As long as the largest family n is not populated, this number is constant over time. A reasonable boundary condition to solve equation (2.2) is to set all concentrations to zero at t = 0, except for c0, which will take a finite value. If we would exclude family 0 from the sum, the total concentration of clusters would be zero at the beginning of the reaction and then increase until it reaches a saturation value. It will remain stable until the first clusters appear in family n.
In article A.1 it is shown that with the solutions of equation (2.2) it is possible to reproduce the general behaviour of the absorption spectra shown in Figure 2.2. Actually through this explanation the coalescence model is not entirely excluded. One can still imagine the existence of small entities such as molecules consisting of few atoms (rings of the form (CdSe)3) that are deposited as building block onto the clusters.
2.3 Structural Analysis
So far the analysis of the MSCs was restricted to optical measurements. In this section some methods for a more detailed analysis will be presented and also some plausible atomistic models for the structure of the MSCs will be shown. The experimental methods used to determine the size of the MSCs comprise X-ray diffraction (XRD), elemental analysis, TEM and mass spectroscopy.
XRD-spectra were recorded on dried samples of MSCs of different sizes. Unfortunately these spectra cannot be interpreted unambiguously to reveal the precise structure of the clusters. The signal, as displayed in the supporting information of article A.1, shows relatively broad peaks. This broadness is partially due to the small size of the clusters and most probably also to effects of surface-reconstruction. [62, 70, 71] Nevertheless, the spectra can be fitted better assuming a cubic zinc-blende (ZB)-structure than a hexagonal WZ-structure. This observation of the ZB-structure is in agreement with the general observation of a higher stability of ZB-structure with respect to the WZ- structure. [72] Especially at the low temperature of the growth it is more probable to find the clusters in this ZB-structure.
A second insight into the structure of the MSCs was gained by an elemental analysis. In principle with this technique the total number of atoms of one species in the sample can be determined. For the measurements, fresh samples of different MSCs were pro- duced and a size-selective precipitation was performed to isolate only the largest clusters present in the solution. As the concentration of the MSCs is unknown, only the ratio Cd:Se in the individual samples could be determined. In all samples a higher concentra- tion of Cd than of Se was found. This imbalance is also found in larger CdSe-nanocrystals with a different technique [73] and can be explained by an overrepresentation of Cd on the surface of the clusters due to the higher affinity of the surfactants to the Cd atoms than to the Se atoms.
TEM-measurements of the MSCs were limited by the difficulty to descry between the actual particles and the modulations of the image due to the support. Only the largest clusters (family 6) could be identified clearly (see Figure 2.6). From the TEM-images a diameter of these clusters could be determined as 2.1nm. Due to the limitations in the
22 Chapter 2. Size Control - Magic Size Clusters
Figure 2.6: Low resolution TEM-images of MSC of family 6. Individual MSCs can be identified in (a) and (b). Their diameter is measured as ca. 2.1nm. (c) On the TEM-grid frequently rod-like structures are observed, which probably represent a matrix of residual or- ganic molecules with MSCs incorporated. Generally these aggregates hinder the observation of individual particles especially when they are of small sizes. Also in (a) and (b) this matrix can be seen as a slight shadow. The scale-bars represent 20 nm.
imaging system of the TEM this value is to be considered as an upper limit.
In previous publications mass-spectroscopy was employed to characterize MSCs of CdSe. [62, 74] Unfortunately this technique does not seem to be appropriate for the MSCs presented in this work. Matrix-assisted laser desorption ionization time of flight (MALDI-TOF) experiments were performed on size-selected and purified samples. As shown in the supplementary information of article A.1 the mass-spectra show peaks relative to clusters of the type (CdSe)13, (CdSe)33 and (CdSe)34 like those found also by Kasuya and co-workers. [62, 74] But actually these clusters were found for any CdSe sample exceeding a certain size (i.e. with an absorption peak at a wavelength of 432nm and greater, families 5 and 6). But the fact that we find the same clusters for any CdSe sample larger than MSCs of family 4 and even in classical nanocrystals with a diameter of ca. 3nm must be interpreted as a general limitation of the technique for the inquiry of our samples. It seems that the clusters actually become reshaped under the strong laser excitation, most probably by stripping off of the outermost atoms and the surfactants. The peak relative to the cluster (CdSe)13appears for clusters of the families 5 and 6 but only very weakly – if at all – for clusters of family 4, whereas the peaks of (CdSe)33 and (CdSe)34 do not appear at all for this sample. From this we can deduce that MSCs of family 4 are smaller than (CdSe)33.
From the absorption spectra one can infer the size of the nanocrystals. In literature one can find a reference curve that attributes a size to the position of the first exciton peak in the absorption spectrum. [75] When assuming a perfect zinc-blende structure the size gives an indication of the number of atoms in the individual MSCs. These results are reported in Table 2.1. The size of the clusters of family 6 determined by this technique correlates nicely with the TEM measurements.
2.3. Structural Analysis 23
CdSe MSC family 5 (exciton peak at 431 nm)
CdSe MSC family 6 (exciton peak at 447 nm)
2000 3000 4000 5000 6000 7000 8000m/z
Intensity (a.u.)
CdSe13
CdSe33, CdSe34
Figure 2.7: Mass spectra of large magic size clusters. Matrix-assisted laser desorption ionization time of flight spectra (MALDI-TOF) were recorded on samples of families 5 and 6. The peak at ca. 2500 mass units can be attributed to clusters of the form CdSe13, the peaks around 6500 mass units are relative to clusters CdSe33 and CdSe34. Mass spectra of larger nanocrystals show a similar pattern (see article A.1).
models for the structure of the MSCs. The measurement of the ratio Cd:Se has turned out to be the most valuable information for this purpose. As mentioned above the imbalance between Cd and Se can be explained by the number of the relative atoms on the surface. A way to rationalize this imbalance in models is to restrict the number of allowed dangling bonds for the surface atoms. If Cd-atoms are allowed to have 2 or less dangling bonds, Se-atoms only 1 dangling bond, clusters can be generated that show a higher number of Cd atoms than of Se atoms. This model reflects the behaviour of the surfactants. We can speculate that the carboxylic acids offers two binding sites for the Cd-atoms, whereas the Se forms only a single bond with TOP. The cluster models were generated by a perl script, which randomly generated clusters of a given size and geometry (spheres, tetrahedra, cubes) in perfect ZB structure. Subsequently Cd surface atoms with 3 dangling bonds and Se surface atoms with 2 or 3 dangling bonds were removed so that the surface atoms of the remaining cluster showed the imposed number of dangling bonds. With this approach ca. 400 different clusters containing less than 200 atoms and having the restricted amounts of dangling bonds were found. Out of these models six clusters fulfilled the stronger restriction that Cd-atoms have either two or no dangling bonds, but not just one single dangling bond. This series actually has sizes similar to those determined by the optical absorption measurements. Furthermore the clusters are based on a tetrahedral architecture. Similar clusters could be identified in previous works by crystallization of the clusters and X-ray diffraction. [60, 61, 76, 77]
24 Chapter 2. Size Control - Magic Size Clusters
Table 2.1: Overview of the characteristics of the different magic size clusters (MSCs).
The positions of the absorption– and fluorescence-peaks are measured on non purified sam- ples. From the position of the absorption peak the diameter of the particles is estimated through a calibration curve from literature. [75] Assuming a perfect ZB-structure the number of atoms per cluster can be estimated from the diameter. The ratio Cd:Se is measured by elemental analysis on purified samples. In the lower part the characteristics of the proposed models are summarised (see Figure 2.8).
Family 1 2 3 4 5 6 Measurements Absorption peak (nm) 327 360 384 406 432 447 Fluorescence peak (nm) – – – 430 450 475 Diameter (nm) 0.9 1.2 1.4 1.6 1.8 1.9 Number of atoms 13 31 53 78 113 135
Measured ration Cd:Se – – – 1.10–1.22 1.28–1.29 1.24–1.26
Proposed models (see Figure 2.8)
Number of atoms 10 29 59 75 102 142
Ratio Cd:Se 1.50 1.23 1.11 1.34 1.04 1.29
In those reported cases, however, the Se- or S-atoms were overrepresented on the surface of the clusters, which is due to the employment of strong ligands for Se or S. Therefore the tetrahedra in those reports can be understood as the inverted structure of those proposed here. In this series (See Figure 2.8) the smallest cluster has the structure of adamantane. The two subsequent clusters are obtained basically by the addition of a triangular layer on the base of the precedent cluster. This geometrical feature would nicely explain the existence of the barriers in the growth-process. To grow from one tetrahedral size to the next, a new layer has to be added. The addition of the less atoms than necessary for the entire layer produces an unstable cluster. Only when a sufficient number of atoms is assembled in the layer, it is likely that the missing atoms are added to form the complete layer before the unstable atoms are removed. This can be understood as a nucleation barrier for the growth of the individual layers.
In the model-structure for larger cluster intermediate clusters between two full tetra- hedra have to be introduced. These clusters (clusters 4 and 6) show small cage-like structures on their four flat surfaces. A possible explanation for these structures is that the entire cluster restructures when this size is reached. In this case not only the barrier for the nucleation of a new shell but also a barrier for the conformational change would have to be overcome to reach this size. Another explanation would be that this structure is intermediate in the sense that not the entire triangular layer has to be deposited onto one facet. Once a certain size is overcome, domains smaller than a full facet could be stable. In the case of cluster 4 in Figure 2.8 this is an adamantane cage in the centre of the triangular facets. One can also imagine that this cage is formed only on one of the
2.3. Structural Analysis 25
Figure 2.8: Possible models for the first five magic size clusters. A series of clusters in ZB-structure that show similar architecture so that the addition of few atoms onto one clus- ter can lead to the subsequent structure. The series meets the observed ratio Cd:Se, the estimated sizes of the clusters.
3 Shape Control
The nanocrystals discussed so far were of almost spherical shape. But already in very small nanocrystals such as the magic size clusters presented in the previous chapter different crystalline facets can be identified. But due to the small size of these facets the deposition of only few atoms onto one of them changes its nature completely. In Figure 3(a) an example of an almost spherical nanocrystals is shown. In this nanocrystal the hexagonal facets can be identified clearly. On larger nanocrystals the faceted shape becomes more obvious, as for instance on PbSe-nanocrystals (see Figure 3(b)). [67] Indeed the growth of perfectly spherical crystals is more an exception than the rule. For instance macroscopic quartz is frequently found in an elongated shape with six large facets that intersect under an angle of 120◦. The tip of the crystal shows a series of facets, all inclined with respect to the long axis of the crystal. Generally the shape of a crystal depends on the relative speeds at which the individual facets grow. Here, the speed of growth of a facet is measured as the speed at which the distance of its centre to the centre of the entire crystal increases. The faster a crystal grows on one facet the more likely to disappear is this facet. [78] To rationalise the differences in the growth speeds let us reconsider the discussion of section 1.4 on the growth process. There the exact nature of the surface or the existence of facets of the growing crystal was not taken into account. The introduction of the critical size (r∗) in the growth process was actually based on the values of the vapour pressure of a flat surface (C∞) and the surface energy (σ), but in equation (1.10) implicitly these values were assumed constant for all facets. In fact these two quantities depend strongly on the exact nature of the facet in consideration. Especially the surface energy of the different facets can
(a) (b)
Figure 3.1: Examples of faceting of nanocrystals. TEM-images of a hexagonally shaped CdSe-dot in WZ-structure (a, picture taken from reference [7]) and of a sample of octagonally shaped PbSe-nanocrystals. (b)
28 Chapter 3. Shape Control
Figure 3.2: TEM-images of nanocrystals with different aspect ratios. Spherical CdTe- nanocrystals (left), rice– and rod-shaped CdSe-nanocrystals
be influenced strongly by the choice of the surfactants. [64, 79–81] If the surfactants bind stronger to one facet than to its neighbouring facets, new monomers are more likely to be incorporated into these neighbouring facets. In other words, those facets onto which the surfactants bind stronger have a lower surface energy, as the area of this facet is extended easier than that of a facet with weak ligands. In literature this relation between growth-speed on the different facets and their individual surface energy is known as the Curie-Gibbs-Wulff-theorem. [44]
This effect is seen on many different nanomaterials, where nanocrystals of different shapes can be synthesised, as for instance cubes can be formed from Cu2O, gold, silver or PbSe [82–84], stars from PbSe and PbS [85, 86], disks of cobalt, silver and iron oxide [87–90], rods or wires of gold [91, 92].
In the wurtzite (WZ)-structure the choice of the growth axis is relatively straight- forward. Through its high symmetry the unique c-axis is distinguished from the other axes. In CdE nanocrystals it serves as the directional axis for the asymmetric growth. [93] In the case of Co-nanocrystals in the -phase growth along this axis is suppressed. [87] But the asymmetric growth is not restricted to materials that expose a hexagonal struc- ture. [94] CaF2-nanorods grow in the cubic fluorite structure along the (111)-axis [95]. PbSe grows in the highly symmetric cubic rock salt structure, but by the help of bulky surfactants a unique growth-direction can be assigned and nanowires are formed. [84] For the materials of interest in this chapter (CdS, CdSe and CdTe) generally the crystalline structure is the WZ-structure. For instance, nanorods of CdSe (as shown in Figure 3.2) preferentially grow in WZ-structure, as can be inferred from XRD-measurements and high-resolution-TEM. [96] A nice demonstration of the existence of the well-developed facets in the rods can be seen in honeycomb-structures, in which the nanorods align vertically and in TEM-images the rods can be seen along the long axis, or parallel to the lateral facets. In this case the hexagonal boundaries of each rod can be observed clearly. [96] It is interesting to notice that for thin nanorods the WZ structure is favoured even in conditions where the bulk material favours the cubic ZB-structure due to the reduced number of dangling bonds on the surface. [97, 98]
29
Figure 3.3: TEM-image of tetrapods of CdTe.The particles consist of four rods that are fused together in one point and that span the tetrahedral angle be- tween two legs. In the projection of the TEM, three legs can easily be identified, the forth legs points
directly upwards. Due to the
longer distance that the electron beam has to travel through the material this leg appears as a dark spot.
intriguing example of such structures is the tetrapod. It is composed of four rods that are fused together in a central core. This shape is observed in several materials, such as ZnO [99], ZnSe ( [100] and article B.2), ZnS [101], CdSe ( [102, 103] and article B.7), CdTe ( [104] and article B.1), Pt [105] and Fe2O3 [106]. Generally, the growth mode of the tetrapods is driven by the same mechanism as the growth of the rods. During growth monomers are deposited onto the high-energy facets of the arms, that is on their tips, whereas deposition of monomers onto the lateral facets of the arms is hindered. The main difference in the growth process resides in the nucleation event and the very early stage of the growth, when the core of the tetrapod is formed, as will be discussed in the following sections and article B.1.
Shape controlled nanocrystals are interesting for many aspects. The nanorods show linearly polarised emission [107] and are proven of advantage in solar cells [19, 108]. The appealing feature of the nanorods is their behaviour as quasi one-dimensional objects. Their quantum-properties can be tuned merely over the diameter, the total length of the nanorods affects its electronic properties only for nanorods with a low aspect ratio. [109] That makes them an appealing system for electronic devices. For instance nanowires1 of different materials can be embedded into electric circuits to act as transistors or other active elements [110, 111], electroluminescence is observed from CdSe-nanorods [112] and CdSe– and CdSe/ZnS-nanorods have been proven to be of advantage as lasing medium [113, 114]. Their symmetry facilitates the alignment of nanorods. In some cases it is sufficient to slowly evaporate the solvent to obtain large areas of aligned rods. [115–118] Better results are achieved when the nanorods are oriented by an electric field. (See references [119–122] and articles B.8 and C.3)
Tetrapods offer the possiblity to expand this spectrum of applications. When three of the arms are contacted electrically, one of the arms can be used as a gate to control the 1The terms nanowires and nanorods are used interchangeably here. There is no strict differentiation between the two terms. A tendency is to call long nanorods, i.e. with a length superior to ca. 500 nm, nanowires.
30 Chapter 3. Shape Control
current through the entire structure. [23] Also, these nanocrystals can be of advantage in solar-cells. They exhibit a large surface on which charges can be separated, and still provide through their complex and extended geometry a pathway to transport charges to an electrode. [123] Furthermore tetrapods exhibit an interesting structure of their excited states. Electrons and holes can be localised in the core-region of in the arms and