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**Submitted:** April 11, 2024 | **Approved:** April 23, 2024 | **Published:** April 24, 2024

**How to cite this article:** Yadav P, Yadav DS, Singh DV. Electronic and Thermo-Dynamical Properties of Rare Earth RE_{2}X_{3} (X=O, S) Compounds: A Chemical Bond Theory. Int J Phys Res Appl. 2024; 7: 048-052.

**DOI:** 10.29328/journal.ijpra.1001083

**Copyright License:** © 2024 Yadav P, et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

**Keywords:** RE_{2}O_{3} and RE_{2}S_{3}; Electronic properties; Mechanical properties; Thermodynamic properties

# Electronic and Thermo-Dynamical Properties of Rare Earth RE_{2}X_{3} (X=O, S) Compounds: A Chemical Bond Theory

####
Pooja Yadav^{1}, DS Yadav^{2*} and DV Singh^{1}

^{1}Department of Physics, Agra College, Agra-282002, UP, India

^{2}Department of Physics, Ch. Charan Singh PG College, Heonra (Saifai), Etawah-206001, UP, India

***Address for Correspondence:** DS Yadav, Department of Physics, Ch. Charan Singh PG College, Heonra (Saifai), Etawah-206001, UP, India,
Email: dhirendra.867@rediffmail.com

The electrical, mechanical, and thermodynamic properties of cubic structured rare earth sesqui-chalcogenides RE_{2}X_{3} (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 (RE_{2}O_{3}), 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, RE_{2}O_{3} 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 RE_{2}S_{3} 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 RE_{2}O_{3} (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 (RE_{2}O_{3}) 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 RE_{2}O_{3} & RE_{2}S_{3} (RE= La-Lu, except for the radioactive element Pm) with C-type bixbyite and Th_{3}P_{4} 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 RE_{2}O_{3} & RE_{2}S_{3} (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 (sp^{3}) 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 K_{1} = 2.5 are the constants, i.e., remain unchanged in different crystals.

A = 39.74 and K_{1} = 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 Nc^{2}, and K_{2} = 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), d_{0} = (d/2) (d is the nearest adjacent distance), and b is 4.6137 for C-type RE_{2}O_{3} and 2.532 for Th_{3}P_{4} type RE_{2}S_{3}. 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, d_{0}, are where these potentials should be assessed. The Thomas-Fermi screening factor e-K_{s}.d_{0} 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:

${k}_{s}=2{a}_{B}^{-0.5}{(3N/\pi V)}^{0.167}$ (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, ε_{0} and ε_{∞}, are taken from separate sources [14,25]. The defined variable S0, which is changeable, is [24]; 0.78 for RE_{2}O_{3} and 0.80 for Re2S3. The defined

${S}_{0}=1-\left(\frac{{E}_{g}}{4{E}_{f}}\right)+\frac{1}{3}{\left(\frac{{E}_{g}}{4{E}_{f}}\right)}^{2}$ (7)

The valence electron plasmon energy is given by the relation^{23} –

$\hslash {\omega}_{P}=28.8\sqrt{\frac{{N}_{eff}d}{M}}$ (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)

${f}_{i}=1-\frac{1}{6}\mathrm{exp}\left(-\Delta {X}^{2}/4\right)$ (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 b_{0} = 4.143 × 10^{4} & b_{1} = 1.034 for cubic structured RE_{2}O_{3}, which depends upon the structure of rare earth sesquioxides and the exponent has values 1.147. In cubic structured RE_{2}O_{3}, In, b_{0} = 4.143 × 10^{4} & b_{1} = 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)

Where d_{ue}and d

_{XY}are the bond lengths of germanium and the RE

_{2}O

_{3}, respectively, ∆H

_{0}= 1190, S = 3.0, and the factor D(

*XY*) is defined as

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

Where *Ē( XY)* is the average of

*E*) and

_{0}(XY*E*) and

_{1}(XY*E*) are higher critical energies of the compound (

_{2}(XY*XY*),

*E*) is the lowest direct energy gap, and b = 0.0467. The values of

_{0}(XY*E*),

_{0}(XY*E*), and

_{1}(XY*E*) can be either taken from the experimental reflectivity data or calculated theoretically using relations given by Neumann

_{2}(XY^{30}.

The values of E_{h}, E_{c}, E_{g}, Ep, and fi that have been examined for RE_{2}O_{3} 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 RE_{2}O_{3} 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 RE_{2}O_{3} 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 Th_{3}P_{4} type RE_{2}X_{3} 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.

- Kitai AK. Oxide phosphor and dielectric thin films for electroluminescent devices. Thin Solid Films. 2003; 445:367.
- Barrera EW, Pujol MC, Diza F, Choi SB. Emission properties of hydrothermal Yb3 +, Er3 + and Yb3 +, Tm3 + -codoped Lu2O3 nanorods: upconversion, cathodoluminescence and assessment of waveguide behavior. Nanotechnology. 2011; 22:075205.
- Andreeva D, Ivanov I, Ilieva L. Gold catalysts supported on ceria doped by rare earth metals for water gas shift reaction: Influence of the preparation method. Appl. Catal. A. 2009; 357:159.
- Pan TM, Hung WS. Physical and electrical characteristics of a high-k Yb2O3 gate dielectric. Appl. Surf. Sci. 2009; 255:4979.
- Zelmon DE, Nothridge JM, Haynes ND. Appl. Opt. 2013; 52:3825.
- Orlovskaya N, Lukich S, Subhash G. Mechanical properties of 10 mol% Sc2O3–1 mol% CeO2–89 mol% ZrO2 ceramics. J. Power Sources. 2010; 195:2774.
- Azimi G, Dhiman R, Kwon H M. Hydrophobicity of rare-earth oxide ceramics. Varanasi KK. Nat. Mater. 2013; 12:315.
- Zinkevich M 2007 Prog. in Material Science 52 7597
- Goldschmidt V M, Ulrich E and Barth T. A Theoretical Study of Binary and Ternary Hydride-Bonded Complexes N(Beh2)...X with N = 1 or 2 and X = K+ or Ca+2. Skrifter Norske Videnskaps-Akadoslo, I: Mat. Naturev, Kl.5. 2011.
- Petit L, Svane A, Szotek Z, Temmerman WM. First-principles study of rare-earth oxides. Phys. Rev. B. 2005; 72:205118.
- Abrashev M V, Todorov N D, Geshev J. Raman spectra of R2O3 (R—rare earth) sesquioxides with C-type bixbyite crystal structure: A comparative study. J. Appl. Phys. 2014; 116:103508.
- Sheng J, Gang B L, Jing L. The Phase Transition of Eu2O3 under High Pressures. Chin. Phys. Lett. 2009; 26:076101.
- Hirosaki N, Ogata S, Kocer C. Ab initio calculation of the crystal structure of the lanthanide Ln2O3 sesquioxides. J. Alloys Compounds. 2003; 351:31-34.
- Xue D, Betzler K, Hesse H. Dielectric constants of binary rare-earth compounds. J. Phys.: Condens. Matter. 2000; 12:3113.
- Rahm M, Skorodumova NV. Phase stability of the rare-earth sesquioxides under pressure. Phys. Rev. B. 2009; 80:104105.
- Remay H. Introduction. Home Inorganic Reactions in Water Chapter. Inorganic Chemistry. 1956; 2:247.
- (a). Jiang S, Liu J, Li X. Structural transformations in cubic Dy2O3 at high pressures. Solid Stat. Comm. 2013; 169:37-41.
- (b). Jiang S, Liu J, Li X. Phase transformation of Ho2O3 at high pressure. J. Appl. Phys. 2011; 110:013526.
- (c). Jiang S, Liu J, Lin C. Pressure-induced phase transition in cubic Lu2O3. J. Appl. Phys. 2010; 108:083541.
- Yadav DS. Electronic properties of aluminum, gallium and indium pnictides. Phys. Scr. 2010; 82:65705.
- Yadav DS, Verma AS. Electronic, optical, and mechanical properties of AII-BVI semiconductors. International Journal of Modern Physics B, 2012, vol. 26, 1250020.
- Singh OP, Gupta VP. Electronic properties of europium chalcogenides (EuO, EuS, EuSe, EuTe). Phys. Stat. Sol. (b). 1985; K
_{1}53:129. - Singh DV, Gupta VP. Bulk Moduli of Sm, Eu, and Yb Monochalcogenides. Phys. Stat. Sol. 1992; (b) K71:171.
- Van-Vechten JA. Quantum Dielectric Theory of Electronegativity in Covalent Systems. I. Electronic Dielectric Constant. Phys. Rev. 1969; 182:891.
- Phillips JC. Bonds and Bands in Semiconductors (New York: Academic). 1973.
- Penn DR. Wave-Number-Dependent Dielectric Function of Semiconductors. Phys. Rev. 1962; 128:2093.
- Zhue VP, Shelykh AI. Sov. Phys. Semiconductor. 1989; 23:245.
- Tubbs MR. A Spectroscopic Interpretation of Crystalline Ionicity. Phys. Stat. Solidi. 1970; 41:61.
- Pauling L. The Chemical Bonds (Ithaca, NY: Cornell University Press). 1960.
- Neumann H. Bulk modulus — volume relationship in alkali halides with rocksalt structure. Cryst. Res. Tech. 1988; 23:531.
- Phllips JC, Van-Vechten JA. Phys. Rev. 1970; B 2:2147.
- Neumann H. Cryst. Res. Tech. 1983; 18:167.