Dear Label_Name,

Greetings! I hope you are well.

I am writing on behalf of an international publishing house, Science and Education Publishing, USA (http://www.sciepub.com).

Science and Education Publishing (SciEP) is a multidisciplinary open access publisher of Journals covering the fields of Science, Technology and Medicine. All SciEP publications are subject to high-quality peer review, editorial and production processes and are freely available online.

Welcome to the April, 2013 edition of SciEP newsletter, providing monthly summaries of articles on science & education related topics of interest to scientist and professionals. The mission of our journals is to provide an open access forum for publishing new findings on issues pertinent to scientific and academic field. We also invite you to submit original research papers, review articles, short reports, or case studies are considered for publication in our Multidisciplinary Journals.

American Journal of Public Health Research
DOI: 10.12691/ajphr-1-1-3

The Implementation of a Tele-Homecare System with Service Oriented Architecture and HL7 Message Transmission Standard
Ching-Sung Wang

American Journal of Medical and Biological Research
DOI: 10.12691/ajmbr-1-1-2

The Influence of Ethanol on Pyruvate Kinases Activity in Vivo, in Vitro, in Silico
Lelevich Sergey Vladimirovich, Khrustalev Vladislav Victorovich, Barkovsky Eugene Victorovich, Shedogubova Tatyana Aleksandrovna

American Journal of Medical Sciences and Medicine
DOI: 10.12691/ajmsm-1-1-1

Clinical and Histological Features in Wegener Granulomatosis
Li Cavoli Gioacchino, Passantino Rita, Bono Luisa, Tortorici Calogera, Ferrantelli Angelo, Giammarresi Carlo, Rotolo Ugo

American Journal of Infectious Diseases and Microbiology
DOI: 10.12691/ajidm-1-1-2

The Curse of Lymphatic Filariasis: Would the Continual Use of Diethylcarbamazine Eliminate this Scourge in Papua New Guinea?
Anne Lanham, Lillian Mwanri

We have released issues of some journals on February 2013.
American Journal of Public Health Research
American Journal of Medical Sciences and Medicine
American Journal of Medical and Biological Research
American Journal of Pharmacological Sciences
American Journal of Clinical Medicine Research
American Journal of Nursing Research
American Journal of Cancer Prevention
American Journal of Medicine Studies
American Journal of Infectious Diseases and Microbiology
American Journal of Cardiovascular Disease Research
American Journal of Biomedical Research
American Journal of Mechanical Engineering
American Journal of Microbiological Research
Journal of Preventive Medicine
American Journal of Food and Nutrition
American Journal of Hypertension Research
American Journal of Sports Science and Medicine
Journal of Cancer Research and Treatment
American Journal of Zoological Research
Journal of Biomedical Engineering and Technology
American Journal of Microbiological Research
Biomedical Science and Engineering

We are pleased to announce that submissions are now being accepted for peer review for all the journals across all our subject areas in SciEP.

Please submit your manuscript via our online submission system:
http://www.sciepub.com/author/signin

To unsubscribe from our mailing list, please click on the following link: unsubscribe

I am looking forward to hearing from you.

Kind regards,

David Rain
Email:join@sciepub.com
Science and Education Publishing (http://www.sciepub.com)

DFT-Calculations of Thermodynamic Parameters of ZnTe, ZnSe, ZnS Crystals

T.O. Parashchuk*, N.D. Freik, P.M. Fochuk

Department of Physics and Chemistry of Solid State

Abstract: Based on the analysis of the crystal and electronic structure of semiconductors ZnX (X=Te,Se,S) in cubic phase there have been built the cluster model for calculation of the geometric and thermodynamic parameters. The method of consideration of the boundary conditions for the proposed cluster models has been presented. Based on the results of ab initio quantum-chemical calculations of the crystal structure of molecular clusters the temperature dependence of formation energy ∆E, formation enthalpy ∆H, Gibbs free energy ∆G, entropy ∆S, specific heat capacity at constant volume CV have been defined. Computer calculations of the thermodynamic parameters were carried out with the help of density functional theory (DFT), using hybrid valence base set B3LYP. Also, in the work have been derived analytical expressions of temperature dependences of the presented thermodynamic parameters, which have been approximated by a quantum-chemical calculation data using mathematical package Maple 14.

Keywords: DFT, cluster model, quantum chemistry, Zinc chalcogenide, thermodynamic parameters

Received January 23, 2014; Revised February 14, 2014; Accepted February 16, 2014

Citation: T.O. Parashchuk, N.D. Freik, P.M. Fochuk, DFT-Calculations of Thermodynamic Parameters of ZnTe, ZnSe, ZnS Crystals. Physics and Materials Chemistry, 2 (1). 14-19. January 2014. doi:10.12691/pmc-2-1-3

1. Introduction

Zinc chalcogenides ZnX (X=Te,Se,S), as semiconductor materials are studied more intensely because of the polymorphic phase transitions and application for optoelectronic devices [1,2]. In particular, they can be used in semiconductor and quantum electronics - solar panels and detectors of Х- and γ- radiation, lasers, infrared receivers [1,3,4].

With its stable properties at large power and high temperatures, semiconductor crystals, nanocrystals and films of ZnS, ZnSe, ZnTe, as pure, as with the content of impurities are in high technological interest [2].

Presented semiconductors are crystallized in such way that each atom of Zn(X) (X = S, Se, Te) is located in the center of a regular tetrahedron; the 4 vertices are the atoms of another element X(Zn). These tetrahedrons form 2 types of structures: sphalerite and wurtzite. Low-temperature modication ZnX (sphalerite) refers to the cubic system, space group F43m. High-temperature modication 2H ZnX (wurtzite) refers to the hexagonal crystal system, space group P6mc. Sphalerite's structure is more stable than wurtzite's structure below the transition temperature, while wurtzite's structure is much more stable above the transition temperature [2].

The values of phase transition temperatures and thermodynamic parameters of ZnX crystals, which are known in the literature, differ substantially. According to [5], the polymorphic transition "sphalerite-wurtzite" for ZnS occurs at TT = 1250 K - 1450 K, and for ZnSe at TT = 1420 – 1713 K. We were unable to find the appropriate information for ZnTe, but at our paper [5] we have gotten the result TT = 1382 K for these crystals. As for the values of enthalpy ΔH, Gibbs free energy ΔG and entropy ΔS, they are mainly defined only for normal conditions and with significant differences (Table 1) [1,3,4,11,15,18].

Also, there is a well-known fact that the properties of group of ZnX crystals should have a clear dependence on the atomic number of non-metals, what grows with increasing of interatomic distance [2]. That, in turn, leads to a decrease in the strength of chemical bonds. As a result there should be viewed monotonic decrease of the formation enthalpy ΔH, formation energy ΔE and Gibbs free energy ΔG in the transition of ZnS→ZnSe→ZnTe. The same is true for the heat capacities of ZnX crystals (Table) [4,12,17].

As can be seen from literature source, the experimental results of the study of thermodynamic parameters of zinc chalcogenides crystals and especially their temperature dependences are insufficient, and existing calculation theories give only a qualitative picture. That’s why the development of alternative methods of investigation, which would have provided a new and add the already existing information on the mechanisms of thermal processes in these crystals is actual. Computer methods of quantum chemistry are visible and effective methods in solving of presented problems.

In this paper, based on ab initio calculations, new approaches to the determination of the important thermodynamic parameters of crystals ZnS, ZnSe, ZnTe, and their temperature dependences have been proposed.

2. Method of Calculation

2.1. Elements of Theory

At rigid molecules approximation (limits of internal rotation and inversion significantly exceed kT) there can be determined a number of thermodynamic parameters: formation energy ∆E and formation enthalpy ∆H, Gibbs free energy ∆G, specific heat capacity at constant volume CV, entropy ∆S by neglecting of the anharmonicity of fluctuations and some other effects.

The formation enthalpy ΔH of crystals is defined as follows [7]:

[picture] | (1)

where Hеlec – electronic part of enthalpy, Hvib – vibrational part of enthalpy, - enthalpy of ground state point, Hrot – rotational part of enthalpy, Htrans – translational part of enthalpy, R – universal gas constant, T – temperature. The formation energy ΔE has been calculated similarly.

Rotary and translational components of enthalpy are presented as follows:

[picture] | (2)

where N – the total number of atoms, k – Boltzmann constant.

[picture] | (3)

Fluctuations enthalpy are a function of vibrational frequencies of the molecules.

The enthalpy of the ground state point:

[picture] | (4)

where h – Planck's constant, - i-th fluctuations frequency.

Vibrational contribution to the enthalpy (depends on T):

[picture] | (5)

Entropy is defined as follows:

[picture] | (6)

where N₀ – the Avogadro constant, n – number of moles of molecules, М – mass of the molecule.

The translational entropy component Strans is:

[picture] | (7)

Rotary Srot and fluctuating Svib components of the entropy are calculated similarly to the components of the enthalpy.

Heat capacity at constant volume with account of previous approximations is determined by the following formula:

[picture] | (8)

The contributions of translational degrees of freedom are calculated without data of quantum-chemical calculations, as dependent on external influences (Т, Р) and molecular weight m.

In the harmonic approximation, according to which the symmetric displacement with respect to the equilibrium point of nuclei leads to the symmetric change of potential energy, the contribution of the vibrational component is given by:

[picture] | (9)

where gi – a degeneracy level of i-th oscillation.

Calculating the contributions of ground state point energy and entropy of individual members of molecular reagents А and products В gives the opportunity to calculate the Gibbs free energy of the crystal at any temperature Т.

[picture] | (10)

Presented ideology of calculation is successfully used in quantum-chemical calculation programs [6].

2.2. Cluster’s Models

Let’s consider the nature of Zn-Te bond (the cluster models of ZnSe and ZnS crystals have been created similarly), taking into account the configuration of the valence electrons of the constituent atoms: Zn-5d¹⁰4s², S-5s²5p⁴. That is, the system Zn-Te has two valence electrons of the metal and four valence electrons of the chalcogen. In the cubic phase of ZnX each atom of the metal (chalcogen) has four neighboring atoms of the chalcogen (metal), from which it follows that two bonds Zn-Te involve three electrons of the chalcogen atom and one electron of the metal atom (two electrons on each bond).

Boundary conditions of clusters were formed on base of the following considerations. Two boundary Tellurium atoms are belonged three electrons in two bonds. That is, there are five electrons in two bonds that are not compensated. To compensate these electrons in the cluster we added Carbon atoms (С), which take four electrons from chalcogen and one uncompensated electron from Hydrogen atom (H). That means, there are five uncompensated electrons on two Tellurium atoms, which are compensated by five electrons of Carbon and Hydrogen atoms.

The saving of geometrical parameters after optimization within 1% error defines the rationality of this choice. The application of presented cluster models allows the calculations of the thermodynamic properties with sufficient accuracy even with using of small clusters.

2.3. Computational Details

The calculation of thermodynamic parameters have been spent with using the software package Firefly (PCGamess) within the density functional theory method (DFT), using hybrid valence electrons basic set B3LYP [5]. Visualization of spatial structures was carried out using Chemcraft.

In calculating of the ΔE, ΔH, ΔS, ΔG there was used the following methods of consideration of the initial conditions, which are illustrated by the example of the formation energy ΔE. Firstly, the formation energy ΔEА of A cluster have been calculated (Figure 1,a) by [7]:

[picture] | (11)

where ∆Е – formation energy; Е - the total energy of the system; Eel - electronic energy of the atoms that make up the system (in atomic state); ∆Еat - atomization energy of atoms. The total energy and electronic systems were taken from the results of the calculation, and all other values – from the reference materials [8,9,10,11].

Figure 1. Clusters model: А (ZnC₂H₂S₄) (а) and В (Zn₄C₆H₆S₁₃) (b) according to the cubic phase β-ZnTe

The formation energy ΔEB of В cluster (Figure 1,b) was calculated similarly. So, from the formation energy of В cluster there are subtracted the triple value of the formation energy of А cluster, that mean, from the formation energy of the cluster consisting of sphalerite crystal fragment and three ligands are subtracted the formation energy of three ligands. This value can be related to real crystal [7]:

[picture] | (12)

Based on the calculated vibrational spectrum, the calculation of the thermodynamic properties of ZnХ crystals at different temperatures have been spent (Figure 2-Figure 6).

In the case of heat capacity, from the value of heat capacity of larger cluster are subtracted the value of the triple heat capacity of smaller cluster. That is, from CV of cluster, consisting of a fragment of the ZnX crystal and three ligands was subtracted CV of three ligands. This value of the molar heat capacity at constant volume can be attributed to of Zinc chalcogenide crystals.

3. Results and Discussion

3.1. The Formation Energy ∆E, Formation Enthalpy ∆H, Gibbs Free Energy ∆G

In Figure 2-Figure 5 there are presented the formation energy ΔE, the formation enthalpy ΔH, Gibbs free energy ΔG and entropy ΔS of ZnX crystals at temperatures from 100 K to 1000 К.

Figure 2. The temperature dependences of the formation energy ΔE for ZnX sphalerite crystals

Analytical expressions of dependence between formation energy ΔЕ and temperature Т, which have been extrapolated from the results of quantum-chemical calculations for ZnX crystals are presented by the following expressions respectively:

[picture] | (13)

[picture] | (14)

[picture] | (15)

Figure 3. Temperature dependences of the formation enthalpy ΔH for ZnX sphalerite crystals

Analytical expressions of temperature dependence of the formation enthalpy ΔH for ZnTe, ZnSe, ZnS crystals are as follows respectively:

[picture] | (16)

[picture] | (17)

[picture] | (18)

Figure 4. Temperature dependences of Gibbs free energy ΔG for ZnTe, ZnSe, ZnS sphalerite crystals

The expressions of temperature dependences of Gibbs free energy ΔG for ZnTe, ZnSe, ZnS crystals, are as follows respectively:

[picture] | (19)

[picture] | (20)

[picture] | (21)

Figure 5. The temperature dependences of entropy ΔS for ZnX sphalerite crystals

Analytical expressions for temperature dependence of entropy ΔS, which have been extrapolated from the results of quantum-chemical calculations for ZnTe, ZnSe, ZnS crystals are represented by the following expressions respectively:

[picture] | (22)

[picture] | (23)

[picture] | (24)

It can be found that with temperature increasing the calculated parameters grow for all temperature range. Thus, there is a rapid grow in Gibbs energy ΔG at the temperature increasing, what is natural for semiconductor crystal of cubic phase (Figure 2-Figure 5). From chemical thermodynamics we know that the formation enthalpy ΔH relates with the Gibbs free energy ΔG by the following equation [17]:

[picture] | (25)

In addition, at low temperatures, the determining factor is the first term (), and at the higher temperature - second term. The result of calculations which we have gotten confirm these approaches (Figure 2-Figure 5).

The table shows the known literature data and our calculated values of the thermodynamic parameters of ZnX crystal, which are in good agreement.

Table 1. Basic structural and thermodynamic properties of ZnX crystals in the cubic phase at normal conditions

3.2. Heat capacity CV

According to [17] temperature dependences of specific heat of crystalline structures are defined by the following functions:

[picture] | (26)

where a, b, c – constant coefficients, which depend on the type of crystal lattice and chemical compounds.

Obtained by us analytical expressions of temperature dependencies of heat capacity at constant volume, which were approximated by quantum-chemical calculation points using a mathematical package Maple 14, are shown by the following equations:

[picture] | (27)

[picture] | (28)

[picture] | (29)

The obtained values of heat capacity at constant volume Cv for different temperatures are shown on Figure 1. It has been performed the comparison of these results with the experimental values given in [12]. Note, that at low temperatures the discrepancy between the experimental results [12] and approximating curves (Figure 6) calculated by us for Сv, is less than 1 %. However, for sufficiently high temperatures, the experimental values of isochoric heat capacity are higher than its theoretical values. This is due to the anharmonicity fluctuations of the real crystal, which cannot be included into account of any theoretical model [13].

Figure 6. Isochoric molar heat capacity of ZnTe, ZnSe та ZnS crystals respectively: lines - approximation of analytic functions; point - a quantum-chemical calculation; dotted line - experiment ZnTe [12]

It should be noted that the presented expressions of the heat capacity temperature dependences (Figure 6) are particularly important for the calculation of the Debye characteristic temperature. This, in turn, provides insight into the analysis of heat transfer processes and phonon interactions with each other and with defects in the crystal structure [13].

Note, that in order to calculate basic thermodynamic functions for the studied compounds at 298,15 К, smoothed heat capacity values were extrapolated to 0 К using the model equations, which include the phonon contribution to the specific heat capacity as a combination of Debye and Einstein functions and electronic component [8]. By changing of the equation parameters, we minimized the standard deviation of the heat capacity values with their smoothed values.

It should also be noted that our results of calculation of the formation energy ΔE, formation enthalpy ΔH, entropy ΔS and Gibbs free energy ΔG provides valuable information about changes in the properties of crystals at high temperatures. The latter, in turn, let us predictably use ZnX crystals at presented temperature range.

4. Conclusion

1. Based on the crystal and electronic structure of cubic ZnX and paid attention on their physical and chemical properties the cluster models for calculating of the thermodynamic parameters of zinc chalcogenides have been proposed. There has been shown method of consideration of the boundary conditions for cluster models of ZnX crystal at cubic phase.

2. There have been defined temperature dependences of the thermodynamic parameters of the ZnX crystal: formation energy ΔE, formation enthalpy ΔH, entropy ΔS and Gibbs free energy ΔG.

3. It is shown that these results can be used for prediction of the properties for ZnX crystals during annealing.

4. From the first principles calculations there have been received the analytical expressions for the temperature dependences of the specific heat capacity of the ZnX crystals in cubic phases at constant volume Cv.

Acknowledgement

Work performed as the part of state budget №01074006768 of Ministry of Education and Science of Ukraine.

References

[1]   | Tan Jia-Jin, Ji Guang-Fu, Chen Xiang-Rong, Gou Qing-Quan, “Phase Transition and Phonon Spectrum of Zinc-Blende Structure ZnX (X=S, Se,Te)” Commun. Theor. Phys., 53, 1160, 2010.
[2]   | Gangadharan, R., Jayalakshmi, V., Kalaiselvi, J., Mohan, S., Murugan, R., Palanivel, B. “Electronic and structural properties of zinc chalcogenides ZnX (X=S, Se, Te)”, Journ. of Alloys Compounds, 359, 22-26, 2003.
[3]   | Qteish, A., Munoz. A. “Stability and structural properties of ZnS and ZnSe under high pressure”, Physica Status Solidi (b), 223, 417-423, 2001.
[4]   | Malloci, G., Chiodo, L. Rubio, A., Mattoni, A. “Structural and Optoelectronic Properties of Unsaturated ZnO and ZnS Nanoclusters”. Journ. of Phys.Chem. 116, 8741-8746, 2012.
[5]   | Ahiska,R., Freik, D., Parashchuk, T., Gorichok I. Quantum chemical calculations of the polymorphic phase transition temperatures of ZnS, ZnSe, and ZnTe crystals. Turkish Journ. of Physics. (In print. Accepted: 25.10.2013).
[6]   | Granovsky, PC GAMESS version 7.0, http://classic.chem.msu.su/gran/gamess/index.htmlJ.
[7]   | David C. Young. Computational Chemistry: A Practical Guide for Applying Techniques to Real-World Problems. John Wiley & Sons, Inc., New York 2001.
[8]   | Hu Cui-E. Zeng Zhao-Yi, Cheng Yan, Chen Xiang-Rong, and Cai Ling-Cang. “First-principles calculations for electronic, optical and thermodynamic properties of ZnS”. Chiness Physics B, 17(10), 3867-3874, 2008.
[9]   | E.C. Hu, L.L. Sun, Y.Z. Zeng, R.X. Chen, “Pressure temperature induced phase transition of ZnS from firsr-principles calculations, Chinese Physics Letters”, 25, 675-678. 2008.
[10]   | Wang, S. Q. First-principles study of the anisotropic thermal expansion of wurtzite ZnS. Applied Physics Letters, 88(6), 061902-061902-3, 2006.
[11]   | William M Haynes, “CRC handbook of chemistry and physics : a ready-reference book of chemical and physical data”, Boca Raton, Fla. : CRC Press 2010.
[12]   | Gavrichev, K.S., Sharpataya, G.A., Guskov, V.N., Greenberg, J.H., Feltgen, T., Fiederle, M., Benz, K.W. “High-temperature heat capacity and thermodynamic functions of zinc telluride” Elsevier. Thermochimica Acta, 381, 133, 2002.
[13]   | Gorbachuk, N.P., Bolgar, A.S., Sidorko, V.R., Goncharuk, L. V. “Heat Capacity and Enthalpy of Bi₂Si₃ and Bi₂Te₃ in the Temperature Range 58-1012 K” Powder Metallurgy and Metal Ceramics 43, 5-6, 284-290, 2004.
[14]   | Brooks, L.S. “The Vapor Pressures of Tellurium and Selenium”, J. Am. Chem. Soc. 74, 227, 1952.
[15]   | Christensen, N. E., Christensen, O. B. “Electronic structure of ZnTe and CdTe under pressure” Phys. Rev. B. 33, 4739, 1986.
[16]   | Nelmes, R.J., McMahon, M.I., Wright, N.G., Allan, D.R. “Structural studies of II–VI semiconductors at high pressure” J. Phys. Chem. Solids, 56, 1995.
[17]   | Abrikosov, N.H. Semiconductor compounds, their preparation and properties, Nauka, Moskov, 1970.
[18]   | C.S. Dinesh, S. Subodh, Y.K. Vijay, Y.K. Sharma, “Preparation and Optical Properties of ZnTe/ZnTe:Cr Bilayer Thin Films”, Intern. Journ. of Recent Research and Review, II,16-20, 2012.