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1. Introduction
Full-Heusler compounds have recently attracted a strong attention for their relatively high Curie temperature and large magnetic moment [1-9], and for their useful applications as ferromagnetic shape memory alloys [10,11,12], spintronic devices [13] and magnetic actuator [14]. Amongst the full-Heusler alloys, the Rh containing compounds, which have recently investigated by many theoretical and experimental researchers [15-28] to determine different properties, was firstly studied by Suits [29] to investigate the magnetic and structural properties for ferromagnetic compounds of the form Rh2MnX where X is Al, Ga, In, Tl, Sn, Pb, and Sb. He found that the exchange is described in terms of competing ferromagnetic Mn-Rh-Mn interactions and antiferromagnetic Mn-Mn interactions.
Afterwards, the electronic structure and magnetic moment of three Rh-based Heusler alloys Rh2MnX, with X = Ge, Sn and Pb have been calculated using the tight binding linear muffin-tin orbital (TB-LMTO) method by Pugacheva et al., [30]. These authors have found that the total magnetic moment increases with increasing atomic number of X. Not later, Jezierski et al., [31] presented the influence of local ordering on the electronic and magnetic properties of Heusler-type alloys Rh2TMSn (TM = Mn, Fe, Co, Ni, Cu) and Rh2MnX (X = Al, Ga, In, Ge, Sb, Pb). The band structure and magnetic moments are calculated by the ab-initio spin-polarized tight binding linear muffin-tin orbital method (TB-LMPTO).
In addition, using the full-potential screened Korringa-Kohn-Rostoker method, Galanakis et al., [32] have studied the full-Heusler alloys based on Co, Fe, Rh, and Ru. They show that many of these compounds show a half-metallic behavior. Later on, Klaer et al., [33] have measured the localized magnetic moments in the Heusler alloy Rh2MnGe by using, the X-ray magnetic circular dichroism (XMCD) of core-level absorption (x-ray absorption spectroscopy, XAS) spectra in the soft x-ray region has been measured for the ferromagnetic Heusler alloy Rh2MnGe at the Rh M3,2 and Mn L3,2 edges. They have found that the orbital moments of the Rh 4d and Mn 3d states are very small.
Recently, Sanvito et al., [34] have investigated the physical properties of Rh2MnTi, Rh2MnZr, Rh2MnHf, Rh2MnSc and Rh2MnZn by using density functional theory. These authors have found that most of the known magnetic HAs are metallic. Faleev et al., [35] have studied the origin of the tetragonal ground state of Rh2-based Heusler compounds for both the regular and inverse structures and various magnetic configurations using the VASP program with project or augmented wave (PAW) potentials and Perdew-Burke-Ernzerhof (PBE) generalized-gradient-approximation (GGA) DFT functional.
More recently, a numerical work on the electronic and magnetic structural parameters of Rh2MnTi crystal was carried out within the density functional theory level as implemented in all electron full-potential linearized augmented plane wave (FP-LAPW) method by Benzoudji et al., [36]. The authors show that the calculated density of states (DOS) and band structure for Rh2MnTi show the absence of energy band gap in their minority-spin channel which moves away half-metallic character. Aguilera-Granja et al., [37] have performed first-principle calculations of the structural, electronic and magnetic properties of Heusler X2MnZ with X = Fe, Co, Ni, Cu, Ru, Rh, Pd, Ag, Pt, Au and Z = Al, Si, Ga, Ge, In and Sn, using density functional theory (DFT) calculations, as implemented in the SIESTA code.
Very latterly, the structural, electronic, elastic, magnetic, and thermodynamic properties of full-Heusler alloys Rh2MnZ (Z = Zr, Hf)) using first-principles calculations were studied by Mentefa et al., [38]. The electronic properties reveled the metallic nature of the Heusler Rh2MnZ (Z = Zr, Hf) alloys. The elastic properties confirmed the elastic stability of the two alloys. They show high rigidity, anisotropic, and little deformation and behave in ductile way. The magnetic properties confirmed the ferromagnetic state of both compounds. In another theoretical study, Güler et al., [39] have investigated the electronic, elastic, optical and magnetic properties of Rh2MnX (X = Ti, Hf, Sc, Zr, Zn) Heusler alloys. Their study show that all investigated alloys are mechanically stable and ductile. Also, they have a typical metallic behavior and show strong ferromagnetic ordering following the magnetic moment rank of Ti > Zr > Hf > Sc > Zn.
The aim of the current work is to better visualize the behavior of the structural stabilities, mechanical, electronic, magnetic and thermodynamic properties of three full Heusler alloys Rh2MnSn, Rh2MnPb and Rh2MnTl which is rarely found in the literature, in particular the thermodynamic and thermal properties which are not available in the literature, in order to provide a good baseline data to experimentalists and to enrich the existing theoretical calculations on these materials for future investigations. For this, we have employed the full-potential linearized augmented plane wave (FP-LAPW) method as implemented in the WIEN2k code, in the framework of the density functional theory (DFT) within the GGA approximation.
The manuscript is organized as follows. Section 2 deals with computational procedure. Section 3 presents the results of the reported calculations and discussion. Finally, a brief conclusion is given in the last section.
2 Theoretical calculation details
In this work, the first-principles calculations are performed within the full potential linearized augmented plane wave (FP-LAPW) method [40] as implemented in the WIEN2k package [41], in the framework of the density functional theory (DFT) [42,43]. The exchange-correlation functional was calculated using the generalized gradient approximation (GGA) in the parameterization of Perdew-Burke-Ernzerhof (PBE) [44]. The plane wave parameter RMT × Kmax = 7 was used, where RMT is the average radius of muffin-tin spheres and Kmax is the largest K vector in the plane wave. To ensure the correctness of the calculations, we have taken Imax = 10 inside of the muffin tin spheres. The Gmax parameter, which defined as the magnitude of largest vector in charge density Fourier expansion, was taken to be 12 a.u-1. The separation energy between the core and the valence states was chosen as -7.0 Ry. The atomic sphere radii used are for Rh2MnSn; RMT(Rh) = RMT(Mn) = RMT(Sn) = 2.41, for Rh2MnPb; RMT (Rh) = 2.39, RMT(Mn) = 4.45, RMT(Pb) = 2.5 and for Rh2MnTl, RMT (Rh) = 2.38, RMT(Mn) = 2.44, RMT (Tl) = 2.5 a.u (atomic unit). The sampling of the Brillouin zone was done with a 12 × 12 × 12 mesh, which gives convergence of 10-4 Ry in the total energy.
3 Crystal structure
Full-Heusler alloys possess an X2YZ generic formula, where X and Y denote transition metals and Z is the main group element. The full Heusler compounds crystallize either in the L21 cubic structure with AlCu2Mn as the prototype or in the inverse Heusler structure (Xa) with CuHg2Ti as prototype. In Fig. 1, we show the crystal structure of Rh2MnZ alloys, where the present structures composed of four interpenetrating face-centered-cubic (fcc) sublattices. In L21 phase with the Fm-3m space group (no. 225), X atoms occupy A (0, 0, 0) and C (1/2, 1/2, 1/2) sites while the Y and Z atoms are located on B (1/4, 1/4, 1/4) and D (3/4, 3/4, 3/4) sites, respectively, in Wyckoff coordinates. In the CuHg2Ti-type full Heusler alloy with the F43-m space group (no. 216), X atoms are located at the non-equivalent A (0, 0, 0) and B (1/4, 1/4, 1/4) sites, Y atom occupies C (1/2, 1/2, 1/2) site and Z atom occupies D (3/4, 3/4, 3/4) site.
Figure 1 Crystal structure of Rh2MnZ (Z = Sn, Pb,Tl) compounds. a) CuHg2Ti and b) AlCu2Mn. These figures are plotted by using XCrySDen [45].
4 Results and discussions
4.1 Structural properties
In the first step of the current study, we have performed calculations on the Rh2MnZ (Z = Sn, Pb, Tl) ternary Heusler alloys, in both full (regular) Heusler (AlCu2Mn-type (L21) and inverse Heusler (CuHg2Ti-type (Xa)) structural phases for non-magnetic (NM) and ferromagnetic (FM) configurations in order to determine the stable structure of the considered compounds. The calculated total energies variation within the GGA approximation versus the volume is plotted in Fig. 2. It is clearly seen from Fig. 2, that the ferromagnetic state is the most stable states as compared to the other states in the AlCu2Mn-type structure for all the studied compounds due to its lowest total energy. This result confirms that L21-type structure agrees quite well with previous experimental works reported by Pillay et al., [17], Dhar et al., [18], and Jha et al., [19], and theoretical results of Jezierski et al., [31] and Sanvito et al., [34] using first-principle calculation methods based on the DFT. The calculated total unit cell energy as a function of unit cell volume is fitted to Murnaghan’s equa- tion of state [46] to obtain ground state properties, such as the equilibrium lattice constants, the bulk modulus B and its pressure derivative B’.
Figure 2 Variation of total energy as a function of lattice constant for both nonmagnetic (NM) and ferromagnetic (FM) states for CuHg2Ti-type and AlCu2Mn-type phases of a) Rh2MnSn, b) Rh2MnPb, and c) Rh2MnTl alloys.
The computed structural parameters for the ferromagnetic Rh2MnZ (Z = Sn, Pb and Tl) compounds using the GGA are given in Table I. Our results for the lattice parameter are in reasonable agreement with previously obtained experimental data and theoretical calculations reported by other researchers for the present investigated compounds.
Table I Optimized equilibrium lattice constant (a), bulk modulus (B), its pressure derivative (B’) and total equilibrium energy E0 for Rh2MnZ (Z = Sn, Pb, Tl) Heusler compounds in non-magnetic (NM) and ferromagnetic (FM) states of both CuHg2Ti and AlCu2Mn-type structures.
| Compound | structure | a(å) | B(GPa) | B’ | E0(Ry) |
|---|---|---|---|---|---|
| Rh2MnSn | CuHg2Ti (FM) | 6.3086 | 151.8352 | 5.7888 | -33816.588243 |
| CuHg2Ti (NM) | 6.2028 | 202.9067 | 5.7793 | -33816.540315 | |
| AlCu2Mn (FM) | 6.2921 | 177.4007 | 4.6825 | -33816.692560 | |
| 6.25 [18] | |||||
| 6.232 [20] | |||||
| 6.23 [23] | 195.55 [23] | 5.55 [23] | |||
| 6.233 [24] | 191.132 [24] | ||||
| 6.252 [29] | |||||
| 6.239 [30] | |||||
| 6.35 [37] | |||||
| AlCu2Mn (NM) | 6.2147 | 198.7040 | 4.6920 | -33816.542129 | |
|
|
|||||
| Rh2MnPb | CuHg2Ti (FM) | 6.4105 | 137.9806 | 5.8149 | -63315.567127 |
| CuHg2Ti (NM) | 6.3001 | 181.3053 | 5.5093 | -63315.503536 | |
| AlCu2Mn (FM) | 6.3976 | 159.1320 | 5.1034 | -63315.653000 | |
| 6.33 [18] | |||||
| 6.271 [20] | |||||
| 6.332 [24] | 167.932 [24] | ||||
| 6.332 [29] | |||||
| 6.403 [30] | |||||
| 6.334 [31] | |||||
| AlCu2Mn (NM) | 6.3172 | 178.2819 | 5.4172 | -63315.497179 | |
|
|
|||||
| Rh2MnTl | CuHg2Ti (FM) | 6.3875 | 141.0870 | 5.0567 | -62036.065625 |
| CuHg2Ti (NM) | 6.2709 | 182.1366 | 5.1346 | -62035.992866 | |
| AlCu2Mn (FM) | 6.3576 | 150.6142 | 5.8206 | -62036.146599 | |
| 6.324 [29] | |||||
| AlCu2Mn (NM) | 6.2794 | 184.0574 | 5.2955 | -62036.000368 |
4.2 Elastic and mechanical properties
The computed three independent elastic constants C11, C12 and C44 of Rh2MnZ (Z = Sn, Pb, Tl), from first-principle calculations, at an ambient pressure compared to other theoretical data are reported in Table II. The bulk modulus calculated from the theoretical values of the elastic constants B = (1/3)/(C11 + 2C12) are also given in Table II. The traditional mechanical stability conditions in cubic crystals at equilibrium, which are known as the Born-Huang criteria [47], are defined in terms of elastic constants as follows: C11 — C12 >0, C44 > 0, C11 + 2C12 > 0 and C12 < B < C11 [48]. As can be seen, the elastic constants in Table II obey the above well known stability conditions, meaning that the cubic L21 phase is stable against elastic deformations at ambient conditions for the three compounds.
Table II Calculated elastic constants (Cij) and Bulk modulus (B) all expressed (in GPa) for Rh2MnZ (Z = Sn, Pb, Tl) alloys in the AlCu2Mn structure ferromagnetic (FM) ground state.
We note that the earlier ab-initio calculation on elastic properties for Rh2MnSn and Rh2MnPb compounds reported by Benkhelifa et al., [24] and our results are in general in substantial agreement. Within best of knowledge, there is no report on the experimental and theoretical elastic constants and moduli available on Rh2MnTl alloy in the literature for comparison. Therefore, our present results can be discussed as purely predictive.
Furthermore, we have derived and listed in Table III, other mechanical parameters namely shear modulus G, Young’s modulus E, Poisson’s ratio v, Zener anisotropy factor A and Kleinman parameter ξ for Rh2MnZ (Z = Sn, Pb, Tl) within GGA, by using the Voigt-Reuss-Hill (VRH) averaging scheme [49,50,51]:
Where GV is Voigt’s shear modulus corresponding to the upper bound of G values and GR is Reuss’s shear modulus corresponding to the lower bound of G values.
Table III The calculated Cauchy pressure Cp, shear moduli GV, GR, G (in GPa) and B/G ratio, Young’s modulus E (in GPa), Poisson’s ratio v, shear anisotropic factor A and Kleinman parameter ξ for the Rh2MnZ (Z = Sn, Pb, Tl) full-Heusler alloys.
| Compound | CP | GV | GR | G | B/G | E | v | A | ξ |
|---|---|---|---|---|---|---|---|---|---|
| Rh2MnSn | 50.02 | 73.68 | 59.05 | 66.36 | 2.60 | 176.48 | 0.32 | 2.65 | 0.76 |
| 81.63 [24] | 56.38 [24] | 69.00 [24] | 2.78 [24] | 184.83 [24] | 0.34 [24] | 3.58 [24] | |||
| Rh2MnPb | 30.33 | 68.03 | 45.86 | 56.94 | 2.52 | 150.91 | 0.32 | 3.74 | 0.79 |
| 82.97 [24] | 34.53 [24] | 58.75 [24] | 2.98 [24] | 158.54 [24] | 0.35 [24] | 7.71 [24] | |||
| Rh2MnTl | 38.37 | 80.70 | 66.99 | 73.84 | 2.35 | 149.05 | 0.31 | 2.44 | 0.72 |
The Cauchy pressure CP = C12 — C44 is an important parameter to identify the type of bonding [52]. If the Cauchy pressure is negative, the bonding is more directional and non-metallic in character, whereas positive value corresponds to a predominant metallic bonding. From Table III, all the studied compounds possess positive Cauchy pressure values, thus the predominant bonding for these compounds is metallic.
We note the higher bulk modulus and Young’s modulus, which is an indication of a strong incompressibility for these compounds. Pugh [53] proposed an approximate criterion by the ratio of B/G to predict the brittle and ductile behavior of materials. Higher (lower) B/G ratio corresponds to ductile (brittle) behavior and the critical value that separates brittle and ductile materials is about 1.75. Therefore, our obtained results of B/G ratio suggest that all the herein studied compounds can be categorized as ductile materials.
The value of the Poisson’s ratio is indicative of the degree of directionality of the covalent bonds. The typical value of the Poisson’s ratio (v) for ionic materials is 0.25 [54]. The data in Table III indicate that Poisson’s ratio vary from 0.313 to 0.329, which reveals that the interatomic forces are central forces [55]. It is clearly seen, from our calculated values of the Zener anisotropy factor A, which is a measure of the degree of elastic anisotropy of the crystal, that our Heusler alloys are elastically anisotropic.
Another important parameter is that of Young’s modulus E which is the usual property used to characterize stiffness of a material. The greater the value of E, the stiffer is the material. From Table III, it is clearly seen that Rh2MnSn is stiffer than Rh2MnPb and Rh2MnTl.
Once the values of young have been calculated, the Young’s modulus E, bulk modulus B and the shear modulus G, we can easily compute the Debye temperature θD from the average sound velocity [56]:
where h is Plank’s constant, k the Boltzmann constant, NA the Avogadro number, ρ the density, M the molecular weight and n is the number of atoms in the molecule. To estimate the average sound velocity in the polycrystalline material, we have used the following equation [57]:
where vt and vl are the transverse and longitudinal elastic sound velocities obtained using the shear modulus G and the bulk modulus B from Navier’s equation [58]:
The melting temperature is used in the heating system and can be theoretically estimated by the following equation [59]:
In Table IV, we summarized the estimated elastic wave velocities, Debye temperature and predicted melting temperature for Rh2MnZ (Z = Sn, Pb and Tl) Heusler alloys in their stable structure. It is clearly seen that Rh2MnSn alloy exhibits the highest Debye temperature compared to those observed for Rh2MnTl and Rh2MnPb. It is also noted that the investigated compounds have higher values of melting temperature.
Table IV The calculated density ρ (in g·cm-3), the transverse, longitudinal and average elastic wave velocities vt, vl and vm (in m·s-1) and the Debye temperatures θD (in K) and melting temperatures Tm (in K) for Rh2MnZ (Z = Sn, Pb and Tl) compounds.
| Compound | ρ | vt | vl | vm | θD | Tm ± 300 |
|---|---|---|---|---|---|---|
| Rh2MnSn | 10.01 | 2552.66 | 4979.38 | 2861.04 | 338.68 | 1705.79 |
| 10.408 [24] | 2574.78 [24] | 5221.64 [24]> | 2890.74 [24] | 316.16 [24] | ||
| 413 [15] | ||||||
| Rh2MnPb | 11.87 | 2190.37 | 4301.16 | 2454.56 | 287.30 | 1476.70 |
| 12.792 [24] | 2143.06 [24] | 4452.98 [24] | 2409.24 [24] | 263.21 [24] | ||
| Rh2MnTl | 12.02 | 2478.67 | 4759.57 | 2775.75 | 327.55 | 1915.49 |
The obtained results for Rh2MnSn and Rh2MnPb alloys are in well agreement with earlier ab-initio calculation. As far as we know, the Rh2MnTl compound is believed to be reported here for the first time.
4.3 Electronic properties
The total and partial density of states as a function of energy for the Rh2MnZ (Z = Sn, Pb, Tl) Heusler alloys at its equilibrium lattice constant calculated using the GGA approximation are depicted in Fig. 3. As seen from Fig. 3, the intersection of both majority and minority spin band structures with the Fermi level clearly reveals that all the herein compounds have metallic character. It is also clearly seen that the contributions to the resulting DOS are mainly formed by the Mn and Rh atoms around the Fermi level.
Figure 3 Total and partial density of states for a) Rh2MnSn, b) Rh2MnPb, and c) Rh2MnTl Heusler alloys in their stable structure types.
The analysis of the densities of electron states shows that there are two principal regions in the spin-up and spin-down states of DOS. The lower valence-bands below -8 eV for Rh2MnSn (below -9 eV for Rh2MnPb, below -7 eV for Rh2MnTl) are merely due to the s-orbitals of the Sn, Pb and Tl atoms, whereas the next region localized between -6.5 and 5 eV for Rh2MnSn (-6 to 5 eV for Rh2MnPb and -5 to 5 eV for Rh2MnTl) arise mostly from the 3d-states of the transition metals Rh/Mn.These obtained results for Rh2MnZ are in fairly good accordance with other theoretical calculations [22,23,24,30,31,37].
4.4 Magnetic properties
The calculated total and partial magnetic moments for the series of Rh2MnZ (Z = Sn, Pb and Tl) in AlCu2Mn-type structure are reported in Table V. As can be seen, the total magnetic moment composes the Rh atoms, the Mn atom, the Z atom, and the interstitial region. It can be clearly seen from Table V that the main magnetic moment resides at the Mn atom, because of the existence of a high exchange splitting between the majority and minority spin states of Mn atom. This behavior is also well known in most Heusler compounds [60]. As listed in Table V, the local spin magnetic moment on the Rh and Mn atoms in these ferromagnetic compounds are positive, while Sn, Pb and Tl atoms have a negative and very small magnetic moment which can be neglectable. Furthermore, the values obtained of total and localized spin magnetic moments for the Heusler alloys under study agree quite well with the available experimental and theoretical data in the literature.
The Curie temperature (TC) is calculated within the mean-field approximation (MFA) as follows [61]:
Where ΔE and KB denote the total energy difference between antiferromagnetic and ferromagnetic configurations and the Boltzmann constant, respectively. The calculated values of the Curie temperature for Rh2MnSn, Rh2MnPb, and Rh2MnTl are also listed in Table V.
Table V Calculated local, interstitial and total spin magnetic moments (in μB), and Curie temperatures (in K) of Rh2MnZ (Z = Sn, Pb, Tl) alloys.
| Compounds | MRh | MMn | MZ | Minterstitial | Mtot | TC |
|---|---|---|---|---|---|---|
| Rh2MnSn | 0.431 | 3.853 | -0.0119 | 0.083 | 4.786 | 547 |
| 0.45 [22] | 3.73 [22] | 4.60 [22] | 435 [22] | |||
| 0.43 [23] | 3.81 [23] | -0.01 [23] | 0.04 [23] | 4.72 [23] | 410 [62] | |
| 0.411 [24] | 3.636 [24] | -0.004 [24] | 0.249 [24] | 4.704 [24] | 420 [24] | |
| 0.38 [30] | 3.77 [30] | -0.016 [30] | 4.51 [30] | 431 [30] | ||
| 0.393 [32] | 3.831 [32] | -0.010 [32] | 4.607 [32] | |||
| 0.310 [37] | 4.075 [37] | -0.039 [37] | 4.74 [37] | |||
| Rh2MnPb | 0.424 | 3.882 | -0.0079 | 0.077 | 4.802 | 463 |
| 0.45 [22] | 3.69 [22] | 4.58 [22] | 423 [22] | |||
| 0.401 [24] | 3.673 [24] | -0.0011 [24] | 0.265 [24] | 4.741 [24] | 338 [65] | |
| 0.43 [30] | 3.79 [30] | 0.006 [30] | 4.66 [30] | |||
| 0.383 [32] | 3.888 [32] | -0.009 [32] | 4.644 [32] | |||
| Rh2MnTl | 0.305 | 3.808 | -0.022 | 0.026 | 4.424 | 103 |
| 0.266 [32] | 3.765 [32] | -0.027 [32] | 4.27 [32] |
4.5 Thermodynamic properties
In this section, we have used the quasi-harmonic Debye model as implemented in the Gibbs2 code [66,67] to inspect the influence of high pressure from 0 to 30 GPa and high temperature from 0 to 1400 K at the behavior of Rh2MnSn, Rh2MnPb and Rh2MnTl full-Heusler alloys being studied here.
Figure 4 shows the variation of volume unit cell and bulk modulus versus pressure under various temperature values. It is clear that for all the herein compounds, the volume decreases (the bulk modulus increases) gradually with increasing pressure. Furthermore, as the temperature increases, the volume increases (the bulk modulus decreases) slowly. Due to the high rate of temperature increase, we can therefore conclude that the effect of temperature on material is dominant and leads to a reduction in hardness. At ambient temperature and 0 GPa pressure, the obtained values of volume for Rh2MnSn, Rh2MnPb and Rh2MnTl compounds are 424.82 (a.u.3), 447.25 (a.u.3) and 440.22 (a.u.3), respectively, and those of bulk modulus are 171.35 GPa, 152.06 GPa and 140.86 GPa, respectively.
Figure 4 Variation of volume unit cell (V) for a) Rh2MnSn, b) Rh2MnPb, c) Rh2MnTl, and bulk modulus (B) for d) Rh2MnSn, e) Rh2MnPb, and f) Rh2MnTl as function of pressure for different temperatures.
In addition, the evolution of Debye temperature (θD) and thermal expansion coefficient α versus temperature at different pressures is displayed in Fig. 5. At 0 K, the Debye temperature increases from 424 K at 0 GPa to 557 at 30 GPa, from 364 K at 0 GPa to 501 at 30 GPa, from 353 K at 0 GPa to 509 at 30 GPa for Rh2MnSn, Rh2MnPb and Rh2MnTl Heusler alloys respectively. It has a negligible variation from 0 to 100 K. In the temperature range from 100 to 1400 K, we distinguish that the thermal expansion coefficient decreases slowly with increments of temperature at a given pressure.
Figure 5 Variation of Debye temperature (θD) for a) Rh2MnSn, b) Rh2MnPb, c) Rh2MnTl, and thermal expansion coefficient (α) for d) Rh2MnSn, e) Rh2MnPb, f) Rh2MnTl as function of temperature for different pressures.
It is also observed from the Fig. 5 that the thermal expansion coefficient increases dramatically while temperature still 200 K. Above this value, it takes another curvature and it increases weakly with raising temperature for all the investigated compounds. It is noted that the effect of temperature becomes less. The thermal expansion coefficient decreases when an external pressure varies from 0 GPa to 30 GPa. This influence is slight for low temperatures but it becomes substantial as the temperature increases. The calculated values of α at zero pressure and room temperature are equal to 3.03 × 10-5K-1, 3.65 × 1-5K-1 and 4.63 × 10-5K-1 for Rh2MnSn, Rh2MnPb and Rh2MnTl, respectively.
Additionally, the temperature effects on the specific heat (CP) at constant pressure and specific heat (CV) at constant volume parameters at different pressures are shown in Fig. 6.
Figure 6 Variation of specific heat (Cp) at constant pressure for a) Rh2MnSn, b) Rh2MnPb, c) Rh2MnTl and specific heat (CV) at constant volume parameters for d) Rh2MnSn, e) Rh2MnPb, f) Rh2MnTl as function of temperature for different pressures.
At low temperatures, the variation of CP and CV is proportional to the Debye law
At zero pressure and room temperature, our obtained results for CV are found to be 92.99, 90.66 and 93.49 J.mol-1·K-1 and for CP are equal to 92.45, 95.42 and 97.06 J·mol-1·K-1 for Rh2MnSn, Rh2MnPb and Rh2MnTl alloys, respectively.
5 Conclusion
We have investigated in detail the full-Heusler compounds Rh2 MnZ (Z= Sn, Pb, Tl) using full-potential linearized augmented plane wave (FP-LAPW) method in the framework of density functional theory (DFT) within PBE-GGA for structural stabilities, elastic, electronic, magnetic and thermodynamic properties. Our calculations show that the AlCu2Mn-type structure is more stable than Hg2CuTi-type structure at the equilibrium volume for the systems under study. The obtained results for equilibrium lattices constant and bulk modulus are in good agreement with the previously calculated results. Also, the plots of density of states establish that Rh2MnZ (Z = Sn, Pb, Tl) compounds possess ferromagnetic and metallic character. Furthermore, the calculated magnetic moments are in qualitative agreement with available experimental and theoretical values. The contribution to magnetic moment in these ferromagnetic compounds comes mainly from Mn atom. The magnetic properties reveal that these compounds have a ferromagnetic coupling between Rh and Mn atoms whereas anti-ferromagnetic coupling between Rh and Sn (Pb, Tl) atoms. Regarding the elastic properties, all compounds studied here satisfy the mechanical stability criteria, confirming the stability of the AlCu2Mn-type structure. These alloys are found to be elastically anisotropic and have a ductile behavior. The values of elastic constants and their elastic moduli parameters are very close to those existing in the literature. We have also calculated the thermodynamic properties including volume unit cell, bulk modulus, Debye temperature, thermal expansion coefficient and specific heat capacities, at a temperature range from 0 to 1400 K and pressure from 0 to 30 GPa within the quasi-harmonic Debye approach implemented in the Gibbs2 code. Finally, we hope that our theoretical results on Rh-based Heusler alloys may provide a good baseline data to experimentalists for future spintronic application.
