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

**T.O. Parashchuk**^{1,}, **N.D. Freik**^{2}, **P.M. Fochuk**^{3}

^{1}Department of Physics and Chemistry of Solid State

^{2}Physics and Chemistry Institute, SHEE “Vasyl Stefanik Precarpathian National University”, Ivano-Frankivsk, Ukraine

^{3}Department of Inorganic Chemistry, Yuriy Fedkovych Chernivtsi National University, Chernivtsi, Ukraine

### 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 C_{V} 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.

### At a glance: Figures

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

*Physics and Materials Chemistry*, 2014 2 (1),
pp 14-19.

DOI: 10.12691/pmc-2-1-3

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

**Copyright**© 2013 Science and Education Publishing. All Rights Reserved.

### Cite this article:

- Parashchuk, T.O., N.D. Freik, and P.M. Fochuk. "DFT-Calculations of Thermodynamic Parameters of ZnTe, ZnSe, ZnS Crystals."
*Physics and Materials Chemistry*2.1 (2014): 14-19.

- Parashchuk, T. , Freik, N. , & Fochuk, P. (2014). DFT-Calculations of Thermodynamic Parameters of ZnTe, ZnSe, ZnS Crystals.
*Physics and Materials Chemistry*,*2*(1), 14-19.

- Parashchuk, T.O., N.D. Freik, and P.M. Fochuk. "DFT-Calculations of Thermodynamic Parameters of ZnTe, ZnSe, ZnS Crystals."
*Physics and Materials Chemistry*2, no. 1 (2014): 14-19.

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### 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 T_{T} = 1250 K - 1450 K, and for ZnSe at T_{T} = 1420 – 1713 K. We were unable to find the appropriate information for ZnTe, but at our paper ^{[5]} we have gotten the result T_{T}_{ }= 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 C_{V}, entropy ∆S by neglecting of the anharmonicity of fluctuations and some other effects.

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

(1) |

where H_{еlec} – electronic part of enthalpy, H_{vib} – vibrational part of enthalpy, - enthalpy of ground state point, H_{rot} – rotational part of enthalpy, H_{trans} – 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:

(2) |

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

(3) |

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

The enthalpy of the ground state point:

(4) |

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

Vibrational contribution to the enthalpy (depends on T):

(5) |

Entropy is defined as follows:

(6) |

where N_{0} – the Avogadro constant, n – number of moles of molecules, М – mass of the molecule.

The translational entropy component S_{trans} is:

(7) |

Rotary S_{rot} and fluctuating S_{vib} 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:

(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:

(9) |

where g_{i} – 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.

(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-5d^{10}4s^{2}, S-5s^{2}5p^{4}. 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 ΔE_{А} of A cluster have been calculated (Figure 1,a) by ^{[7]}:

(11) |

where ∆Е – formation energy; Е - the total energy of the system; E_{el }- electronic energy of the atoms that make up the system (in atomic state); ∆Е_{at} - 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 (ZnC

_{2}H

_{2}S

_{4}) (а) and В (Zn

_{4}C

_{6}H

_{6}S

_{13}) (b) according to the cubic phase β-ZnTe

The formation energy ΔE_{B} 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]}:

(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 C_{V} of cluster, consisting of a fragment of the ZnX crystal and three ligands was subtracted C_{V} 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:

(13) |

(14) |

(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:

(16) |

(17) |

(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:

(19) |

(20) |

(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:

(22) |

(23) |

(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]}:

(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 C**

_{V}According to ^{[17]} temperature dependences of specific heat of crystalline structures are defined by the following functions:

(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:

(27) |

(28) |

(29) |

The obtained values of heat capacity at constant volume C_{v} 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 С_{v}, 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 C_{v}.

### Acknowledgement

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

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