Persistence of magic cluster stability in ultra-thin semiconductor nanorods

Abstract

The progression from quasi zero-dimensional (Q0D) nanoclusters to quasi one-dimensional (Q1D) nanorods, and, with increasing length, to nanowires, represents the most conceptually fundamental transition from the nanoscale to bulk-like length scales. This dimensionality crossover is particularly interesting, both scientifically and technologically, for inorganic semiconducting (ISC) materials, where striking concomitant changes in optoelectronic properties occur.^1,2 Such effects are most pronounced for ultra-thin³ ISC nanorods/nanowires, where the confining and defective nature of the atomic structure become key. Although experiments on ISC materials in this size regime have revealed especially stable (or “magic”) non-bulk-like Q0D nanoclusters,^4,5 all ISC Q1D nanostructures have been reported as having structures corresponding to bulk crystalline phases. For two important ISC materials (CdS and CdSe) we track the Q0D-to-Q1D transition employing state-of-the-art electronic structure calculations demonstrating an unexpected persistence of magic cluster stability over the bulk-like structure in ultra-thin nanorods up to >10 nm in length. The transition between the magic-cluster-based and wurtzite nanorods is found to be accompanied by a large change in aspect ratio thus potentially providing a route to nano-mechanical transducer applications.

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The impressive scientific and technological advances resulting from the intense research effort into inorganic semiconducting (ISC) nanosystems is exemplified by the continuing developments based on nanoscale CdS. Inspired by the technological promise of its tunable (opto)electronic properties,^1,2 recent work on 1D nano-CdS has given rise to a wide range of device applications such as lasers,⁶ transistors⁷ and waveguides.⁸ Enhancement of finite size effects in 1D nano-CdS systems such as defect-induced optical response and quantum confinement,⁹ can be achieved through restricting the length to form a nanorod and/or by reducing the diameter of nanowires or nanorods to the ultra-thin, or strongly-confined regime. For ISC Q1D nanosystems this size range can be defined as that where the dimensions of a nanorod or nanowire are significantly below the Bohr exciton radius (∼3 nm for CdS and ∼5 nm for CdSe) where one expects the greatest deviation from bulk properties. A number of recent experimental studies have reported the preparation of 1D CdS^10,11 and CdSe^12,13nanorods with diameters <5 nm. Thus far, however, all such ultra-thin CdS and CdSe nanorods have had reported diameters >1.5 nm and, where measured (e.g. by high-resolution transmission electron microscopy (HRTEM) and X-ray diffraction), seemed to possess the bulk wurtzite (wz) atomic structure with the (001) direction aligned along the length of the rods. Relatively thick CdS and CdSe nanorods and nanowires (≥30 nm diameter) can be grown directly from the vapour phase from pure material powders,¹⁴ but typically ultra-thin Q1D nanosystems are made from colloidal growth and assembly of ligated clusters in solution (i.e. oriented attachment³). Although ligands are important for shape and size control, they can also affect the atomic structure of ISC nanosystems¹⁵ and thus for studying the intrinsic size-dependent behaviour of pure ISC materials it is desirable that they should be removed. Below, we concentrate our discussion mainly on ligand-free Q0D and Q1D nanosystems of CdS, although we note that strongly analogous results are also found for the corresponding CdSe systems.

Considering the wealth of experimental studies and, further, the role of 1D nano-CdS as a model system for studying the fundamental physics of anisotropic quantum confinement, surprisingly few detailed calculations have been performed on infinite CdS nanowires,^16–18 and, as far as we are aware, none on CdS nanorods. In contrast, computational modelling studies of Q0D CdS nanosystems are relatively abundant and have tended to proceed via two distinct routes: (i) top-down, where nanoparticle atomic structures are derived from cutting fragments from either the wz or zincblende (zb) bulk phases,^19,20 or (ii) bottom-up, where the most energetically stable nanocluster structures are sought without recourse to bulk crystalline structural stabilities.^5,21 Largely due to the use of its simplifying structural assumption, the former approach has permitted studies of size-dependent optical and electronic properties of wz- or zb-structured (CdS)_Nnanoparticles over a wide size range.^19,20 Technically, due to the explosion in the number of possible structures with increasing size, the bottom-up approach to modelling CdS nanoclusters is significantly more difficult and the ground-state energy minimum structures of (CdS)_N nanoclusters are reasonably well established up to only N = 16 with some suggestions for 28 < N < 35.^4,5 Unlike the assumed crystalline structures adopted in the top-down approach none of the reported ground-state nanocluster structures in this latter size regime are found to be wz-like or zb-like. Recent work has also suggested that wz-structured CdS nanoparticles up 2 nm in diameter are unstable with respect to amorphisation.²² In Q1D nanowires and high-aspect-ratio nanorods the surface/volume ratio is significantly lower than in Q0D systems of the same composition and we may expect a more rapid convergence to bulk-like crystalline structures. That this may not be the case for ultra-thin nanowires was first proposed in 1998 by Tosatti et al. for elemental metal systems using simple atomistic models.²³ For compound materials of nanotechnological interest, such as the numerous types of ISC, however, neither experiment nor theory has reported any tendency for ultra-thin Q1D nanosystems to deviate from bulk-like structures.

In an effort to bridge the gap between experiment and theory, with respect to the inherent behaviour of ultra-thin Q1D nanosystems of pure ISC materials, we employed large state-of-the-art scale density functional theory (DFT) calculations to follow the Q0D-to-Q1D transition for ultra-thin CdS. In order to tackle both ends of this range we have combined both bottom-up and top-down modelling approaches by calculating the stability of both small nanoclusters and infinite nanowires. The main feature of our approach, however, is that we explicitly follow the transition between these extremes by computing how the stabilities of cluster-assembled nanorods vary with increasing length. Specifically, we have investigated the energetic stability and atomistic and electronic structure of ultra-thin nanorods and nanowires assembled from experimentally detected (CdS)₁₃/(CdSe)₁₃ magic clusters (i.e. clusters exhibiting pronounced abundance peaks in cluster beams^4,5) with a diameter 1.0–1.2 nm, with respect to the corresponding bulk-like wz-structured Q1D systems. Of note is that the prominent high abundance of clusters with 13 units is observed both in laser ablation experiments producing ligand-free clusters,^4,5 and in solution-based nucleation experiments⁴ where the clusters are ligated. Although the structure of the magic cluster(s) corresponding to these peaks cannot, as yet, be directly ascertained from experiment, following previous studies,^4,5 we assume that it most likely corresponds to the bare (CdS)₁₃/(CdSe)₁₃ isomer with the lowest energy.

To calculate the relative stabilities and properties of nanorods and nanowires built from wz and magic nanocluster building blocks we employed periodic density functional (DF) theory using the PW91 implementation²⁴ of the Generalized Gradient Approach (GGA) form of the exchange–correlation potential using the VASP code.²⁵ A plane wave basis set with a kinetic energy cut-off of 415 eV was used, with the effect of the atomic core on the valence electron density taken into account by the projector augmented wave (PAW) approach.^26,27 The calculations for all cluster isomers and nanorods were carried out by placing each system inside a large enough box so as to make the interaction between repeated images negligible in all directions (10 Å was found to be sufficient). The infinite nanowires, similarly spaced in all directions perpendicular to the axes of the wires, were also considered. For the infinite nanowire a 1 × 1 × 9 Monkhorst–Pack mesh²⁸ of special k-points was employed, whereas for the finite nanorods and clusters Γ-point calculations were performed. In all cases the atomic structure of the system was fully relaxed until the forces were smaller than 0.01 eV Å⁻¹. In constructing the infinite nanowires, a magic cluster or wz cluster was initially placed in a periodic cell such that the axis of symmetry was aligned along the z-direction with a z-spacing of a suitable bond length. In order to prevent artificial geometric constraints due to the use of only one cluster as a repeat unit in the subsequent structural optimisations of the nanowires a two-cluster supercell (i.e. 52 atoms) was employed. Firstly, the nanowires were optimised (both the internal atomic structure and supercell length) to their closest energy minimum to the as-constructed structure (cluster-assembled nanowires). Subsequently, we mechanically annealed the cluster-assembled nanowires by gradually compressing and stretching them (by up to 30% of their original length) by appropriately varying the z-dimension of the supercell of the initial cluster-assembled nanowires; for every fixed z-value optimising the nanowire structure. Every time a structural change occurred (causing a new minimum energy atomic configuration), the annealing procedure was again performed around the minimum energy configuration of the newly obtained structure. This process was repeated until stretching and compressing the nanowires did not change their minimum energy structure. The resulting nanowires are referred to as annealed nanowires below.

In Fig. 1 we show the structures and calculated relative energetic stabilities of three selected (CdS)₁₃/(CdSe)₁₃ cluster isomers: a) the likely ground-state “magic” cluster, b) a cage-like cluster, and c) a wz cluster fragment. The (CdS)₁₃ magic cluster has been predicted by calculations to have the form of a distorted cage filled by a single 4-coordinated sulfur atom bonded to four 4-coordinated cadmium atoms giving rise to a C₃ structure.^4,5 Particularly interesting for the present study is the fact that the magic cluster structure lies ∼1 eV below the wz nanocluster. Our calculations further confirm that the magic cluster structure is also the likely ground state for CdSe with a very similar energetic stability over the corresponding wz-structured isomer. The cage-like cluster isomer is an example of the numerous cluster isomers having energetic stabilities between that of the wz cluster and the likely ground-state magic cluster.^5,21 Although, in principle, one could consider many Q1D nanosystems based on these isomers, herein, we concentrate on comparing arguably the most important clusters (i) the wz cluster (giving rise to wz-structured Q1D nanosystems as observed for many Q1D CdS/CdSe nanosystems experimentally), (ii) the most stable cluster isomer, assumed to correspond to the magic-cluster abundance peak in experiments.

Fig. 1 Optimised structures of three N = 13 nanocluster isomers, where the upper and lower rows correspond to different views of the same three clusters: a) magic-, b) cage-, and c) wurtzite-based clusters. Below each cluster the relative energy (in eV) is given for both CdS and CdSe compositions with respect to the energy of the magic-cluster. As for all figures yellow balls represent sulfur or selenium atom positions and green balls cadmium atom positions. All reported energies of clusters, nanorods and nanowires refer to binding energies with respect to constituent spherical non spin-polarised atoms.

In previous studies we showed that stable clusters of other materials (e.g. ZnO, LiF) when considered as nanosized building blocks, can, when appropriately assembled, lead to predictions of novel stable bulk polymorphs.^29,30 Here, although the symmetry of the magic (CdS)₁₃ cluster does not easily lend itself to assembly in three dimensions, we may take advantage of its axial C₃ symmetry to form ultra-thin 1D nanowires and nanorods by forming Q1D stacks of axially aligned clusters. The wz (CdS)₁₃ isomer also has three-fold symmetry allowing us to also construct axially aligned ultra-thin nanowires and nanorods with the wz structure (with their long axis aligned with the (001) direction in the wz crystal as found in experiments) of exactly the same atomic composition as in the magic-cluster case for direct comparison. It was found in both cases considered that annealed nanowires were significantly more stable than cluster-assembled wires. The wz cluster-assembled nanowire was found to be a shallow, relatively high lying, energy minimum having a layered hexagonal structure (see Fig. 2) analogous to that found in bulk boron nitride (h-BN) which has also been predicted theoretically to be stable for ultra-thin nanorods and nanowires of ZnO³¹ and in thicker ZnO nanorods under tensile strain.³² We note that this relatively dense phase should be differentiated from a similar low-density hexagonal phase with larger interlayer spacing,^33,34 which was not observed in our investigation. Upon annealing, the hexagonal layered nanowire transforms into new structure with a deeper lying energetic minimum having a more bulk-like wz structure with a correspondingly longer (+28%) unit cell length (Figs. 2, 3a1 and 3b1). For the magic-cluster-assembled nanowire, the annealing procedure proceeds via an intermediate nanowire structure and finally leads to a more stable nanowire structure having a slightly shorter (-6%) unit cell length (Figs. 2, 3a2 and 3b2) but which is still significantly longer (+20%) than the annealed wz nanowire. Unlike for (CdS)₁₃ nanoclusters, but following experimental observations for thicker CdS nanowires and nanorods, Fig. 2 clearly shows that the wz structure is the most stable structure for (CdS)₁₃-based nanowires. The comparable diameters of the wz and magic-cluster-based ultra-thin nanowires results in similarly strong quantum confinement with an increase in the band gap with respect to the bulk wz band gap by a factor of 2.5 (see Fig. 4). In contrast, the absolute energy levels of the annealed wz nanowire are deeper than for the magic-cluster-based nanowire probably due to the energetically preferred atomic structure of the former.

Fig. 2 Energy (in eV/CdS) versus length curves for the infinite CdS nanowires. Total energy of the cluster-assembled and annealed nanowires constructed from the (CdS)₁₃ wz and magic clusters as a function of the unit cell length (half the super cell length used in the calculations) along the nanowirez-direction. The arrows indicate the energetically downhill annealing path. The inset figures show the structure of a unit cell of the cluster-assembled and final annealed nanowires.

Fig. 3 Summary of the structural features of annealed wz and magic-cluster-based CdS nanowires and nanorods. Perspective views of the annealed wz- (a1) and magic-cluster-based (a2) nanowires, cross-section of the of the annealed wz (b1) and the magic-cluster-based (b2) nanowire/nanorod, radial distribution function of the annealed wz (c1) and magic-cluster-based (c2) [(CdS)₁₃]₆nanorod, atomic structure of the annealed wz (d1) and the magic-cluster-based (d2) [(CdS)₁₃]₆nanorods (the (CdS)₁₃ repeat unit is highlighted in each case). Quoted distances include the radii of two sulfur ions to better compare with estimates made by HRTEM. The two distances in b2 indicate the maximum and minimum of the oscillatory diameter of the annealed magic-cluster-based nanorods and nanowires.

Fig. 4 Summary of the electronic structure of annealed wz- and magic-cluster-based CdS nanowires and nanorods. Calculated energy levels (eV) of CdS systems (from left to right): annealed wz [(CdS)₁₃]_nnanorod, infinite annealed wz nanowire, wz bulk crystal, infinite annealed magic-cluster-based nanowire, annealed magic-cluster-based [(CdS)₁₃]_nnanorod. Blue denotes occupied levels (sulfur s–p states) and red unoccupied levels (cadmium s states). The gaps of the nanowires are shown to be red-shifted by a 2.5× multiplicative factor with respect to the bulk CdS wz bandgap (ΔE_wz) due to quantum confinement. We note that, as is typical for DFT calculations, the band gaps are systematically underestimated and thus the figure provides an indication of relative changes in electronic structure (i.e. Q1D with respect to bulk) only.

In order to analyze the transition from the most stable 01D nanosystem for (CdS)₁₃ (i.e. the magic-cluster structure) to the most stable infinite nanowire (i.e. bulk-like wz-structured) we considered [(CdS)₁₃]_nnanorods of all four CdS nanowire structures considered above (i.e. wz- and magic-cluster-based, both annealed and cluster-assembled) from the size of a one-cluster unit (n = 1 or 26 atoms) to a six-cluster unit (n = 6 or 156 atoms). In Fig. 5 we show the change in total energy of the four nanorods per CdS unit with increasing length. Interestingly, unlike for the nanowires, for up to n = 3, the cluster-assembled wz nanorod was found to be more stable than the nanorod with the annealed wz structure. For all considered sizes, however, the annealed magic-cluster nanorod was found to be most energetically stable. As the length of the nanorods increases, the energy difference between the annealed wz and magic-cluster-based nanorod structures gradually reduces implying that a transition to the wz structure occurs for larger n. In order to predict at what length this transition occurs, we have extrapolated the energy versus length trends of the magic-cluster-based and wz nanorods by fitting the data points to an inverse power law (see inset to Fig. 5, fitted with R² >0.99). We find that the persistence of the excess stability of the annealed magic nanorods with respect to the annealed wz nanorods lasts until approximately 13 cluster units; equating to 338 atoms and to more than 10 nm length with respect to the length of the annealed magic nanorod. We have also calculated the energy difference between annealed wz and magic-cluster-based [(CdSe)₁₃]₆nanorods which is very similar to that found for CdS indicating analogous behaviour in both Q1D systems. Relative energies of all nanorods and nanowires are given in Table 1. To understand these trends we have examined the atomic and electronic structure of the nanorods.

Table 1 Calculated energies (eV per formula unit) of all considered nanorods and nanowires of CdS and CdSe

Composition	wz cluster assembled	wz cluster annealed	Magic cluster assembled	Magic cluster annealed
Cd13S13	−5.516	—	−5.593	—
Cd26S26	−5.620	—	−5.649	−5.674
Cd39S39	−5.633	−5.593	−5.679	−5.704
Cd52S52	−5.646	−5.649	−5.692	−5.721
Cd65S65	−5.652	−5.678	−5.702	−5.731
Cd78S78	—	−5.701	−5.709	−5.738
Cd91S91	—	−5.718	—	—
CdS nanowire	−5.720	−5.818	−5.747	−5.780
Cd13Se13	−5.062	—	−4.988	—
Cd78Se78	—	−5.177	—	−5.140
CdSe nanowire	—	−5.211	—	−5.246

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Fig. 5 Energy (eV/CdS) of the cluster-assembled and annealed [(CdS)₁₃]_nnanorods constructed from the (CdS)₁₃ wz and magic-cluster units as a function of n. The inset shows extrapolated energetic relative stability of the wz and magic-cluster-based annealed [(CdS)₁₃]_nnanorods as a function of n with respect to the energy of the calculated energy of the infinite annealed wz nanowire. The horizontal grey line shows the energy of the infinite magic-cluster annealed nanowire.

The higher crystallinity of the annealed wz [(CdS)₁₃]₆nanorod is clear from the radial distribution function (RDF) taken with respect to the position of a central sulfur atom showing prominent peaks at well-determined distances (see RDF for the annealed wz [(CdS)₁₃]₆ in Fig. 3c1 with respect to the structure shown in Fig. 3d1). In contrast, the corresponding RDF of the annealed magic-cluster-based [(CdS)₁₃]₆nanorod (see Fig. 3c2 with respect to structure shown in Fig. 3d2). This lower crystallinity is also manifested in the oscillatory diameter of the annealed magic-cluster-based Q1D nanosystems (Fig. 3b2) due to the recognisable magic-cluster (CdS)₁₃ repeat units (see highlighted segment in Fig. 3d1). In contrast, the annealed wz Q1D nanosystems have a much more regular stacked structure with a uniform diameter (see Fig. 3d1). This crystalline layering has been observed by HRTEM in CdS and CdSe nanorods of larger diameters than those studied here.^10,13 Based upon our results we predict that such layering will be absent in well-annealed bare CdS and CdSe nanorods of diameters <1.5 nm and lengths <13 nm. Of potential technological importance, such annealed magic-cluster-based nanorods would also have significantly higher aspect ratios than wz nanorods of the same composition (∼50% higher in the case of [(CdS)₁₃]₆). Under pressure cycling, ultra-thin CdSe nanorods have previously been shown to exhibit dramatic structural transitions between the six-fold coordinated rocksalt phase and the four-fold coordinated wz phase³⁵ often leading to fracturing. In the present case, although accompanied by a large aspect ratio change, the atomic-level transition is much more subtle and may thus be more robust for repetitive utilization in nano-mechanical transducers. We also note that we have confirmed the experimental observation that rocksalt nanorods are only stable under external pressure for the ultra-thin regime by attempts to optimise ultra-thin rocksalt-structured nanorods. We found in all cases that the such nanorods spontaneously relaxed into distorted non-cubic structures (usually with final structures similar to wz) which were always significantly less stable than the other nanorods considered herein.

Experimentally, CdS and CdSe nanorods possessing a layered wz structure oriented along the (001) direction are known to have very large electric dipole moments along the length of the rod³⁶ which are thought to be important for their self-assembly.³⁷ Maintaining a large dipole is not energetically favourable and it is known in similarly structured infinite 2D wz nanoslabs (where such a dipole is unsustainable) in other materials, that the atomic structure can reconstruct in ways to drastically reduce the moment across the slab.^33,34 In our case we estimate the dipole moment in the annealed wz CdS nanorods to be reduced by approximately 5% when going to the annealed magic-cluster-based CdS nanorods indicating that dipole reduction is not a significant structure-directing influence in the ultra-thin Q1D systems considered.

Along their length, both magic-cluster and wz annealed nanorods have the same atomic nearest neighbour coordination per (CdS)₁₃ unit: four four-coordinated and nine three-coordinated sulfur atoms and four four-coordinated and nine three-coordinated cadmium atoms. At their ends, termination induces three two-coordinated cadmium atoms in both annealed nanorod types and a further three two-coordinated sulfur atoms in the annealed wz nanorods only. Two-coordinated sulfur atoms appear to be an inherent terminating structural feature of bare annealed wz nanorods which cannot be repaired by local reconstruction. In order to retain a wz-like structure, while avoiding such under-coordination of the sulfur atoms, wholesale reconstruction seems to be the only option. Such a reconstruction leads to nanorods with a hexagonal layered structure and the resulting lower under-coordination of the sulfur atoms is a likely reason why such nanorods are more stable than annealed wz nanorods for n <4 (see Fig. 5). Electronically, both the wz- and magic-cluster-based nanorods have a large number of states in the region of the gap of their respective infinite nanowire counterparts due to their defective end terminations. In line with the higher under-coordination in the annealed wz nanorods, the energy level spectrum is destabilised with respect to that of the annealed magic-cluster-based nanorods (see Fig. 4). We can view the annealed magic-cluster nanorods as structurally reconstructed versions of the annealed wz nanorods which helps to heal the less energetically favourable sulfur terminations in the latter. This reconstruction, although stabilising with respect to the nanorod ends also causes the internal atomic structure of the nanorod to be non-wz-like. Thus, with increasing length, a greater and greater percentage of the atoms in the annealed magic-cluster nanorods will be found in a structure known to be energetically less stable than the wz structure in the infinite limit and the percentage of atoms involved in an energetically preferable terminating reconstruction will become correspondingly lower. Evidently, at some length this situation will no longer be more stable than an annealed wz structure with a less favourable terminating end reconstruction. At this point, structural transition between the annealed magic-cluster structure and the annealed wz nanorod will be energetically favourable (at a length of ∼10 nm, as predicted by our calculations). We note that this strong dependence of the overall CdS/CdSe ultra-thin nanowire structure, and, furthermore, aspect ratio, on the terminating surface atomic/electronic structure may provide a means for influencing the predicted nanorod structural crossover by reversible attachment of suitable ligands.

In summary, we have compared the stability of ultra-thin bare CdS nanowires and nanorods having the bulk wz structure with those based on the assembly of particularly stable (CdS)₁₃ magic-cluster building blocks using first principles calculations. Although the wz structure is energetically favoured in relatively thick CdS nanorods and in the limit of infinitely long ultra-thin nanowires, in ultra-thin nanorods the magic-cluster-based structure is found to be persistently more energetically stable than the correspondingly sized bulk-like wz nanorods up to a length of 10 nm. This nanoscale structural transition is also predicted to be found in ultra-thin Q1D CdSe and in both materials should be experimentally verifiable by HRTEM. As the length-dependent transition is also accompanied by a large change in the aspect ratio of the nanorods, if physically realized, this effect may find potential application in nano-mechanical transducers. We believe that our explicit demonstration of the persistence of the stability non-bulk like atomic structures in Q0D nanoclusters to Q1D ultra-thin nanorods in two nanotechnologically important ISC materials is likely to be general to many other materials. We hope that our work may help to encourage experimental efforts in this area to fabricate ultra-thin Q1D nanosytems (particularly with diameters ∼1 nm) in a range of materials, many of which we predict will have novel non-bulk-like atomic structures and potentially useful new nanoscale properties.

Acknowledgements

We acknowledge support from the Spanish Ministry of Education and Science (grant FIS2008-02238), time on the MareNostrum supercomputer (Barcelona Supercomputing Center/Centro Nacional de Supercomputación) and grants from the Thailand Research Fund (to WS and JL) and the National Science and Technology Development Agency (NSTDA Chair Professor and NANOTEC Center of Excellence). Support from the Thai Commission on Higher Education, Ministry of Education under Postgraduate Education and Research Programs in Petroleum and Petrochemicals, and Advanced Materials is also acknowledged.

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