Structural and Electronic Properties of Sin, Gen, and SinGen Clusters

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Structural and Electronic Properties of Sin, Gen, and SinGen Clusters Habib ur Rehman,† Michael Springborg,* and Yi Dong‡ Physical and Theoretical Chemistry, University of Saarland, 66123 Saarbr€ucken, Germany ABSTRACT: Using a combination of a parametrized density-functional method for the calculation of the total energy of a given structure and a genetic-algorithm method for the unbiased determination of the structure of the lowest total energy, we have determined structural, energetic, and electronic properties of Sin, Gen, and SinGen clusters with 2-44 atoms. With various specifically developed descriptors, particularly stable clusters are identified, and the structures, overall shapes, bonding patterns, and structural similarities are analyzed and quantified. Finally, the energies of the HOMO and the LUMO are analyzed, and their energy difference is compared with the stability of the clusters.

1. INTRODUCTION Nanoparticles, with spatial extensions of up to some 10 nm and containing between some 10s and several 10 000s of atoms, are below the thermodynamic limit. This implies that their properties depend in a highly nontrivial way on the size of the systems; i.e., simple scaling laws do not apply. A physical explanation of this is that surfaces make out a non-negligible part of the total system so that the structures of the surfaces are important for the properties. In addition, when the spatial extensions of the nanoparticles become comparable with the exciton radii, finite-size effects influence the electronic properties. The properties of the nanoparticles depend often very strongly, although hardly predictably, on their size. For practical applications, this does open up the possibility of tuning the materials properties simply by choosing nanoparticles of the proper size. This is obviously only then possible once a correlation between size and property has been established. Thus, very many, experimental and/or theoretical, studies have been devoted to the properties of nanoparticles; see, e.g., refs 1-5, and references therein. Because of the lack of scaling laws, such studies require that a whole range of nanoparticle sizes is being investigated which easily becomes very demanding. Since it is difficult to characterize such nanoparticles experimentally, theoretical studies have become very important in the development of the understanding of the size dependence of the properties. However, with theoretical methods a couple of problems show up that for more conventional studies are less severe. First, if the system of interest contains N nuclei and M electrons, the r 2011 American Chemical Society

computational costs for calculating the properties for one single structure scale as the size of the system to some power, i.e., like Mk or Nk, where k is some power from 2 and upward. For systems for which directional chemical bonds are important (which is the case for the ones of the present study), electronic-structure calculations are needed, which means that k typically is 3 or larger. Thus, in this case it should be obvious that for just intermediately large nanoparticles the calculation of the properties for just one single geometry may become computationally very costly. Independent of the scaling of the computational needs as a function of the size of the system, another, complementary, problem causes additional complications. This problem is related to the fact that even for nanoparticles containing just one type of atom the number of nonequivalent minima on the total-energy surface as a function of structure grows exponentially with the size of the system. For nanoparticles with more than one type of atom, the additional existence of so-called homotops6,7 increases the computational demands enormously. Homotops for, e.g., ApBq clusters are defined as clusters with the same size, composition, and geometric arrangement but differing in the way in which the A- and B-type atoms are arranged. Since it is mandatory to have realistic structures for the nanoparticles if structure-property relations shall be identified, these complications mean that (1) efficient, often approximate, methods for the calculation of the total energy for a given Received: September 26, 2010 Revised: January 19, 2011 Published: February 22, 2011 2005

dx.doi.org/10.1021/jp109198r | J. Phys. Chem. A 2011, 115, 2005–2015

The Journal of Physical Chemistry A structure are to be used and (2) efficient, special-purpose methods for the unbiased determination of the structure of the global total-energy minimum shall be employed. Even if issue 1 may lead to some inaccuracies in the results, the method shall be so accurate that realistic structures for the largest part of the nanoparticles shall be obtained. In this paper we shall present results of a theoretical study of the properties of Sin and Gen monatomic clusters as well as of SinGen clusters. We consider nanoparticles with up to N = 44 atoms, whereby N = n for the pure clusters and N = 2n for the mixed clusters. The purpose is to identify general structureproperty relations, and therefore, we have met the two complications above by applying an approximate density-functional method for issue 1 and genetic algorithms for issue 2. Also, because of the importance of the macroscopic materials for a number of technological applications, silicon and germanium semiconductor clusters as well as mixed Si-Ge and Si-C clusters have been a topic for a large number of experimental and theoretical studies (see, e.g., refs 1-3). However, as we shall see, most of those have considered only a smaller number of sizes and/or structures, although in some theoretical studies more accurate methods have been used. However, due to this limitation it is often difficult to extract general trends about the size dependence of the properties. Of the systems of the present study, silicon clusters, Sin, have been the topic of most experimental and theoretical studies, although there also exist some studies on germanium clusters, Gen. In one of the first studies on these systems, Martin and Schaber reported experimental abundance spectra for Siþ n and 8 9 Geþ n clusters. Chesnovsky et al. measured the photodetachment thresholds of Sin and Gen clusters, i.e., essentially vertical electron affinities. More studies have been devoted to the structural, vibrational, and electronic properties of Sin clusters with n from 4 to 7.10-14 By combining ion-mobility experiments with theoretical calculations, Jarrold and co-workers have presented geometries for charged Sin clusters with n up to 55.15-19 In another study, M€uller et al.20 compared experimental and theoretical photoelectron spectra for Sin clusters with n up to 20. On the other hand, Antonietti et al.21 reported photodissociation spectra for Siþ n Xe clusters with n up to 13. These spectra provide information on the photoabsorption spectra of the Siþ n clusters. Also, Lyon et al.22 studied experimentally the properties of Siþ n Xe clusters. They measured vibrational properties and, by comparing with theoretical results for different structures, could identify 23 the structures of the Siþ n clusters. Finally, Fielicke et al. studied the vibrational properties of neutral Sin clusters with n = 6, 7, and 10. We see that most experimental studies consider charged clusters. Therefore, it is relevant to emphasize that it is known16,18,19,24 that the structures of neutral and charged Sin clusters are not at all identical. Nevertheless, we shall below compare the available experimental information, also for the charged clusters, with our results for the neutral clusters. On the theoretical side, Sieck et al.25 used a method similar to ours for clusters with n = 25, 29, 35, 71, and 239 atoms. For each size, more different structures, obtained through moleculardynamics simulations, were considered. A similar total-energy method, combined with an unbiased structure-optimization method, was used by Jackson et al.26 for n = 20-27. Genetic algorithms together with a semiempirical total-energy method were used by Bazterra et al.27,28 for n up to 60. The resulting structures were ultimately refined through parameter-free density-functional calculations. On the other hand, Ramos et al.29

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studied the optical properties of large, passivated silicon and germanium nanoparticles for which the structure was assumed to be a part of the crystal. Yoo et al.30,31 considered Sin clusters with up to 45 atoms. They applied parameter-free density-functional methods for various, specifically constructed structures. A similar approach was used by Ma et al.32 who considered fullerene-like structures for 30 e n e 39. Different total-energy methods in combination with an unbiased structure-optimization method were used by Hellmann et al.33 for n up to 19. Also, the study of Ghasemi et al.34 focused on a comparison of different totalenergy methods for selected silicon-based structures, including some clusters. Structural changes for some few selected Sin clusters were studied by Tsetseris et al.35 using parameter-free density-functional calculations. In their study, Tereshchuk et al.36 used a semiempirical total-energy method for specifically constructed structures for Sin clusters with n up to 65. Genetic algorithms together with a tight-binding total-energy method were used by Zhao et al.37 to compare the structures of Sin and Gen clusters for n = 25-33. In three related studies, Qin et al.38-40 studied Sin and Gen clusters. Many fewer studies have been devoted to Gen clusters. Besides those mentioned above, we mention the work by Pizzagalli et al.41 who used a combination of semiempirical and parameter-free methods for selected structures for some few values of n. Wang et al.42 used genetic algorithms in combination with a parametrized tight-binding total-energy method for n = 2-15. The structures were subsequently refined with parameter-free density-functional calculations. Density-functional calculations for different structures of Ge20 were carried through by Li et al.43 Heteroatomic semiconducting clusters have been much less studied. Selected structures for mixed clusters containing C, Si, and/or Ge with up to 10 atoms in total were studied by Li et al.44 using parameter-free density-functional calculations. Results of a similar approach were presented by Bing et al.45 for neutral and charged Si-Ge clusters with up to 7 atoms in total. Wielgus et al.46 studied Si-Ge clusters with 5 atoms, whereas Huo and co-workers47,48 studied the properties of SinCn clusters with n = 1-15 using parameter-free density-functional calculations for selected structures. Finally, Walker et al.49 performed parameterfree density-functional calculations for selected structures of larger Si-Ge nanoparticles. Thus, only few unbiased studies cover a larger size range, making the identification of general trends difficult. Our study is aimed at filling this gap. Simultaneously, the more accurate studies mentioned above will provide an important control for the results of the present study, and also with a comparison with experimental results on charged clusters useful information is obtained. The present work is an extension of our earlier one50,51 in which we considered only specifically constructed structures of Si-Ge nanoparticles. Finally, a brief account of a few parts of the present study was presented earlier.52 The paper is organized as follows. In section 2 our computational approach is briefly described. Section 3 is devoted to the results, and a brief summary is offered in section 4.

2. COMPUTATIONAL APPROACH Our computational approach combines a simplified description of the total energy as a function of structure with an unbiased method for structure optimization. Since the systems of our interest contain directional chemical bonds, the former method includes an explicit description of the electronic orbitals. Our 2006

dx.doi.org/10.1021/jp109198r |J. Phys. Chem. A 2011, 115, 2005–2015

The Journal of Physical Chemistry A

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approach is very similar to what we have used earlier for other types of nanoparticles.53-55 As a compromise between accuracy and computational demands we have used the so-called DFTB (density-functional tight-binding) method of Seifert and co-workers.56-58 The DFTB method is based on the density-functional theory of Hohenberg and Kohn in the formulation of Kohn and Sham. Within the DFTB approach, the total energy relative to that of the noninteracting atoms is given as Etot =

occ

∑i εi - ∑j ∑m εjm þ 2 j6∑¼k Ujk ðj RBj 1

R Bk jÞ

ð1Þ

where εi is the energy of the ith orbital for the system of interest and εjm is the energy of the jth orbital for the isolated mth atom. Moreover, Ujk is a short-range pair potential between atoms j and k that is adjusted so that results from parameter-free densityfunctional calculations on two-atomic systems as a function of the interatomic distance are accurately reproduced. The elements of the Hamilton and overlap matrices, i.e., Æχm1n1|Ĥ|χm2n2æ and Æχm1n1|χm2n2æ with χmn being the nth atomic orbital of the mth atom, are obtained from calculations on diatomic molecules. The Hamilton operator contains the kineticenergy operator as well as the potential: 2 ^ ¼ - p r2 þ V ð B rÞ ð2Þ H 2m The latter is approximated as a superposition of the potentials of the isolated atoms

Vð B rÞ ¼

∑m Vm ðj Br -

Bm jÞ R

ð3Þ

and we assume that the matrix element Æχm1n1|Vm|χm2n2æ vanishes unless at least one of the atoms m1 and m2 equals m. Only the 3s and 3p functions of Si and the 4s and 4p functions of Ge were explicitly included in the calculations, whereas all other electrons were treated within a frozen-core approximation. Thus, within this approach all information that enters the calculations is extracted from the properties of diatomic molecules. These can, in turn, be determined from accurate densityfunctional calculations. It is obvious that the approach we are using has been designed for the smallest possible systems Si2, Ge2, and SiGe. We have verified the capability of this method to do calculations for larger systems by studying some characteristics of bulk Si and Ge. The experimental lattice constants of crystalline Si and Ge are 5.43 and 5.66 Å, respectively, whereas our calculations give 5.46 and 5.71 Å, respectively, i.e., within less than 1% of the experimental values. The experimental band gaps of bulk Si and Ge are 1.12 and 0.66 eV, respectively, and our calculated values are 1.097 and 0.65 eV, respectively, which are also close to the experimental values. Here, the standard problem of density-functional calculations to yield too small band gaps seems to be absent, mainly due to the fact that our basis set is minimal in size. For an unbiased structure determination we combine the DFTB method with genetic algorithms. The genetic algorithms are based on the principles of natural evolution and are, therefore, also called evolutionary algorithms59,60 and have been found to provide an efficient tool for global geometry optimizations. The version of the genetic algorithms that we are using is as follows. A population of P initial structures is chosen randomly (these clusters are called parents), and each structure is relaxed to the

nearest total energy minimum. By cutting each parent randomly into two parts a next set of P structures is obtained by interchanging (“mating”) these two parts and allowing the resulting “children” to relax, too. Comparing the energies of the 2P clusters of both sets, those P with the lowest total energies are chosen to form the set of parents for the next generation. This procedure is repeated for many hundred generations until the lowest total energy is unchanged for a large number of generations. In principle, it cannot be excluded that the resulting structures do not belong to total-energy minima. However, due to the way the structures are constructed (i.e., repeatedly invoking random numbers) and since, in the present study, most of the structures are of low symmetry, it is extremely unlikely that this is the case. Therefore, we have not checked that the structures indeed belong to total-energy minima and not to saddle points. In the present study we have considered pure Sin and Gen clusters with up to n = 44 atoms as well as stoichiometric SinGen clusters with up to n = 22 atom pairs. The total number of atoms is thus N = n for the pure Sin and Gen clusters and N = 2n for the SinGen clusters. P was set equal to 3 in all calculations, and dependent on the size of the system we submitted up to around 10 independent sets of calculations. Also dependent on the size, the total number of generations in each set of calculations was up to several hundred.

3. RESULTS As results of the calculations we obtain structure, total energy, and the spatial distribution plus energy of the electronic orbitals. A further challenge is, accordingly, to extract chemical/physical information from this quite large amount of information. We shall here present and discuss our approach by separating the discussion into those of energetic, structural, and electronic properties. A. Energetic Properties. In Figure 1 we show the binding energy per atom (i.e., the negative of the total energy per atom) as a function of the total number, N, of atoms for the three systems of our study. The figure shows immediately that Si clusters in general have a stronger bonding than is the case for the Ge clusters, and the SiGe clusters lie in between. This is consistent with the experimental values for the cohesive energy of the crystalline elements (i.e., 4.64 and 3.88 eV/atom for Si and Ge, respectively61). Moreover, for any type of clusters, separately, the binding energy is roughly monotonically increasing, implying that any cluster is more stable than the sum of two fragments of it as long as stoichiometry is kept for all parts. A more careful inspection of the figure reveals that each of the three curves possesses some structure, suggesting that certain clusters are more stable than others. In order to identify those it is useful to consider the so-called stability function that is defined as Δ2 ðnÞ ¼ Etot ðn þ 1Þ þ Etot ðn - 1Þ - 2Etot ðnÞ

ð4Þ

where Etot(k) is the total energy of the cluster with k units. This function, which has maxima and minima for particularly stable and unstable clusters, respectively, is shown in Figure 2. It is seen that particularly stable Si clusters are found for clusters with N = 7, 16, 34, 38, and 42 atoms; for Ge clusters for N = 5, 16, 26, 39, and 43; and for SiGe clusters with N = 10, 14, 24, 28, and 36 atoms. Only few values are common to more types of clusters, suggesting that the notion “stability” is very specific for the type of cluster. 2007


The Journal of Physical Chemistry A For Sin clusters there exist some other results regarding particularly stable cluster sizes. Thus, N = 7 and 14;27 29 and 33;30,31 21 and 32;28 26, 29, and 31;37 and 4, 7, 10, 12, and 2938 have been reported. Many fewer results have been reported for Gen clusters: 26, 29, and 31;37 and 4, 7, 10, 14, 16, 18, 21, and 2342. In their abundance experiments, Martin and Schaber8 observed strong peaks for Genþ clusters for n = 6, 10, 14, 15, and 18, whereas for Sinþ clusters the dominant feature was the low abundance for n = 14. For Sin-, Icking-Konert et al.12 reported higher abundances for n = 7, 10, 15, and 18. We are not aware of any results for SiGe clusters, but Hou and Sing47 have reported particularly stable (SiC)n clusters with N = 4, 12, and 18 atoms. By comparing those values with each other as well

Figure 1. Binding energy per atom for Sin, Gen, and SinGen clusters as a function of the total number of atoms.

Figure 2. Stability function for Sin, Gen, and SinGen clusters as a function of the total number of atoms.

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as with ours only few values of N are common for the same system. Both inaccuracies in the applied total-energy method, effects due to the additional charge for the experimentally studied systems, and the difficulties in identifying the structure of the global total-energy minimum in the theoretical studies can be responsible for those discrepancies. Another property that both is experimentally relevant and can provide information on the stability of the clusters is the fragmentation. For each cluster size, characterized by the number of units, n, we seek that value 0 < m e n/2 for which Δfrag EðnÞ ¼ Etot ðn - mÞ þ Etot ðmÞ - Etot ðnÞ

ð5Þ

is smallest. The cluster with that value of units has Nmin atoms and the corresponding energy ΔfragE is a lower bound for the energy that is required in splitting the cluster into two fragments. It is a lower bound since in an experiment the cluster may not split into the two most stable clusters in their ground states. The resulting values for Nmin are shown in the upper part of Figure 3, and the lower part of this figure shows the corresponding energy ΔfragE. For Si clusters we see that the values Nmin = 7 and 16 occur particularly frequently, in agreement with our finding that these structures are particularly stable. For the other systems there is a less clear correlation between the values of Nmin and those of particularly stable clusters, although Nmin = 5 is found for Ge. Another interesting feature is that the fragmentation energy in general decreases as a function of N, which is a consequence of the general shape of the total-energy per atom which becomes increasingly flat as a function of N. Our results are in general agreement with those of Qin et al.38 for Sin clusters, but are markedly different from our earlier results for Nin clusters62 for which Nmin was found to be Nmin e 3 for clusters with N up to 150 atoms. B. Structural Properties. For the smallest clusters, depicting their structures is useful. Therefore, we show in Figures 4, 5, and 6 the structures for Sin, Gen, and SinGen, respectively, with between 3 and 10 atoms. As we discuss further below, already for these quite small clusters there is a tendency for the Ge clusters to be more open and have lower-coordinated atoms than what is found for the other two systems. Moreover, only the 4-atomic clusters are found to have similar structures. For the Sin clusters with n = 5-7 it is interesting to notice that the stablest structures are those of a bipyramid with either 3, 4, or 5 atoms forming the basis. A somewhat distorted version of this

Figure 3. Energetically most likely fragmentation channel for Sin, Gen, and SinGen clusters as a function of the total number of atoms. The upper parts show the size of the smallest fragment of the cluster, and the lower parts show the corresponding energy. 2008



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Figure 4. Optimized structures of Sin clusters with n ranging from 3 (top, left) to 10 (bottom, right).

Figure 5. Optimized structures of Gen clusters with n ranging from 3 (top, left) to 10 (bottom, right).

Figure 6. Optimized structures of SinGen clusters with n = 2, 3, 4, and 5 (from left to right).

structure is found for the Si3Ge3 cluster, which becomes even more distorted when passing to Ge6. For the Sin clusters our results for n = 4-7 are very close to those obtained theoretically by Bazterra et al.27 as well as to the earlier experimental results.10,11,13,16,18,21-23 Also, for n = 8-10 there are close similarities, but also some differences in the details. Similar results are found when comparing with the studies of Qin et al.,38 Lu et al.,40 and Ghasemi et al.34 For the Sin clusters with n = 8-10 our structures resemble those obtained earlier in experimental and theoretical studies,16,18,21-23 although we cannot exclude that details may differ between the results of the different studies. As mentioned above, much less information on the structures of Gen clusters is available. Wang et al.42 presented a listing of the structures for n e 10 which makes a direct comparison difficult. It does, however, look as if their structures are more similar to those of the Sin clusters than what we find. For Si2Ge2 Li et al.44 presented a very similar structure with, however, the surprising difference that the Si and Ge atoms have been interchanged. Since Si-Si bonds are stronger than Ge-Ge bonds (as also found in our study; cf. Figure 1), our structure

appears to be more realistic. For Si4Ge4 our structure resembles that of Li et al. Moreover, for this both studies find Si-Si bonds but not Ge-Ge bonds. Our prediction for the structure of Si2Ge2 is supported by the results of Bing et al.45 who found the same structure. On the other hand, for Si3Ge3 we find a different structure than Bing et al. who, however, did not carry a complete geometry optimization through. Finally, when comparing with the structures of the SinCn clusters that were presented by Huo and coworkers47,48 much larger differences are observed. For the Si-C systems, a strong preference for clustering of the C atoms is found. For the larger clusters we show the structures only for the particularly stable sizes, i.e., n = 16, 34, 28, and 42 for Sin clusters, n = 16, 26, 39, and 43 for Gen clusters, and n = 7, 12, 14, and 18 for SinGen clusters in Figures 7, 8, and 9, respectively. It is clear that the structures have low symmetry. Moreover, the structures are built up of compact parts, each often containing around roughly 10 atoms but sometimes being smaller. These parts are less strongly bonded with each other. This finding is consistent with the dissociation pattern of Figure 3. Furthermore, for the smaller clusters the overall structure is that of an elongated, cigar-shaped, oblate object, whereas the larger clusters are more spherical. 2009



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Figure 7. Optimized structures of the particularly stable larger Sin clusters with n = 16, 34, 38, and 42 (from left to right).

Figure 8. Optimized structures of the particularly stable larger Gen clusters with n = 16, 26, 39, and 43 (from left to right).

Figure 9. Optimized structures of the particularly stable larger SinGen clusters with n = 7, 12, 14, and 18 (from left to right).

At first, we define the center of a given cluster with N atoms R B0 ¼

1 N R Bi N i¼1

∑

ð6Þ

where R Bi is the position of the ith atom. The radial distance of this atom is then

Figure 10. Radial distances (in au) for the different systems and atoms. In the figure, the notation “A in B” means the radial distances for the A atoms in the B clusters. In each panel, for a specific number of atoms, N, each little line stands for at least one atom with that value of the radial distance.

The same structural trends were found in other theoretical studies on Sin clusters.25-28,30,31,33,35,37,38,40 Also, experimental studies (in many cases supplemented with theoretical calculations) on Sin clusters have predicted that the structures consist of larger, compact building blocks that are less strongly bonded with each other.15,16,18,19,21,22 There are, nevertheless, differences in the structures. These can first of all be due to differences in the charges of the clusters, but also to kinetic effects in the experiments, and to various inaccuracies in the theoretical studies. The general bonding pictures are, however, found in all studies. Finally, similar results as ours were found in the few other studies on Gen clusters.37,39,40,42 Further information on the structural properties can be obtained from the radial distances that are defined as follows.

r ij ¼ j R Bi - R B0 j ð7Þ ri ¼ j B In Figure 10 we show the radial distances for all systems. The two upper panels show those of the pure Sin and Gen clusters, whereas the two lower ones show those for the SinGen clusters for which we have considered the two atom types separately. For the Gen clusters we see that for roughly 22 e N e 31 the radial distances split into more groups each having values in different ranges. This is, e.g., the case when the atoms form onionlike structures, although in the present case, the structures are not roughly spherical. A similar, although less pronounced, behavior is seen for Sin clusters for n > 32. For the other sizes of the Sin and Gen clusters such a regular pattern is not seen and the structures are, accordingly, much less ordered. For the SinGen clusters we recognize a trend toward having Si atoms at slightly lower values of the radial distance, i.e., they are mainly found in the inner parts, whereas the Ge atoms are more frequently found in the outer parts of the clusters. In our earlier work50 we analyzed different Si-Ge clusters for which we made a specific assumption about their structure. Analyzing the total energies we predicted that the Si atoms would prefer to be in the inner parts of the clusters, in perfect agreement with the results here. An overall shape can be identified by calculating the eigenvalues, IRR, of the 3 3 matrix with the elements ∑isiti.62 Here, si and ti are 2010



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Figure 11. Left panels show the eigenvalues IRR (in au), normalized by N-5/3, for the different clusters. In the right panels, the Si and Ge atoms of the Si-Ge clusters have been considered separately, and the normalization has been done with n-5/3. The solid curves show the three different eigenvalues, and the dashed line shows their average.

the x, y, or z components of the radial vectorBr i (see eq 7) for the ith atom, and the summation runs over either all atoms of a given Sin, Gen, or SinGen cluster or over only the Si or Ge atoms of a given SinGen cluster. For a jellium model these eigenvalues will scale as the number of atoms, N or n, to the power 5/3, and therefore, we have normalized the eigenvalues IRR by N-5/3 or n-5/3. Two small and one large eigenvalue implies that the cluster is cigar-shaped (prolate) whereas two large and one small eigenvalue suggests a disk-shape (oblate). Finally, spherical systems are found if all three eigenvalues are identical. Here, “small” and “large” is relative to the average value. Moreover, the average value gives an estimate of the spatial extension of a cluster. The results are shown in Figure 11. None of the clusters have an essentially spherical shape, although some of the smallest Sin clusters with n = 5, 7, and 10 are almost spherical according to our classification procedure. Similarly, the Gen cluster with n = 5 is roughly spherical. Moreover, all Gen and SinGen clusters have a cigar-like shape. On the other hand, a transition from prolate to oblate structures is seen for Sin clusters for n = 32. That such a transition takes place in this size range is consistent with the findings of Sieck et al.25 who considered Si clusters with 25, 29, and 35 atoms and used a related procedure for identifying the shape. When looking at the average value for the pure clusters, this seems to pass through a maximum for 30 e n e 40, suggesting that these clusters are the spatially most extended ones. Jarrold and co-workers17,19 have used ion-mobility experiments for Siþ n and Sin clusters to identify the overall shape of the clusters. For the former they observed a transition from prolate structures to more compact ones for n somewhere between 24 and 34, whereas for the latter a similar transition was found for slightly larger values of n. Our results in Figure 11 show that the largest differences in the normalized eigenvalues IRR/n5/3 indeed are found for n up to around 25, suggesting that these clusters are the least compact ones, which is at least qualitatively in agreement with the experimental findings. When comparing the Si and the Ge atoms of the SinGen clusters, it is clearly seen that the values of IRR for the Si atoms in general are lower than those for the Ge atoms. This confirms the results from our analysis of the radial distances: the Si atoms tend to be found in the inner parts of the clusters, and the Ge atoms in the outer parts. Moreover, in this case, the Si atoms form an overall cigar-like structure for all cluster sizes. Next we shall discuss the interatomic bonds and properties that can be derived from those. At first, we shall for each cluster

Figure 12. Pair correlation function g(r) for (from top to bottom) Si, Ge, and Si-Ge clusters. g has been normalized to have a maximal value of 1, and r is given in au.

type identify an interatomic distance, d0, which separates bonded pairs of atoms (their mutual distance is less than d0) from nonbonded pairs. To this end we define a scaled pair correlation function: gðrÞ ¼ K 3

∑n i6∑¼j f ðr - rnij Þ

ð8Þ

Here, K is a constant that assures that the maximum value of g equals 1. Moreover, the n summation is over all clusters of a given type, whereas the i and j summations are over all atoms of a given clusters. rnij is the distance between the ith and jth atom of the nth cluster. Finally, f(s) is a narrow, normalized Gaussian with a full width at half-maximum of 0.01 au. The resulting function is shown in Figure 12 for the three different cluster types. It is clearly seen that each curve has a fairly narrow maximum for small r. For the Sin clusters we find for larger r essentially only a broad feature, whereas for the germanium containing clusters a second maximum for a slightly larger r is observed. The first maximum corresponds to the typical nearest-neighbor distance in the three different types of clusters, i.e., 5.03 au for Sin clusters, 6.73 au for Gen clusters, and 5.02 au for SinGen clusters. Above the first maximum, there is a range for which g(r) becomes very close or identical to 0. We shall use the midpoint of this range in defining d0 which, thus, becomes dSi0 ¼ 5:35 au 0 ¼ 5:85 au dGe 0 dSiGe ¼ 5:85 au

ð9Þ

However, only for the Gen clusters g(d0) = 0. Using these values we can next calculate the relative occurrence of different coordinations for the atoms of the clusters, i.e., o(nAB). Thereby, nAB is the number of B atoms around an atom of type A. The results are shown in Figure 13 where we have used three different values for d0, i.e., those of eq 9 as well as those multiplied by either 0.95 or by 1.05. Some smaller differences from these different values are observed, but the main trends 2011



Figure 13. Function o(nAB) that describes how many B neighbors an A atom has in the different types of structures. In the panels, the notation “B to A in C” represents the B and A atoms in C clusters. The three different curves in each panel are obtained by using three slightly different values for determining whether two atoms are bonded.

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Figure 14. Different similarity functions. The structures that are being compared are given in the individual panels. For further details, see the text.

subsequently are sorted. Then, we calculate " #1=2 NðN - 1Þ=2 dA , i dB, i 2 2 - 0 q¼ NðN - 1Þ i ¼ 1 dA0 dB

∑

ð10Þ

where A and B equal Si, Ge, or SiGe, and the interatomic distances have been scaled by the lengths of eq 9. From this, we define a similarity function62 according to S¼

Figure 15. Results of the common neighbor analysis for the three different types of clusters. In each panel, each curve shows the relative occurrence for a given (klm) 6¼ (000). For the largest clusters (N = 44) the (klm) values are (from above) (100), (200), and (211) for Sin clusters, (100) and (200) for Gen clusters, and (100), (200), and (211) for the SinGen clusters.

remain. These are that, even for the pure clusters, Ge atoms tend to be lower coordinated (hardly above 4-fold coordinated) than Si atoms. The fact that Si atoms have high coordinations in the clusters has also been observed by Sieck et al.25 Moreover, this difference between Si and Ge can also explain why the Sin clusters are more compact than the Gen clusters. Also for the Si-Ge clusters, there are only few Ge-Ge bonds, whereas most often heteroatomic Si-Ge bonds are found, but also some Si-Si bonds exist. We shall next use the interatomic distances in discussing structural similarity between different clusters. First we compare two clusters with the same number of atoms, N. For each of those we calculate all [N(N - 1)]/(2) interatomic distances that

1 1þq

ð11Þ

that approaches 1 (0) if the two structures are structurally very similar (different). For the Si-Ge clusters we do not distinguish between atom types. We shall also compare two clusters of the same material but of neighboring sizes. For the pure clusters we compare the clusters with n and n - 1 atoms. To this end, we consider the n different structures that can be obtained by removing one atom from the n-atomic cluster and keeping the structure of the remaining n - 1 atoms fixed. For each of those we use the approach above in comparing with the structure of the cluster with n - 1 atoms, and from the n different values of S we choose the largest value. Equivalently, for the mixed Si-Ge clusters, we consider all the n2 structures that can be obtained from a SinGen cluster by removing one Si and one Ge atom. The resulting structure is compared with that of the Sin-1Gen-1 cluster without distinguishing between the atom types, and out of the n2 different values for S the largest is kept. Figure 14 shows the results. It is immediately seen that there is a strong similarity between the structures of clusters with neighboring sizes and of the same material. It is more interesting to observe that the structures of the mixed Si-Ge clusters are much more similar to those of the Si clusters of the same size than to those of the Ge clusters. As a maybe logical consequence of this, the structures of the Si and Ge clusters become increasingly different as a function of size. The former finding is in agreement with other results of the present study. Si-Si bonds are stronger than Si-Ge bonds that in turn are stronger than Ge-Ge bonds. Therefore, the systems seek to maximize the number of Si-Si 2012



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Figure 16. Energies of the HOMO and LUMO orbitals for the three types of clusters as functions of their number of atoms.

bonds, making the Si atoms those that dictate the structure of the mixed clusters. Further information about the structure can be obtained from a so-called common neighbor analysis.63-65 With this, for each pair of atoms one relates three integers, (klm). Here, k is the number of neighbors that are common to the two atoms, l is the number of bonds between those, and m is the longest unbroken sequence (measured in number of bonds) of bonds among the l bonds. Finally, one considers the relative occurrence of different (klm) 6¼ (000). In order to identify neighbors and bonds we use the threshold values of eq 9. Moreover, for the Si-Ge clusters we do not distinguish between different types of bonds. Figure 15 shows the results for the three different cluster types as a function of the number of atoms in the clusters. In each panel, the curves show the relative occurrence of a given (klm). For the largest clusters, (klm) = (100) has far the largest relative occurrence. (klm) = (100) is the only (klm) 6¼ (000) for the crystalline materials with a diamond or zinc blende structure, so the large occurrence for this suggests that the cluster structures are developing a crystal-like structure. However, in particular for the Sin clusters we also find other values of (klm), i.e., values that indicate more common neighbors than found in the crystal. This is consistent with the findings above, that the cluster structures also have many higher-coordinated atoms; i.e., the clusters are in some sense more compact than what is found for the crystal. This tendency is much less pronounced for the Gen clusters, and also for the mixed SinGen clusters, (klm) = (100) is dominating. For the smaller clusters, many other (klm) are found, supporting these clusters having structures that are very different from those of the crystals. C. Electronic Properties. The energies of the frontier orbitals, i.e., of the HOMO and the LUMO, are important for applications of the materials in electronic and opto-electronic devices. Simultaneously, their size variation can give further information on the properties of the clusters. Therefore, in Figure 16 we show those energies as functions of the sizes of the clusters. The figure shows that the HOMO and the LUMO in general appear at lower energies for Sin clusters than for Gen clusters. For

Figure 17. Left part shows the stability function as a function of the HOMO-LUMO energy gap, whereas the right part shows the binding energy per atom as a function of the HOMO-LUMO energy gap. Notice that the Si2 and Ge2 clusters have a vanishing HOMO-LUMO gap and are not shown in the right part.

the mixed SinGen cluster the energy of the LUMO is quite close to that of the Gen cluster with the same number of atoms, whereas that of the HOMO appears at an energy between those of the HOMO of the Sin and the Gen clusters with the same number of atoms. Moreover, for all three types of clusters, the energy of the HOMO has a slightly stronger size dependence than that of the LUMO. Furthermore, there is an overall tendency for the HOMO-LUMO gap to decrease as a function of increasing size of the system, although the energies of both frontier orbitals show a significant oscillatory behavior. All findings suggest that both frontier orbitals have significant contributions from the atoms at the surfaces, whereby the oscillatory behavior can be explained. However, of the two frontier orbitals, the HOMO may have larger contributions from the atoms of the inner parts of the clusters than is the case for the LUMO. This would explain the stronger dependence of this energy on cluster size. Finally, since the mixed clusters have a larger number of Ge atoms near the surface, the similarity between the LUMO energies of the Gen and the SinGen clusters can also be explained by assuming that the LUMO has significant contributions from the surface atoms. Also in their study on Sin clusters, O~na et al.28 found that the HOMO-LUMO gap decreases as a function of size although, similar to our results, it showed non-negligible oscillations. This was also the case for the study of Zhao et al.37 who compared Sin and Gen clusters. Also in agreement with our results, they found a larger gap for the Sin clusters than for the Gen clusters and, moreover, that the HOMO and LUMO of the former appeared at lower energies than those of the latter. However, in general their orbitals appeared at lower energies than what we find, and 2013


The Journal of Physical Chemistry A moreover, their HOMO-LUMO energy gap is larger than ours. Wang et al.42 have reported experimental and theoretical values for the energy gap and the ionization potential of Gen clusters. These suggest that our values for the former are more realistic than are those of Zhao et al., whereas their values for the latter are more realistic. Actually, density-functional calculations with the currently used approximate density functionals are known to lead to orbital energies at too high energies. Both Cheshnovsky et al.9 and M€uller et al.20 have reported experimental values for vertical electron detachment energies for Sin clusters, which are roughly the ionization potentials for the neutral Sin clusters in the structures of the anions. In turn, the ionization potential should be the (negative) HOMO orbital energy. Then, we find that our values in Figure 16 lie somewhat lower than the experimental values. On the other hand, Liu et al.16 have calculated the vertical and nonvertical ionization potentials directly by comparing the total energies of the neutral and positively charged clusters and found somewhat larger values than ours. In an earlier work66 we have found a strong correlation between the HOMO-LUMO gap and stability for various IIVI and III-V semiconductor clusters whose structure was assumed to be similar to a roughly spherical cut-out of the crystal. This could be interpreted as being a manifestation of the hard-and-soft-acids-and-bases (HSAB) principle.67 We shall here consider the possibility of having such a correlation for the clusters of the present study. However, the concept “stability” may not be unique. We may, thus, consider, e.g., either the stability function or the binding energy per atom as a means for identifying stability. In Figure 17 we show the stability function as well as the binding energy per atom as a function of the HOMO-LUMO energy gap for all clusters of the three types of the present study. It is clear that there is no correlation between the stability function and the HOMO-LUMO energy gap, whereas there is a clear (although not perfect) correlation between the binding energy and the HOMO-LUMO energy gap in particularly for the large clusters (that have the largest binding energy per atom and the smallest HOMO-LUMO energy gap). It may thus be suggested that the correlation merely is a consequence of both being essentially monotonic functions of size.

4. CONCLUSIONS In this work we have carried through a systematic study of the energetic, structural, and electronic properties of Sin, Gen, and SinGen clusters with 2-44 atoms. Due to the very large number of total-energy evaluations that such a study requires we have used a parametrized density-functional method for the totalenergy calculation for a given structure combined with a geneticalgorithm method for the determination of the structure of the global total-energy minimum. Although the approximations of such an approach may lead to inaccuracies, we are convinced that our general conclusions are valid. This is supported by the comparison we have made throughout the paper with the (somewhat scattered) information from other studies on these clusters. Despite their chemical similarities, the pure Sin and Gen clusters possess some surprising differences. Thus, the Si atoms are often higher than 4-fold coordinated, which is not the case for the Ge atoms. For the mixed clusters, the lower energy of GeGe bonds leads to the occurrence of only few such bonds,

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whereas Si-Si and Ge-Si bonds are dominating. The structures of all three systems consisted of less strongly connected building blocks each containing roughly 10 atoms. This has as one consequence that the energetically favored dissociation patterns correspond to the splitting off of fairly large parts, and as another consequence that the clusters in most cases are oblate. An exception is the Sin clusters for which a transition from oblate to prolate shapes is found for clusters with slightly more than 30 atoms. The stronger Si-Si bonds mean that the mixed clusters show a tendency for the Si atoms to occupy the inner parts and the Ge atoms to be found in the outer parts. Moreover, these clusters possess a structure that is more similar to that of the pure Si clusters than to that of the pure Ge clusters. Finally, the HOMO and LUMO energies depend in a highly nontrivial way on cluster size, although an overall decrease in the HOMO-LUMO gap as a function of increasing size can be observed. The former can be related to surface effects, and the latter to an overall quantum-size effect. Attempting to correlate this gap with the stability of the clusters showed that such a correlation depends strongly on the definition of the notion “stability”.

’ AUTHOR INFORMATION Corresponding Author

*E-mail: [email protected]. Notes † E-mail: [email protected]. ‡ E-mail: [email protected].

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Structural and Electronic Properties of Sin, Gen, and SinGen Clusters

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