Author's Accepted Manuscript

Thermodynamic parameters of CdTe crystalsin the cubic phase

Dmytro Freik, Taras Parashchuk, Bohdana Vo-lochanska

PII: S0022-0248(14)00339-XDOI: http://dx.doi.org/10.1016/j.jcrysgro.2014.05.005Reference: CRYS22246

To appear in: Journal of Crystal Growth

Received date: 31 March 2014Revised date: 5 May 2014Accepted date: 7 May 2014

Cite this article as: Dmytro Freik, Taras Parashchuk, Bohdana Volochanska,Thermodynamic parameters of CdTe crystals in the cubic phase, Journal ofCrystal Growth, http://dx.doi.org/10.1016/j.jcrysgro.2014.05.005

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Thermodynamic parameters of CdTe crystals in the cubic phase

Dmytro Freik, Taras Parashchuk, Bohdana Volochanska

Physics and Chemistry Institute,  

SHEE “Vasyl Stefanyk Precarpathian National University”, Ivano-Frankivsk, Ukraine  

*Corresponding author: taras-parashchuk@i.ua

 Abstract. Based on the analysis of the crystal and electronic structure of CdTe crystals in the

cubic phase cluster models have been built for calculation of the geometric and  

thermodynamic parameters. According to density functional theory (DFT) and using the  

hybrid B3LYP functional the temperature dependence of formation energy ∆E, formation  

enthalpy ∆H, Gibbs free energy ∆G, entropy ∆S, specific heat capacity at constant volume CV  

and pressure CP have been defined. 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. The results of ab initio calculations compared with experimental date.

Keywords: A1 Crystal structure; A1 Quantum-chemical calculations; A1 Thermodynamic  

properties; B2 Semiconducting II-VI materials  

1. Introduction

Cadmium telluride is a perspective material for production on the base a number of highly efficient

devices of nuclear power, solar technology, optical and acoustic electronics, X-and gamma-ray

detectors, substrates for epitaxial growth [1],[2],[3]. Also, it is used in converters of solar energy, in

cells of TV commutation networks and in a number of other Solid State Electronics devices [2].

Cadmium telluride, with a melting point 1092ºC [4], is the most fusible material of A2B6

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compounds group. It usually crystallizes in the sphalerite structure with а = 6,481 Å [4], d(A-B) =

2,8 Å, d(A-A) = 4,58 Å [5], which is stable to temperatures 1000 К [6]. Thus, each atom A (B) is

located in the center of a regular tetrahedron, the 4 tops of which are the atoms of another

element В(A).

Wurtzite modification in its purest form in bulk crystals are not obtained, but it appears at high

pressures and temperatures [5], and condensates in thin film [3]. Estimated size of the hexagonal

phase equal to: а = 4,57 Å, с = 7,47 Å [4], which is well consistent with the parameters calculated

from the cubic lattice constant [5].

Effective using of cadmium telluride crystals needs the information about the thermodynamic

parameters. We should add, a particularly important factor is the behavior of these parameters with

temperature change, which in the literature are not sufficiently studied, and available results differ

(Table [3],[5],[7],[8]).

In this paper, based on ab initio calculations, using known crystallographic parameters, the new

approaches to the determination of the important thermodynamic parameters of sphalerite CdTe

crystal and their temperature dependences have been proposed. And the experimental studies of

isobaric heat capacity CP have been spending.

2. Calculations of Thermodynamic Parameters

2.1. Models of Clusters

In the CdTe sphalerite structure, the chemical bonding between two atoms Cd-Te, taking into

account the configuration of the valence electrons Cd-5s2, and Te-5p4, carried out by the

participation of three electrons of the chalcogenide atom and one electron of the metal atom.

For calculations of the thermodynamic parameters we proposed two cluster models: A – "small"

(Fig. 1,a) and B – "large" (Fig. 1,b).

At "small" cluster of CdTe sphalerite structure (Cd +4Te) the compensation of dangling bonds was

implemented by four electrons of carbon atoms C and one electron of hydrogen atom H, which

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corresponds to the formula CdC2H2Te4 (fig.1,a).

Except of three structures of previous cluster the model of "large" cluster B (Fig. 1,b) includes Cd

and Te atoms, which corresponds to the formula Cd4C6H6Te13 (fig.1,b).

The saving of geometrical parameters after optimization (finding of minimum of potential energy)

within 1% error defines the rationality of this choice. The application of presented cluster models

allows the calculation of the thermodynamic properties with sufficient accuracy even with using of

small clusters.

2.2. Calculation Methods

With using of the rigid molecule approximation [9] the formation enthalpy H of the crystals is

defined as:

( ) ( ) ( )0elec vib vib rot transH H +H +E T +H T +H T +RT≈ , (1)

where Hеlec – electronic component of enthalpy, Hvib – vibrational component of enthalpy, 0vibH –

enthalpy of zeropoint vibrations, Hrot – rotational component of enthalpy, Htrans – translational

component of enthalpy, R – universal gas constant, T – temperature. Similarly, there was calculated

formation energy ΔE.

The entropy of the crystal, in general, is the sum of the components:

)trans rot vib elec 0ΔS=S +S +S +S nR ln(nN 1]− −⎡⎣ , (2)

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

After calculating contributions of zero point energy and entropy of certain members of the

molecules of reagents A (Cd) and B (Te), we can calculate the Gibbs free energy of the crystal at

the given temperature T.

( )

A B i ji j B

A B A B A Bvibr vibr rot rot trans trans

1 1G H H h h2 2

T S S S S S S .∈Α ∈

Δ = − + ν − ν −

− − + − + −

∑ ∑ (3)

For calculations of ΔE, ΔH, ΔS, ΔG we used the following method taking into account of the initial

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conditions, which is illustrated by the formation energy ΔE. At first we calculated the formation

energy ΔEА of the cluster А (fig.1,a) according [9]:

A el atE E E EΔ = − +∑ ∑ , (4)

where Е - the total energy of the system; Eel- electronic energy of the atoms that formed the system

(in atomic state); Еat - atomization energy of atoms. The total and electronic energy of the system

were taken from the results of the quantum-chemical calculations, and all other values – from the

reference materials [6],[10].

The formation energy ΔEB of В cluster (Fig. 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 [9]:

B AΔE ΔE 3 ΔE= − ⋅ . (5)

Based on the calculated vibrational spectrum, the calculation of the thermodynamic properties of

CdTe crystals at different temperatures have been spent (Fig. 2-4).

2.3. Details of the calculation

The first step for the quantum-chemical calculating of the cluster properties was the

determination of the lowest energy configuration. All calculations started with SCF convergence,

geometry optimization and after obtaining a stable minimum, the frequencies were calculated. The

calculations were carried out using density functional theory, using bases in Stevens-Basch-Krauss-

Jasien-Cundari (SBKJC) [11] parameterization. In this basic set only the valence electrons which

are directly involved in chemical bonding are considered. This basis set was chosen due to our

previous experience with this basis set in several vibrational studies carried out in this

group [12],[13]. DFT calculations were performed with the use of Becke’s three parameter hybrid

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method [14] with the Lee, Yang, and Parr (B3LYP) gradient corrected correlation functional [15]

using the PCGamess program packages [16]. The visualization of spatial structures was carried out

using Chemcraft.

3. Results and discussion

3.1. The thermodynamic parameters

In Fig. 2.3 are presented the change of the formation energy ΔE, formation enthalpy ΔH, Gibbs free

energy ΔG and entropy ΔS of CdTe crystals as a function of the temperature from 20 K to 800 К.

Their analytical expressions can be represented by dependences:

E(T) 0,0608 T 240, 25Δ = ⋅ + ; (6)

H(T) 0,0442 T 240,25Δ = ⋅ + ; (7)

ΔS(T) = 43,743 ln(T) 101,28− ; (8)

G(T) 0,3995 T 246,24Δ = ⋅ + . (9)

The table shows the literature date and our calculated values of geometrical and thermodynamic

parameters at normal conditions. It can be seen that the calculated lattice parameter of the cluster

models is consistent with the known literature data, and we can conclude that the calculation results

of the thermodynamic parameters of cluster models are coordinate to the real crystals values.

From the table we can also see that the obtained thermodynamic parameters correlate with

known literature data. It is evident that there is shown a growth of the presented characteristics

throughout all temperature range (Fig. 2,3). Thus there is a rapid increase of Gibbs free energy ΔG

at high temperatures, which is natural for semiconductor crystals in the cubic phase (Fig. 2).

3.2. Specific heat capacity at constant volume CV and pressure CP

Heat capacity at constant volume CV, using indicated approximation, is determined by the following

formula:

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V V(trans) V(rot ) V(vib)C C C C= + + . (10)

The contributions of the translational degrees of freedom are calculated without data of quantum-

chemical calculations, because their depending of external influences (T, P) 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:

i

i

hc2 2 kT

i iV(vibr ) 2hci

kT

g ehcC RkT

1 e

ν−

ν−

ν⎛ ⎞= ⎜ ⎟⎝ ⎠ ⎡ ⎤

−⎢ ⎥⎣ ⎦

∑ . (11)

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

As for the calculation of the heat capacity of the presented cluster method, from the heat capacity of

larger cluster CV (CP) subtracted triple value of the heat capacity of the smaller cluster. That is, from

CV (CP) of cluster, consisting of a fragment of the crystal CdTe and three ligands subtracted CV (CP)

of three ligands. This value of the molar heat capacity at constant pressure and volume respectively

can be attributed to cadmium telluride crystal.

According to [22] temperature dependence of the heat capacity of the crystal structures determined

by the following function:

-3 5 -2C = a+ b 10 T c 10 T⋅ − ⋅ , (12)

where a, b, c – constant coefficients that depend on the type of crystal structure and of chemical

compounds.

Obtained by us analytical expressions of temperature dependencies of heat capacity at constant

volume CV and at constant pressure CP, which were approximated by quantum-chemical calculation

points using a mathematical package Maple 14, are shown by the following equations:

3 5 2

VC 37,8883 23, 470 10 T 0, 2397 10 T− −= + ⋅ − ⋅ , (13)

3 5 2

PC 54,3336 23, 2460 10 T 0,7116 10 T− −= + ⋅ − ⋅ . (14)

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The received values of heat capacity at constant volume Cv and constant pressure CP at different

temperatures are shown in Fig. 4. It has been performed the comparison these results from

experimental values measured by us.

Synthesis of cadmium telluride was performed in quartz ampoules by fusion of elements cadmium

(CD-0000) and tellurium (TV-4) (according to the certificate, the content of the basic substance not

less than 99.9999% and 99.9997%, respectively) taken in stoichiometric ratios, up to 10-4 g.

Preliminary elements further purified by zone melting. The concentration of background impurities

in the source component does not exceed 10-5 weights. Received CdTe samples periodically

checked by the contents of uncontrolled impurities by atomic absorption analysis, secondary ions,

Auger- and laser mass spectrometry.

Measurements of isobaric heat capacity at the temperature range 13-300 K were carried out on the

KU-300. Temperature ranges (300 - 600 K) have been studied by differential cooling calorimeter

Parkin-Elmer. The operation principle of which is based on a comparison of the heat flow of the

sample and the standard. Calorimetric measurements were carried out on crystalline samples

weighing 12 mg and dimensions of 2x2x4 mm.

Temperature and energy calibration of the calorimeter were carried out in the melt of pure metals

In, Sn, Bi, Al, Cu and with known exact temperature and enthalpy of melting.

Note that at low temperatures, the experimental values of isobaric heat capacity well coincide with

the calculated approximation curve. Values at temperatures above 300 K are low relative to theory.

The latter is due from anharmonic vibrations of real crystal, which it is difficult to take into account

in the theoretical model [23].

Also, you should note that for the calculation of basic thermodynamic functions of the studied

compounds at 298,15 K, smoothed heat capacity values were extrapolated to 0 K using the model

equations that include the phonon contribution to the specific heat as a combination of Debye and

Einstein functions, and as electronic component [24].

It should be noted that the temperature dependences of heat capacity (Fig. 4) are particularly

8 

important in the calculation of the Debye characteristic temperature, which, in turn, provides insight

about the analysis of heat transfer processes of phonons and phonon interactions with each other

and with defects in the crystal structure [23],[25].

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 provide valuable information about changes in

the properties of crystals at high temperatures. The latter, in turn, provides growing of CdTe crystals

with predictable properties and gives the ability to use the crystals effectively throughout the

temperature range.

4. Conclusions

1. Based on the crystal and electronic structure of cubic CdTe and paid attention on their

physical and chemical properties, the cluster models for calculating of the thermodynamic

parameters of cadmium telluride have been proposed. There has been shown method of

consideration of the boundary conditions for cluster models of CdTe crystal at cubic phase.

2. There have been defined temperature dependences of the thermodynamic parameters of the

CdTe crystals: formation energy ΔE, formation enthalpy ΔH, entropy ΔS and Gibbs free

energy ΔG. These results can be used to predict properties of CdTe crystals during

annealing.

3. From the first principles calculations there have been received the analytical expressions for

the temperature dependences of the specific heat capacity of the CdTe crystals in cubic

phases at constant volume CV and and at constant pressure CP.

4. Experimental calorimetric study of the temperature dependence of heat capacity for CdTe

crystals at constant pressure was carried out and compared these results with quantum-

chemical calculation results. Based on the coincidence of theoretical and experimental points

we can conclude about the adequacy of the proposed cluster models.

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Acknowledgement

This research is sponsored by NATO's Public Diplomacy Division in the framework of “Science for

Peace” (NATO.NUKR.SFPP 984536).

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Table

Basic structural and thermodynamic properties of cubic phase CdTe

Parameter Literature references Calculations

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Lattice constant, а, Ǻ

6.379, 6.754, 6.479[17];

6.48[4], [17], [18];

6.48 [19];

6.49 [20];

6,481(exp.); 6,470 [5]

6,46

Distance between oppositely atoms,

Ǻ 2.806(exp.); 2.8649(calc.) [5] 2,76

The enthalpy of formation, -ΔH298,15,

kJ/mol

(286,604 ± 2,511) [3];

214,221[5];

282,42 [8]

237,4

Gibbs energy, ΔG298,15, J/mol·K -287,190+188,0290 T [21] 360,6

Entropy S298,15 ́, J/mol·K 191,63[3]; 185,07 [8] 155,2

Heat capacity, CV,298,15, J/mol·K 38.9 [7] 44,4

Fig. 1. Model of clusters А (CdC2H2Te4) (а) and В (Cd4C6H6Te13) (b) respectively of CdTe cubic

phase.

Fig. 2. Temperature dependences of the energy ΔЕ and the enthalpy ΔH of formation, Gibbs free

energy ΔG for CdTe sphalerite crystals.

Fig. 3. Temperature dependence of entropy ΔS for CdTe sphalerite crystals.

Fig. 4. Temperature dependence of the isochoric СV - □ and isobaric CP - ◊ heat capacities and

experimental values СP - Δ of CdTe crystals.

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а)

b)

Fig. 1 (а,b)

14 

Fig. 2

 

Fig. 3.

Fig. 4.

15 

We constructed cluster model for CdTe.

Using DFT we have determined the thermodynamic parameters.

We obtained the experimental temperature dependence of the specific heat capacity.

Experimental and theoretical results are consistent.