Density functional theory study on the structural properties and energetics of ZnmTen microclusters

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Physica E 40 (2008) 2921–2930 www.elsevier.com/locate/physe

Density functional theory study on the structural properties and energetics of Znm Ten microclusters Rengin Peko¨z, S- akir Erkoc-  Department of Physics, Middle East Technical University, 06531 Ankara, Turkey Received 6 February 2008; received in revised form 13 February 2008; accepted 13 February 2008 Available online 4 March 2008

Abstract Density functional theory calculations with B3LYP exchange-correlation functional using CEP-121G basis set have been carried out in order to elucidate the structural properties and energetics of neutral zinc telluride clusters, Znm Ten (m þ np6), in their ground states. The geometric structures, binding energies, vibrational frequencies and infrared intensities, Mulliken charges on atoms, HOMO and LUMO energies, the most possible dissociation channels and their corresponding energies for the clusters have been considered. r 2008 Elsevier B.V. All rights reserved. PACS: 36.40.Mr; 36.40.Qv; 61.46.Bc; 71.15.Mb; 73.22.f Keywords: ZnTe clusters; DFT method

1. Introduction Zinc telluride (ZnTe) is an important semiconductor material because of its low cost, large band gap (2.26 eV) and promising applications in optoelectronic devices [1]. Moreover, it is found in 1997 by Winnewisser et al. [2] that ZnTe thin films could be used in optoelectronic detection of THz radiation. In the literature, there have been many theoretical and experimental studies on homonuclear zinc and tellurium clusters. However, up to our knowledge, there is no study on heteronuclear ZnTe clusters. Schautz et al. [3] applied quantum diffusion Monte Carlo (QDMC) method with large-core pseudopotentials for Zn2 and other dimers. Flad et al. [4] investigated the bonding of Znm (m ¼ 226) with an ab initio method using a coupled-cluster with an effective core potential basis (CCSD(T)/ECP). Katırcıog˘lu and Erkoc- [5] investigated the structural and electronic properties of the ground state of the small Znm Sn (m þ np4) clusters with DFT calculations at B3LYP level Corresponding author. Tel.: +90 312 210 32 85; fax: +90 312 210 50 99. E-mail address: [email protected] (S- . Erkoc- ).

1386-9477/$ - see front matter r 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.physe.2008.02.010

with CEP-4G basis set. Znm Cdn (m þ np3) microclusters were investigated by performing DFT calculations at B3LYP level with CEP-121G basis set in Ref. [6]. Michaelian et al. [7] investigated the global minimum structures of isomers of Zn clusters with magic numbers (13, 38, 55, 75, 147). Wang et al. [8] studied the structural and electronic properties of zinc clusters (up to 20 atoms) by using DFT method with generalized gradient approximation (GGA). Stiles and Miller [9] performed experimental (high resolution infrared laser spectroscopy) and ab initio calculations at MP2 and CCSD(T) levels to identify the structural determinations of zinc atoms included in clusters. Iokibe et al. [10] used ab initio/MP2 method with LANL2DZ basis set and DFT/LSDA method with functionals B3LYP and PW91 to obtain the stable structures of zinc clusters up to 32 atoms. One of the experimental studies was performed by Czajkowski et al. [11] in 1990 to observe the first excitation spectrum of zinc dimer. On the other hand, Balasubramanian and Dai [12] computed the several electronic states of Te3 cluster by employing ab initio complete active space multiconfiguration self-consistent field (CASMC SCF) calculations. The geometries, vibrational frequencies and energies of Te3 clusters were investigated by Goddard et al. [13] using pure

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and hybrid DFT methods at DZP or larger basis sets and they concluded that isosceles and equilateral triangle structures were almost degenerate. Pan [14] investigated the structural and electronic properties of Te clusters up to eight atoms performing DFT calculations with GGA at different levels (BP, BLYP, and PW91) using STO-3x basis set. Hassanzadeh et al. [15] performed an experiment on adsorption spectra on Ten (n ¼ 224) clusters and observed the frequencies and low-energy structures. In the present study, neutral and isolated Znm Ten (m þ np6) clusters as well as Znm and Ten clusters have been investigated in order to obtain the structural properties and energetics in their ground states. The emphasis has been given to the heteronuclear ZnTe microclusters. The optimizations and the vibrational frequency calculations have been performed by applying DFT method within B3LYP exchange-correlation functional using CEP-121G basis set. The clusters having positive vibrational frequencies are considered in the calculations because positive values mean that the clusters are found to be on the local minima in the relevant potential energy surface, while negative vibrational frequencies indicate that it is a transition state. This paper is organized as follows: In Section 2, a brief explanation of the calculational method is given. In Section 3, the obtained results have been presented and discussed, and a brief comparison have been made with available literature data. Some concluding remarks are pointed out in Section 4. 2. Method of calculation In the present study, density functional theory [16] calculations at B3LYP [17] level with compact effective potential (CEP-121G) [18,19] basis set have been used in order to investigate the ground state structural properties and energetics of isolated neutral Znm Ten (m þ np6) clusters. All the calculations have been performed using Gaussian 03 [20] package program. Both the homonuclear (Znm and Ten ) and heteronuclear (Znm Ten ) clusters have been considered. The exchange term of B3LYP consists of Becke’s threeparameter functional and the correlation term consists of VWN3 (Vosko, Wilk, Nusair) local correlation functional [21] and LYP (Lee, Yang, Parr) correction functional [22]. CEP-121G basis set is extensively used in the study of compounds containing heavy elements [18,19]. The basis sets of CEP theory are consistent within the lanthanide series, second and third row metals. Since the number of electrons in the studied systems is even, the optimizations are performed at the restricted Hartree–Fock (RHF) level. The requested convergence on the energy was 106 a.u. and the root-mean-square (RMS) density matrix was 108 a.u. in order to obtain satisfactory results on the optimizations and the frequency calculations. 3. Results and discussions In the literature, homonuclear zinc [3–11] and tellurium [12–15] clusters have been studied, however, up to our

knowledge, their combination has not been investigated yet. In the following sections, the current calculation results will be presented and a brief comparison will be made with the other calculation results for the homonuclear Zn and Te microclusters. 3.1. Geometric structures and energetics 3.1.1. Homonuclear Zn clusters Zn2: The present calculations predicted the binding energy and the bond distance of the ground state of zinc ˚ respectively (2 in Fig. 1). dimer to be 0.013 eV and 3:56 A, This very small binding energy indicates that van der Waals interaction is dominant on the bounding properties. The present results along with other predictions are shown in Table 1. Other studies have found binding energies lower than 0.03 eV but still higher than the present prediction of 0.013 eV. Although relativistic effects are included in CEP121G basis set, it takes into account only the valance electrons, which relatively simplifies the calculations from computation point of view with respect to GGA type methods. The small difference in binding energy between CEP-121G and GGA methods could be due to the exclusion of core electrons in CEP-121G basis. On the other hand, other methods yield bond lengths larger than the present result, the only exception being Ref. [8]. The vibrational frequency (33 cm1 ) calculated in this work is closer to the experimental one (25:7 cm1 ). A more detailed information on Zn dimer can be found in Ref. [23] and the various results are given in Table 1. Zn3 : The global minimum structure is found to be an equilateral triangle with D3h symmetry by the present method and by the other calculations. The B3LYP/CEP121G method calculated the bond length and the vibra˚ and 53:1 cm1 tional frequency as, respectively, 3:21 A (3 in Fig. 1). The present vibrational frequency is in good agreement with the result of Ref. [8] (59:9 cm1 ). The present bond length is in accordance with the GGA [8] and MP2 [9] calculation data. The linear structure for Zn3 was optimized and the energy of this structure was found to be 0.06 eV higher than the ground state energy with an imaginary vibrational frequency, which corresponds to a transition state. The results are presented in Table 2. Zn4 : Four different initial geometries (a Y-shape structure, a tetrahedral structure, a rhombus-like structure, and two right angle triangles with the same base) are optimized in order to find the lowest-energy structure of Zn4 cluster. The minimum energy structure for Zn4 cluster is a tetrahedron structure (4a in Fig. 1) with Td symmetry. ˚ and The bond length in the tetrahedron structure is 2:90 A, the maximum vibrational frequency is 87:8 cm1 , which has lower energy (DE ¼ 0:26 eV) than a rhombus structure (4b in Fig. 1) with D2h symmetry and bond length 3.36 eV. Wang et al. [8] and Flad et al. [4] also found the minimum energy structure as a tetrahedron with the bond lengths ˚ and the vibrational frequencies 94:5 and 2:84 and 2:94 A, 81:2 cm1 , respectively (see Table 3).

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Fig. 1. (Color online) The geometries of the homonuclear zinc clusters at the minimum energy configurations with the corresponding parameters.

Table 1 Spectroscopic constants of the diatoms Zn2, Te2 and ZnTe (binding ˚ and the energy, E b , is in eV, equilibrium interatomic separation, re , is in A, fundamental frequency, oe , is in cm1 ). Diatom

Eb

re

oe

Table 2 Some calculation results of the homonuclear trimers Zn3 and Te3 (structure, binding energy E b (in eV), equilibrium bond lengths a ¼ b (in ˚ the bond angle y (in degree), and the vibration with maximum A), amplitude o (in cm1 ))

Method [Ref.] Trimer Structure

Zn2 Zn2 Zn2 Zn2 Zn2

0.013 0.024 0.022 0.027

Zn2 Zn2

o0.06 0.034

Te2 Te2 Te2 Te2 ZnTe

4.111 3.62–4.04 3.94 1.834

3.56 3.88 4.11 3.37 3.83–4.19

33.0 25.0 21. 43.1

4.51 4.80

25.7

2.70

220.6

2.64 2.56 2.42

246.0 241.7

B3LYP/CEP-121G QDMC [3] Ab initio/CC [4] DFT/GGA [8] MP2-CCSD(T)/6311++G(d,p) [9] MP2/LANL2DZ [10] Exp. [11] B3LYP/CEP-121G DFT/BLYP-PW91 [14] DFT/BP [14] Exp. [15] B3LYP/CEP-121G

Zn5 : The initial geometries optimized for the Zn5 cluster are a triangular bipyramid, a square pyramid, a butterfly shape, a pentagon, an envelope, and a screw. The present calculations predict that there are two isomers with very close energies. The trigonal bipyramid structure (5a in Fig. 1) with symmetry D3h is more favorable compared with the structure shown in 5b in Fig. 1 with an energy difference of 0.003 eV (see Table 3 for frequency values).

Eb

a

o

Method [Ref.]

60 60 60 60

53.1 B3LYP/CEP-121G 28.2 Ab initio/CC [4] 59.9 DFT/GGA [8] MP2/CCSD(T) [9]

Zn3

0.073 3.21 0.090 3.75 0.219 3.08 3.38/ 3.95 Equilateral p0.09 4.28

60

Te3 Te3

Equilateral Equilateral

60 60

Te3

Isosceles

2.67

113.2

Te3

Equilateral

2.80

60

Te3

Isosceles

2.64

Te3 Te3

Isosceles Equilateral

MP2/LANL2DZ [10] B3LYP/CEP-121G Ab initio/CASSCF [12] Ab initio/CASSCF [12] B3LYP/321+G(d) [13] B3LYP/321+G(d) [13] DFT/PW91 [14] DFT/PW91 [14]

Zn3 Zn3 Zn3 Zn3

Equilateral Equilateral Equilateral Equilateral

y

3.071 2.92 2.83

6.90 6.93

198.5

237.0

115.1 235.0

60

213.5 210.0

Zn6 : The considered initial geometries of Zn4 cluster are a square bipyramid, a triangular prism, a pentagonal pyramid, a hexagon, and a planar structure. The lowestenergy structure is a rectangular bipyramid (6 in Fig. 1)

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Table 3 Calculated vibrational harmonic frequencies (in cm1 ) and the corresponding non-zero infrared intensities (in km/mol) (given in parenthesis) for the lowest energy structures of homonuclear Znm and Ten (m; n ¼ 426) clusters Species

Vibrational frequencies (infrared intensities)

Zn4 Te4 Zn5

72.89 52.54 33.45 82.02 19.79 148.03 7.54 52.78 10.98 125.99

Te5 Zn6 Te6

(0.02) (0.08) (0.44) (3.38) (0.08) (0.43)

73.24 87.33 33.61 82.43 64.41 166.97 30.90 52.82 39.56 137.71

(0.08) (0.34)

(0.04) (0.04)

80.88 137.96 33.97 91.32 75.67 167.46 33.27 58.41 54.55 151.24

(0.12)

(0.23) (0.64) (0.35) (0.10) (3.91)

80.97 (0.12) 141.31 (0.29) 45.41

81.41 (0.12) 141.31 (0.29) 45.87

87.79 171.93 60.70

79.99

126.79 (0.77)

130.60 (0.34)

40.00 59.04 58.33 153.86

44.25 63.64 63.55 165.13 (0.70)

46.67 103.01 82.74 (0.03) 166.87 (0.14)

Fig. 2. (Color online) The geometries of the homonuclear tellurium clusters at the minimum energy configurations with the corresponding parameters.

with symmetry D4h . The equilateral triangle structure with the three zinc atoms positioned at the center of the edges has higher energy (DE ¼ 0:28 eV) than the ground state structure but this geometry has four imaginary vibrational frequencies thus it is a transition state (see Table 3). 3.1.2. Homonuclear Te clusters Te2 : The B3LYP/CEP-121G calculation predicted the bond length and the vibrational frequency of the Te dimer ˚ and 220:6 cm1 (2 in Fig. 2). The present data are as 2:70 A in good agreement with the experimental results, the difference between the experimental and the present calculational data for bond length and frequency is about 5% and 12%, respectively. Another DFT calculation with PW91 functional [14] obtained the vibrational frequency of Te2 to be 231 cm1 and at the same study with ˚ BP functional the bond length was calculated as 2:64 A. Table 1 displays both the present results and the available literature data.

Te3 : According to the present calculations, the ground state of Te3 is found to be an equilateral triangle (3 in Fig. 2) having D3h symmetry with binding energy ˚ and the vibrational 3.07 eV, the bond distance 2:92 A, 1 frequency 198:5 cm . The linear geometry was found to be 1.35 eV above the ground state. Balasubramanian and Dai [12] found the equilateral and isosceles (C2v ) triangle structures to be nearly degenerate and the bond length difference was less than 9% as can be seen from Table 2. Pan [14] also found that the energy difference between the Te3 clusters with the symmetries D3h and C2v was very small, hence concluded that this cluster had two isomers. Goddard [13] showed that the energies of isosceles and equilateral triangle geometries to be nearly the same with the bond lengths less and the frequencies more than the present data. More information about Te3 cluster can be found in Table 2 for comparison. Te4 : The initial geometries are the same with Zn4 cluster. The lowest-energy structure is a 3D bent rhombus

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Fig. 3. (Color online) The geometries of the Znm Ten (m þ n ¼ 2; 3; 4) clusters (Zn is purple and Te is orange) at the minimum energy configurations with the corresponding parameters.

(4 in Fig. 2) with C2v symmetry. The bond length is equal ˚ the bending angle is 33:4 and the maximum to 2:94 A, frequency is 171:9 cm1 (see Table 3). Pan [14] predicted the ground state structure of this cluster as a square and a trapezoid and the corresponding frequencies are 233.9 and 234:1 cm1 , respectively. The experimental frequency result is equal to 243 cm1 [15] which is larger than the present result. Te5 : The same initial structures with Zn5 clusters are optimized and two different structures are obtained. The global minimum predicted by the present calculation is an envelope (5a in Fig. 2) with Cs symmetry. The other optimized structure (5b in Fig. 2) has Td symmetry and has much higher energy (DE ¼ 2:65 eV) than the lowest-energy structure (see Table 3). Pan [14] has also found the global minimum structure to be an envelope structure with the vibrational frequency 180:7 cm1 which is in reasonable agreement with the present result (167:5 cm1 ). Te6 : The optimized geometries are the same with those of Zn6 clusters. The ground state has C2v symmetry which is shown in 6 in Fig. 2 and its maximum vibrational frequency is equal to 153:9 cm1 (see Table 3). Pan [14] predicted the lowest-energy structure to be a distorted hexagon with D3d symmetry and the vibrational frequency as 168:8 cm1 . The disagreement between present results and Pan’s results for 4- and 6-atom Te clusters could be due to the difference in basis sets. A more detailed calculation may be required for these clusters.

Table 4 Trimers of ZnTe2 and Zn2Te at minimum-energy configurations A–B–C

a

b

y

o1

o2

o3

o4

Te–Zn–Te Zn–Zn–Te

2.51 2.54

2.51 2.40

79.16 180.0

101.57 29.07

202.87 29.07

235.15* 110.32

283.86*

˚ bond angle y is in degree, and vibrational Bond lengths a and b are in A, frequencies oi are in cm1 . The geometries are shown in Fig. 3. The asterisked frequencies represent the vibrations with maximum amplitude.

3.1.3. Heteronuclear ZnTe clusters ZnTe: The calculated binding energy of ZnTe by CEP121G basis function is found to be 1.83 eV, the interatomic ˚ and the vibrational frequency is distance is 2:42 A, 241:7 cm1 (2 in Fig. 3 and Table 1). Zn2 Te: The initial geometries of the Zn2Te cluster before the optimization procedure are an equilateral triangle, a linear structure with tellurium atom at the center and a linear structure with tellurium atom at the one side. The ground state is found to be a linear structure with Te atom at the edge (3a in Fig. 3 and Table 4). The distance between ˚ which is less than Zn atoms and Zn–Te is equal to 2:54 A, ˚ which is almost the same with the the dimer, and 2:40 A, ZnTe, respectively. The other linear structure with the Te atom at the center (3b in Fig. 3) is higher in energy than the ground state structure by 0.02 eV. ZnTe2 : The optimizations are started with the similar initial geometries with Zn2Te. The lowest-energy structure

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Table 5 Calculated vibrational harmonic frequencies (in cm1 ) and the corresponding non-zero infrared intensities (in km/mol) (given in parenthesis) for the lowest energy structures of heteronuclear Znm Ten (4pm þ np6) clusters Species

Vibrational frequencies (infrared intensities)

Zn2Te2 ZnTe3 Zn3 Te ZnTe4

71.81 22.66 63.49 47.73 150.45 23.70 167.98 16.04 162.93 7.69 151.93 29.63 131.55 10.95 103.09 37.14 126.51 35.35 114.53 53.63 126.32

Zn4 Te Zn3 Te2 Zn2Te3 ZnTe5 Zn5 Te Zn2 Te4 Zn4 Te2 Zn3 Te3

(2.35) (1.06) (0.01) (0.03) (0.72) (0.17) (4.74) (0.32) (0.46) (0.19) (0.57) (1.97) (0.31) (2.69) (1.74)

(1.69)

131.67 88.67 89.77 53.95 161.23 43.04 200.87 27.81 188.30 33.10 194.38 43.18 141.61 42.63 123.08 50.84 141.57 61.76 139.57 53.78 126.50

(1.82) (2.34) (1.64) (0.27) (0.38) (2.25) (0.20) (11.05) (0.34) (15.30) (0.17) (1.29) (0.20) (0.04) (0.25) (0.21)

(1.69)

136.76 141.06 101.45 68.91 273.01 61.33 206.59 56.32 211.48 39.45 240.11 47.28 150.53 58.21 135.31 54.85 155.73 74.81 171.69 68.87 151.23

(5.49) (0.55) (1.75) (0.09) (11.73) (0.69) (0.38) (0.10) (0.80) (6.96) (0.38) (1.45) (0.14) (0.36) (0.19) (0.06)

(2.13)

for ZnTe2 cluster is found to be an isosceles triangle (3c in Fig. 3 and Table 4) which has lower energy (DE ¼ 0:28 eV) than a linear structure with Zn atom at the center (3d in Fig. 3). Zn2 Te2 : For this cluster, five different initial geometries (a tetrahedron, a rhombus, a square, a linear structure, and a Y-shape geometry) are optimized and three different optimized structures are obtained. The lowest-energy structure is a rhombus structure (4a in Fig. 3 and see Table 5 for frequencies) with D2h symmetry that has lower energy than an isosceles trapezoid (4b in Fig. 3) with symmetry C2v (DE ¼ 1:07 eV) and a planar isosceles triangle with one Zn atom externally bounded to the other Zn atom (4c in Fig. 3) which has C3v symmetry (DE ¼ 1:48 eV). Zn3 Te: Five different initial geometries (a linear geometry, a parallelogram, a rhombus, a tetrahedron, and an equilateral triangle with an external additional zinc atom) are performed to find the lowest-energy structure of Zn3 Te cluster. It is found that the 2D distorted rhombus structure (4d in Fig. 3 and see Table 5 for frequencies) with C2v symmetry is more stable when compared with a linear geometry (4e in Fig. 3, DE ¼ 0:12 eV) and an isosceles triangle structure with an additional tellurium atom with C2v symmetry (4f in Fig. 3, DE ¼ 0:16 eV). ZnTe3 : The initial geometries are the same with that of the Zn3 Te where Zn(Te) atoms replaced by Te(Zn) atoms. The lowest-energy structure is a planar distorted rhombus structure (4g in Fig. 3 and see Table 5 for frequencies) with symmetry C2v . Another local minima is found to be a planar structure (4h in Fig. 3) with Cs symmetry which is 0.93 eV above the lowest-energy structure and the other

183.35 141.89 (0.04) 166.78 (0.51) 76.59 (2.65) 63.84

213.30 186.84 (2.14) 173.54 (0.17) 135.17 (2.16)

255.70 (29.84) 266.71 204.37 148.87 (0.76)

128.67

130.69 (0.23)

73.46 (1.05)

99.56 (5.24)

148.02 (2.17)

73.59 (2.94)

107.63 (2.25)

148.76 (2.77)

63.30 164.74 83.69 183.31 67.19 269.88 81.99 272.84 76.18 295.06

75.96 284.81 88.17 237.85 96.86 295.40 113.08 290.66 117.49 295.07

62.29 155.73 68.74 151.17 59.40 155.85 75.31 222.03 76.15 253.33

(0.99) (0.99) (0.05) (0.12) (1.12) (3.56) (38.95) (0.44)

(1.73) (1.74) (0.03) (6.81) (1.92) (5.15) (10.91) (0.44) (35.66)

(0.76) (11.34) (0.04) (18.33) (0.43) (31.30) (4.44)

(35.67)

local minima is 1.28 eV above the rhombus structure with a zigzag geometry (4i in Fig. 3). Zn4 Te: Four different geometries (two trigonal bipyramids with Te atoms different locations, a square pyramid, and a butterfly) are initially optimized and the lowestenergy structure for Zn4 Te is found to be a pentagon (5a in Fig. 4) with D5h symmetry which has a lower energy than a three isosceles triangles with the same basement (5b in Fig. 4) having D3h symmetry with 0.34 eV. The frequencies are given in Table 5. ZnTe4 : The optimized geometries are similar to the Zn4 Te clusters. The lowest-energy structure is a 3D distorted pentagon (5c in Fig. 4) which is lower in energy than the planar structure (5d in Fig. 4) by 1.10 eV. The frequencies are given in Table 5. Zn3 Te2 : The initial geometries which are similar to the geometries of Zn4 Te cluster yielded two different structures. The global minimum structure is found to be a trapezoid with an external additional zinc atom (5e in Fig. 4). The envelope structure (5f in Fig. 4) with Cs symmetry is found to be the other local minima with 0.33 eV above the global minimum structure. The frequencies are given in Table 5. Zn2 Te3 : All the initial geometries which are similar to the geometries of Zn4 Te cluster resulted in only one optimized structure: a quadrilateral with two sides of equal length with an additional external zinc atom (5g in Fig. 4). The frequencies are given in Table 5. Zn5 Te: Four initial geometries, two square bipyramids with zinc atom positioned differently, a triangular prism and a pentagon pyramid, are optimized and two different structures with very close energies are obtained. The 3D

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Fig. 4. (Color online) The geometries of the Znm Ten (m þ n ¼ 5; 6) clusters (Zn is purple and Te is orange) at the minimum energy configurations with the corresponding parameters.

lowest-energy geometry which is shown in 6a in Fig. 4 is lower than the 2D pentagon with an additional zinc atom (6b in Fig. 4) in energy difference by 0.04 eV. The frequencies are given in Table 5. ZnTe5 : The initial structures for ZnTe5 cluster are the same with those of Zn5 Te cluster and these geometries result in one structure with positive vibrational frequency (see Table 5). 3D distorted hexagon (6c in Fig. 4) with D3d symmetry is found to be the lowest-energy structure for this cluster. Zn4 Te2 : The initial geometries for the optimization are chosen as two square bipyramids with tellurium atoms positioned differently, a triangular prism and a hexagon. In the case of Zn4 Te2, hexagonal structure (6d in Fig. 4) is

found to be more stable than a rhombus structure with two zinc atoms bounded to the other zinc atoms (6e in Fig. 4) with the energy difference is equal to 0.48 eV. The frequencies are given in Table 5. Zn2 Te4 : Similar geometries are optimized with those of the Zn4 Te2 cluster. The lowest-energy structure for Zn2 Te4 cluster is found to be a 3D triangle with one Te atom is out of the plane of the other atoms (6f in Fig. 4) and the corresponding symmetry is D3h . The frequencies are given in Table 5. Zn3 Te3 : The starting geometries for this cluster are a planar and 3D hexagon, a triangle with Zn atoms placed in the middle of the sides, a W-shaped structure, and an octahedron. The distorted planar triangle (6g in Fig. 4)

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with D3h symmetry is found as the global minimum structure. The Zn–Te bond distances, the obtuse angles and the acute angles found in the structure are the same. The frequencies are given in Table 5. 3.2. HOMO–LUMO energies The highest occupied molecular orbitals (HOMOs), the lowest unoccupied molecular orbitals (LUMOs) and the frontier molecular orbital energy gaps (LUMO–HOMO difference in energy, E g ) of the clusters are given in Table 6. Since the number of electrons is even for both the Zn and Te atoms, there are only a-states for the molecular orbital energies. Fig. 5 shows the trend of the MO energy gaps for the studied clusters. While the E g of homonuclear zinc clusters decreases as the number of atoms increases in the system, the homonuclear tellurium clusters show a zigzag increment in the E g . Moreover, E g of the Zn clusters is slightly higher than that of the Te clusters. For the stoichiometric clusters (having equal number of different atoms, i.e. Znm Tem ) the E g increases, however, for the other types of clusters, which are not mentioned previously, there is not a proper increase or decrease in the E g as the number of atoms increases. In general, the average binding energy per atom for the clusters is defined by E b ¼ ½EðZnm Ten Þ  m  EðZnÞ n  EðTeÞ=ðm þ nÞ. For Zn and Te clusters, E b decreases (increases in magnitude) with increasing the number of

Fig. 5. The molecular orbital energy gaps (E g ) of Znm Ten clusters are presented, where ðm; nÞ represents the cluster type.

Table 6 Calculated HOMO, LUMO energies (in a.u.), HOMO–LUMO gap energies (E g ) and the average binding energies, i.e. E b =atom (E b ) (in eV) of the studied systems (see Figs. 5 and 6) Species

HOMO

LUMO

Eg

Eb

Zn2 Zn3 Zn4 Zn5 Zn6 Te2 Te3 Te4 Te5 Te6 ZnTe Zn2Te2 Zn3 Te3 Zn2Te Zn3 Te Zn4 Te Zn5 Te Zn3 Te2 Zn4 Te2 ZnTe2 ZnTe3 ZnTe4 ZnTe5 Zn2Te3 Zn2Te4

0.223 0.221 0.224 0.208 0.210 0.204 0.218 0.209 0.216 0.219 0.210 0.212 0.227 0.190 0.185 0.181 0.182 0.185 0.217 0.203 0.272 0.216 0.218 0.192 0.225

0.041 0.058 0.070 0.069 0.074 0.182 0.138 0.145 0.134 0.146 0.156 0.120 0.097 0.139 0.122 0.102 0.111 0.118 0.099 0.143 0.134 0.128 0.133 0.122 0.130

4.952 4.435 4.190 3.782 3.701 0.599 2.177 1.742 2.231 1.986 1.469 2.503 3.537 1.388 1.714 2.150 1.932 1.823 3.211 1.632 3.755 2.395 2.313 1.905 2.585

0.006 0.029 0.096 0.089 0.091 2.055 2.394 2.465 2.662 2.672 0.917 1.729 2.058 0.798 0.698 0.672 0.598 1.228 1.364 1.613 2.140 2.357 2.459 1.779 2.251

Fig. 6. The average binding energy (E b ) of Znm Ten clusters are presented, where ðm; nÞ represents the cluster type.

atoms, which is in accordance with the results of Ref. [10]. The binding energy per atom increases as the size of the clusters get larger for the Znm Ten (m ¼ n) clusters. It is worth to mention that binding energy of Te rich clusters are considerably larger than that of the Zn rich clusters. For interested readers, the E b variation with the number of atoms for the considered clusters is represented in Fig. 6. 3.3. Mulliken charge populations and dipole moments Calculated Mulliken charges on each atom and the dipole moments for the heteronuclear clusters are presented in Table 7. Since zinc is metallic and tellurium is non-metallic, we expect that zinc shows a cationic property and tellurium shows an anionic property. As can be seen from Table 7, zinc atoms are positively charged and tellurium atoms are negatively charged with a few

ARTICLE IN PRESS R. Peko¨z, S- . Erkoc- / Physica E 40 (2008) 2921–2930 Table 7 Calculated Mulliken charges (Q) and the dipole moments (D.M., in Debye) for the lowest energy structures of heteronuclear Znm Ten clusters

Table 8 Fragmentation data of the most stable Znm Ten microclusters Cluster

QTe

Cluster QZn ZnTe ZnTe2 Zn2Te Zn2Te2 ZnTe3 Zn3 Te Zn3 Te2 Zn2Te3 Zn4 Te ZnTe4 Zn3 Te3 Zn4 Te2 Zn2Te4 Zn5 Te ZnTe5

0.156 0.099 0.193 0.071 0.045 0.116 0.142 0.183 0.093 0.050 0.041 0.051 0.058 0.072 0.019

(2)

(2) (3) (4) (2)

0.156 0.050 0.041 0.234 0.071 0.028 0.025 (2) 0.165 0.076 0.009 0.115 0.045 0.054 0.007 (2) 0.171 0.027 0.041 0.102 0.030 0.038 (2) 0.015 (2) 0.117 0.016

2929

DCs

DE d

Cluster

2Zn Zn2 þ Zn Zn3 þ Zn Zn4 þ Zn Zn5 þ Zn 2Te Te2 þ Te 2Te2 Te3 þ Te2 2Te3 Zn+Te ZnTe2 þ Zn Zn2 Te3 þ Zn

0.01 0.07 0.30 0.06 0.10 4.11 3.07 1.64 2.02 1.67 1.83 2.08 3.45

Zn2Te Zn3 Te Zn4 Te Zn5 Te Zn3 Te2 Zn4 Te2 ZnTe2 ZnTe3 ZnTe4 ZnTe5 Zn2 Te3 Zn2 Te4

DCs

DE d

ZnTe þ Zn Zn2 Te þ Zn Zn3 Te þ Zn Zn4 Te þ Zn ZnTe2 þ Zn2 Zn2 Te2 þ Zn2 Zn þ Te2 Zn þ Te3 Zn þ Te4 Zn þ Te5 Zn2 þ Te3 Zn2 Te2 þ Te2

0.56 0.39 0.57 0.23 1.29 1.25 0.73 1.38 1.92 1.44 1.70 2.48

D.M.

(2) (2) 0.008 (2) 0.112 (2) 0.030 (2) (3) (2)

0.002 (2)

0.029 (3) 0.002 (4)

4.19 3.34 7.91 0.00 2.08 3.25 6.05 5.36 4.89 1.31 0.00 0.00 1.15 3.90 1.34

exceptions, in other words, there is a charge transfer from zinc atoms to tellurium atoms. Although the excess charge on each atom is relatively small, it can be concluded that the bonding is covalent character. For the case of stoichiometric clusters, as the cluster size increases the charge on each atom decreases and the bond length increases. Dipole moment is the indicator of the symmetry property of the structure. If the system is symmetric then the dipole moment is equal to zero which is valid for Zn2Te2, Zn3 Te3 , and Zn4 Te2 clusters. Furthermore, when the dipole moment is zero, the charges distributed on each atom are equal to each other. 3.4. The dissociation (fragmentation) channels and dissociation energies Besides the information regarding the energetics, geometries and the vibrational frequencies, the lowest possible energy dissociation channels are important to understand the clusters. Moreover, the knowledge of the fragmentation pathways of the clusters is used in vapor deposition and adatom adsorption on surfaces. In the present calculations, possible dissociation channels of the studied clusters are considered and the most probable ones with the corresponding energies are presented in Table 8. A few possible dissociation channels for a general case which are taken into account are given by Am Bn ! mA þ nB

(1)

Am Bn ! Am þ Bm

(2)

Am Bn ! Am2 þ 2A þ Bm1 þ B

(3)

The fragmentation energy of the fragmentation channel of Eq. (1) is defined as the difference of the energies of the both sides of the reaction, i.e., DE d ¼ EðAm Bn Þ  mEðAÞ

Zn2 Zn3 Zn4 Zn5 Zn6 Te2 Te3 Te4 Te5 Te6 ZnTe Zn2Te2 Zn3 Te3

! ! ! ! ! ! ! ! ! ! ! ! !

! ! ! ! ! ! ! ! ! ! ! !

The most probable dissociation channels (DCs) are given and the corresponding dissociation energies (DE d ) are in eV.

nEðBÞ. For this type of reaction, DE d is named as the binding energy of the system and it is always negative in value. The most probable dissociation channel has the maximum value (minimum in magnitude) of DE d . As can be seen in Table 8, the dissociation pathways show different properties for zinc and telluride clusters. While the homonuclear Zn clusters dissociate one by one (Znm ! Znm1 þ Zn), the homonuclear Ten clusters prefer to dissociate as dimers for np4 and as trimers for n ¼ 5 or 6. When the number of Zn and Te atoms are the same in a cluster, zinc atoms separate one by one in the most probable dissociation pathway. In the zinc rich clusters (Znm Ten , m4n), again zinc atoms prefer to dissociate one by one for n ¼ 1, and in the form of dimers for the other cases. On the other hand, in the tellurium rich clusters, when there is only one Zn atom in the system the most possible dissociation is as follows: ZnTem ! Zn þ Tem where m ¼ 5. For Zn2 Te3 cluster, the dissociation is in the form of zinc dimer and tellurium trimer and Zn2 Te4 cluster dissociates leading to Te dimer. 4. Summary We have studied the ground state structures of homonuclear Zn and Te clusters and their combinations up to six atoms applying DFT with B3LYP exchange correlation at CEP-121G basis set. The global minimum geometries are predicted by considering the lowest-energy structures and the corresponding positive vibrational energies. For the lowest-energy structures of homonuclear Znm and Ten clusters, the transitions from planar to 3D structures are found when m and n are equal to 4. On the other hand, in the case of heteronuclear clusters, 3D structures are obtained for m; n ¼ 5. The frontier molecular orbital energy gap calculations predicted that the binding energy per atom decreased with increasing the number of atoms in the clusters. The dissociation channels are also studied and

ARTICLE IN PRESS 2930

R. Peko¨z, S- . Erkoc- / Physica E 40 (2008) 2921–2930

found that the homonuclear zinc clusters dissociate one by one and the homonuclear tellurium clusters dissociate in the form of dimers and trimers. In the mixed clusters zinc atoms dissociate first. Acknowledgments The authors would like to thank METU for partial support through the project METU-BAP-2006-07-02-0001. One of the authors (R.P.) would like to thank TU¨BITAK for partial support provided through BAYG scholarship. The authors would like to thank Altug˘ O¨zpineci for critically reading the manuscript. References [1] P.C. Kalita, K.C. Sarma, H.L. Das, J. Assam Sci. Soc. 39 (1998) 117. [2] C. Winnewisser, P.U. Jepsen, M. Schall, V. Schiya, H. Helm, Appl. Phys. Lett. 70 (1997) 3069. [3] F. Schautz, H.-J. Flad, M. Dolg, Theoret. Chem. Accounts 99 (1998) 231. [4] H.-J. Flad, F. Schautz, Y. Wang, M. Dolg, A. Savin, Eur. Phys. J. D 6 (1999) 243. [5] S- . Katırcıog˘lu, S- . Erkoc- , J. Mol. Struct. (Theochem.) 546 (2001) 99.

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