Ab-initio calculations of the structural, mechanical, electronic, magnetic and thermoelectric properties of Zr2RhX (X= Ga, In) Heusler alloys

Journal Pre-proof Ab-initio calculations of the Structural, Mechanical, Electronic, Magnetic and Thermoelectric Properties of Zr2RhX (X= Ga, In) Heusler alloys

Marah J. Alrahamneh, Jamil M. Khalifeh, Ahmad A. Mousa PII:

S0921-4526(19)30821-X

DOI:

https://doi.org/10.1016/j.physb.2019.411941

Reference:

PHYSB 411941

To appear in:

Physica B: Physics of Condensed Matter

Received Date:

31 July 2019

Accepted Date:

06 December 2019

Please cite this article as: Marah J. Alrahamneh, Jamil M. Khalifeh, Ahmad A. Mousa, Ab-initio calculations of the Structural, Mechanical, Electronic, Magnetic and Thermoelectric Properties of Zr2 RhX (X= Ga, In) Heusler alloys, Physica B: Physics of Condensed Matter (2019), https://doi.org/10. 1016/j.physb.2019.411941

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2019 Published by Elsevier.

Journal Pre-proof

Ab-initio calculations of the Structural, Mechanical, Electronic, Magnetic and Thermoelectric Properties of Zr2RhX (X= Ga, In) Heusler alloys Marah J. Alrahamneha, Jamil M. Khalifeha,*, Ahmad A. Mousab a Department

of Physics, The University of Jordan, Amman 11942, Jordan East University, Amman 11831, Jordan

b Middle

Abstract We investigate the ground state properties namely structural, mechanical, electronic and magnetic properties and thermoelectric behavior of the inverse Heusler alloys Zr2RhGa and Zr2RhIn. Ab-initio calculations are carried out through the WIEN2k package. Both Zr2RhGa and Zr2RhIn are found to show a ferromagnetic half-metallic behavior with a total magnetic moment 2µB, which agrees with the generalized Slater-Pauling rule. The formation energy for both Zr2RhGa and and Zr2RhIn Heusler alloys are calculated and found to be equal to 0.129929 Ry and -0.110681 Ry, respectively, which means in principle they could be synthesized in laboratories. We further investigate the stabilites of the two alloys by calculating the the mechanical and elastic properties which showed that both alloys are mechanically stable. Using the mean _eld approximation we were able to estimate the Curie temperature which turned out to be 536.28 K for Zr2RhGa and 972 K for Zr2RhIn. Thermoelectric behavior is studied using the BoltzTrap code, both systems show good thermoelectric properties for the spin down channel, however utilizing the two-current model the properties are not good enough for conventional thermoelectric materials. Keywords: Heusler alloys; Mechanical properties; Electronic structure; Magnetic properties; Half-metallic materials *Corresponding author: [email protected] (Jamil M. Khalifeh) 1

Journal Pre-proof

1. Introduction Heusler alloys are not novel; they are more than 100 years old. They were discovered by Friedrich Heusler in 1903 [1]. He reported that Cu2MnAl compound shows ferromagnetic behavior despite the fact that non of its constituenting elements is ferromagnetic. Generally, Heusler alloys are classified into two structures, the Full Heusler alloy with the general formula X2YZ and the halfHeusler alloy with formula XYZ, here X, Y are transition group elements and Z is an sp element [2]. Full-Heusler alloys have two structural types; both consist of four interpenetrating fcc sublattices. The so called "regular" full Heusler alloys (L21) which crystallizes in cubic form with symmetry Fm-3m, no. 225 with Cu2MnAl as a prototype. This structure contains three inequivelent Wyckoff positions, incorporating four atoms per unit cell. Both X atoms occupy the 8c (0.25, 0.25, 0.25) position, the Y atom 4a (0, 0, 0) and the Z atom occupies the 4b (0.5, 0.5, 0.5). The second type is the inverse Heusler alloy (XA), in which the valance of the Y transition-metal atom is larger than the X atom. In this case the Heusler alloy crystallizes in cubic form with symmetry F-43m, no. 216 with Hg2CuTi as a prototype. X atoms occupy the positions 4a (0,0,0) and 4d (0.75, 0.75, 0.75), Y and Z atoms occupy the positions 4b (0.5, 0.5, 0.5) and 4c (0.25, 0.25, 0.25),respectively. The sequence of the atoms along the diagonal is X-X-Y-Z. Since their discovery Heusler alloys had attracted a lot of attention [3], [4], [5], [6] especially after the theoretical discovery of half-metallic materials in 1983 [7]. de Groot discovered that the half-Heusler alloy NiMnSb is half-metallic using first principles calculations based on density functional theory. Since then Heusler alloys became the heart of research for new half-metallic materials. Half-metallic materials have the distinct behavior in which the two spin channels exhibit different behaviors, the majority (Spin-up) channel behaves as a metal, whereas the Fermi level of the minority (Spin-down) channel lies in an energy gap exhibiting a 2

Journal Pre-proof

semiconducting behavior, this leads to a 100% spin polarization at the Fermi level. Full spin-polarized current is essential in maximizing the efficiency of spintronic devices [8]; this makes half-metallic materials ideal candidate for spintronic devices applications. In this paper, we intend to investigate the mechanical, electronic, and magnetic properties as well as the thermoelectric behavior of the Zr2RhGa and Zr2RhIn Heusler alloys. 2. Method of calculation We preformed ab-initio calculations using the full-potential linearized augmented plane wave (FP-LAPW) method [9] within density functional theory (DFT) [10] as implanted in the WIEN2k package [11]. The exchange correlation potential (VXC) within the generalized gradient approximation is obtained using the Perdew-BurkeErnzerhof parametrization of (PBE-GGA) [12]. In this method, the crystal is separated into two regions; the non-overlapping muffin-tin (MT) spheres surrounding the sites of the atoms, and the interstitial regions between these MT spheres. The muffin-tin radius for all atoms is equal to 2.5 a.u. Inside the muffintin spheres the wave functions are expanded in terms of spherical harmonics up to lmax=10, and the largest vector in Fourier expansion of the charge density is Gmax=14(a.u)-1. In the interstitial region the plane wave cut off parameter (Kmax)2=(8/RMT )2. A 141414 k-points are used for all properties calculation except for thermoelectric properties. In thermoelectric properties we use 505050 k-point and the BoltzTraP package [13]. The self-consistent calculations are considered to be converged only when the convergence tolerance of energy and charge are less than 10− 4 Ry and 10− 4 electron, respectively. The elastic constants and mechanical properties are obtained using the IRelast package 3

Journal Pre-proof

implanted within the WIEN2k package [14] [15]. In this package mechanical and elastic properties of crystals with different symmetries are calculated using the energy approach. In this approach, second-order derivative (E"(ε)) of the polynomial fit of energy (E = E(ε)) versus strain at zero strain (ε = 0) is used to carry out calculations [16].

3. Results and discussion 3.1. Structural properties Investigations of Zr-based Heusler alloys usually adopt the inverse Heusler structure [17, 18, 19, 20] stemming from the rule based on the number of valance electrons of the two transition metals X and Y. If atom Y has more valance electrons than atom X, atom X will occupy the positions 4a (0, 0, 0) and 4d (0.75, 0.75, 0.75), while the Y atom occupies the position 4b (0.5, 0.5, 0.5) and the XA structure is obtained. The other case where X has more valance electrons than the Y atom, both X atoms occupy the 8c (0.25, 0.25, 0.25) position, and the Y atom occupies the 4a (0, 0, 0) position, the L21 structure is obtained. In this section we investigate the inverse structure (XA) of both Zr2RhGa and Zr2RhIn Heusler alloys as shown in Fig.1. Structural optimization is performed using minmization method for the total energy of Zr2RhGa and Zr2RhIn in terms of varying unit cell volume to obtain the equilibrium lattice parameter. The obtained total energy as a function of volume is fitted using Murnaghan's equation of state [21]. The obtained values are plotted as a function of the volume in Fig.2 and Fig.3 Figures 2 and 3 show the Energy-Volume diagrams of Zr2RhGa and Zr2RhIn for the different magnetic phases, respectively. It is clear from the figures that the ferromagnetic phase is the most stable configuration for the Heusler alloys under 4

Journal Pre-proof

study. The equilibrium lattice constants of Zr2RhGa and Zr2RhIn obtained from minimizing the total energy are listed in Table 1 along with other equilibrium parameters. We note that the Bulk modulus of Zr2RhIn decreases as compared to Zr2RhGa which means that Zr2RhIn is more compressible compared to Zr2RhGa. To the best of our knowledge no experimental values are available for the lattice parameter of Zr2RhGa and Zr2RhIn. A previous theoretical work by Wang et.al (2016) reported results close to our own. 3.2. Formation and cohesive energies To further investigate the structural stability of the Zr2RhGa and Zr2RhIn inverse Heusler alloys, the formation and cohesive energies are calculated. The cohesive energy Ec is expressed as: 𝑎𝑡𝑜𝑚 2𝑅ℎ𝑋 𝐸𝑐 = 𝐸𝑍𝑟 ―[2𝐸𝑎𝑡𝑜𝑚 + 𝐸𝑎𝑡𝑜𝑚 ] 𝑡𝑜𝑡 𝑍𝑟 𝑅ℎ + 𝐸𝑋

(1)

𝑎𝑡𝑜𝑚 2𝑅ℎ𝑋 where 𝐸𝑍𝑟 is the total energy of Zr2RhX (X= Ga, In) alloys and 𝐸𝑎𝑡𝑜𝑚 𝑡𝑜𝑡 𝑍𝑟 , 𝐸𝑅ℎ ,

are the energies of Zr, Rh, and X (X= Ga, In) as an isolated atoms. We 𝐸𝑎𝑡𝑜𝑚 𝑋 calculated the cohesive energies for the two alloys; the values are shown in Table 1. The negative values of the cohesive energy indicate the chemical stability for both alloys. The formation energy (Ef) is calculated using the following formula: 𝐵𝑢𝑙𝑘 𝐵𝑢𝑙𝑘 2𝑅ℎ𝑋 𝐸𝑓 = 𝐸𝑍𝑟 ―[2𝐸𝐵𝑢𝑙𝑘 𝑡𝑜𝑡 𝑍𝑟 + 𝐸𝑅ℎ + 𝐸𝑋 ]

(2)

𝐵𝑢𝑙𝑘 2𝑅ℎ𝑋 where 𝐸𝑍𝑟 is the total energy of Zr2RhX (X= Ga, In) alloys and 𝐸𝐵𝑢𝑙𝑘 𝑡𝑜𝑡 𝑍𝑟 , 𝐸𝑅ℎ ,

where (X= Ga, In) are the energies of Zr, Rh, Ga and In in hcp, fcc, 𝐸𝐵𝑢𝑙𝑘 𝑋 orthorhombic and body center tetragonal structures, respectively. The formation energy results for both compounds are shown in Table 1. The negative formation energies indicate the stability of both compounds; thus they could be synthesized in laboratory using appropriate techniques. 5

Journal Pre-proof

In order to investigate the stabilities of Zr2RhGa and Zr2RhIn further, we study the elastic and mechanical properties in the next section. 3.3. Mechanical properties The mechanical behavior of a material reflects its response to deformation in relation to an applied stress [22]. The most elementary mechanical property is elasticity, which describes the linear reversible deformation of a solid due to a very small stress. The fundamental material parameters which describe the elastic behavior of solids are the elastic constants. Elastic constants can provide useful information about the bonding character as well as the structural stability of the solid [23]. The elastic properties of Zr2RhGa and Zr2RhIn Heusler alloys with XA structure are obtained using the IRelast package implanted within the WIEN2k software's package [14][15]. The package effectively calculates the elastic properties of crystals with different symmetries [15] using the energy approach discussed by Stadler [24]. A complete background knowledge of the elastic constant method employed in the present work can be found in the literature [15]. The total number of independent elastic constants depends on the symmetry of the crystal. In particular, for Zr2RhGa and Zr2RhIn Heusler alloys, which have cubic symmetry, there are only three independent elastic constants. These constants are C11, C12, and C44. The elastic constants of Zr2RhGa and Zr2RhIn are calculated, the results are listed in Table 2. From the single crystal constants one can determine whether the compound is mechanically stable or not. For a cubic crystal there are three generally accepted elastic stability criteria given by Born and Huang [25] C11 -C12 > 0

C11 + 2C12 > 0

C44 > 0

(3)

From Table 2, it is clear that these criteria are verified for both Zr2RhGa and Zr2RhIn. This means that Zr2RhGa and Zr2RhIn are mechanically stable. We did 6

Journal Pre-proof

not find theoretical results on the Zr2RhGa and Zr2RhIn to compare with, so we expect our work to serve as a reference for future work. Using the elastic constants listed in Table 2, one can give practical characterization of the mechanically stable Zr2RhGa and Zr2RhIn Heusler alloys elastic properties using different parameters. These parameters are: the Bulk modulus B, Shear modulus G, Young's modulus E, Poisson's ratio ν and Anisotropy A. The crystal bulk modulus B, Eq. (4), which represents the elastic modulus for the isotropic volume change. The Shear modulus G, as in Hill's average, Eq. (5) [26], which consists of Voigt Shear modulus GV, Eq. (6) [27] and Reuss Shear modulus GR, Eq. (7) [28] values, are calculated using the elastic constants listed in Table 2, the results are listed in Table 3.

𝐵= 𝐺=

𝐶11 + 2𝐶12

(4)

3 𝐺𝑅 + 𝐺𝑉

𝐺𝑉 =

(5)

2 𝐶11 ― 𝐶12 + 3𝐶44

(6)

5 5(𝐶11 ― 𝐶12)𝐶44

(7)

𝐺𝑅 = 4𝐶44 + 3(𝐶11 ― 𝐶12)

Voigt's Shear modulus provides an upper limit, while the Reuss Shear modulus provides a lower limit. It is found, however, that the experimentally measured modulus lie between them [26]. That's why the Hill's Shear modulus is used to obtain the other elastic parameters. The Young's modulus E, also known as the elastic modulus, can be thought of as a measure of materials resistance to elastic deformation or its stiffness. The greater the modulus the stiffer the material. Using Hill's elastic moduli B and G, the Young's modulus E is calculated as in Eq. (8), the result is listed in Table 3. 7

Journal Pre-proof

9𝐵𝐺

(8)

𝐸 = 3𝐵 + 𝐺

The anisotropy of a crystal can be calculated as in Eq. (9) [29]. Perfect isotropic crystals are characterized by a value of unity, while anisotropic crystals have a value that differs from one. The value of A listed in Table 3 indicates that both Zr2RhGa and Zr2RhIn are anisotropic. 2𝐶44

(9)

𝐴 = 𝐶11 ― 𝐶12

In order to determine whether a material is ductile or brittle, Pugh's ratio B/G empirical criterion gives an indication about the material nature so that; high values of B/G is usually associated with ductility while low values with brittleness [30]. If B/G is greater than 1.75, the material is ductile, otherwise the material has a brittle nature. The values of B/G are listed in Table 3. The obtained values show that Zr2RhGa and Zr2RhIn have a ductile nature. Using Hill's elastic moduli B and E, Poisson's ratio ν is calculated using Eq. (10). Poisson's ratio helps to determine the ductile and brittle nature of the material; if the value of υ is less than 1/3 brittle nature is detected, otherwise the material is ductile [31]. According to this, and the value of ν listed in Table 3, Zr2RhGa and Zr2RhIn are ductile in agreement with the result obtained from Pugh's ratio B/G.

ν=

3𝐵 ― 𝐸 6𝐵

(10)

Besides the fact that Poisson's ratio ν is related to the brittlity and ductility of any material, it also gives information about the kind of bonds in the compound. For ionic materials, ν is approximately equal to 0.25 or more, while for covalent materials it is less than 0.25 [16]. The values of ν for Zr2RhGa and Zr2RhIn are 8

Journal Pre-proof

listed in Table 3, the calculations show that ν > 0.25 which indicate an ionic bonding nature. Another indicator which can also predicts the bond types in a material is the sign of the Cauchy pressure (C"= C12-C44). For compounds where the covalent bonds are dominant C" is negative, while for ionicly bonded compounds it is positive. The Cauchy pressure result shown in Table 3 predicts that the ionic bonds are dominant in both compounds which agree with the results we got from Poisson's ratio. Unfortunately, as far as we know, there are no data available related to these properties in the literature for Zr2RhGa and Zr2RhIn Heusler alloys. Therefore, the obtained results can be considered as completely predictive and could serve as reference for future research on those alloys. 3.4. Electronic properties The electronic properties of the Zr2RhGa and Zr2RhIn alloys are obtained using the optimized lattice parameters. Fig.4 shows the spin-resolved density of states (DOS) of the Zr2RhGa and Zr2RhIn inverse Heusler alloys. The half metallic nature of the two alloys is quite clear as displayed in the figure, the spin up bands show the usual metallic behavior, whereas the spin down bands exhibit a semiconducting behavior with a gap at the Fermi-level. The existence of the gap leads to 100% spin polarization at the Fermi-level. The spin polarization is given by [32]: 𝜌↑(𝐸𝐹) ― 𝜌↓(𝐸𝐹)

(11)

𝑃 = 𝜌↑(𝐸𝐹) + 𝜌↓(𝐸𝐹) × 100%

Where ρ↑(EF) and 𝜌↓(𝐸𝐹) are the spin dependent density of states at the Fermi level EF. The 100% spin-polarization at EF makes both Zr2RhGa and Zr2RhIn alloys suitable for spintronic devices, since a full spin polarized current is of great importance to maximize the efficiency of these devices [33]. 9

Journal Pre-proof

In addition, from figure 4, obviously, the basic DOS features are not affected much by X in the system, so that there is a negligible impact of Ga and In on the shape of the DOS, and they are different only in the occupation of the empty orbitals; Ga accommodates the charges more than In, shifting up the Fermi level in Zr2RhGa than Zr2RhIn, this result was reported in a previous study by Wang et. al. [34]. The spin resolved band structures as well as the atom resolved density of states for both spin channels for Zr2RhGa and Zr2RhIn inverse Heusler alloys are shown in Fig. 5 and Fig. 6. From Figures 5a and 6a the overlap between the valance bands and the conduction bands is evident, which is a sign of metallic nature. The spindown channel band structures are shown in Fig. 5b and Fig. 6b. We note that the Fermi level is located within a band gap, this is a normal semiconducting behavior. The band gaps for both alloys are clearly indirect since (VBM) is located along Γ and X high symmetry points, while (CBM) is located at L, where the indirect band gaps for Zr2RhGa and Zr2RhIn are 0.63281 eV and 0.6428 eV, respectively. These results indicate the half-metallic nature of Zr2RhGa and Zr2RhIn inverse Heusler alloys, with Half-metallic gap equal to 0.46858 eV for Zr2RhGa and 0.36713 eV for Zr2RhIn. Zr2RhGa alloy has the widest half-metallic gap which makes it the most stable alloy in practical applications as compared to Zr2RhIn. Fig. 5a, Fig. 6a, Fig. 5b and Fig. 6b also show the contributions of the different orbitals of Zr, Rh, Ga and In atoms. Moreover, it confirms the correlation between the features in the band structure and the density of stats, i.e the highest density of states occurs when there are bands with low curvature. For instance, from Fig. 5b and Fig. 6b the flat band along XK segment corresponds to “s” orbitals from Ga and In atoms for both alloys. For both compounds the occupied bonding states are mostly from Rh-d orbitals, and the anti-bonding unocupied states are mostly from Zr-d orbitals for both spin channels. Fig. 5a and Fig. 6a show that the DOS at the 10

Journal Pre-proof

Fermi level is mainly due to the d- orbitals of the Zr and Rh atoms. Fig. 5b and Fig. 6b show that the band gap in the spin down channel is located between the bonding d-states of the Rh atom and the anti-bonding d-states of the Zr[1] atom. The origin of the gap for Heusler alloys with inverse structure is discussed in [17] and [35].

3.5. Magnetic properties The ground state of Zr2RhX (X= Ga, In) is ferromagnetic, with a total magnetic moment equals to 2µB. The integer value of the magnetic moment is an important characteristic of half-metallic materials. It should be noted that both alloys follow the Generalized Slater-Pauling rule, which gives the total magnetic moment (Mt) in terms of the total valance electrons of the compound (Zt) Eq. (12) [35]. Zr2RhX (X=Ga, In) have total valance electrons Zt equal to 20, in this case the total magnetic moment is 2µB which is what we got from our ab-initio calculations shown in Table 4. Mt = Zt - 18

(12)

The contributions of different atoms to the total magnetic moment are listed in Table 4. From the Table it is clear that the magnetic moment is mainly due to the Zr atoms. The Curie temperature TC is obtained for Zr2RhGa and Zr2RhIn structure, within the mean field approximation (MFA) as [36, 37]: 2

(13)

𝑇𝐶 = 3𝐾𝐵∆𝐸

Here ΔE and KB are the energy difference between antiferromagnetic and ferromagnetic phases and the Boltzmann constant, respectively. The total energy difference ΔE for Zr2RhGa is equal to 5.09687 mRy, which gives a Curie temperature equal to 536.28 K. For Zr2RhIn ΔE is equal to 9.236 mRy which gives 11

Journal Pre-proof

a Curie temperature equal to 972 K. Both alloys possess Curie temperature higher than room temperature which means they are appropriate for spintronic applications at room temperature. 3.6. Thermoelectric properties Semiconductors with flat conduction band are predicted to be good for thermoelectric materials. The band structures of Zr2RhGa and Zr2RhIn Heusler alloys confirm the existence of a flat conduction band in the spin down channel along the Γ-X symmetry points, which also exhibits a semiconducting nature Fig. 5b and Fig. 6b. The spin-polarized transport properties mainly: Seebeck coefficient (S) Fig. 7, electrical conductivity σ/τ Fig. 8, electronic thermal conductivity ke/τ Fig. 9 of Zr2RhGa and Zr2RhIn are obtained using the BoltzTrap code [13]. Fig. 7(a) illustrates that the value of the Seebeck coefficient increases with increasing temperature for the majority channel of Zr2RhGa and Zr2RhIn Heusler alloys. On the other hand, from Fig. 7(b) one can see the inverse relationship between S and temperature for the minority channel for both alloys. The electrical conductivity σ/τ of the majority channel Fig. 8(a) decreases with increasing temperature for both Zr2RhGa and Zr2RhIn Heusler alloys, this decrease is a typical behavior of metallic materials, as expected from the majority channel. Fig. 8(b) shows σ/τ for the minority channel, which increases with increasing temperature. Fig. 9(a) shows the relationship between ke/τ and temperature for the majority channel. In addition, Fig. 9(b) shows the relationship between ke/τ and temperature for the minority channel. The values of the Seebeck coefficient S, electrical conductivity σ/τ, and electronic thermal conductivity ke/τ of both spin channels for Zr2RhGa and Zr2RhIn Heusler alloys at 300 K are listed in Table 5.

12

Journal Pre-proof

The energy conversion efficiency in thermoelectric materials is determined by the dimensionless figure of merit ZT [38]. The figure of merit benchmark value for many conventional thermoelectric materials, such as Bi2Te3 is equal to 1 [39]. The spin-polarized dimensionless figure of merit ZT for Zr2RhGa and Zr2RhIn Heusler alloys is shown in Fig.10. The behavior of ZT with increasing temperature is similar for both alloys, however it is quite different for different spin channels. Fig. 10(a) shows that the value of ZT increases with temperature for the spin up channel, while for the spin down channel shown in Fig. 10(b) the value of ZT decreases with increasing temperature. The values of ZT at room temperature are listed in Table .5. These values of ZT take into account the electronic thermal conductivity only, if we were to obtain the lattice thermal conductivity the values of ZT will decrease even further. For Heusler alloys the value of ZT is limited by the large thermal conductivity, which could be reduced by partial substitution of other elements on the Z atom site [2] [40] [41]. We utilize the two-current model to study the variation of transport properties with respect to the temperature for both spin channels. The total Seebeck coefficient, within the two-current model, is given by Eq. (14) [42, 43, 44]. 𝑆=

𝜎(↑)𝑆(↑) + 𝜎(↓)𝑆(↓) 𝜎(↑) + 𝜎(↓)

(14)

Where σ(↑), σ(↓), are the electrical conductivities for the spin up and spin down channels, respectively. S(↑), S(↓) are the Seebeck coefficients for the spin up and spin down channels, respectively. Fig. 11 illustrates the behavior of the total S within the two current model with temperature for Zr2RhGa and Zr2RhIn Heusler alloys. The values of the total Seebeck coefficient at ambient temperature are listed in Table 5. In the two-current model, the total electrical conduction is obtained by considering the currents from both spin channels [45]. This means that the total electrical 13

Journal Pre-proof

conductivity σ/τ as well as the total electronic thermal conductivity ke/τ both are obtained by summing the corresponding values for both spin channels [44]. The variation of the total σ/τ and ke/τ with the temperature are shown in Fig. 12 and Fig. 13, respectively for both Zr2RhGa and Zr2RhIn Heusler alloys. The values of σ/τ and ke/τ at room temperature are listed in Table 5. Using the values of S, σ/τ and ke/τ obtained using the two-current model we calculate the figure of merit ZT within this model for the two Heusler alloys Zr2RhGa and Zr2RhIn. Fig. 14 shows the temperature dependence of the total figure of merit ZT for the two systems. The values of ZT at room temperature are listed in Table 5. We observe that these values are so much smaller than the benchmark of 1 for thermoelectric materials. Thus within the two-current the present systems of study are not suitable for conventional thermoelectric applications. 4. Conclusion First-principles calculations are used to investigate the ground state properties; mainly structural, mechanical, magnetic and electronic as well as the thermoelectric behavior of Zr2RhGa and Zr2RhIn inverse Heusler alloys. Both Zr2RhGa and Zr2RhIn Heusler alloys show a ferromagnetic half-metallic behavior with a total magnetic moment 2µB which agrees with the generalized SlaterPauling rule. The mechanical stability of both systems is also investigated. We found that these two alloys are mechanically stable. The mechanical properties of the stable Zr2RhGa and Zr2RhIn Heusler alloys, suggest a ductile anisotropic material where the ionic bonds are dominant. The Curie temperature for Zr2RhGa and Zr2RhIn Heusler alloys are calculated. It turned out to be equal to 536.28 K and 972 K for Zr2RhGa and Zr2RhIn, respectively. This means that the two 14

Journal Pre-proof

systems are suitable for spintronic applications at room temperature. The transport properties of the Zr2RhGa and Zr2RhIn alloys are also studied. The transport properties of both alloys are quite similar with the minority channel showing bright future in thermoelectric devices applications, due to the value of figure of merit ZT, which is found to be equal to 0.835 and 0.97 for Zr2RhGa and Zr2RhIn, respectively. However, when we use the two current model to investigate the collective behavior of the variation of transport properties with temperature, it turns out that the half-metallic Zr2RhGa and Zr2RhIn alloys are not suitable for thermoelectric applications.

References [1] F. Heusler, Verh. Dtsch. Phys. Ges 5 (1903) 219. [2] T. Graf, C. Felser, S. S. Parkin, Simple rules for the understanding of heusler compounds, Progress in solid state chemistry 39 (1) (2011) 1-50. [3] S. M. Azar, B. A. Hamad, J. M. Khalifeh, Structural, electronic and magnetic properties of Fe3-xMnxZ (Z=Al, Ge, Sb) heusler alloys, Journal of Magnetism and Magnetic Materials 324 (10) (2012) 1776-1785. [4] N. Mahmoud, J. Khalifeh, B. Hamad, A. Mousa, The effect of defects on the electronic and magnetic properties of the Co2VSn full heusler alloy: Ab-initio calculations, Intermetallics 33 (2013) 33-37. [5] S. M. Azar, A. A. Mousa, J. M. Khalifeh, Structural, electronic and magnetic properties of Ti1+xFeSb heusler alloys, Intermetallics 85 (2017) 197-205. [6] N. T.Mahmoud, J.M. Khalifeh, A. A.Mousa, H. K. Juwhari, B. A. Hamad, The energetic, electronic and magnetic structures of Fe2-xCoxVSn alloys: Ab-initio calculations, Physica B: Condensed Matter 430 (2013) 58-63. [7] R. De Groot, F. Mueller, P. Van Engen, K. Buschow, New class of materials: half-metallic ferromagnets, Physical Review Letters 50 (25) (1983) 2024. 15

Journal Pre-proof

[8] A. Hirohata, J. Sagar, L. Lari, L. R. Fleet, V. K. Lazarov, Heusler-alloy films for spintronic devices, Applied Physics A 111 (2) (2013) 423-430. [9] P. Blaha, K. Schwarz, P. Sorantin, S. Trickey, Full-potential, linearized augmented plane wave programs for crystalline systems, Computer Physics Communications 59 (2) (1990) 399-415. [10] W. Kohn, A. D. Becke, R. G. Parr, Density functional theory of electronic structure, The Journal of Physical Chemistry 100 (31) (1996) 12974-12980. [11] P. Blaha, K. Schwarz, G. Madsen, D. Kvasnicka, J. Luitz, wien2k, An augmented plane wave+ local orbitals program for calculating crystal properties. [12] J. P. Perdew, K. Burke, M. Ernzerhof, Generalized gradient approximation made simple, Physical review letters 77 (18) (1996) 3865. [13] G. K. Madsen, D. J. Singh, Boltztrap. a code for calculating band-structure dependent quantities, Computer Physics Communications 175 (1) (2006) 67-71. [14] M. Jamal, Irelast and 2dr-optimize packages are provided by m. jamal as part of the commercial code wien2k (2014). [15] M. Jamal, M. Bilal, I. Ahmad, S. Jalali-Asadabadi, Irelast package, Journal of Alloys and Compounds 735 (2018) 569-579. [16] M. Jamal, S. J. Asadabadi, I. Ahmad, H. R. Aliabad, Elastic constants of cubic crystals, Computational Materials Science 95 (2014) 592-599. [17] M. J. Alrahamneh, A. A. Mousa, J. M. Khalifeh, First principles study of the structural, electronic, magnetic and thermoelectric properties of Zr2RhAl, Physica B: Condensed Matter 552 (2019) 227-235. [18] X.Wang, T. Lin, H. Rozale, X. Dai, G. Liu, Robust half-metallic properties in inverse heusler alloys composed of 4d transition metal elements: Zr2RhZ (Z= Al, Ga, In), Journal of Magnetism and Magnetic Materials 402 (2016) 190-195. [19] Z.-Y. Deng, J.-M. Zhang, Half-metallic and magnetic properties of fullheusler alloys Zr2CrZ (Z= Ga, In) with Hg2CuTi-type structure: A first principles study, Journal of Magnetism and Magnetic Materials 397 (2016) 120-124.

16

Journal Pre-proof

[20] X. Wang, Y. Cui, X. Liu, G. Liu, Electronic structures and magnetism in the Li2AgSb-type heusler alloys, Zr2CoZ (Z= Al, Ga, In, Si, Ge, Sn, Pb, Sb): A firstprinciples study, Journal of Magnetism and Magnetic Materials 394 (2015) 50-59. [21] F. Murnaghan, The compressibility of media under extreme pressures, Proceedings of the National Academy of Sciences 30 (9) (1944) 244-247. [22] W. D. Callister, D. G. Rethwisch, Materials science and engineering, 9th Edition, John Wiley & Sons NY, 2013, pp. 168-210. [23] P. Lazar, D. R. Naturarum, Ab initio modelling of mechanical and elastic properties of solids, na, 2006. [24] R. Stadler, W. Wolf, R. Podloucky, G. Kresse, J. Furthmuller, J. Hafner, Ab initio calculations of the cohesive, elastic, and dynamical properties of CoSi2 by pseudopotential and all-electron techniques, Physical Review B 54 (3) (1996) 1729. [25] M. Born, K. Huang, Dynamical theory of crystal lattices, Clarendon press, 1954. [26] R. Hill, The elastic behaviour of a crystalline aggregate, Proceedings of the Physical Society. Section A 65 (5) (1952) 349. [27] W. Voigt, Lehrbuch der kristallphysik (teubner, leipzig, 1928), Google Scholar 962. [28] A. Reuss, A. reuss, z. angew. math. mech. 9, 49 (1929)., Z. Angew. Math. Mech. 9 (1929) 49. [29] C. Zener, Elasticity and anelasticity of metals, University of Chicago press, 1948. [30] S. Pugh, Xcii. relations between the elastic moduli and the plastic properties of polycrystalline pure metals, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 45 (367) (1954) 823-843. [31] V. Sharma, G. Pilania, Electronic, magnetic, optical and elastic properties of Fe2YAl (Y=Ti, V and Cr) using first principles methods, Journal of Magnetism and Magnetic Materials 339 (2013) 142-150. 17

Journal Pre-proof

[32] R. Fetzer, M. Aeschlimann, M. Cinchetti, Spin-resolved photoemission spectroscopy of the heusler compound Co2MnSi, in: Heusler Alloys, Springer, 2016, pp. 51-86. [33] K. Inomata, N. Ikeda, N. Tezuka, R. Goto, S. Sugimoto, M. Wojcik, E. Jedryka, Highly spin-polarized materials and devices for spintronics, Science and Technology of Advanced Materials 9 (1) (2008) 014101. [34] X. Wang, Z. Wang, G. Zhang, J. Jiang, Insight into Electronic and Structural Reorganizations for Defect-Induced VO2 Metal−Insulator Transition, J. Phys. Chem. Lett. 8 (13) (2017) 3129-3132. [35] S. Skaftouros, K. Ozdogan, E. Sasioglu, I. Galanakis, Generalized slaterpauling rule for the inverse heusler compounds, Physical Review B 87 (2) (2013) 024420. [36] N. Kervan, S. Kervan, O. Canko, M. Atis, F. Taskin, Half-metallic ferrimagnetism in the Mn2NbAl full-heusler compound: a first-principles study, Journal of Superconductivity and Novel Magnetism 29 (1) (2016) 187-192. [37] B. G. Yalcin, Ground state properties and thermoelectric behavior of Ru2VZ (Z= Si, Ge, Sn) half-metallic ferromagnetic full-heusler compounds, Journal of Magnetism and Magnetic Materials 408 (2016) 137-146. [38] W. Liu, Q. Jie, H. S. Kim, Z. Ren, Current progress and future challenges in thermoelectric power generation: From materials to devices, Acta Materialia 87 (2015) 357-376. [39] G. Mahan, Introduction to thermoelectrics, APL Materials 4 (10) (2016) 104806. [40] Q. Shen, L. Chen, T. Goto, T. Hirai, J. Yang, G. Meisner, C. Uher, Effects of partial substitution of Ni by Pd on the thermoelectric properties of ZrNiSn-based half-heusler compounds, Applied Physics Letters 79 (25) (2001) 4165-4167. [41] C. S. Lue, C. Chen, J. Lin, Y. Yu, Y. Kuo, Thermoelectric properties of quaternary heusler alloys Fe2VAl1-xSix, Physical Review B 75 (6) (2007) 064204.

18

Journal Pre-proof

[42] A. Botana, P. M. Botta, C. De la Calle, A. Pineiro, V. Pardo, D. Baldomir, J. Alonso, Non-one-dimensional behavior in charge-ordered structurally quasi-onedimensional Sr6Co5O15, Physical Review B 83 (18) (2011) 184420. [43] H. Xiang, D. J. Singh, Suppression of thermopower of NaxCoO2 by an external magnetic _eld: Boltzmann transport combined with spin-polarized density functional theory, Physical Review B 76 (19) (2007) 195111. [44] S. Sharma, S. K. Pandey, Applicability of two-current model in understanding the electronic transport behavior of inverse heusler alloy: Fe2CoSi, Physics Letters A 379 (38) (2015) 2357-2361. [45] N. F. Mott, Electrons in transition metals, Advances in Physics 13 (51) (1964) 325-422.

19

Journal Pre-proof

List of Tables Table 1: Lattice constant (a), Bulk modulus (B), Bulk modulus derivative with respect to pressure (B΄), Total energy (Etot), Cohesive energy (Ec) and Formation energy (Ef) of Zr2RhGa and Zr2RhIn alloys. Table 2: Elastic constants (C11, C12, C44) in GPa for Zr2RhGa and Zr2RhIn compounds. Table 3: Bulk (B), Shear (G, GV , GR), and Young's moduli in GPa, Poisson's ratio ν, Anistropy ratio (A), Pugh's ratio (B/G) and Cauchy Pressure (C") in GPa, for Zr2RhGa and Zr2RhIn alloys. Table 4: Total and local magnetic moments for Zr2RhX (X= Ga, In) inverse Heusler alloys in units of µB. Table 5: The thermoelectric properties of Zr2RhGa and Zr2RhIn inverse Heulser alloys at 300 K for both spin channels and the two current model at constant relaxation time τ.

20

Journal Pre-proof

List of Figures Figure 1. XA structure of Zr2RhGa and Zr2RhIn Heusler alloys. Figure 2. Total energy per atom as a function of volume for Zr2RhGa. Figure 3. Total energy per atom as a function of volume for Zr2RhIn. Figure 4. Spin-resolved Density of states for Zr2RhGa and Zr2RhIn inverse Heusler alloys. Figure 5. Spin resolved Band structure and atom resolved density of states for Zr2RhGa inverse Heusler alloy for (a) the majority (b) the minority spin channel. Figure 6. Spin resolved Band structure and atom resolved density of states for Zr2RhIn inverse Heusler alloy for (a) the majority (b) the minority spin channel. Figure 7. The Seebeck coefficient S with temperature for Zr2RhGa and Zr2RhIn for both (a) Majority spin (b) Minority spin, for constant relaxation time τ. Figure 8. The electrical conductivity σ/τ with temperature for Zr2RhGa and Zr2RhIn for both (a) Majority spin (b) Minority spin, for constant relaxation time τ. Figure 9. The electronic thermal conductivity ke/τ with temperature for Zr2RhGa and Zr2RhIn for both (a) Majority spin (b) Minority spin, for constant relaxation time τ. Figure 10. The dimensionless figure of merit ZT with temperature for Zr2RhGa and Zr2RhIn for both (a) Majority spin (b) Minority spin, for constant relaxation time τ. Figure 11. Temperature dependence of the total Seebeck coefficient S for Zr2RhGa and Zr2RhIn inverse Heusler alloys in the two-current model for constant relaxation time τ.

21

Journal Pre-proof

Figure 12. Temperature dependence of the total electrical conductivity σ/τ for Zr2RhGa and Zr2RhIn inverse Heusler alloys in the two-current model for constant relaxation time τ. Figure 13. Temperature dependence of the total electronic thermal conductivity for Zr2RhGa and Zr2RhIn inverse Heusler alloys in the two-current model for constant relaxation time τ. Figure 14. Temperature dependence of the total dimensionless figure of merit ZT for Zr2RhGa and Zr2RhIn inverse Heusler alloys in the two-current model for constant relaxation time τ.

22

Journal Pre-proof Author Contributions Section:

We contributed equally to the manuscript including the calculations, the analysis and writing.

Journal Pre-proof Please note that there is no conflict of interest.

Journal Pre-proof

Figure 1. 1

Journal Pre-proof

Figure 2.

2

Journal Pre-proof

Figure 3.

3

Journal Pre-proof

Figure 4.

4

Journal Pre-proof

(a)

(b)

Figure 5.

5

Journal Pre-proof

(a)

(b)

Figure 6.

6

Journal Pre-proof

(a)

(b) Figure 7. 7

Journal Pre-proof

(a)

(b) Figure 8.

8

Journal Pre-proof

(a)

(b) Figure 9.

9

Journal Pre-proof

(a)

(b) Figure 10.

10

Journal Pre-proof

Figure 11.

11

Journal Pre-proof

Figure 12.

12

Journal Pre-proof

Figure 13.

13

Journal Pre-proof

Figure 14.

14

Journal Pre-proof

Table 1: a(Å)

B (GPa)

B΄

Etot (Ry)

Ec (Ry)

Ef (Ry)

Zr2RhGa 6.6213

122.4287

3.8381

-27855.741329

-3.16098

-0.129929

Zr2RhIn

118.3140

4.4097

-35734.073244

-3.11317

-0.110681

6.8055

Table 2: C11

C12

C44

Zr2RhGa

141.9432

114.0308

72.0346

Zr2RhIn

140.6290

110.3889

77.2141

Table: 3 B

G

GV

GR

E

ν

A

B/G

C"

Zr2RhGa 123.334 37.918 48.802 27.034 103.180 0.360 5.161 3.252 41.9962 Zr2RhIn 120.468 40.796 52.376 29.217 109.973 0.347 5.106 2.953 33.1748

Journal Pre-proof

Table 4: mtot

mZrA

mZrB

MRh

MX

Intersitial

Zr2RhGa

2

0.85914 0.45267 0.10011 0.01346

0.57479

Zr2RhIn

2

0.83882 0.46635 0.06025 0.00465

0.63063

Table 5:

Zr2RhGa

Zr2RhIn

Spin-up Spin-down Two-current Spin-up Spin-down Two-current

S (µVK-1) 5.65 -305.8 2.4 7.7 -753.4 7.65

σ/τ (1020Ω-1m-1s-1) 3.49 0.03 3.52 2.92 2.2 10-4 2.9

ke/τ ZT (1015Wm-1K-1s-1) 2.52 2.4 10-4 0.12 0.835 2.64 2.3 10-4 2.13 2.45 10-3 3.9 10-3 0.97 2.13 2.41 10-3