Electronic, elastic and optical properties of divalent (R⁺²X) and trivalent (R⁺³X) rare earth monochalcogenides

Abstract

Based on plasma oscillations theory of solids, simple relations have been proposed for the calculation of bond length, specific gravity, homopolar energy gap, heteropolar energy gap, average energy gap, crystal ionicity, bulk modulus, electronic polarizability and dielectric constant of rare earth divalent R⁺²X and trivalent R⁺³X monochalcogenides. The specific gravity of nine R⁺²X, twenty R⁺³X, and bulk modulus of twenty R⁺³X monochalcogenides have been calculated for the first time. The calculated values of all parameters are compared with the available experimental and the reported values. A fairly good agreement has been obtained between them. The average percentage deviation of two parameters: bulk modulus and electronic polarizability for which experimental data are known, have also been calculated and found to be better than the earlier correlations.

1 Introduction

During the last few decades, an extensive research work has been done to understand the electronic, elastic and optical properties of rare earth materials because of their high electrical and thermal conductivities. Rare earth ions doped glasses crystallize in rare earth monochalcogenides (REX, RE = La, Ce, Pr, Nd, Sm, Eu, Gd, Tb, Dy, Ho, Er Tm, Yb, Lu; X = S, Se, Te) with rocksalt structure having 4f electrons. The presence of these 4f electrons in compounds are mainly responsible for their potential applications in the areas of glass-making, grinding alloys, composite lasers, electro-optic, electronic, opto-electronic and non-linear optical devices. The magnetic properties of these materials also help scientists to use rare earths in the form of fast light beam in addressing memory system of computers, in magneto-optic modulators, magnetic field activated electronic switches, spintronics and spin filtering devices [1–7]. The natural and free rare earth atoms have the electronic configuration: (Xe) 4fⁿ5d^0,16 s². The divalent rare earths have the outer electronic structure 4f¹⁴5d⁰6 s² in which outer two ‘s’ electrons contribute as valence electrons. In rare earth monochalcogenides, these two electrons fill the valence band derived from the ‘p’ state of the anion. However the trivalent state, the outer electronic structure of rare earth changes from 4f¹⁴5d⁰6 s² to 4f^(14−X)5d^X6 s². In a crystal, rare earths are in ionic form and the removal of 4f electrons let the ion shrink. Hence, the ionic radii of rare earths vary noticeably with their valence: for example Sm²⁺: 1.16 Å, Sm³⁺: 0.98 Å, and Tm²⁺: 1.04 Å, Tm³⁺: 0.87 Å. As a result, there is a variation of the lattice constant of the REX as one moves from one rare earth to the other in rare earth series. The divalent rare earth (R⁺²X) compounds are semiconductors, which have a significantly larger lattice constant than the trivalent (R⁺³X) compounds, which are metals. Most of the rare earth atoms are divalent but generally become trivalent in the metallic state. Butcher et al. [8] have shown that trivalent lanthanum monochalcogenides (LaS, LaSe and LaTe) are superconductors near 1 K. The superconducting transition temperature and electronic specific heat coefficient increase as we move from mono sulfide (LaS) to mono telluride (LaTe), whereas the Debye temperature decreases from LaS to LaTe.

Crystal ionicity of materials is one of key parameters in describing the problems related to elastic constants, heats of formation, bulk modulus, cohesive energy and crystal structure. Various theoretical explanations have been given to calculate the ionicity of semiconducting materials. Phillips and Van Vechten (PV) [9, 10] have proposed dielectric theory for the calculation of ionic and covalent energy gaps, and hence the average energy gap, ionicity and covalency of binary semiconductors. Levine [11] has extended the dielectric theory of PV to ternary and multiband crystals and proposed bond charge model for the calculation of these parameters also considering the effect of d electrons. Verma [12] and Yadav [13] have recently proposed the modified form of PV model for the calculation of ionic energy gap of R⁺²X and R⁺³X monochalcogenides. Charifi et al. [14] have evaluated the elastic parameters C₁₁, C₁₂, C₄₄, B and G of B1(NaCl) and B2(CsCl) structures of REX using full-potential linearized-augmented plane wave (FP-LAPW) scheme in the frame of the generalized gradient approximation (GGA) and effect of pressure on these parameters. Binary rare earths (R⁺²X) crystallize in the face centered cubic NaCl-type structure and show phase transition from B1(NaCl) to B2(CsCl) under high pressure. The theoretical and experimental investigations of high pressure structural behavior of lanthanum monochalcogenides have also been studied by Vaitheeswaran et al. [15]. Jayaraman et al. [16] have calculated bulk modulus of rocksalt type divalent and trivalent REX using effective valence product of the cation and the anion. The luminance and decay analysis of Eu³⁺ and Pr³⁺ ions doped lead telluride glasses for different concentrations in laser applications have been studied by other workers [17, 18]. The author [19, 20] have recently proposed simple relations, based on plasma oscillations theory of solids, for the calculation of bond length, covalent and ionic energy gaps of binary ionic rocksalt crystals. In this paper, we extend our earlier co-relations for the calculation of bond length (d), specific gravity (ρ), homopolar energy gap (E_h), heteropolar energy gap (E_c), average energy gap (E_g), crystal ionicity (f_i), bulk modulus (B), electronic polarizability (α_e) and dielectric constant (ε_∞) of R⁺²X and R⁺³X rare earths. The calculated values of all parameters are compared with the available experimental values of B and α _e in few compounds where the experiments are performed and the reported values of other parameters. Reasonably good agreement has been obtained between them.

2 Calculations

According to Phillips and Van Vechten [9, 10], the average energy gap (E _g ) of A _m B _n binary crystals can be separated into covalent (E _h ) and ionic (E _c ) parts as E ² _g  = E ² _h  + E ² _c . The fraction of ionic (f_i) and covalent (f_c) characteristics of the individual bonds can be defined as $f_{i} = E_{c}^{2} /E_{g}^{2}$ and f _c  = E ² _h /E ² _g . The generalized expression for E _h and E _c for an A _m B _n compound can be written as [9, 10]:

$$E_{h} = 39.74/d^{2.48}$$

(1)

and

$$E_{c} = 14.4b\left[ {\frac{{Z_{A} }}{{r_{o} }} - \frac{n}{m}\frac{{Z_{B} }}{{r_{o} }}} \right]e^{{ - K_{s} r_{o} }}$$

(2)

where d is the bond length, r₀ = d/2, b is the prescreening constant and ${\text{e}}^{{ - {\text{K}}_{\text{s}} {\text{r}}_{0} }}$ the Thomas–Fermi screening factor. The above Eq. (1) shows the inverse relation between E_h and d^2.48. However, Eq. (2), signifies the difference between the screening Coulomb potential of atoms A and B having core charges Z _A and Z _B . These potentials are to be evaluated at the covalent radii ${\text{r}}_{0}$. Only a small part of the electrons are in the bond, the rest screen the ion cores, reducing their charges by Thomas–Fermi screening factor, which affects the chemical trends in a compound. The screening factor, as well as the bond length (d), both are related to the effective number of valence electrons in a compound. The plasmon energy also depends on the effective number of valence electrons in a compound. This shows that there must be a correlation between the physical process which involves the ionic and covalent contributions to the average energy gap (E _g ) and the plasmon energy (ħω _p) of a compound. Based on this, the authors [19, 20] have proposed simple relations for the calculation of E _h and E _c of binary and ternary semiconductors. Recently, Verma [12] and Yadav [13] have modified PV theory and proposed the following relation for the calculation of covalent energy gap of R⁺²X and R⁺³X rare earth chalcogenides:

$$E_{h} = 40.468/d^{2.50}$$

(3)

where the numerator of PV Eq. (2), i.e., 39.74 has been modified by 40.468 and the denominator d ^2.48 by d ^2.50. Further Verma [12] and Yadav [13] have proposed empirical relations for the calculation of E _c , E _g , f _i, B, α _e and ε _∞ for rare earth chalcogenides, based on ionic charges, nearest neighbor distance and plasmon energy (ħω _p ), and the values of ħω _p have been calculated using the well-known relation:

$$\hbar \omega_{p} = 28.8\sqrt {\frac{{{\rm Z}\rho }}{W}}$$

(4)

where Z is the effective number of valence electrons taking part in plasma oscillations, ρ is the specific gravity and W the molecular weight. The experimental values of ρ are still not known for many R⁺²X and R⁺³X rare earths for which Verma [12] and Yadav [13] have calculated the values of ħω _p . Their calculations show that they have used our earlier relations [19] and back fitted the data of the bond length‘d’ in Eq. (5) for the calculation of ħω _p . In this paper, we extend our earlier correlations developed for 3-parameters d, E _h and E _c of NaCl crystals to 8-parameters d, E _h , E _c , ρ, E _g , f _i , B and α _e for R⁺²X and R⁺³X rare earths. The relation proposed for the calculations of nearest neighbor distance (bond length d) for R⁺²X and R⁺³X can be written as:

$$d = C\left( {\hbar \omega_{p} } \right)^{{ - {\raise0.7ex\hbox{$2$} \!\mathord{\left/ {\vphantom {2 3}}\right.\kern-0pt} \!\lower0.7ex\hbox{$3$}}}}$$

(5)

where C is the constant equals to17.669 and 18.369, respectively, for R⁺²X and R⁺³X rare earths.

Using Eqs. (4) and (5), we get the following relation for the calculation of specific gravity (ρ) of rare earth compounds:

$$\rho = \kappa \left( {\frac{W}{{d^{3} }}} \right)$$

(6)

where κ is the constant and equals to 0.83130 and 0.83029, respectively, for R⁺²X and R⁺³X rare earths. The calculated values of ρ are listed in column 2 of Tables 2 and 3, respectively, for R⁺²X and R⁺³X rare earths along with a few available experimental data for which experiments are performed and presented in parenthesis.

Using Eqs. (1) and (5), the covalent energy gap (E _h ) of R⁺²X and R⁺³X rare earths can be expressed as:

$$E_{h} = K\left( {\hbar \omega_{p} } \right)^{1.6533}$$

(7)

where K is the constant and equals to 0.03207 and 0.02913, respectively, for R⁺²X and R⁺³X rare earths.

Further, we propose the following simple relations based on best fit of the data for the calculation of E _c , E _g , f _i , B and α _e for R⁺²X and R⁺³X rare earth monochalcogenides:

$$E_{c} = K_{1} \exp \left[ {K_{2} \left( {\hbar \omega_{p} } \right)} \right]$$

(8)

$$E_{g} = K_{3} + K_{4} \left( {\hbar \omega_{p} } \right)$$

(9)

$$f_{i} = K_{5} - K_{6} \left( {\hbar \omega_{p} } \right)$$

(10)

$$B = - K_{7} + K_{8} \left( {\hbar \omega_{p} } \right) + K_{9} \left( {\hbar \omega_{p} } \right)^{2}$$

(11)

$$\alpha_{e} = K_{10} - K_{11} \left( {\hbar \omega_{p} } \right) + K_{12} \left( {\hbar \omega_{p} } \right)^{2}$$

(12)

where K ₁ to K ₁₂ are the constants and their numerical values are listed in Table 1 for R⁺²X and R⁺³X rare earths. The values of these constants are obtained by simulating the known values of E _c , E _g , f _i , B and α _e and the calculated values of plasmon energy (ħω _p) using MATLAB software. We have also calculated the values of dielectric constant of R⁺²X and R⁺³X compounds using the relation proposed by Penn et al. [21]:

$$\varepsilon_{\infty } = 1 + \left[ {{{\left( {\hbar \omega_{p} } \right)^{2} } \mathord{\left/ {\vphantom {{\left( {\hbar \omega_{p} } \right)^{2} } {E_{g}^{2} }}} \right. \kern-0pt} {E_{g}^{2} }}} \right]$$

(13)

Table 1 Numerical values of the constants used in Eqs. (8) to (12)

3 Results and discussion

Based on plasma oscillations theory of solids, eight simple relations have been proposed for the calculation of various electronic, elastic and optical properties of R⁺²X and R⁺³X rare earth monochalcogenides. Using proposed Eqs. (5) to (12), the values of 8-parameters d, ρ, E _h, E _c , E _g , f _i , B and α _e have been calculated and listed in Tables 2 and 3, respectively, for R⁺²X and R⁺³X rare earths. The values of ε_∞ have also been calculated using Eq. (13) and listed in Tables 2 and 3. The calculated values of d from Eq. (6) and ionic radii data [22] are also listed in column 6 of Tables 2 and 3. The specific gravity (ρ) of 9 divalent and 20 trivalent rare earths have been calculated for the first time using Eq. (5) and listed in column 3 of Tables 2 and 3 along with the available experimental values. Our calculated values are in good agreement with the available experimental values. The average percentage deviations for B and α _e for which the experimental values are known have also been estimated using the relation, Percentage deviation = [(|Experimental values-Calculated values|)/Experimental values] × 100 and presented in the bottom row of Tables 2 and 3. In the case of B, the average percentage deviation of Eq. (11) has been estimated to be 2.318% against the earlier estimation of 5.989% for R⁺²X, and 8.345% for R⁺³X against the earlier estimations of 7.302%. However, in the case of α _e , the average percentage deviation of Eq. (12) has been found to be 3.876% against the earlier estimation of 5.056% for R⁺²X, and 0.375% for R⁺³X against the earlier values of 5.760%. In almost all cases except one, our percentage deviation of B and α _e is less than the earlier estimations. The percentage deviation of other parameters is not calculated due to unavailability of experimental data. However, our calculated values are in good agreement with the reported and known values. The main advantage of the present models is the simplicity of the formulas, which do not require any experimental data except the plasmon energy of the compound while the earlier models require the experimental values of Thomas–Fermi screening factor, ionic charges and bond length in their calculation, which are not known for many compounds. The proposed Eq. (7) for the calculation of E_h is based on PV Eq. (1), which further shows that dielectric theory of PV still holds good for binary crystals and gives better results than the modified models proposed by Verma [12] and Yadav [13].

Table 2 Density (ρ), plasmon energy (ħω _p ), bond length (d), energy gaps (E _h , E _c , E _g ), ionicity (f _i ), bulk modulus (B), electronic polarizability (α _e ) and dielectric constant (ε _∞ ) of divalent(R⁺²X) rare earth monochalcogenides

Table 3 Density (ρ), plasmon energy (ħω _p ), bond length (d), energy gaps (E _h , E _c , E _g ), iconicity (f _i ), bulk modulus (B), electronic polarizability (α _e ) and dielectric constant (ε _∞ ) of trivalent (R⁺³X) rare earth monochalcogenides

4 Conclusions

Thus, one can calculate the values of d, ρ, E _h , E _c , E _g , f _i , B, α _e and ε _∞ of R⁺²X and R⁺³X groups of rare earth monochalcogenides from their plasmon energy data. The predictive nature of proposed equations is of great importance in predicting the values of these parameters for new compounds of these families. The lower percentage deviation shows the significant improvement over the earlier models, which further demonstrate the soundness of the proposed models.