J. Phys. Chem. C 2008, 112, 6307-6312
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Size-Dependent Properties of Hollow ZnS Nanoclusters Sougata Pal, Biplab Goswami, and Pranab Sarkar* Dept. of Chemistry, VisVa-Bharati UniVersity, Santiniketan -731235, India ReceiVed: NoVember 29, 2007; In Final Form: February 9, 2008
We present the results of our theoretical calculations on structural and electronic properties of hollow ZnnSn clusters as a function of the size of the clusters. Our results include the variation of radial distribution, Mulliken population, electronic energy levels, and band gap as a function of size for both single-layer and double-layer hollow ZnS nanoclusters. The band gap of both single- and double-layer hollow clusters passes through a minimum when studied as a function of the size of the cluster. We have also found that clusters with symmetric charge distribution have higher band-gap values compared to clusters with asymmetric charge distribution. We also compared the properties of the hollow clusters with those of bulk-like clusters.
I. Introduction In recent years, there has been widespread interest in the study of semiconductor nanomaterials.1-5 The interest in semiconductor nanomaterials stems from the fact that new properties are acquired at this length scale and it is equally important that these properties change with the size as well as shape of the nanocrystals. These size- and/or shape-dependent properties of semiconductor nanocrystals, which arise from the quantum confinement effect promise new inventions and new materials with new interesting properties. The past couple of decades have witnessed an exponential growth of activities in this field driven both by the excitement of understanding new science and by the potential hope for technological applications and economic impacts. The largest activity in this field at this time has been in the synthesis of new nanoparticles of different sizes and new shapes. The small size and large diversity in shapes of nanostructures are particularly attractive for exploring many unique and novel properties. Much progress is currently underway directed toward the growth and synthesis of nanocrystals of a particular shape.6-10 Recently, studies on the synthesis and properties of both metal and semiconductor hollow nanocrystals become very popular because hollow nanocrystals offer possibilities in material design for applications in catalysis, nanoelectronics, nano-optics, drug delivery systems, and as building blocks for lightweight structural materials.11-21 The ability to manipulate the structure and morphology of hollow materials on a nanometer scale would enable greater control of local chemical environments. In particular, the hollow nanocrystals of ZnS have been the subject of recent study22-31 for their potential applications in high hydrogen storage, medical diagnostics, drug delivery, and as a second-order template because of their large capacity and solubility. Thus, Ma et al.22 reported ZnS hollow nanospheres that were fabricated by employing triblock copolymers as soft templates. Zhang et al.23 reported the synthesis of hollow ZnS nanospheres by templating with in situ generated bubbles at low temperature. They have shown that UV-vis absorption spectra exhibit alarge blue shift because of quantum confinement effects. Very recently, Wolosiuk et. al24 and Yan et. al25 have fabricated hollow ZnS architectures through the sacrificial template route. Although * To whom correspondence is addressed. E-mail:
[email protected].
there are lots of experimental studies on the synthesis and characterization of hollow ZnS nanocrystals, theoretical studies addressing the electronic structure of hollow ZnS semiconductor nanocrystals are scarce.32,33 However, extensive theoretical studies, in particular, the evolution of electronic structure as a function of the size of the nanocrystal are of crucial importance because it allows one to both investigate fundamental physics and to optimize nanostructured devices sometimes offering new results to be verified experimentally. In what follows, we propose here to study the electronic structure of hollow ZnS semiconductor nanocrystals. We shall consider both single-layer and double-layer hollow nanocrystals and address various structural and electronic properties, namely radial distribution, Mulliken population, density of states, and band gap of these hollow ZnS nanocrystals and also their variation as a function of the size of the nanocrystals. The article is organized as follows. In section II, we briefly outline the density-functional method as used in the present work. We devote section III to display the results of our calculation on various finite clusters and section IV contains a brief summary of our findings. II. Theoretical Method We have employed in this work the parametrized densityfunctional tight-binding(DFTB) method of Porezag et al.,34-36 which has been described in detail elsewhere and, therefore, here shall be only briefly outlined. The approximate DFTB method is based on the density-functional theory of Hohenberg and Kohn in the formulation of Kohn and Sham.37,38 In this method, the single-particle wave functions Ψi(r) of the KohnSham equations are expanded in a set of atomic-like basis functions φm with m being a compound index that describes the atom at which the function is centered, the angular dependence of the function, as well as its radial dependence. These functions are obtained from self-consistent density functional calculations on the isolated atoms, employing a large set of Slater-type basis functions. The effective Kohn-Sham potential Veff(r b) is approximated as a simple superposition of the potentials of the neutral atoms Veff(r b) ) ∑jV0j (|r b- B Rj|) Furthermore, we make use of a tight-binding approximation, so that 〈φm|V0j |φn〉 is non-vanishing only when φm and/or φn is centered atR Bj.
10.1021/jp711296a CCC: $40.75 © 2008 American Chemical Society Published on Web 04/02/2008
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Figure 1. Optimized geometries of (a) Zn57S57, (b) Zn86S86, (c) Zn116S116 single-layer hollow clusters, (d) (Zn16S16)Zn57S57, and (e) (Zn16S16)Zn86S86, (Zn16S16)Zn116S116, double-layer hollow clusters.
Then, the binding energy is approximated as the differences between the single-particle energies of the occupied orbitals of the compound and those of isolated atoms augmented with shortrange pair potentials, occ
EB ≈
1
Ujj′(|R Bj - B Rj′|) ∑i i - ∑ ∑j jm + 2 j∑ m * j′
(1)
where i is the energy of the ith orbital for the system of interest and jm is the energy of the jth orbital for the isolated mth atom. Finally Ujj′(|R Bj - B Rj′|) is a pair potential and is determined from exact density-functional calculations on diatomics, that is, in our study on the ZnS, Zn2, and S2 molecules. In our calculations, only the 3d and 4s electrons of zinc and 3s and 3p electrons of sulfur are explicitly included, whereas the other electrons are treated within a frozen-core approximation. As the computational method is parametrized, therefore, to test its transferability to larger systems we first performed calculations on infinite, periodic, crystalline structures. This led to optimized lattice
constant 5.46 Å for the zinc-blende crystal and lattice constants of 3.84 and 6.27 Å for wurtzite crystal. The corresponding experimental values for the zinc-blende and wurtzite structures are 5.41, 3.811, and 6.134 Å, respectively. Thus, the structural parameters of the infinite crystalline material are well reproduced. As a further test of the accuracy, we also calculated the bulk moduli for the two crystal structures and found 84 and 76 GPa for zinc-blende and wurtzite structure, respectively. The experimental value for zinc-blende bulk ZnS, 77.1 GPa, is in good agreement with our value. III. Results and Discussions We have started with predefined hollow structures of ZnnSn clusters of different sizes and optimized these structures by using the method of conjugate gradient and/or steepest descent method by allowing all of the atoms to move. In Figure 1 we have shown the optimized structure of only few selected single (a-c) and double-layer (d-f) hollow ZnS clusters. We have made a comparison of the total energy/pair of ZnnSn clusters of different
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TABLE 1: Total Energies of ZnnSn Clusters of Different Shapes
cluster size
energy/pair of hollow structure (eV)
energy/pair of zinc-blende structure (eV)
energy/pair of Wurtzite structure (eV)
10 16 37 57 68 86
-184.331 -184.334 -184.420 -184.439 -184.393 -184.402
-184.257 -184.160 -184.195 -184.372 -184.346 -184.350
-184.239 -184.134 -184.373 -184.384 -184.214 -184.306
stoichiometries and also of different shapes in Table 1. From the values in energy/pair in Table 1, we see that hollow clusters are energetically more stable than the bulk-like clusters of similar size. Another feature that is evident from the table is that the energy difference between bulk-like clusters and the hollow clusters decreases with increasing cluster size. One of the interesting features of the hollow clusters is that all of the atoms are on the surface and the average coordination number of both zinc and sulfur is three compared to the average coordination number four in bulk-like clusters. The average Zn-S bond lengths in the single-layer hollow structure varies between 2.22 and 2.32 Å with the majority of the bonds of 2.24 Å in length. This length is shorter compared to 2.34 Å in zincblende and wurtzite clusters, which is primarily a consequence of the lower coordination numbers in the clusters compared to bulk. The average Zn-S bond length in the inner layer of the double-layer hollow cluster is 2.34 Å and that of the outer layer is 2.28 Å. These bond lengths are consistent with earlier reports on hollow ZnS nanoclusters.32 We have shown in Figure 2 the binding energies of the hollow, zinc-blende and wurtzite clusters as a function of the size of the clusters. The binding energy of the hollow clusters is relatively higher than that of bulk-like clusters of similar size, implying the higher stability of hollow clusters compared to bulk-like clusters. From the values in Table 1, it is also evident that the energy differences between hollow clusters and those of zinc-blende and wurtzite clusters first increase with increasing cluster size and then decreases. The
higher stabilities of hollow clusters compared to bulk-like clusters may be related to symmetry of the clusters. The bulklike clusters have many unsaturated bonds and also reduced symmetry compared to hollow clusters, which have less unsaturated bonds. The smaller hollow clusters are relatively more stable compared to larger clusters because of the presence of a large number of lower coordinated atoms. To discuss the different structural and electronic properties, we first define the radial distribution as follows. For each atom of the ZnnSn cluster, we calculate the distance to the center of the cluster,
B R0 )
1
2m
∑
2n nj ) 1
B Rj
(2)
where the summation runs over all atoms of the cluster. Subsequently, the radial distance for the jth atom is defined as
Bj - B R 0| rj ) |R
(3)
Figure 3 shows the radial distributions of some selected single-layer (left panel) and double-layer (right panel) hollow ZnS clusters. The figure shows the radial distances for both zinc and sulfur atoms of the optimized structure. The initial geometry of clusters were nearly spherical, but, in optimized geometry, the structure is little deformed and elongated. From the figure, it is evident that the number of zinc and sulfur atoms are more in the middle of the clusters than at the two ends. Inspecting the radial distribution of the zinc and sulfur atoms separately(not shown), it is found that zinc and sulfur atoms that were on the same plane in the initial geometry are now a little displaced, sulfur being moved outward and zinc atoms moved inward. The radial distribution of the double layer (right panel of the figure) suggests that for smaller size clusters there are enough interactions between the atoms in the inner layer and the outer layer. However, for large sized clusters this interaction is less, and the distribution of zinc and sulfur atoms show similar behavior to that of the single layer.
Figure 2. Variation of binding energies as a function of the size of the clusters for (a) single-layer hollow, (b) zinc-blende, and (c) wurtzite ZnS clusters.
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Figure 4. Radial distribution of the Mulliken population for the same set of clusters as shown in Figure 2. Figure 3. Radial distribution of zinc and sulfur atoms for single-layer (left panel) and double-layer (right panel) hollow ZnS clusters of different sizes: (a)Zn10S10, (b)Zn37S37, (c)Zn68S68, (d)Zn86S86, (e) Zn116S116, (f) (Zn16S16)Zn57S57, (g) (Zn16S16)Zn68S68, (h) (Zn16S16)Zn86S86, (i) (Zn16S16)Zn98S98, and (j) (Zn16S16)Zn116S116.
In Figure 4, we have shown the Mulliken populations for the individual atoms as a function of their radial distance as defined in eq 3. Only the valence electrons are included, that is, for the neutral atoms these numbers would be 12 for zinc and 6 for sulfur, and we would like to mention that the analysis based on Mulliken population is very much qualitative in nature. From this figure, it is also evident that the outermost atoms in the clusters are sulfur atoms, whereas zinc atoms are lying relatively in the inner part of the clusters, and this feature is more prominent for smaller sized clusters than for those of larger size. For both single-layer and double-layer hollow clusters (especially smaller), there is nearly same charge transfer throughout the whole region of the clusters. This feature is in sharp contrast to that of bulk clusters,39 namely zinc-blende and wurtzite clusters where there is relatively large charge transfer in the surface region compared to the inner part of the clusters. The same charge transfer throughout the whole region of the clusters is because of the fact that all of the atoms in the singlelayer hollow clusters are on the surface and have the same kind of bonding. To have a quantitative estimate of the chargetransfer, we have calculated the values of average charge transfer from zinc to sulfur for different sized clusters, and it is found that the average charge transfer first decreases with the size of the cluster and then increases again. This difference in the magnitude of the charge transfer has far reaching consequences on the optoelectronic properties of the clusters and will be
discussed in the subsequent section. However, for double-layer hollow clusters there are relatively large charge transfers in the outer layer compared to inner layer, and the average charge transfer follows the same trend as that of single-layer hollow clusters. In Figure 5, we have shown the total density of states, obtained by broadening the individual electronic states slightly with Gaussian for same set of clusters as in Figure 2. The general features are more or less the same for all clusters. The bands corresponding to 3s functions of sulfur lie in the range between -19 and -16 eV, the one corresponding to zinc 3d lie in the range between -12 and -10 eV, and the uppermost occupied band between -9 and -6 eV is formed mainly as a result of the sulfur 3p and partly by zinc 4s functions. One interesting feature of the DOSs of hollow ZnS nanocrystals is that there are no states in the band-gap region, which was found in bulklike clusters.39 For the double-layer hollow clusters, the general features of DOSs are the same as those of single-layer clusters. The inner layer has a major contribution to the valence band region. Figure 6 shows the variation in the HOMO-LUMO energy gap of hollow clusters (∆) as a function of the cluster size (ZnS pair). For a comparison, we also show the variation of bandgap values of zinc-blende (O) and wurtzite (/) clusters. The values of the band gap of smaller and large-sized hollow clusters are higher than those of zinc-blende, wurtzite clusters. This result suggests that the quantum confinement effect is more important for smaller hollow clusters compared to bulk-like clusters. The calculated band-gap values of hollow clusters are close to the calculated values of Hamad et al.32 if we notice that a shift of 1.5 eV (the difference between the experimental and calculated
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Figure 5. Density of states (DOSs) for the same set of clusters as shown in Figure 2.
band-gap values of the bulk crystal) has been given to each calculated values. Our calculated band-gap values are a little lower compared to the experimental values of Zhang et al.,23 and this may be because of the underestimation of band-gap values in density-functional calculation. The band gap of singlelayer hollow clusters decreases with the size of the cluster, consistent with the quantum-size effects; however, after a certain size it starts to increase. The increase in band-gap values after a certain size of the clusters may be related to the presence of curvature-induced σ-π hybridization,40,41 a phenomenon that is more familiar with the tubular form of nanostructures. The hollow clusters and tubular form of nanostructures are similar in view in that in both cases all the atoms are on the surface and possess a certain degree of buckling (inward and outward movement of all of the zinc and sulfur atoms, respectively). However, it is worth mentioning that the nature of buckling in hollow structures are much more complicated than in the tubular structures since in later case the buckling is only along xy plane. The presence of buckling in hollow structures induces the curvature-induced σ-π hybridization. The minimum in the figure is a result of a compromise between the two competing effects, that is, quantum confinement effects (which cause a decrease in band gap with an increase in size) and curvatureinduced σ-π hybridization (which causes an increase in band gap with size). However, the band-gap values of bulk-like clusters showed an overall decreasing trend with the cluster size. The band-gap values of double-layer hollow clusters show behavior similar to those of single-layer hollow clusters. The band-gap values of single-layer hollow Zn57S57 and Zn116S116
Figure 6. Variation of the band gap as a function of the size of the clusters for both (a) single-layer hollow (4), zinc-blende (O), wurtzite (/), and (b) double-layer hollow ZnS clusters.
clusters are 2.970 and 4.128 eV respectively and those of doublelayer hollow cluster (Zn16S16)Zn57Sn57 and (Zn16S16)Zn116S116 are 2.901 and 4.131 eV, respectively. So, we may conclude that, for the smaller cluster, the presence of an inner layer essentially causes the band gap to decrease, but for larger clusters the presence of an inner layer has little effect on the band gap. Hamad et al.33 also reported lower band-gap values of doublelayer hollow clusters compared to single-layer hollow clusters. A careful analysis of the average values of the Mulliken population and the values of the band gap for clusters of different sizes reveals an interesting correlation between the two; the higher the charge transfer, the higher the band gap will be. So, the variation of band-gap values with the size of the cluster follows the same trend as that of the Mulliken charge transfer from zinc to sulfur. A careful analysis of the Mulliken population of an individual atom in a particular cluster shows that the magnitude of charge transfer from zinc to sulfur follows a distribution pattern. The magnitude of the charge transfer from zinc to sulfur in a particular cluster show many different values. Those clusters that have a large number of different kinds of charge-transfer values from zinc to sulfur have lower band-gap values, and we can interpreted this by saying that clusters having less symmetric charge distribution have lower band-gap values. On the other hand, a cluster having more symmetric charge distribution has higher band-gap values.
6312 J. Phys. Chem. C, Vol. 112, No. 16, 2008 IV. Conclusions Inspired by recent promising experimental studies on hollow ZnS nanocrystals, we here performed a theoretical study on the electronic structure of hollow ZnnSn clusters as a function of the size of the nanocrystallite. From the study of both the total and binding energies, it is seen that hollow clusters are more stable than that of bulk-like clusters. The study of Mulliken population suggests that the average values of charge transfer are higher for hollow clusters compared to those of bulk-like clusters and they first decrease with the size of the cluster and then increase, for the cluster size studied here. The band gap of both single-layer and double-layer hollow clusters passes through a minimum when studied as a function of the size of the cluster. The band-gap values we have calculated are in good agreement with those of experimental values of Zhang et al.23 We have found that those clusters that are have a more symmetric charge distribution have higher band gaps compared to clusters with asymmetric charge distribution. In total, our study reveals some new and interesting properties of hollow ZnS nanocrystals, and we believe our theoretical results will stimulate the experimentalists for further exploration of this nanomaterial. Acknowledgment. The authors would like to express sincere thanks to Prof. Said Hamad, Davy Faraday Research Laboratory, U.K., for giving us the coordinates of their hollow ZnS clusters. The financial supports from CSIR, Govt. of India [01(2148)/ 07/EMR-II] and UGC, New Delhi (through SAP), Govt. of India, through research grants are gratefully acknowledged. S.P. is grateful to CSIR, New Delhi, for the award of Senior Research Fellowship (SRF). This article is dedicated to my (P.S.) teacher Prof. S. P. Bhattacharyya, IACS, Jadavpur, India, on the happy occasion of his 60th birthday. References and Notes (1) Heath, J. R.; Shiang, J. J. Chem. Soc. ReV. 1998, 27, 65. (2) Alivisatos, A. P. J. Phys. Chem. 1996, 100, 13226. (3) Hoffman, M. R.; Martin, S. T.; Choi, W.; Bahnemann, D. W. Chem. ReV. 1995, 95, 69. (4) Henglein, A. Topics in Current Chemistry, Vol. 143; Springer: Berlin, 1988. (5) Burda, C.; Chen, X.; Narayanan, R.; El-Sayed, M. A. Chem. ReV. 2005, 105, 1025. (6) Murray, C. B.; Norris, D. J.; Bawendi, M. G. J. Am. Chem. Soc. 1993, 115, 8706. (7) Peng, X. et al. Nature 2000, 404, 59.
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