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Quantum chemical study of the structure and properties of isotopically pure lead chalcogenides

In the present work the theoretical methods B3LYP/SDD, GGA and BP86/TZ2P were used for quantum-chemical calculations of lead chalcogenides. It is shown that these levels of theory are applicable for assessment of their geometric parameters, Raman and IR spectra and thermodynamic characteristics. It is shown that there are correlations between the experimental and calculated characteristics of lead sulphide, selenide and telluride. The influence of different isotopes of lead, sulphur, selenium and tellurium on the thermodynamic parameters and the Raman spectra for the lead chalcogenides is shown.


Introduction
Lead chalcogenides compounds of PbS, PbSe, PbTe are narrow band-gap semiconductors used in infrared detectors, thermoelectric devices and photoresistors. In spite of the fact that the electronic and optical properties of lead chalcogenides were studied long ago and are rather well known, in these materials they continue to reveal interesting properties. For example, PbSe semiconductors display unusually high temperatures of continuous emission that are of great interest for engineering infra-red lasers that work at room temperatures [1]. Data from recent research indicates that elements of highsensitivity receivers in the distant infra-red range  microns) based on lead alloy chalcogenides can successfully compete with germanium and silicon [2]. Unlike those elements, lead chalcogenides possess considerable radiation resistance caused by their high density, stabilizing Fermi's level. Moreover lead chalcogenides are interesting as objects of research, namely studying the influence of isotope composition of semi-conductor materials on their physical properties. Recently interesting results in this direction were obtained in papers [3,4].
Several attempts have been made to develop theoretical settlement models using various approaches, in order to accurately reproduce the properties of lead chalcogenides. Previously, geometrical, spectral and thermodynamic characteristics of sulphur and lead were calculated by density functional method using various values of functionals and basic conditions, such as B3LYP (GAUSSIAN), BP86 (ADF), GGA (FLAPW) [5].
The purpose of this work is the analysis of applicability of the theory of a functional of density to an assessment of frequency shifts in Raman spectra and thermodynamic characteristics of chalcogenides with various isotope compositions of lead, selenium and tellurium, and also an elementary cell PbS, PbSе and PbTe.

Results and discussion
To assess the "quality" of the calculations it is necessary to rely on some physico-chemical properties of similar compounds, for which sufficiently accurate experimental values are known. As experimental values, we used the bond lengths of compounds of lead, the frequency of stretching and deformation vibrations in the IR spectra of two-and four-coordinated lead compounds, as well as the band gap of crystalline halides, and the entropy and enthalpy of formation and dissociation energies of compounds of lead in the gas phase (Table 1). From the results of the calculation it follows that the valent angles of Hal-Pb-Hal optimized in the GAUSSIAN program are close to the experimental values [10] (about 98-100 °), and the bond lengths of Pb-Hal (Hal=F, Cl, Br, I, Me) Pb (II) and Pb (IV) compounds have a good correlation ratio ( fig. 1).    In addition, the frequencies calculated by the two methods of IR and Raman spectra of lead halides are close to the experimental values ( Fig. 2, 3) with a coefficient of correlation of 0.998 and a standard deviation of 9 %.
We observed quite reliable linear correlation (1)(2)(3)(4)(5)(6)(7)(8), which showed that the calculations for lead halides by the PM3 and B3LYP/SDD methods correctly describe not only the geometry, but also the thermodynamic parameters such as enthalpies (ΔH) and entropies (S) of formation, and the dissociation energies (D 0 ). The dissociation energy of the lead compounds was determined as the difference between the energy of the compound and the energies of their constituent atoms, as amended by the enthalpy of formation.
The calculated dissociation energy was related to the number of bonds, compared with the experimental average binding energy of lead-halogen [15]. It should be noted that if the entropy falls on the curve for compounds Pb (II) and Pb (IV), then the relationship between the calculated and experimental enthalpies of formation and dissociation energies are observed separately for the coordination of various lead compounds on the basis of the two methods of calculation GAUSSIAN and ADF programs: The values of the energy distance between the occupied and unoccupied molecular orbitals ΔE(B3MO-HCMO) calculated by both methods agree well with the band gap in crystalline chloride, bromide and iodide of lead [17], with a coefficient of correlation 0.999 and a standard deviation of 0.02 eV.
By analogy with the sulphur compounds [5], for selenium and tellurium compounds were used as a "quality assessment" on the base on the bond lengths, frequencies in the vibrational spectra, and experimental values of the band gap of crystalline chalcogenides, as well as the dissociation energy of selenium and tellurium in the gaseous phase (Table 2). On the other hand, the results of the calculations imply that bond lengths optimized by the B3LYP/SDD method for the compounds of selenium and tellurium are proportional to the experimental values [17], and there is a good correlation between them: For selenium compounds R(exp.) = 0.957 R(calc.) r = 0.995; s = 0.04; n = 18 (9) For tellurium compounds R(exp.) = 0.969 R(calc.) r = 0.993; s = 0.04; n = 10 (10)  [18] (Fig. 4, 5): For selenium compounds ω(exp.) = 59 + 0.996 ω(calc.) r = 0.993; s = 57; n = 91 (11) For tellurium compounds ω(exp.) = 21 + 1.07 ω(calc.) r = 0.993; s = 27; n = 67 (12) It should be noted that correlation coefficients close to one of the calculated parameters in (9)(10)(11)(12) indicate the high reliability of the use of the SDD basis set.    We have also obtained quite reliable linear correlation (13 and 14), which suggests that the calculations for compounds of selenium and tellurium by the B3LYP/SDD and BP86/TZ2P+ methods correctly describe not only the geometry, but also the thermodynamic parameters, in particular the dissociation energy (D 0 ). The energy of dissociation of selenium and tellurium compounds was determined as the difference between the energy of the compound and the energies of their constituent atoms, as amended by the enthalpy of formation. The calculated dissociation energy, relative to the number of bonds, was compared with the experimental average binding energy of selenium-selenium and tellurium-tellurium [19] (Table 3).
As a result we obtained a correlation ratio based on the two methods of calculation using the GAUSSIAN and ADF programs (Fig. 6, 7 It should be noted that our calculation using the FLAPW program for crystalline PbS, PbSe and PbTe gave the best value of the lattice constants at 6.01 Å, 6.23 Å and 6.57 Å, close to the experimental values of 5.94 Å, 7.1 Å and 6.75 Å, respectively. One of the most important characteristics of semiconductors is the value of the energy gap. The calculation in the solid phase for PbS, PbSe and PbTe gave values of 0.5, 0.44 and 0.70 eV and in the gas phase 0.38, 0.36 and 0.28 eV using the GAUSSIAN program. The experimental width of a singlecrystal PbS is 0.41 eV [17]. Thus, we can assume that the calculations for the gas phase and crystal lead chalcogenides gave values fairly close to the experimental values for the energy characteristics. Deviation of the calculated energy gap in gas phase from the FLAPW calculation for PbTe apparently is due to the inaccuracy of the parameters for Te by the PM3 method. Based on the calculations in the solid phase (FLAPW) and in the gaseous phase by ADF we can estimate a bond in terms of ionic (electrostatic) and covalent (orbital) components. FLAPW data indicate a mixed ionic-covalent type (Fig. 8). The ionic component is created by the charge polarization in the direction of Pb → S, Pb → Se and Pb → Te in an ideal ion model of the charge states of Pb 2+ Hal 2-. In reality, the effective charges of the atoms are smaller due to the covalent contribution. The covalent constituent is formed by the partially overlapping Halnp -Pb6s, 6p states. The covalent constituent can be illustrated as in Fig. 8, which shows the overlapping of isoelectronic surfaces in the direction Pb-(S, Se, Te). In the ADF program the energy of the molecule is split into several contributions. Energy members can be identified by the three main components of the chemical bond: repulsive Pauli, electrostatic attraction and covalent interactions. Frenking and coworkers [20] suggested that ΔE elstat. could be used to assess the strength of the electrostatic binding and ΔE orb. for covalent binding. These results for the calculation of lead chalcogenides in the gaseous phase also indicate that the chemical bond is composed of ionic and covalent components with a predominance of the latter (65% for S, 79% for Se and 56% in the case of Te).

4.Conclusions
Thus, the results suggest that the use of the density functional method in various versions with different functionals and basic conditions, such as B3LYP (GAUSSIAN), BP86 (ADF), GGA (FLAPW) presents an opportunity to evaluate not only the spectral isotope shifts, but also the thermodynamic parameters of the structures of lead chalcogenides containing different isotopes of lead, selenium and tellurium.