Struct Chem (2015) 26:573–585 DOI 10.1007/s11224-014-0518-z
ORIGINAL RESEARCH
Tin content effect on the structural and energetic properties of lead telluride clusters Yonas Mulugeta • Hagos Woldeghebriel
Received: 18 June 2014 / Accepted: 12 September 2014 / Published online: 8 October 2014 Ó Springer Science+Business Media New York 2014
Abstract Density functional theory calculations are performed to investigate tin content effect on the structural and energetic properties of lead telluride clusters. Tin atoms prefer to be doped on the exterior sites rather than the interior sites of lead telluride clusters. The geometry of the clusters is not affected due to the amount of tin content except for a slight distortion observed by changes in bond length. A considerable change has been observed in the HOMO–LUMO gap of the clusters with increasing number of tin atoms for corresponding size, but the band inversion which is observed in the bulk is not observed on the clusters in these size ranges. The change in binding energy between doped clusters is almost negligible for the same number of tellurium atoms, it is observed to be constant or there is a change of up to 0.02 eV with each successive doping of tin atoms. Both adiabatic and vertical detachment energies are observed to increase with increasing number of doping tin atoms. The fragmentation energy of losing PbTe dimer changes with an even–odd alternation, clusters with even number of Te atoms are more stable because they need higher energy to dissociate as compared to the other clusters. Some clusters are found to be magic (more stable) because of their enhanced binding, HOMO– LUMO gap, and fragmentation energies, typical examples are Pb4-mSnmTe4 (m = 1–4) clusters.
Electronic supplementary material The online version of this article (doi:10.1007/s11224-014-0518-z) contains supplementary material, which is available to authorized users. Y. Mulugeta (&) Department of Physics, Aksum University, Aksum, Ethiopia e-mail:
[email protected] H. Woldeghebriel Department of Physics, Mekelle University, Mekelle, Ethiopia
Keywords Lead tin telluride clusters Energetic properties DFT
Introduction Since the properties of a cluster can be controlled by changing its size and composition, it is possible to design cluster-based materials having desirable traits. Isovalentdoped semiconductor clusters, especially those doped with group IV metals can provide tunable building blocks for novel nano-structured materials and devices. Varying the doping concentration can control the gap between the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO) of the isovalentdoped clusters [1]. In 1964, the first mid-infrared p–n junction laser was made by Mukherjee et al. [2] using Pb1-xSnxTe, and since then, efficient mid- and far-infrared IV–VI compound diode lasers have been fabricated, finding their main applications for remote sensing of gaseous pollutants in trace gas sensing devices, toxic gas analysis systems, human breath analysis in medical diagnostics, and industrial process control. Recent progress in epitaxial growth techniques has led to the fabrication of lead salt midinfrared diode lasers operating at high temperature. In fact, the lead tin telluride alloy system has been investigated for many years and applied mainly in the fabrication of infrared photodetectors and diode lasers can cover the whole 3–30 lm wavelength region and feature the highest operation temperature of conventional infrared band gap lasers by choosing appropriate chemical compositions [3]. The energy gap of bulk PbTe is 190 meV and pure SnTe corresponding to x = 1 in Pb1-xSnxTe is a p-type semiconductor with energy gap of 200 meV [4], both the two
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tellurides (Pb and Sn) possess the same face centered-cubic symmetry with nearly matched lattice constants, SnTe = 0.6480 nm and PbTe = 0.6462 nm, and both PbTe and SnTe crystallizes in rocksalt (rs) structure [5]. It is well known that this two tellurides, PbTe and SnTe, can be allowed to combine and form ternary alloys Pb1-xSnxTe that have narrow band gap semiconductors characterized by an energy gap that is linearly proportional to the tin concentrations. According to the band inversion BI model [6], the Pb1-xSnxTe energy gap (Eg) initially decreases as the Sn composition increases and vanishes for an intermediate alloy composition. Further increasing the Sn composition, the Eg starts to increase, with the band edge states inverted, up to the SnTe value. They observed that Sn composition for which the band inversion should occur varies from x = 0.35 to x = 0.65 as the temperature increases from 4 to 300 K. Despite the importance of lead tin telluride in the ultrasmall size regime, investigations have been limited. Work which has been reported includes molecular beam epitaxial growth of Pb1-xSnxTe quantum dots investigated by Koike et al. [5]. Ferreira et al., studied experimental observation of band inversion in the PbSnTe system [3] and Lakshmi et al., investigated increase in the thermoelectric power produced by mechanically alloyed Pb1-xSnxTe [7]. Inspired by the experimental results of tin-doped lead telluride crystal, we have investigated the effect of tin content on the structural and electronic properties of lead tin telluride clusters. To our knowledge, there is no investigation on lead tin telluride clusters, thus starting from the lowest energy structures of the pure (PbTe)n (n = 1–20) previously reported in [8], we considered substitutional doping of Sn for Pb in these clusters to observe the changes in structure and electronic properties of lead tin telluride clusters.
Computational method All the calculations described in this work are performed with the Vienna Ab Initio Simulation Package (VASP) [9]. The electron–ion interactions are described by ultrasoft pseudopotentials [10], based on a plane-wave expansion method employing the local density approximation (LDA) for exchange and correlation term [11]. The cutoff energy used in the plane-wave expansion is 143.58 eV. Structural optimizations are performed using the conjugate gradient (CG) method where the components of forces on atoms are ˚ , the structural optimizations are taken to about 0.004 eV/A be converged. The self consistent field (SCF) calculations are carried out with a convergence in the total energy of 10-4 eV. The reciprocal space is sampled by the C point. A
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sufficiently large simulation cell is chosen so that the minimum distance from the cluster boundary to the cell ˚ . We use 20 9 20 9 20 A ˚ 3 simulaboundary is about 5 A ˚ tion box if the length of the cluster is less than 10 A 3 ˚ otherwise we use 30 9 30 9 30 A Simulation box.
Results and discussion All the cluster structures considered in this work are obtained from the most stable structures of (PbTe)n, (n = 1–20) clusters [8]. We substituted tin atoms in place of the lead atoms on the most energetically preferred site of (PbTe)n. In order to search for the most energetically favorable Sn-doped PbnTen structures, all possible substitution of Sn for Pb sites were considered for the relatively small size clusters. The vertical detachment energies (VDE) were calculated as the energy difference between the anion and neutral clusters at the same structures as those of the anions, and the adiabatic detachment energies (ADE) were obtained as the energy difference between the cluster anions and the optimized neutral clusters using the anion structures as initial structures [12]. Our aim was to examine the change in structure and energetic properties of these clusters as a function of the number of doping tin atoms. The binding energy per atom, Eb, of Pbn-mSnmTen (n = 1–20, 2 m B n) clusters is calculated from the following equation [13]: Eb ¼
nEðTeÞ þ ðn mÞEðPbÞ þ mEðSnÞ 2n EðPbnm Snm Ten Þ : 2n
We have calculated the fragmentation/dissociation energies of Pbn-mSnmTen (n = 2–20, 2m B n) clusters along losing of PbTe pathways Efg(PbTe) using the following equation [14]: Efg ðPbTeÞ ¼ EðPbnm1 Snm Ten1 Þ þ EðPbTeÞ EðPbnm Snm Ten Þ Structural and energetic properties of Pbn-1SnTen, (n = 2–20) clusters The substitution of a single Pb atom with Sn atom in the lowest energy structure of lead telluride clusters does not affect the geometry of the lead telluride clusters [8] except for slight distortion, the distortion in the geometry of Pbn-1SnTen cluster is may be due to the difference in bond length between Sn–Te and Pb–Te bonds. The resulting low-lying structures (the most preferred doping sites of the tin atom) of Pbn-1SnTen clusters with corresponding chemical composition are presented in Fig. 1. There are
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Fig. 1 Optimized geometry of Pbn-1SnTen, (n = 2–20) clusters with corresponding chemical composition
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Fig. 2 Optimized geometry of Pbn-2Sn2Ten, (n = 4–20) clusters with corresponding chemical composition
nineteen clusters in total, the smallest cluster being PbSnTe2. The effect of substituting one lead atom with one tin atom of (PbTe)n, (n = 2–20) clusters on their electronic properties is summarized in Figs. 3, 4, 5, 6, and 7. In Fig. 3, we show the HOMO–LUMO gap of Pbn-m SnmTen (m = 0–4) clusters as a function of the number of tellurium atoms (n). In general, the HOMO–LUMO gap of Pbn-1SnTen clusters decreases as compared to the undoped
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(PbTe)n clusters [8] for the same number of Te atoms. There is a maximum decrease of HOMO–LUMO gap observed in Pb3SnTe4 cluster which is 0.22 eV, in contrast to the increase in the HOMO–LUMO gap of Pb2SnTe3 cluster by 0.07 eV with the exception of this a very small change in HOMO–LUMO gap was observed on the quadrangular prism structures such as Pb5SnTe6 and Pb9 SnTe10 clusters. The binding energy values as a function of
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Fig. 3 HOMO–LUMO gap of Pbn-mSnmTen (m = 1–4) clusters as a function of the number of Te atoms (n), graph for m = 0 is included for comparison
Fig. 4 Binding energy per atom of Pbn-mSnmTen, (m = 1–4) clusters as a function of number of Te atoms (n), graph of m = 0 is included for comparison
Fig. 5 The adiabatic detachment energy of Pbn-mSnmTen (m = 1–4) clusters as a function of the number of Te atoms (n), graph of m = 0 is also included for comparison
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Fig. 6 The vertical detachment energy of Pbn-mSnmTen (m = 1–4) clusters as a function of the number of Te atoms (n), graph of m = 0 is included for comparison
Fig. 7 The fragmentation energy of losing PbTe unit of Pbn-m SnmTen (m = 1–4) clusters as a function of the number of Te atoms (m), graph of m = 0 is included for comparison
the number of tellurium atoms (n) are plotted in Fig. 4. Generally, the binding energy per atom increases with the increasing number of Te atoms of the same species, but the increase is almost saturated for relatively large clusters. It can be observed in the same figure that the binding energy shows an odd even alternation with increasing size. Though the binding energy increases with the increasing m (number of Sn atoms in the cluster), the change between m and m ? 1-sized clusters is almost negligible or is a constant about 0.03 eV, the maximum being at PbSnTe2 cluster which is 0.05 eV. The adiabatic detachment energy (VDEs) and the vertical detachment energy (ADEs) values as a function of the number of tellurium atoms (n) are plotted in Figs. 5 and 6, respectively. Both adiabatic and VDE increase after doping almost for all clusters, where the maximum change is observed at n = 5 clusters. The fragmentation energy of
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losing PbTe dimer, shown in Fig. 7, changes with an even– odd stability pattern with clusters of even number of tellurium atoms being greater stability than that of the neighboring odd number of tellurium atoms. Hence, we understand that the thermodynamic stability of clusters with even number of tellurium atoms increases while with odd numbers decrease. For example, clusters at n = 4 are more stable as they need higher energy to dissociate and clusters at n = 5 and 17 are less stable because they can easily dissociate with less energy. Stability of some clusters can also be described based on the values of HOMO– LUMO gap energy Fig. 3 (n = 3 less stable and n = 4 more stable) and binding energy, Fig. 4 (n = 4 more stable and n = 5 less stable).
Structural and energetic properties of Pbn–2Sn2Ten, (n = 4–20) clusters Doping with two tin atoms in the most preferred configurations of (PbTe)n (n = 4–20) clusters does not lead to significant structural distortion of the Pbn-2Sn2Ten, (n = 4–20) clusters as compared to the undoped (PbTe)n clusters, but there is a very small change in Sn–Te bond length for Pbn-1SnTen, (n = 4–20) clusters as compared to Pb–Te bond length in the undoped clusters. All geometries of the doubly doped clusters are similar to the corresponding singly doped lead telluride clusters. The resulting low-lying structures and the most preferred doping sites of the tin atom of Pbn-2Sn2Ten clusters with corresponding chemical composition are presented in Fig. 2. There are a total of seventeen clusters and the smallest cluster is Pb2 Sn2Te4. The effect of substituting two lead atoms with two tin atoms of (PbTe)n, (n = 4–20) clusters on their electronic properties such as HOMO–LUMO gap, binding energy, detachment energy, and fragmentation energy losing of PbTe molecule is summarized in Figs. 3, 4, 5, 6, and 7. The HOMO–LUMO gap of Pbn-2Sn2Ten, (n = 4 - 20) clusters are found to be less than that of Pbn-1 SnTen, (n = 2 - 20) clusters (Fig. 3). It is observed that as the number of doping tin atoms increases to two, the change in energetic properties decrease, for example there is a maximum decrease of 0.06 eV in HOMO–LUMO gap as cluster changes from Pb7SnTe8 to Pb6Sn2Te8 and almost no change is observed on quadrangular prism structures such as Pb4Sn2Te6 and Pb8Sn2Te10 clusters as shown in Fig. 3. The change in binding energy upon doping is almost negligible for the same number of Te atoms. In most clusters, the change is only 0.02 eV which is observed to be the maximum on Pb2Sn2Te4 cluster as shown in Fig. 4. Both adiabatic and VDE increase for all clusters upon doping, a maximum change being on Pb2Sn2Te4 cluster as shown in Figs. 5 and 6.
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Structural and energetic properties of Pbn-3Sn3Ten, (n = 6 - 20) clusters In order to understand the variation of structural and electronic properties of lead tin telluride clusters further on the amount of tin content, we increase the doping content up to ten. Doping with three tin atoms in the most preferred configurations of (PbTe)n (n = 6 - 20) clusters does not change the geometry of the lead telluride framework. However, in most clusters there is a very small change in bond length between the atoms from Pbn-2Sn2Ten (n = 4 - 20) clusters. This is may be due to the fact that the Sn atom has a valence shell isoelectronic to the Pb atom. All geometries of these clusters are similar to the corresponding doubly doped clusters presented in the previous sections. The resulting low-lying structures (the most preferred doping sites of the tin atom) of Pbn-3Sn3Ten clusters with corresponding chemical compositions are presented in Fig. 8. There are a total of fifteen clusters, and the smallest cluster is Pb3Sn3Te6. The effect of substituting three lead atoms with three tin atoms in (PbTe)n, (n = 6 - 20) clusters on their electronic properties such as HOMO–LUMO gap, binding energy, vertical and ADE, and fragmentation energy of losing PbTe molecule is summarized in Figs. 3, 4, 5, 6, and 7. The HOMO–LUMO gap of Pbn-3Sn3Ten, (n = 6–20) clusters generally decrease as compared to Pbn-2Sn2Ten, (n = 4–20) clusters (Fig. 3). It is observed that as the number of doping tin atoms increased to three, the change in energetic properties decreases, for example there is a maximum decrease of 0.04 eV in HOMO–LUMO gap as the cluster changes from Pb9Sn2Te11 to Pb8Sn3Te11. Similar to the singly and doubly doped clusters in the previous sections, almost no changes were observed on quadrangular prism structures such as Pb3Sn3Te6 and Pb7Sn3Te10 clusters. The change in binding energy due to doping is almost negligible in most of the clusters, and it is observed to be constant or changes by about 0.01 eV for each successive doping of tin atoms (m), the gap narrows with the increasing Te atoms in the cluster. Both adiabatic and VDE increase for almost all clusters, where a maximum change is observed on Pb4Sn3Te7 cluster (Figs. 5, 6). The fragmentation energy of losing PbTe dimer changes with an even–odd alternation, and it is observed that the fragmentation energy of clusters with even number of tellurium atoms is greater as compared to the odd numbers (Fig. 7). Structural and energetic properties of Pbn-4Sn4Ten, (n = 8–20) clusters Upon increasing the number of doping tin atoms to four in the most preferred configurations of (PbTe)n, (n = 8–20)
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Fig. 8 Optimized geometry of Pbn-3Sn3Ten, (n = 6–20) clusters with corresponding chemical composition
clusters do not lead to a change on the geometry of the clusters, but in most clusters there is a very small structural variation as compared to that of Pbn-3Sn3Ten (n = 6-20) clusters which is observed from Sn–Te and Pb–Te bond lengths. There are a total of thirteen clusters considered in these species, the smallest cluster being Pb4Sn4Te8 (Fig. 9). The effect of substituting four lead atoms with four tin
atoms in (PbTe)n, (n = 8–20) clusters on their electronic properties such as HOMO–LUMO gap, binding energy, vertical and ADE and fragmentation energy of losing PbTe dimer are summarized in Figs. 3, 4, 5, 6, and 7. Similar to the previous sections in most clusters the HOMOLUMO gap of Pbn-4Sn4Ten, (n = 8 - 20) clusters decreases as compared to Pbn-3Sn3Ten, (n = 6–20).
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Fig. 9 Optimized geometry of Pbn-4Sn4Ten, (n = 6–20) clusters with corresponding chemical composition
We found that as the number of doping tin atoms increase from three to four, the change in electronic and structural properties decreases, for example, there is a maximum change of 0.03 eV in HOMO–LUMO gap as the size of the cluster changes from Pb7Sn3Te10 to Pb6Sn4Te10 cluster. Unlike the previous doped clusters, a maximum
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change of HOMO–LUMO gap is observed on the quadrangular prism structure (Fig. 3). In most clusters, the change in binding energy between successive tin doping is almost negligible, and it is observed to be constant or varied by an amount of 0.01 eV (Fig. 4). Both adiabatic and VDE increase for almost all clusters, with a
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Fig. 10 Optimized geometry of Pbn-5Sn5Ten and Pbn-6Sn6Ten, (n = 10–20) clusters with corresponding chemical composition
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maximum change being observed on Pb4Sn4Te8 and Pb12Sn4Te16 clusters (Figs. 5, 6). The fragmentation energy of losing PbTe dimer changes as an even–odd alternation. In line with this, it is observed that the fragmentation energy of clusters that have even number of tellurium atoms is greater than those with odd numbers (Fig. 7). Structural and energetic properties of Pbn-5Sn5Ten and Pbn-6Sn6Ten, (n = 10–20) clusters
Fig. 11 Binding energy per atom of Pbn-mSnmTen, (m = 5–9) clusters as a function of the number of Te atoms (n), graph of m = 0 is also included for comparison
Fig. 12 The adiabatic detachment energy of Pbn-mSnmTen (m = 5–9) clusters as a function of the number of Te atoms (n), graph of m = 0 is also included for comparison
Fig. 13 The vertical detachment energy of Pbn-mSnmTen (m = 5–9) clusters as a function of the number of Te atoms (n), graph of m = 0 is also included for comparison
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For most clusters, beyond m = 4, the substitute site of the extra tin atoms is on the four-fold coordinated lead atoms. Thus, there is a significant structural distortion observed on the Sn–Te bond length for the five and six tin atom-doped lead telluride clusters. The geometries of these clusters are almost similar to the corresponding four tin atom-doped clusters presented in the previous sections. The resulting low-lying structures (the most preferred doping sites of the tin atom) of Pbn-5Sn5Ten and Pbn-6Sn6Ten clusters with corresponding chemical composition are presented in Fig. 10. There are a total of eleven clusters for five tin atomdoped and nine clusters for the six tin atom-doped clusters with the smallest clusters being Pb5Sn5Te10 and Pb6Sn6 Te12, respectively (Fig. 16). The effect of substituting five and six lead atoms with tin atoms of (PbTe)n, (n = 10–20) clusters on their electronic properties such as HOMO– LUMO gap, binding energy, vertical and ADE, and fragmentation energy of losing PbTe dimer are summarized in Figs. 11, 12, 13, 14, and 15. Generally, it is observed that the HOMO–LUMO gap decreases with increasing number of doping tin atoms, but there are certain clusters where the HOMO–LUMO gap
Fig. 14 HOMO–LUMO gap of Pbn-mSnmTen (m = 5–9) clusters as a function of the number of Te atoms (n)
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Fig. 15 The fragmentation energy on losing of PbTe unit of Pbn-m SnmTen (m = 5–9) clusters as a function of the number of Te atoms (n), graph of m = 0 is also included for comparison
remains constant and even greater in some other clusters as shown in Figs. 3 and 14. The maximum change of HOMO– LUMO gap, 0.07 eV, upon doping is observed on Pb11 Sn5Te16 and Pb13Sn6Te19 clusters as shown in Fig. 14. Similar to the previous doped clusters, the change in binding energy upon doping is almost negligible in most of the clusters. The binding energy values as a function of the number of tellurium atoms are plotted in Fig. 11. Both adiabatic and VDE increase almost for all the clusters, a maximum change being observed on Pb11Sn5Te16 and Pb10Sn6Te16 clusters (Figs. 12, 13). The fragmentation energy of losing PbTe molecule changes as an even–odd alternation. Similar to the previous clusters, it is observed that the fragmentation energy of clusters that have even number of tellurium atoms is slightly less than those with odd number of tellurium atoms, the maximum change being on Pb11Sn5Te16 cluster. The fragmentation energy of losing PbTe molecule as a function of the number of tellurium atoms are plotted in Fig. 15. Structural and energetic properties of Pbn-mSnmTen, (n = 14–20, m = 7–10) clusters For all clusters of these species, the substituting site is fourfold coordinated lead atoms. The resulting low-lying structures (the most preferred doping sites of the tin atom) of seven, eight, nine, and ten tin atom-doped clusters with corresponding chemical symbol are presented in Fig. 16. There are a total of seven Pbn-7Sn7Ten and five Pbn-8 Sn8Ten structures where the smallest ones are Pb7Sn7Te14 and Pb8Sn8Te16, respectively. The effect of substitution of more than six lead atoms with tin atoms of (PbTe)n, (n = 14–20) clusters, on their electronic structure properties such as HOMO–LUMO gap, binding energy, vertical and ADE, and fragmentation energy on losing of PbTe dimer are summarized in Figs. 11, 12, 13, 14, and 15.
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In Fig. 14, we show the HOMO–LUMO gap as a function of the number of tellurium atoms. In all of the tindoped clusters of these species, the HOMO–LUMO gap generally shows small variation in the second digit for the same number of tellurium atoms. The maximum change in the HOMO–LUMO being observed when the cluster is changed from Pb12Sn8Te20 to Pb11Sn9Te20 cluster. Similar to the previous tin-doped clusters, in most clusters, the change in binding energy is almost negligible. The binding energy values as a function of the number of tellurium atoms are plotted in Fig. 11. Similar to the previous doped clusters, both adiabatic and vertical detachment energy show a slight increment or remains constant for almost all Pbn-7Sn7Ten clusters, a significant change being observed only in Pb10Sn7Te17 cluster. The fragmentation energy on losing of PbTe dimer is almost similar in Pbn-8Sn8Ten and Pbn-9Sn9Ten clusters, hence we understand that there is no difference in thermodynamic stability between eight and nine tin atom-doped lead telluride clusters. In general, for all tin-doped clusters, due to electro negativity difference between Pb and Sn, the bond lengths of Sn– Te bonds are smaller than Pb–Te bonds. There is a considerable change observed on the HOMO–LUMO gap of the clusters, the HOMO–LUMO gap decrease with increasing the number of doped tin atoms, this is may be due to the difference in electron affinity between lead and tin atoms and electron affinity have a relation on the energy band of the clusters [15]. As shown in Fig. 11, doping enhances the binding energy of the clusters for corresponding number of Te atoms, the gap narrows with increasing cluster size. But, it can be easily observed in the same figure that the change in binding energy is almost negligible in the doped clusters for corresponding number of Te atoms, the change is about 0.02 eV for most tin-doped clusters. Both adiabatic detachment energy (ADEs) and vertical detachment energy (VDEs) increase as the number of doping tin atoms increase, this is also due to the difference in electron affinity between lead and tin atoms. Similar to the previous clusters, the fragmentation energy of losing PbTe molecule changes as an even–odd alternation.
Conclusions In this work, tin-doped lead telluride clusters are studied using density functional theory. Lead telluride nanoparticles are applied for thermo-electric device, infrared detectors, and thermal imaging. When these materials are doped sufficiently, the power generated of harvesting waste heat and detecting the infrared energy emitted by objects is increased. All these applications have relations with HOMO–LUMO gap, we can lower the HOMO–LUMO gap of (PbTe)16 cluster by 0.24 eV with doping eight tin atoms,
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Fig. 16 Optimized geometry of Pbn-mSnmTen, (n = 14–20, m = 7–10) clusters with corresponding chemical composition
but the band inversion which is observed in the bulk is not observed on the clusters at this size. Thus, one might expect a significant change in HOMO–LUMO gap of lead
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tin telluride clusters with appropriate tin composition when they undergo a structural transition from a two-dimensional-layered structure to a bulk-like structure.
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In conclusion, tin atoms prefer to be doped on the exterior sites rather than the interior sites of the lead telluride clusters. The geometry of the clusters is not affected by the amount of tin content except for a slight structural distortion which is observed by the bond length difference. All energetic properties are proportional to the number of doping tin atoms (m), for most clusters the HOMO–LUMO gap decrease as the number of doping tin atoms (m) increase. The binding energy of clusters is observed to be enhanced upon doping an indication of increased stability. The change in binding energy between doped clusters for the same number of Te atoms is almost negligible; it is about 0.02 eV for each successive doping of tin atoms. Both ADEs and VDEs increase for all clusters as the number of doping tin atoms (m) increase. The fragmentation energy of losing PbTe dimer changes as an even–odd alternation, it is observed that the fragmentation energy of clusters with even number of tellurium atoms is greater than those with neighboring odd number of tellurium atoms as the number of doping tin atoms (m) increase. Hence, we understand that the thermodynamic stability of clusters with even number of tellurium atoms increases as compared to the odd number of Te atoms. Some clusters are found to be more stable (example at n = 4) as they need higher energy to dissociate and some clusters are less stable (example at n = 5 and 17) because they can easily dissociate with less energy. The stability of such clusters is confirmed through their binding energy and HOMO– LUMO gap values. The most stable clusters are termed as ‘‘magic’’ clusters.
585 Acknowledgments The author acknowledges the Mekelle University Physics Department for providing computational facilities. Y. Mulugeta gratefully acknowledges the Ministry of Education of the Ethiopian Government for financial support.
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