Electronic and Thermo-Dynamical Properties of Rare Earth RE₂X₃ (X=O, S) Compounds: A Chemical Bond Theory

The electrical, mechanical, and thermodynamic properties of cubic structured rare earth sesqui-chalcogenides RE₂X₃ (RE = La-Lu, X = O, S) are examined in this work using the chemical bond theory of solids. For these materials, the values of the homopolar gaps (E_h), ionic gaps (E_c), and average energy gaps (E_p) have been assessed. It has been discovered that the calculated values of the homopolar gap (E_h) and average energy gap (E_p) are in great agreement with the values derived from the Penn and Phillips models. The electrical, mechanical, and thermodynamic properties of these materials (RE₂O₃), such as their bulk modulus and heat of formation, have been estimated using the bond ionicity values. The computed values accord very well with the theoretical results that have been published thus far.

PACS No.: 71.20.E_h, 71.15.Mb, 61.50.K_s, 71.15.Mb

Because of its numerous technological applications in the fields of electroluminescence, cathode-luminescence source, crystals for chemical organic reactions, high-K gate dielectrics, optical components of high power lasers, oxygen ion conducting electrolyte in solid oxide fuel cells, and materials with strongly hydrophobic surfaces, RE₂O₃ with C-type bixbyite crystal structure has received a lot of attention recently [1-7]. Each rare earth atom contributes three electrons to the extremely electronegative O ions in rare earth sesqui-oxides, with the remaining 4f electrons remaining firmly localized at the rare earth site. Larger oxygen coordination numbers are found in the lighter lanthanides because the f-electrons are less firmly connected to the parent atom's nucleolus. Because of the interaction between valence electrons and localized 4f electrons, these materials exhibit several abnormal physical features. Because localized magnetic moments readily hybridize with valence and conduction electrons, valence fluctuation states also exist in these compounds despite their insulating nature and lack of carriers. Because of the Coulomb correlation effect, the 4f band splits into two sub-bands that are separated by 6–12 eV, which results in RE₂S₃ insulators. The crystal structures of these materials are known to fall into three different polymorphic [8] forms: (1) A-type, hexagonal, and, most of the time, space group P3m1 (2)- B-type, monoclinic; typically belongs to space group C2Im (3)- Cubic C-type, typically belonging to space group Ia3. Goldschmidt et al. conducted the first thorough investigation of the rare earth sesquioxides in 1925 [9], and his initial phase classifications (A, B, and C-type) are still in use today. Using the tight-binding linear muffin-tin orbital (TB-LMTO) method and the self-interaction corrected local spin density (SIC-LSD) methodology, Petit, et al. [10] conducted a first-principles investigation on rare earth oxides, namely RE₂O₃ (RE = Ce to Ho). Many attempts have been made in the past few years [11-17] to comprehend the electrical, optical, mechanical, and thermodynamic properties of rare earth oxides (RE₂O₃) using a variety of techniques. Authors [18,19] have effectively used the modified dielectric theory of solids to study the electrical, optical, and mechanical properties of binary semiconductors in the II-VI and III-V groups. Using the modified dielectric theory of solids, we have computed the electrical, thermodynamic, and mechanical properties of RE₂O₃ & RE₂S₃ (RE= La-Lu, except for the radioactive element Pm) with C-type bixbyite and Th₃P₄ type structure in this study [20,21]. To the best of my knowledge, however, the modified PVV theory of solids has not yet been used to study the electrical, thermodynamic, and mechanical properties of RE₂O₃ & RE₂S₃ (RE = La-Lu, except the radioactive element Pm). For these materials, the values of homopolar gaps (E_h), ionic gaps (E_c), and average energy gaps (E_p) are examined using this concept to obtain greater agreement. We can ascertain these criteria to find these materials' Phillips ionicity. Utilizing the deduced ionicity value, the bulk-modulus and formation heat are examined. The heat of formation and bulk-modulus values thus obtained are in excellent agreement with those reported in the literature thus so far [12,15-17].

To decompose the average energy gap (E_p) between bonding and anti-bonding (sp³) hybridized orbitals into contributions from symmetric and anti-symmetric parts by the potential within the unit cell, the average energy gap (E_p) can be split into heteropolar or ionic part (E_c) and homopolar or covalent part (E_h) using the modified dielectric theory of solids [20,21]. These contributions take the following form: E_c stands for heteropolar or ionic contribution, and E_h for homopolar or covalent contribution.

$E_p^2=E_h^2+E_c^2$ (1)

The covalent part E_h depends on the nearest neighbor separation d_AB as follows:

$E_h=A{d}_{AB}^{-K_1}$ (2)

Where A = 40.468 eV(A°)2.5 and the exponent K₁ = 2.5 are the constants, i.e., remain unchanged in different crystals.

A = 39.74 and K₁ = 2.48 were similar values found by Phillips and Van-Vechten [22]. The following relation can be used to determine the ionic contribution:

$E_c=K_2{d}_0^{-1}\cdot e^{-k_{s\cdot }{d}_0}$ (3)

Where b is an adjustable parameter that depends on coordination number 22 around the cation, i.e., b = 0.089 Nc², and K₂ = be2(Z_A-Z_B) is a numerical constant. Z_A and Z_B are the valence states of atoms A and B, respectively. Nc is the average coordination number, K_s is the Thomas Fermi Screening Parameter (TFSP), d₀ = (d/2) (d is the nearest adjacent distance), and b is 4.6137 for C-type RE₂O₃ and 2.532 for Th₃P₄ type RE₂S₃. According to the physical interpretation of equation (3), E_c is the difference between the Screened Coulomb Potentials of atoms A and B with core charges Z_A and Z_B. The covalent radii, d₀, are where these potentials should be assessed. The Thomas-Fermi screening factor e-K_s.d₀ reduces the charge of the ion cores by screening out the remaining electrons, which influences the chemical trend of a compound. Only a small portion of the electrons are in the bond. This screening factor is connected to the effective number of free electrons in the valence band along with the bond length. As a result, the number and length of bonds emerging from the cations determine the values of E_c and E_h. Ten electrons per molecule were taken into consideration for determining the value of K_s, which is defined as follows:

$k_s=2a_B^{-0.5}{(3N/\pi V)}^{0.167}$ (4)

Where a_B is Bohr radius.

The E_h, E_c, and E_p values for these materials have been determined by using the aforementioned relations (1)–(4). Phillips models [23] and Penn [24] can also yield the values of E_h* and. Ep*. The following form represents E_h* following the Phillips model:

$E_h^{*}=\frac{\hslash {\omega }_p{S}_0}{\sqrt{{\epsilon }_0-1}}$ (5)

And E_p* using the Penn model, defined as

$E_p^{*}=\frac{\hslash {\omega }_p{S}_0}{\sqrt{{\epsilon }_{\infty }-1}}$ (6)

Where the valence electron plasmon energy is represented by ħωp, and the static and optical dielectric constants, ε₀ and ε_∞, are taken from separate sources [14,25]. The defined variable S0, which is changeable, is [24]; 0.78 for RE₂O₃ and 0.80 for Re2S3. The defined

$S_0=1-\left(\frac{E_g}{4E_f}\right)+\frac{1}{3}{\left(\frac{E_g}{4E_f}\right)}^2$ (7)

The valence electron plasmon energy is given by the relation²³ –

$\hslash {\omega }_P=28.8\sqrt{\frac{N_{eff}d}{M}}$ (8)

where N_eff – effective no. of the valence electrons, d-density, and M-molecular weight of the material.

Phillips ionicity $(f_1=E_c^2/E_g^2)$ has been assessed for each of these materials to have an additional check on the E_c and E_h values. The results are compared with those derived from the Tubbs ionicity model [26] and Pauling ionicity model [27], which are defined as:

$f_i=E_P/\hslash {\omega }_P\cdot S_0$ (9)

$f_i=1-\frac{1}{6}\exp \left(-\Delta X^2/4\right)$ (10)

Where ∆X represents the difference in electro-negativity between the O and S atoms and RE (La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er, Tm, Yb, and Lu). Table 1 makes it abundantly evident that there is a fair amount of consistency between the different ionicities.

Bulk-modulus

In terms of cell volume, the bulk modulus using the Neumann technique [28] has been determined using the computed values of crystal ionicity as

$B=B_0{V}^{-n}$ (11)

is the constant depends upon the covalence and b₀ = 4.143 × 10⁴ & b₁ = 1.034 for cubic structured RE₂O₃, which depends upon the structure of rare earth sesquioxides and the exponent has values 1.147. In cubic structured RE₂O₃, In, b₀ = 4.143 × 10⁴ & b₁ = 1.034 rely on the structure of rare earth sesquioxides, and the exponent has a value of 1.147, the constant is dependent on the covalence FC.

Heat of formation

It is possible to write the heat of formation [29,30] of rare earth sesquioxide using the bond ionicity values that were obtained above-

$\Delta H_f=\Delta H_0{\left(\frac{d_{Ge}}{d_{XY}}\right)}^{s}D(XY)f_{i,XY}$ (12)

$D(XY)=1-b{\left(\frac{E_2(XY)}{\overline{E}(XY)}\right)}^2$

Where Ē(XY) is the average of E₀(XY) and E₁(XY) and E₂(XY) are higher critical energies of the compound (XY), E₀(XY) is the lowest direct energy gap, and b = 0.0467. The values of E₀(XY), E₁(XY), and E₂(XY) can be either taken from the experimental reflectivity data or calculated theoretically using relations given by Neumann³⁰.

The values of E_h, E_c, E_g, Ep, and fi that have been examined for RE₂O₃ compounds based on the current investigation are listed in Table 1. The values of fi for various materials have been researched and determined using the Phillips ionicity model, utilizing Equation (1-4). The results are compared with the values derived from the Tubbs and Pauling ionicity model and are displayed in Table 1. There is good agreement between the bond ionicity values of various materials. We have calculated the bulk modulus (B, in GPa) and heat of formation (-∆Hf, in KJ/mole) of RE₂O₃ using different ionicities, and the results are displayed in Table 2. Table 2 makes it quite evident that the computed values of B and H from the several ionicities we used show a decent degree of agreement with the other existing theoretical conclusions. Therefore, we believe that the values generated from Phillips ionicity are more appropriate than the values derived from Tubb's and Pauling's ionicity models.

For cubic-structured rare earth sesqui-oxides and sulfides, the values of E_h, E_c, and Ep have been examined using the modified dielectric theory of solids. It has been demonstrated that the examined values agree with the values found in the Penn and Phillips models. The computed data above have been further examined by deriving Phillips ionicity from them. We can calculate these C-type RE₂O₃ compounds' bulk modulus (B) and heat of formation (-∆Hf) using the estimated values of Phillips ionicity. While there is a significant difference between our predicted bulk modulus values and the published experimental data, the heat of formation values of these materials are in good agreement accord with previously published literature values. Thus, we conclude that the chemical bond theory of solids can be used for both cubic and Th₃P₄ type RE₂X₃ compounds in light of the aforementioned data.

Credit authorship contribution statement

Pooja Yadav: Writing an original draft, Review of Literature, Dhirandra Singh Yadav: Methodology, Conceptualization, Formal analysis, Data curation, Supervision, Review & editing: Data presentation, D V Singh: Ideas, Final writing.

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