Venugopalan , Electronic Structure ,Magnetic and Optical properties of Co2TiZ(Z-B,AL,Ga,In), Journal of Magnetism and Magnetic Materials (2017). 30. Xiao-Ping Wei, Weiwei Sun, Ya-Ling Zhang, Xiao-Wei Sun, Ting Song, Ting Wang, Jia-Liang Zhang, Hao Su, Jian-Bo Deng, Xing-Feng Zhu, Investigations on electronic, Fermi surface, Curie temperature and optical properties of Zr2CoAl, Journal of Solid State Chemistry 247 97 (2017). 31. K. Koepernik, H. Eschrig, Phys. Rev. B 59 1743 (1999). 32. I. Opahle, K. Koepernik, H. Eschrig, Phys. Rev. B 60 14035 (1999). 33. Perdew J P, Burke S and Ernzerh of M Phys. Rev. Lett 77 3865 (1996). 34. H. J. Monkhorst , I. D. Pack , Phys. Rev. B 13 5188 (1976). 35.M. Jamal, IRelast Package is provided by M. Jamal as part of the commercial code WIEN2k . Available from:http://www.wien2k.at/reg.user/unsupported; (2014) . 36. F. D. Murnaghan , proe. Natl. Acad. Sci. U.S.A. 30 244 (1944). 49 37.Galanakis I. , Dederichs P. H. , Papanikolaou N. , Slater-Pauling behavior and origin of the half-metallicity of the full-Heusler alloys, Phys. Rev. B,66 1744 29 (2002) . 38. Z. W. Huang, Y. H. Zhao, H. Hou, and P. D. Han, Physica B 407 1075 (2012) . 39. M. Born and K. Huang (Oxford;Clarendon) (1956). 40. Reuss, Z. Angew. Math. Mech. 9 49 (1929). 41. D. M. Teter, MRS Bull. 23 22 (1998). 42. S. Pugh , London ,Edinburgh ,Dublin Philosophical Magazine, J Sci 45 823 (1954). 43. C. Zener, Elasticity and Anelasticity of Metals, University of Chicago Press, Chicago (1948). 44. P. Ravindran, L. Fast, P. A. Korzhavyi, and B. Johansson, J. Appl. Phys. 84 4891 (1998). النجاح الوطنية ةجامع كمية الدراسات العميا الخصائص التركيبية وااللكترونية والمغناطيسية والمرونية Co2TiSn & Zr2TiSn:الطبيعية والمعكوسة هزلر اتلمركب تخدام الجهد التاماسب عدادإ ضحى نصر محمد أبو بكر شرافإ محمد أبو جعفر د.أ. كمية ب ,لمتطمبات الحصول عمى درجة الماجستير في الفيزياء استكماالا هذه األطروحةقدمت .فمسطين -نابمس النجاح الوطنية, في جامعة ,الدراسات العميا 2102 ب الطبيعية هزلر اتلمركب تركيبية وااللكترونية والمغناطيسية والمرونيةالخصائص ال استخدام الجهد التامب Co2TiSn & Zr2TiSn:والمعكوسة اعداد ضحى نصر أبو بكر اشراف محمد أبو جعفر د.أ. الممخص الطبيعية ىزلر اتوااللكترونية والمغناطيسية والمرونية لمركبالخصائص التركيبية فحص تم الوظيفية عن طريق استخدام الجيد النظرية الكثافة ( Zr2TiSn&Co2TiSn ) المعكوسة (DFT) والجيد التام المزيد ( ذو الموجات المستوية الخطيةFP-LAPW والتقريب التدريجي ) .WIEN2kضمن اطار برنامج (PBE-GGAالمعمم ) ( Bومعامل الصالبة ) (a( لحساب ثابت الشبكة )GGAتم استخدام التقريب التدريجي المعمم ) جونسون لتحسين وأيضا تم استخدام نظام بيكي (ʹBومشتقة معامل الصالبة بالنسبة لمضغط ) .فجوة الطاقة من أىم نتائج ىذه الدراسة: العكسي يمتمكان الخاصية المعدنية. Co2TiSnالطبيعي و Zr2RhGaتبين أن المركبين .1 بيعي يمتمكان خاصية النصف الط Co2TiSnالعكسي و Zr2RhGaن تبين أن المركبي .2 .معدنية خصائص العكسي والطبيعي ذاتCo2TiSn العكسي و Zr2RhGaتبين أن المركبات .3 مغناطسية. تبين ان النتائج التي اوجدناىا قريبة من النتائج العممية والنظرية االخرى. .4 الطبيعي مستقران Co2TiSnالعكسي و Zr2RhGaاصية المرونة أن المركبين من خالل خ .5 ميكانيكيا. ج الطبيعي ىش Co2TiSnالعكسي قابل لمسحب والطرق ومركب Zr2RhGaتبين أن مركب .6 لالنكسار. .الطبيعي يمتمكان روابط أيونية Co2TiSnالعكسي و Zr2RhGaتبين أن المركبين .7