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Magic Numbers in a One-Dimensional Nanosystem: ZnS SingleWalled Nanotubes Norawit Krainara,†,‡,§,∥ Jumras Limtrakul,‡,§,∥ Francesc Illas,† and Stefan T. Bromley*,†,⊥ †

Departament de Química Física & Institut de Química Teòrica i Computacional (IQTCUB), Universitat de Barcelona, C/Martí i Franquès 1, E-08028 Barcelona, Spain ‡ Department of Chemistry and NANOTEC Center for Nanoscale Materials Design for Green Nanotechnology, Kasetsart University, Bangkok 10900, Thailand § Center for Advanced Studies in Nanotechnology and Its Applications in Chemical, Food and Agricultural Industries, Kasetsart University, Bangkok 10900, Thailand ∥ PTT Group Frontier Research Center, PTT Public Company Limited, 555 Vibhavadi Rangsit Road, Chatuchak, Bangkok 10900, Thailand ⊥ Institució Catalana de Recerca i Estudis Avançats (ICREA), 08010 Barcelona, Spain ABSTRACT: The structure and stability of a range of zigzag (n,0) and armchair (n,n) ZnS SWNTs of increasing diameter are investigated theoretically using density functional theory. Zigzag and armchair SWNTs are found to be most energetically stable when considerably structurally reconstructed with respect to the corresponding metastable relatively smooth SWNTs. A strong relation between the average Zn− S(Zn)−Zn dihedral angles in the SWNTs and their relative energetic stability points to the out-of-plane tendency of the highly polarizable S anions as the driving force behind the observed distortions. The internal energetic strain in reconstructed armchair ZnS SWNTs is found to gradually monotonically decrease toward that in a reconstructed single 2D ZnS sheet with increasing diameter. This tendency follows that in previously reported calculations for both armchair and zigzag SWNTs of C, BN, MoS2, and TiO2. For reconstructed zigzag ZnS SWNTs, however, the general decrease in energetic strain with increasing diameter possesses clear oscillations, with (3a,0) SWNTs possessing the highest stability and largest band gap. We compare this discrete pattern of stability with that often encountered for particularly stable nanoclusters possessing “magic” numbers of atoms. As far as we are aware, this is the first report of “magic” numbers being predicted to occur in a 1D nanosystem.

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INTRODUCTION The realization of exotic low-dimensional (0D−2D) allotropes of carbon at the molecular 0D level (e.g., fullerenes1) and with 1D or 2D periodicity (e.g., single-walled nanotubes (SWNTs)2,3 and graphene,4 respectively) has triggered a revolution in the field of materials science. Taking inspiration from the structural richness exhibited by such carbon-based materials, one may enquire whether other materials may display similar behavior. As for carbon in the case of graphite, the thermodynamically driving force to form quasi-2D bulk crystal structures consisting of weakly interacting layers in a number of other materials (e.g., BN, MoS2) has greatly assisted experimental efforts to prepare low-dimensional nanosystems with numerous potential applications.5,6 For the specific case of SWNTs, formed by a single 2D sheet of atoms or constituent molecular units, examples have been experimentally prepared in the cases of BN7 and MoS2.8 More generally, a theoretical modelling approach gives one the freedom to investigate if materials which do not inherently form layered bulk polymorphs are: (i) structurally stable as single 2D sheets and/or SWNTs and, (ii) if so, whether they also follow similar © 2013 American Chemical Society

trends to those of systems observed in experiment. For a group of six materials having the dense nonlayered wurtzite structure (AlN, BeO, GaN, SiC, ZnO, and ZnS), theoretical studies have predicted nanoscale thin films to energetically prefer to exhibit a polymorphic structure consisting of relatively weakly bound planar hexagonal sheets, each analogous to the layered hexagonal form of BN.9 In the case of ZnO, this prediction has been experimentally confirmed for ultrathin films containing 1−3 atomic layers supported on Ag(111).10 For ZnO 11 and SiC 12 calculations have also shown that unsupported monatomic sheets should be most energetically stable when perfectly planar. A slightly different situation is encountered for elemental Si and Ge where theory predicts that the single-sheet graphene-like structures (silicene and germanene) seem to be unstable with respect to a slight puckering of the sheet with atomic displacements of ±0.22 Å (Si) and ±0.32 Å (Ge) with respect to a perfect plane.13 The existence of this Received: July 31, 2013 Revised: September 25, 2013 Published: September 25, 2013 22908

dx.doi.org/10.1021/jp407625p | J. Phys. Chem. C 2013, 117, 22908−22914

The Journal of Physical Chemistry C

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From our wide structural overview we extract stability versus diameter relationships for all ZnS SWNTs considered and compare them with those previously calculated for other SWNT systems (e.g., C,30 BN,30 TiO2,31 MoS2,32CeO2,33 SiTiO334). Unlike in these latter examples, where the energetic strain (i.e., energy relative to the corresponding 2D sheet) tends to vary in a smooth fashion with increasing SWNT diameter for both armchair and zigzag SWNTs, we find significant regular variations in the zigzag ZnS SWNT strain energy with increasing diameter. These peaks and troughs of energetic stability occur at well-defined SWNT sizes and are analogous to the magic stability peaks well-known for 0D nanoclusters with increasing size. Further, the pattern of energetic stability is also followed by a similar variation in the band gaps of the zigzag ZnS SWNTs. Geometric analysis reveals an underlying link between the detailed structures of the SWNTs and their stability. As far as we are aware, this is the first time that a pattern of magic stability with change of size has been predicted for a 1D nanosystem.

slight distortion is supported by molecular beam epitaxy synthesis of silicene on a silver substrate 14 and in experimentally observed Si SWNTs.15 That silicon is found as the dense diamond Si polymorph under ambient thermodynamic conditions also indicates, in principle, that the fabrication of 2D sheets and SWNTs may be extended to many other materials that do not tend to exhibit layered bulk phases. In previous work we employed electronic structure calculations to investigate single hexagonal sheets and a zigzag (12,0) SWNT of the technologically important II−IV chalcogenide semiconductor material zinc sulfide (ZnS).16 In this study, unlike the above examples, we predicted that a 2D sheet and the SWNT would undergo significant energy lowering structural reconstructions with respect to a perfectly planar 2D sheet or a slightly puckered SWNT, respectively (relative atomic displacements of ±1.95 Å). We note that studies prior to ours only reported ZnS SWNTs possessing relatively smooth structures with a very small amount of puckering17−20 which are, in fact, metastable local minima. We attributed the cause of this large structural distortion to the high out-of-plane anionic polarizability which we expect to be a common feature of numerous materials containing large anions (e.g., chalcogenides). We note that for chalcogenides such as MoS2 the 1:2 stoichiometry allows for the structurally distorting tendency of the S anions to move out-of-plane to be compensated by having S ions symmetrically protruding either side of a planar array of Mo ions. In the case of ZnS, however, the planar single sheet energetically prefers to reconstruct and to take upon a structure reminiscent of the outermost atomic layer of the (112̅0) surface of ZnS wurtzite.16 We further showed that the application of compressive and/or tensile strain can reversibly switch between the reconstructed ZnS sheet and SWNT systems and their smoother structured higher laying energy minima. This mechanically induced transition between two structural stable states is also coupled to significant direct band gap changes.16 We also note that, similar to experimentally prepared inorganic fullerenes of numerous materials (e.g., C, BN, MoS2),21 cagelike clusters of ZnS have been theoretically predicted to be energetically favored over dense clusters up to (ZnS)50.22,23 Clearly, ZnS represents a fundamentally interesting case study of a representative 2D sheet-based system that displays significantly different structural behavior to other known 2D materials. Moreover, ZnS itself is a very technologically interesting material where many experimental nanotechnological advances are being made.24−26 Considering, for example, that nanoscale ZnS structures have impressive optoelectronic properties27 and the continuing advances in ZnS synthesis nanofabrication and nanomanipulation (e.g., synthesis of arrays of single crystalline ZnS nanosheets28 and nanotubes24 and manipulation of the tensile properties of ZnS nanowires29), a more in-depth study of the fundamental properties of ZnS nanostructures is both timely and valuable. Here, significantly extending our previous study considering the 2D hexagonal ZnS sheet and a single ZnS SWNT, we present a detailed analysis of the energetics and structure of 19 ZnS SWNTs of both zigzag (0,n), n = 3−18, and armchair (m,m), m = 3−11, types, following the vector naming convention for carbon SWNTS. In all cases, significantly reconstructed SWNT structures are energetically preferred over relatively smooth slightly puckered structures. For all but the smallest two SWNT diameters considered, we find that the reconstructed zigzag SWNTs are the most energetically stable.

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COMPUTATIONAL METHODOLOGY The optimized structures and properties of armchair and zigzag ZnS SWNTs were investigated by means of a periodic density functional theory (DFT) approach using the Vienna Ab initio Simulation Package (VASP) code.35 All calculations used the PW9136 implementation of the generalized gradient approach form of the exchange-correlation potential and a plane wave basis set with a kinetic energy cutoff of 500 eV. The effect of the inner cores of S and Zn on the valence electron density was taken into account by the projector augmented wave method.37 The interaction between systems arising from the use of threedimensional periodic symmetry was reduced up to a negligible degree by ensuring a 15 Å separation between repeated images. A 1 × 1 × 9 Monkhorst−Pack38 mesh of special k-points was used throughout to carry out the numerical integration in reciprocal space of the one-dimensional periodic SWNTs. The initial geometries of the SWNTs were formed from rolling the smooth form of the 2D ZnS graphene-like hexagonal sheet following the appropriate roll-up vector. The number of ZnS units used in the supercells was 4m for the (m,m) armchair SWNTs and 2n for the (n,0) zigzag SWNTs so that, for example, both (3, 3) and (6, 0) supercells contained 12 ZnS units. Both the internal atomic structure and supercell dimensions were relaxed until forces acting on the atoms were smaller than 0.005 eV/Å. We employed three methods in order to search for low-energy reconstructed SWNT structures starting from the as-constructed smooth SWNTs. First, following the experimental effectiveness of mechanical annealing of small systems,39 we used our simulated mechanical annealing technique.16,40 Hereby, the SWNTs were gradually compressed and stretched in a stepwise manner along their length by up to ±15% while optimizing the atomic positions at each step. Each time a structural change occurred, the procedure was repeated around the newly obtained minimum-energy configuration until no further change in structure occurred. Second, based upon a structural analysis of the most stable reconstructed SWNTs found from the first method, we constructed new SWNTs following “building” principles which appeared to lead to stable reconstructed ZnS SWNTs. This method entailed identifying characteristic repeated structural subunits in particularly stable SWNTs and using them as bottom-up building blocks to construct candidate new SWNT structures which were then optimized using DFT. Lastly, in 22909



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energetic strain with increasing diameter with the most stable limit being the flat single sheet. Clearly, the structural energetics of relatively smooth ZnS SWNTs is significantly different than that in those materials where the most stable 2D single sheet is perfectly planar. For finding lower energy reconstructed zigzag ZnS SWNTs, the simulated mechanical annealing method was found to be most successful in the case of the (18,0) ZnS SWNT. Specifically, under a 4% length compression of the asconstructed smooth (18,0) a new more stable reconstructed structure was obtained. The main effect of the compressioninduced reconstruction was to symmetrically displace a proportion of the S atoms toward the axial center of the tube, which, in turn, modified the cross section of the SWNT from circular to octagonal. The reconstructed (18,0) SWNT maintained 3-coordinated Zn and S centers whereas the bond angles changed considerably. The cross section of the reconstructed (18,0) ZnS SWNTs is shown in Figure 2.

order to test the structural stability of the energy minima found using the first two methods, and potentially discover new minima, we used microcanonical molecular dynamics (MD) runs with suitably parametrized classical interatomic potentials41 using the GULP code.42 The SWNTs were first preheated to a range of initial temperatures between 500 and 2000 K. For each initial temperature, the system was then left to evolve for runs of 25−50 ps, during which various sample snapshots of structural configurations were taken and their structures separately energy-minimized at a classical level of theory, and then further optimized using DFT. Tests using larger supercells with more than one repeating unit along the length of the SWNTs using DFT-based simulated mechanical annealing and/or classical MD runs did not reveal any lower energy structures.

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RESULTS AND DISCUSSION We start by considering the results of the DFT electronic structure calculations corresponding to zigzag and armchair SWNTs constructed directly from flat hexagonal ZnS sheets. We note that although these SWNTs are based on flat hexagonal ZnS sheets, the resulting SWNTs show some small deviations from perfect smoothness. This small radial anion− cation buckling (±0.1 Å) in such nonreconstructed ZnS SWNTs has been noted previously16 and has also been observed in calculations of BN SWNTs.30 Our results predict that the energetic strain (relative to the flat ZnS sheet) in such relatively smooth ZnS SWNTs increases with their diameter indicating that the flat sheet is not the most stable limiting case for this system (see Figure 1) and that flat sheet-based SWNTs with the highest curvature are the most stable. Interestingly, for these planar-sheet-based ZnS systems the energetics of both armchair and zigzag SWNTs follow the same trend. For C, BN, and composites thereof,30 MoS2,32 and TiO2,31 the calculated energetic stability versus diameter trend for both armchair and zigzag SWNTs follows a distinct monotonic decrease in

Figure 2. Constituent subunits (Zn7S7: blue; Zn4S4: red) found in the (18,0) ZnS SWNT (left) and the reconstructed single sheet16 (bottom right) found from simulated mechanical annealing. We show how new SWNTs can be constructed using such subunits with the example of the lowest energy (12,0) SWNT found (top right).

Figure 1. Relationship between strain energy relative to the most energetically favorable reconstructed ZnS single sheet16 (eV/ZnS) and the diameters of the smooth and reconstructed armchair and zigzag ZnS SWNTs. The inherent strain in a planar 2D sheet is indicated by the dashed line. The magic (3a,0) zigzag SWNTs are also labeled.

Within the structure of the reconstructed (18,0) SWNT one can identify a number of Zn4S4 subunits (highlighted in red in Figure 2), which can, in turn, be found in the structure of the reconstructed single ZnS sheet as reported in a previous study.16 By directly joining these separate Zn4S4 subunits, one can derive the reconstructed 2D ZnS sheet. The reconstructed ZnS (18,0) SWNT actually possesses curved Zn7S7 subsections which can be viewed as two Zn4S4 subunits with a shared ZnS bridge (highlighted in blue in Figure 2). Both structural subunits were used to create trial reconstructed geometries for the other reconstructed zigzag SWNTs. In Figure 2, for example, a reconstructed ZnS (12,0) SWNT was created by connecting together four Zn7S7 subunits. The other reconstructed ZnS zigzag SWNTs of various sizes can be formed by varying the number of Zn7S7 subunits which can be either directly fused or spaced via monomeric ZnS bridges and/or Zn4S4 subunits. For each zigzag SWNT diameter a number of such constructed nanotubes were tested. In each case, after optimization with DFT calculations the reconstructed form was retained, which was always more energetically more stable than the corresponding slightly buckled smooth form. No more stable zigzag SWNTs were found from the classical MD tests. In the case of the smallest diameters, internal wall-to-wall steric effects slightly destabilized the as-built SWNT geometries due to inner S atom repulsion. The reconstructed and smooth 22910



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structures for a selection of zigzag SWNTs are reported in Figure 3, which clearly shows that the topology of the reconstructed (9,0), (12,0), and (15,0) ZnS SWNTs is reminiscent of the one found by mechanical annealing for the (18,0) case.

Figure 4. Structures of the lowest energy reconstructed armchair ZnS SWNTs found: (a) (3,3), (b) (5,5), (c) (6,6), (d) (7,7), (e) (9,9), and (f) (11,11).

Figure 3. Geometries of the magic reconstructed (9,0), (12,0), (15,0), and (18,0) ZnS SWNTs (a−d) and the corresponding (9,0), (12,0), (15,0), and (18,0) smooth SWNTs (e−h).

For the armchair ZnS SWNTs, as for the zigzag SWNTs, the simulated mechanical annealing method was most effective at finding a low-energy reconstructed structure for the larger (9,9) and (11,11) nanotubes. Unlike the corresponding (18,0) zigzag SWNT, the most energetically stable armchair (9,9) SWNT found, for example, has a 6-fold hexagonal cross section (see Figure 4). As for the zigzag SWNTs, a building unit could be identified from which a number of structures for different diameters could be tested. In the case of the armchair SWNTs, the reconstructed nanotube structures can be thought of as being constructed from connecting simple Zn2S2 units. The resulting reconstructed (n,n) ZnS SWNTs for n = 3, 5, 7, 9, and 11 are shown in Figure 4. The building scheme is also shown schematically in Figure 5 for the most energetically stable (3,3) and (5,5) armchair SWNTs found. As in the case of the zigzag SWNTs, all reconstructed nanotube structures thus constructed were found to be both structurally stable when optimized using DFT and all more energetically stable than correspondingly sized slightly buckled smooth armchair SWNTs. Similarly to the set of reconstructed zigzag SWNTs, no more stable armchair SWNTs were found from the classical MD runs. Compared to the near-planar SWNTs, the energetics of the reconstructed zigzag and armchair SWNTs have quite distinct trends of energetic strain with both now tending to stabilize with increasing diameter (see Figure 1). With respect to the planar ZnS, all SWNTs considered in this study have a negative strain energy (i.e., tube formation from the flat ZnS sheet is always energetically favorable). Such negative strain behavior has also been predicted for other theoretical SWNT systems (e.g., CeO2,33 SrTiO334). As the strain energy is calculated as

Figure 5. Schematic examples showing how the union of simple Zn2S2 building units can be used to construct reconstructed armchair SWNTs: (a) (3,3) and (b) (5,5).

the energy difference between a 2D sheet and a SWNT, it is essential to ensure that one carefully searches the potential energy landscape of the structures of both the SWNTs and the 2D sheet as thoroughly as possible. In particular, employing a metastable structure of the 2D sheet could lead to artificially low, and even negative, strain energies being erroneously predicted. In the present case of ZnS, in our previous work we found that a significantly structurally reconstructed 2D sheet was 0.084 eV/ZnS more stable than the perfectly planar sheet.16 This sheet bears some resemblance to a reconstructed bulk wurtzite ZnS surface layer but cannot simply be derived from a 2D layer from the bulk. When one uses this low-energy reconstructed single sheet as a reference for calculating strain energies, one finds that all ZnS SWNTs have positive (albeit essentially zero in some cases) strain. The strain energies of all considered ZnS SWNTs are shown in Figure 1. For the smallest diameters considered (between ∼0.6 and 0.9 nm) the armchair SWNTs are most stable. For larger diameters 22911



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(∼0.9−2.1 nm), however, the energetic strain of the armchair SWNTs does not further decrease and the zigzag SWNTs become more stable. Of particular interest, however, is the starkly contrasting oscillatory variation in the stability trend of the zigzag SWNTs which has clear dips for (9,0), (12,0), (15,0), and (18,0), suggesting generally higher relative stabilities for (3a,0). For the (12,0) and the (15,0) SWNTs, in particular, the strain energy is essentially zero with respect to the reconstructed 2D sheet. This pattern of strain is unlike the smoothly varying stability trends relative to diameter calculated for numerous SWNTs of other materials (see above) but is strongly reminiscent of that often observed for 0D nanoclusters in beams where particularly prominent abundance peaks are assigned to specifically sized clusters with a so-called “magic” number of atoms. As far as we are aware, this is the first report of such magic stabilty in a 1D nanosystem. The appearance of magic numbers in nanocluster systems can be influenced by a number of factors (e.g., connectivity and geometric closures for carbon fullerenes,43 closed-shell atomic packings,44 and/or electronic shell closure45 for metal clusters, and even possibly magnetism in nanoclusters containing openshell transition metal atoms46). As mentioned above, the root cause of the higher stability of strongly reconstructed singlesheet-based nanosystems for ZnS appears to be the out-ofplane preference of the large polarizable S anions. This results in slightly larger Zn−S bond lengths (0.1−0.2 Å) and reduced Zn−S(Zn)−Zn dihedral angles (see Figure 6) compared to the

Figure 7. Relationship between the deviation of the average Zn− S(Zn)−Zn dihedral angle from 109.47° (Δϕ) and the relative SWNT energy.

are particularly compatible with the natural atomic displacements of the wrapped-up ZnS sheet. We note, however, that the (9,0) and (18,0) zigzag SWNTs have a slightly larger average dihedral deviation from 109.47° (2.53° and 3.53°, respectively), which are also observed in the non-magic-sized (13,0) and (11,0) SWNTs, respectively. Thus, although the magic stabilities of the (3a,0) SWNTs seems to be linked with their structure, a more complete structural descriptor is required to capture the subtle yet definite energetic variations observed. In order to further investigate possible implications of the magically stable zigzag SWNTs structures, we have also extracted the band gaps of all SNWTs. In order to avoid possible artifacts due to the well-known problems of GGA exchange-correlation functionals to properly describe the band gap of narrow band systems,47−49 we have chosen to simply report the values relative to the band gap of the ground state zinc blende polymorph of ZnS calculated using a consistent DFT setup. From the set of results summarized in Figure 8 it is clear that the band gap of all the ZnS SWNTs explored in the present work is approximately 30−60% greater than that of bulk zinc blende. However, a closer inspection of Figure 8 also reveals that although all the near-planar SWNTs and all reconstructed armchair SWNTs have similar energy gaps for all diameters, the more energetically stable zigzag SWNTs all possess higher gaps. In addition, the energy gaps for the reconstructed zigzag SWNTs oscillate with those having the highest gaps also being the most energetically stable, and this occurs precisely for the (3a,0) structure with a = 3−6. In addition to further confirming the magic status of these particular sizes and types of ZnS SWNT, this regular pattern of energy gap variation (and its underlying structural cause) may have more general implications for band gap engineering of ZnS nanosystems for devices.

Figure 6. Schematic showing the Zn−S(Zn)−Zn dihedral angle (ϕ).

corresponding values in the near-planar SWNTs. The latter effect in particular leads to S anions being on the apex of trigonal pyramids with respect to their three Zn neighbors. In contrast, the smaller Zn cations tend to remain in locally planar arrangements with respect to their three S neighbors. In order to further analyze this structural effect, in Figure 7 we plot the relative energetic stability of all smooth and reconstructed armchair and zigzag SWNTs with respect to their average Zn− S(Zn)−Zn dihedral angles. The plot shows that there is a robust linear correlation (R2 = 0.94) between these two quantities which holds for all sizes and types of SWNTs considered. Within the 108°−148° range of observed angles, SWNTs with an average Zn−S(Zn)−Zn dihedral angle closer to the perfect tetrahedral angle of 109.47° are found to be more energetically stable. In particular, we note that all the zigzag SWNTs have the smallest average Zn−S(Zn)−Zn dihedral angles, all laying between 108° and 114°. Of the zigzag SWNTs, the (12,0) and (15,0) tubes have the smallest average dihedral deviation from 109.47° (−1.47° and 1.53°, respectively), which points toward the possibility that the reconstructed structures of these magic-numbered SWNTs 22912



ACKNOWLEDGMENTS

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REFERENCES

This work was supported by the Spanish Ministerio grants (FIS2008-02238 MAT2012-30924) and Generalitat de Catalunya grants (2009SGR1041 and XRQTC) and, in part, by grants from the National Science and Technology Development Agency (NSTDA Chair Professor and NANOTEC Center for the Design of Nanoscale Materials for Green Nanotechnology), the Kasetsart University Research and Development Institute (KURDI), the Commission on Higher Education, Ministry of Education (“the National Research University Project of Thailand (NRU)”, and “Postgraduate Education and Research Programs in Petroleum and Petrochemicals and Advanced Materials”). N.K. thanks the Office of the Higher Education Commission Thailand for supporting him with a grant under the program Strategic Scholarships for Frontier Research Network for the PhD Program Thai Doctoral degree and the Graduate School of Kasetsart University for his research. Time on the MareNostrum supercomputer (Barcelona Supercomputing Center) is also acknowledged.

Figure 8. Electronic band gap versus diameter for smooth and reconstructed armchair and zigzag ZnS SWNTs. The reported band gap values are relative to that of bulk zinc blende ZnS on the righthand scale. The percent increase of the gap with respect to that of zinc blende is given on the left-hand scale. The magic (3a,0) zigzag SWNTs are also labeled.

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CONCLUSIONS In this work we have used periodic density functional theory based calculations to explore the structure and energetic strain of a range of zigzag and armchair ZnS SWNTs of increasing diameter. Simulated mechanical annealing15,40 and inspiration from the resulting low-energy SWNTs and our previously reported reconstructed ZnS single sheet16 were used to search for low-energy SWNT structures. The most energetically stable structures found for both zigzag and armchair SWNTs are considerably structurally reconstructed with respect to the corresponding metastable relatively smooth SWNTs. The stabilizing structural reconstruction observed seems to mainly stem from the out-of-plane tendency of the highly polarizable S anions as is evidenced by the strong relation found between the average Zn−S(Zn)−Zn dihedral angles and relative energetic stability. Although the strain energy of reconstructed armchair SWNTs decreases monotonically with diameter, following the trend calculated for C, BN, and MoS2, TiO2 SWNTs, in the case of the reconstructed ZnS zigzag SWNTs the general decrease in strain with diameter possesses clear oscillations. The most stable zigzag SWNTs are found to be regularly spaced with respect to diameter and of type (3a,0), with the (12,0) and (15,0) SWNTs, in particular, having essentially zero internal strain. The special stability of (3a,0) ZnS SWNTs is also present in the electronic structure with oscillations in the band gap with the gap reaching maxima for the (3a,0) SWNTs. This discrete pattern of stability is reminiscent of the situation often encountered for magically stable nanoclusters. As far as we are aware, this is the first time that such “magic” numbers have been encountered in a 1D nanosystem.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected] (S.T.B.). Notes

The authors declare no competing financial interest. 22913



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