THE ELECTRONIC AND THERMODYNAMIC PROPERTIES OF TERNARY RARE EARTH METAL ALLOYS

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INTRODUCTION
A lot of investigation has been done on the equal-atomic ternary rare-earth intermetallic compounds, owing to their various interesting properties in the field of material science.In recent years, intermetallic compounds have received a lot of attention from scientists due to their distinctive properties.They exhibit numerous structural properties [1][2].There is a focus on structural, electronic, magnetic, and thermodynamic properties, such as fermionic, heavy-fermions [3][4], and half-metallic behavior in rare-earth intermetallic compounds such as SmInZn, SmInCd and SmTlZn [5][6], and the giant magneto-resistance device is an industrial application of heavy rare earth compounds [7][8][9].These equal-atomic type compounds of rare earth have been investigated seriously in the last few years [10][11][12][13][14][15].Sm-based compounds have been studied with theoretical and experimental significance [16][17][18][19][20].It is seen from this study that Sm-based compounds can be used for refrigeration.This is based on the magneto-caloric effect (MCE) [21].The SmInZn, SmInCd and SmTlZn compounds show many interesting applications in spintronics devices, magnetic sensors, unconventional superconductors, met-magnetism, magnetic randomaccess memories, and spin glass [21][22].In this calculation, we used the space group P63/mmc (number-194) for the SmInZn, SmInCd and SmTlZn rare-earth compounds, and their crystal structures are similar to those of CaIn2 at low temperatures [21].The calculations have been executed with the help of the first principle method based on DFT, which is exerted in WIEN2K software package [23].We have investigated the electronic and thermodynamic characteristics of SmInZn, SmInCd, and SmTlZn compounds using the FP-LAPW technique and GGA approximation [24,25].Here, structural and electronic properties are described, such as the energy Vs volume curve (which shows the stability of material), band diagrams, and, total and partial density of states.Thermodynamic behaviors of the SmInZn, SmInCd and SmTlZn compounds have been investigated with the help of GIBBS2 software.It's based on the Quasi Harmonic Debye (QHD) model [26].In thermodynamics behavior, we have investigated some parameter like bulk modulus of materials (B 0 ), specific heat (Cv), Grüneisen parameter (γ), Debye temperature (θ D ), thermal expansion coefficient (α) and entropy (S).

COMPUTATIONAL METHOD
This article has been presented for first-principal calculations using full-potential approach based on DFT, it is utilized by the software WIEN2K [27][28], to investigate the electronic and thermodynamic properties of SmInZn, SmInCd, and SmTlZn.In this modern DFT approach method; exchange-correlation energy potential has played an important role in the SCF process based on GGA [29].In the SCF process, we take the value of cut-off parameter Kmax = 7/R MT [where Kmax is the magnitude of largest reciprocal lattice vector and R MT is the minimum radii of atoms in BZ (Brillouin zone)] and R MT values have been taken 2.50 a.u.for Gd and Sb atoms and 2.46 a.u.for Ni atom.In SCF process, 47 k-mess points and 2000 k-point was used in the irreducible parts of full BZ and cut-off energy −6.0 Ry.We have chosen the energy convergence 0.0001ec.The energy terms used in this process gives information about valence and core atomic states.The total energy Vs volume curve has been fitted in the Brich-Murnaghan equation of state (EOS) [30].Thermodynamics behaviors of the SmInZn, SmInCd and SmTlZn compound have been investigated, with the help of GIBBS2 software.It's based on the Quasi Harmonic Debye (QHD) model.

Structural and electronic properties
The stability curves (volume vs energy curve) can be used to calculate the structural characteristics of an intermetallic compound.The bulk modulus of material (B 0 ), pressure derivative bulk modulus of material (B ), and lattice constant (a 0 ) parameters were derived from structural characteristics.Figures 1(a), 1(b), and 1(c) depict the energy vs. volume curve (i.e., material stability curve) for SmInZn, SmInCd, and SmTlZn respectively, as fitted by the Brich-Murnaghan equation of state (EOS).Using the optimization method and revealing that compound SmInZn is stable with a minimum energy value (E 0 = -72456.829501Ry) at the minimum volume (V 0 = 1017.8983a. u 3 ).The optimization method revealed that compound SmInCd is stable with a minimum energy value (E 0 = -87656.386376Ry) at the minimum volume (V 0 = 1111.7033a. u 3 ) while SmTlZn is stable with a minimum energy value (E 0 = -130079.217602Ry) at the minimum a b c volume (V 0 = 1047.2396a. u 3 ).After calculating the energy Vs.volume curve (i.e., material stability curve), we get the values of lattice constant (a 0 ), bulk modulus of material (Bo) and pressure derivative of bulk modulus of material (B ).(See Table 2).These structural parameters are obtained using the GGA approximation.The value of the lattice constant we determined from our calculations is supported by some research on these compounds [32].At present, we are discussing here spin polarized electronic characteristics such as energy bands and DOS (i.e., density of states).The calculated energy bands for majority and minority spin channels along the high symmetry directions Г, Σ, M, K, Λ and Δ in the first Brillouin zone for SmInZn, SmInCd, and SmTlZn compounds were obtained using GGA approaches, which are shown in Figures 2 (a) to 2 (f) respectively.Figures 2(a It is clear from these graphs that most of the energy levels in the valence bands lie between -8.0 eV and 0.0 eV and conduction bands lies above 0.0 eV, Whereas the Fermi level is assumed to at the origin.It can be seen from Figure 2(a,b,c,d,e,f) Due to the electronic arrangement of Tl and In [Xe] 4f 14 5d 10 6s 2 6p and [Kr] 4d 10 5S 2 5p 1 , the bands that were present in at roughly-09.0 eV in SmTlZn, SmInZn and SmInCd for spin up have been disappear in spin down bands, as can be seen in Figs.2(a)-(f) (i.e.these bands in exist due to Zn bands).

Spin up Spin down Spin down
Additionally, the valence band and conduction bands both cross the Fermi level, so there is no band gap.Thus, three compounds (SmInZn, SmInCd, and SmTlZn) indicative metallic properties.The Sm-f orbitals are mainly to blame for the formation of these bands.Now we are going to talk about the total densities of states, also known as T-DOS, for providing metallic character to these compounds.In addition, the profile of the partial densities of states (P-DOS) for the SmInZn, SmInCd, and SmTlZn compounds have been calculated with the help of the GGA approach and are plotted in graphs in Figure 3 (a-i) for SmInZn, Figure 4 (a-i) for SmInCd, and Figure 5 (a-i) for SmTlZn.The Fermi level at the origin can be seen in Fig. 3(a-i).It has been found that three sharp peaks are at around 0, -1 and -7 electron-volts (eV) in the spin-up state below the Fermi level and also one peaks are obtained at around +4 electronvolts(eV) above the Fermi level.It has been found that one sharp peak is at Fermi level in a spin-down state.According our observations, the Zn-d states are responsible for the peak that occurs at around -7 eV below the Fermi level for both channels.A small contribution of In-s and In-d states.A second peak at about 0 eV and -1 eV is caused by the spinup state of Sm-f state and a minor contribution from In-d and Zn-p states.Now we discuss another peak at around +4e V is caused by spin down state of Sm-f state above the Fermi level.The Fermi level at the origin can be seen in Fig. 4(a-i).It has been found that three sharp peak is at around 0 eV, -1 eV spin up state and -9 electron-volts (eV) in the both state below the Fermi level and also one peak is obtained at around +4 electron-volts(eV) in the spin down state above the Fermi level.We notice that a sharp peak around -9 eV is caused by the Cd-d states with small contribution In-s and In-d state for both channels.A second peak at about 0 eV and -1 eV is caused by the spin-up state of Sm-f states and a minor contribution from In-p and Cd-p states.Now we discuss another peak at around +4e V is caused by spin down state of Sm-f state above the Fermi level.The Fermi level at the origin can be seen in Fig. 5(a-i).It has been found that four sharp peaks are at around -9 eV and -13 electron-volts (eV) in both states below the Fermi level, and also one peak found around 0 eV in the spin up state at Fermi level.And one peak is obtained at around +4 electron-volts (eV) in the spin down state above the Fermi level.We observe that the peak of around -9 eV for both channels below the Fermi level is caused by the Zn-d state with the hybridization of the Zn-s and Tl-s states.Another peak at around -13 eV is caused by the both state of Tl-d states and a minor contribution from Zn-p and Zn-s states.The

Thermodynamics properties
The QHD model, implemented in the Gibbs software package was used to calculate the thermodynamic properties for SmInZn, SmInCd, and SmTlZn compounds within wide range of temperature (0-400 K).Figures 6(a-f) depict temperature dependent thermodynamics characteristics.In thermodynamic properties, we have extracted some important parameters, such as entropy (S), thermal expansion coefficient (α), bulk modulus of material (B), specific heat at constant volume (CV), and Debye temperature (θ D ).The variation of the bulk modulus with temperature is shown in Fig. 6(a).These graphs show that as the temperature increases, the bulk modulus (B), decreases.The bulk modulus (B), and compressibility (C), are inversely proportional.So, we can say that as the temperature increases, the compressibility (C), of these compounds will also increase.As a result, Figures 6(a) demonstrates also that as the temperature rises, the three compounds (SmInZn, SmInCd, and SmTlZn) become more compressible, or flexible.Other thermodynamic properties, such as the Debye temperature, provide interesting information about the elastic nature of materials.The relationship between the Debye temperature and temperature is depicted in Figures 6(b), which also demonstrates that for all three compounds.
The Debye temperature (θ D ) gradually decreases as the temperature increases.Additionally, Figs.6(b) show that SmInZn has a larger θ D value than SmInCd and SmTlZn, indicating that SmInZn is stiffer than those two metals (as stiff material has a high Debye temperature).
The anharmonicity of crystal lattices under vibrational motion is described by the Grüneisen parameter (γ).Figures 6(c) indicate how the Grüneisen parameter (γ) changes with temperature and show that it rises with rise in temperature.In the quasi-harmonic Debye model, γ is a function of volume which is depends upon lattice parameters.The dimensions of the lattice and, hence, the volume, grow as the temperature rises so γ also increase with rise in temperature.As a result, increasing temperature raises crystal anharmonicity and the value of γ.A crucial factor that offers crucial insight into the vibrational characteristics and microscopic structure of a crystal is specific heat (CV).
The variation of specific heat (C V ) is shown in Figures 6 (d) which illustrates how the computed heat capacity at constant volume (C V ) responds to temperature.At temperature ≈ 300 K, it's clear that C V follows the Debye model relationship (C V proportional to T 3 law), and beyond T > 300 K, the Dulong-Petit limit is approaching.The thermal expansion coefficient (α) varies with temperature, as shown in Figures 6(e). it is clear that α increases sharply with temperature when it gets close to 300K.The impact of temperature on the thermal expansion coefficient (α) reduces when temperature rises and exceeding 300 K. when the atoms are moving in a variety of ways.The amount of disorder or irregularity in an atom's or molecule's motion inside a thermodynamic system is described by the entropy(S) Figures 6(f) demonstrate how the entropy (S) for SmInZn, SmInCd, and SmTlZn changes with temperature.Figures 6(f) show that the entropy is zero at absolute zero temperature and that the vibrations inside the thermodynamics system increase as the temperature rises.As a result, the entropy of the system starts to increase rapidly.

CONCLUSIONS
In the current chapter, the DFT approach has been used to examine the electronic properties and quasi-harmonic Debye model has been applied to calculate the thermodynamic characteristics of the SmInZn, SmInCd, and SmTlZn compounds.The following important conclusions have been drawn from the calculations.
SmInZn, SmInCd, and SmTlZn are well stable in hexagonal phase.Equilibrium volume increases from SmInZn → SmTlZn → SmInCd due to increasing lattice parameter from SmInZn → SmTlZn → SmInCd.All three compounds confirm metallic nature due to dominancy of Sm-f spin up bands at the Fermi level.Sm-f spin up bands provide metallic character to these materials.Fermi energies EF = 0.52524 electron-volts (eV) for SmInZn, E F = 0.48818 electron-volts (eV) for SmInCd and E F = 0.54729 electron-volts (eV) for SmTlZn.
Bulk modulus characteristics show that strength of SmInZn, SmInCd, and SmTlZn compounds decreases with temperature and become more flexible with increasing the temperature.The impact of temperature on Debye temperature, θ D was found to be small on SmInZn and SmInCd while impact of temperature on θ D is larger in SmTlZn.Thus, SmInZn was found to be stiffer than SmInZn and SmInCd.All three compounds show deviation from harmonic oscillator and SmInZn show larger deviation from harmonic oscillator character compared to SmInCd and SmTlZn.α increases with increasing the temperature due to expansion of unit cell dimensions and hence volume with temperature.The amount of disorder was found to be zero at 0K and increases rapidly with increasing the temperature in these compounds.

Figure 2 .
Figure 2. Band diagram for (a) spin up for SmInZn (b) spin down for SmInZn (c) spin up for SmInCd (d) spin down for SmInCd (e) spin up for SmTlZn (f) spin down for SmTlZn
is caused by spin up state of Sm-f state at Fermi level.Another peak around +4 eV is caused by spin down state of Sm-f state above the Fermi level.Fermi energy has been found that E F = 0.52524 electron-volts (eV) for SmInZn, E F = 0.48818 electron-volts (eV) for SmInCd and E F = 0.54729 electron-volts (eV) for SmTlZn.

Table 2 .
Some structural parameters calculated using the GGA approximation