Low-energy Structures and Electronic Properties of Large-Sized Si

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C: Physical Processes in Nanomaterials and Nanostructures

Low-energy Structures and Electronic Properties of Large-Sized Si Clusters (N = 60, 80, 100, 120, 150, 170) N

Di Wu, Xue Wu, Xiaoqing Liang, Ruili Shi, Zhe Li, Xiaoming Huang, and Jijun Zhao J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/acs.jpcc.8b00427 • Publication Date (Web): 02 May 2018 Downloaded from http://pubs.acs.org on May 2, 2018

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The Journal of Physical Chemistry

Low-energy Structures and Electronic Properties of Large-sized SiN Clusters (N = 60, 80, 100, 120, 150, 170)

Di Wu,1,2 Xue Wu,1 Xiaoqing Liang,1 Ruili Shi,1 Zhe Li,1 Xiaoming Huang,3 Jijun Zhao1a)

1

Key Laboratory of Materials Modification by Laser, Ion and Electron Beams (Dalian University of

Technology), Ministry of Education, Dalian 116024, China 2

School of Science, Shenyang Aerospace University, Shenyang 110136, China

3

School of Ocean Science and Technology, Dalian University of Technology, Panjin 124221, China

Abstract Global search based on semi-empirical and first-principles calculations has been performed to determine the lowest-energy structures of large-sized silicon (Si) clusters: SiN (N=60, 80, 100, 120, 150, 170). We found that the stuffed-cage structural motif is dominant for these large clusters, where the ratio of the number of surface atoms to core atoms agrees well with a previously proposed space-filling model up to N=120 and the structures of outer cages contain not only the majority of the pentagonal and hexagonal rings but also a few seven-membered or eight-membered rings. Triple-layered stuffed cage structures are found for Si150 and Si170. Based on the lowest-energy structures, the physical properties of Si clusters (including binding energy, ionization potential, adiabatic detachment energy, electronic gap, and photoelectron spectrum) are computed and compared with experimental data. For the first time, our theoretical study provides a fundamental picture for large Si clusters up to 170 atoms.

a)

Corresponding author. E-mail : [email protected]

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1. INTRODUCTION The continuous miniaturization of Si-based microelectronics down to the nanoscale regime requires deep understanding of fundamental physical and chemical properties of nanostructured Si. Especially, Si clusters have attracted attentions because their potential applications for the Si cluster-based new functional devices.1-5 So far, the experimental studies of Si clusters have been done by many groups.6-16 One fundamental issue in Si cluster research is to understand the most stable structures with a given number of Si atoms. However, it is difficult to directly determine the structures of gas-phase clusters in experiments. It has been shown that the comparison between experimental data and theoretical simulations can provide key information on the atomic structures of clusters.17-18 In the last thirty years, extensive theoretical explorations of small-sized Si clusters (SiN) for N≤20 have been carried out.14, 19-27 It is noteworthy that the ground state structure of Si10 is a tetracapped trigonal prism.21 Yoo et al. predicted that the TTP (tricapped-trigonal-prism Si9) to six/six motif (sixfold-puckered hexagonal ring Si6 plus six-atom tetragonal bipyramid Si6) transition occurs at Si16.19-20 Structures of medium-sized Si clusters SiN for 20≤N≤50 have been also investigated by many researchers using various methods.27-34 It has been confirmed that a prolate-to-near-spherical shape transition of SiN occurs at N=27.8, 27-30, 32 For 27≤N≤45, almost all low-lying near-spherical clusters SiN adopt “stuffed-cage” like structures where the cages are homologous to carbon fullerenes.27, 30, 32-34 The fullerene-like cages are most likely generic cage motifs to form low-lying stuffed-cage Si clusters beyond the size N > 27.28 The large-sized Si clusters are in the transition stage from the microscopic phase to the bulk phase. But the previous theoretical investigations rarely focus on the large-sized Si clusters beyond 50 atoms because of the enormous amount of computations. In particular, the number of the local minima on the potential energy surface increases exponentially with increasing cluster size, which makes it difficult to determine the most stable structure. It was reported that the lowest-energy structures of the large-sized Si clusters in the range of 50-80 atoms are stuffed fullerene cages.35-42 Beyond this size range, Qin et al. proposed that the most stable structure for Si172 is a bucky-diamond


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structure.43 Yang et al. presented that Si clusters in the range of 80-100 atoms have onion-like geometries. They also found that for Si136, a bucky-diamond structure with an inner diamond core of 40 atoms is more stable than the onion-like and bulk-like structures, whereas for Si220 cluster, the bulk-like structure is more stable than other structures.44 Despite these limited studies without any global search on the potential energy surface, the lowest-energy structures of large-sized silicon clusters (N>80) and their physical properties remain ambiguous and controversial. In particular, it is not known whether the “stuffed-cage” structural pattern still continues in this size range. In this investigation, we explore the lowest-energy structures of Si clusters in the range of 60-170 atoms via an unbiased search using a genetic algorithm (GA) combined with non-orthogonal tight-binding (NTB) method, refined by a biased basin-hopping (BH) search coupled with density functional theory (DFT). A number of unprecedented structures have been found for these large Si clusters, and the most preferred structural motif still belongs to “stuffed-cage” pattern. Based on these structures, the electronic properties have been also computed and compared with experimental data.

2. THEORETICAL METHODS We adopt a two-step optimization approach to seek the lowest-energy geometrical structures for large-sized SiN clusters (N=60, 80, 100, 120, 150, 170). First, a global search was conducted for all SiN clusters using a genetic algorithm25,

31, 45

incorporated with a

non-orthogonal tight-binding model46, followed by a semi-local search using basin-hopping method47-48 coupled with DFT. The validity and efficiency of this strategy has been demonstrated by a previous study.28 The first step is an unbiased search of the global minimum on the potential energy surface by means of GA-NTB method，which has been used in previous studies of medium-sized SiN clusters.28, 32, 40 The details of GA simulation can be found in our review articles.45, 49 For each cluster, we keep 16 parents in the population and performed over 10000 GA iterations to ensure



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the global minimum on the potential energy surface described by the NTB model. Then, the resulted lowest-energy isomeric structures are called the NTB global minima. In the second step, we adopt the NTB global minima as the starting points to search for lower-energy structure by means of the BH method based on DFT local relaxations.47-48 For each cluster, BH-DFT search lasts for 5000 iterations. Essentially, BH is a Monte Carlo method plus energy minimization, which converts the potential energy surface of a cluster into a multidimensional

“hills

and

basins”

by

the

acceptance

probability

equation

as

ratio=exp[(Eprevious−Enew)/kBT], here Enew and Eprevious are the energy of new cluster and previous point, respectively, kB is the Boltzmann constant, and T is the adjustable temperature coefficient. The BH method can hop over “hills” and find the lowest-energy cluster by the random mutation operations. Six types of mutation operations are adopted here: (1) driving each atom with a small random displacement, (2) moving only one atom by a small random displacement, (3) rearranging several selected atoms by small step and keeping other atoms fixed, (4) randomly selecting one interior atom and moving it onto the surface, (5) moving randomly one exterior atom to nearby another exterior atom, (6) putting one exterior atom into the core region of structure. According to our DFT calculations, the total energies of the lowest-energy structures from BH-DFT search are reduced by 5.88 eV~10.22 eV relative to the NTB global minima from GA-NTB search, indicating that the BH part of the search is capable of escaping from the vicinity of the NTB structure and is able to locate the correct DFT structure. All-electron DFT calculations are performed using the DMol3 package50-51 without symmetry constraint. The general gradient approximation (GGA) with PBE functional52 and double numerical basis including d-polarization function (DND) are used. Self-consistent field calculations have been done with a convergence criterion of 10−6 Hartree on the total energy and real-space orbital cutoff of 6.5 Å. The convergence criteria of 10-5 Hartree for energy, 0.002 Hartree/Å for force, and 0.001 Å for maximum displacement were adopted for geometry optimization. To ensure all the global minimum structures for SiN clusters are real local minimum on the potential energy surface, we have computed the vibrational frequencies for the SiN clusters


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with lowest-energy structures and found no negative frequency. To examine the consistence between NTB and DFT potential energy surfaces for the present large-size Si clusters, we have picked up 25 isomers from NTB-GA search of Si60 and computed their single-point energies using DFT. The relative energies of all these isomers from NTB and DFT calculations are plotted in Figure S1 of Supporting Information. Despite the moderate difference in exact values of relative energies from NTB and DFT, the overall trend is reasonable. Therefore, GA-NTB search is able to make a rough sampling of the potential energy surface and filter out the energetically unfavorable structural motifs, yielding reasonable initial structures for further DFT-BH refining. Due to the extreme complexity of the potential energy surface of these large-sized clusters and the semi-empirical nature for the local minimization method used in the GA optimizations, it should be mentioned that the present optimization may not be able to locate the true ground state structures. Hence, we have assessed our two-stage search scheme by carrying out five independent basin-hopping searches for Si60 cluster using different initial configurations from GA-NTB search. We found that all these five Si60 isomers adopt the same structural motif and their relative energies fall in a narrow range of 0.65 eV as shown in Figure S2. In principle, the lower energy structures would be obtained from more comprehensive global search (either with longer GA/BH iterations or fully DFT instead of tight-binding), which is however beyond our computational resource. Nevertheless, the general trend about the structural evolution of these Si clusters revealed from our present simulation is still valid, even though we cannot guarantee the true global minima are reported here.

3. RESULTS AND DISCUSSION 3.1. Geometrical structures of silicon clusters There are too many SiN clusters within the entire size range, i.e., N = 60-170. Therefore, we have to choose selected sizes (N=60, 80, 100, 120, 150, 170) as the representatives of these large-size SiN clusters. Using the global search scheme mentioned above, the most stable



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structures of these SiN clusters (N=60, 80, 100, 120, 150, 170) are presented in Figure 1. All of them can be considered as stuffed cage geometries. According to the numbers of atoms in the core and outer shell for each cluster, they can be characterized as Si14@Si46, Si20@Si60, Si28@Si72, Si36@Si84, Si44@Si106 and Si50@Si120, respectively. Also note that the largest Si cluster (i.e., Si170) has a diameter of about 18 Å. In a previous study of SiN clusters (N=40, 45, 50), the number of atoms in core and outer shell can be estimated by a simple space-filling model proposed by Zhao et al.32 In this model, m is the number of stuffed atoms inside a spherical cage with n atoms, and their relationship can be expressed by a simple function as m = 0.00515 n 2 + 0.03071n − 1.5525

(1)

Figure 2 plots m versus n for the lowest-energy structures of SiN clusters, along with the curve predicted by Eq. (1). Good agreement was observed up to N=120, suggesting that the optimal stuffed/cage ratio for a near-spherical clusters is still valid simply by considering the space-filling factor for most of the present SiN clusters. In other words, the “stuffed-cage” structural motif, which was found for medium-sized SiN clusters with N=27−50,28,

32

still

dominate at least in the size range of N=60−120. At ambient conditions, bulk Si crystal adopts a diamond lattice structure. As displayed in Figure 3, the core structures in the present stuffed-cage Si clusters can be viewed as fragments of bulk diamond lattice with different lattice orientations and some extent of disorder. Essentially, the structural motifs of cores seem to be a series of distorted “adamantane-like” fragments, similar to previous observations for SiN clusters (N=41−50),53 whereas the core structures are severely distorted owing to formation of sp3 bonds with atoms on the outer cage. Interestingly, the cores of Si150 and Si170 themselves are stuffed-cage structures and can be characterized as Si2@Si42 and Si2@Si48, respectively, as illustrated in Figure 4. In addition to the majority of the pentagonal and hexagonal rings, Si42 shell contains one seven-membered ring, and Si48 shell contains five seven-membered rings and one four-membered ring. In other words, Si150 and Si170 can be characterized as triple-layered stuffed-cage structures as Si2@Si42@Si106 and Si2@Si48@Si120. The triple-layered structures might also explain the deviation of stuffed/cage


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ratio from Eq.(1) at Si150 and Si170. Because sp2 hybridization is unfavorable in Si, unlike the perfect fullerene cages such as C60, the outer cages of lowest-energy large-sized Si clusters are very rough and buckled in order to form sp3 bonds with the inner core atoms. The outer cages mixed with sp2 and sp3 bonds of Si46, Si60, Si72, Si84, Si106 and Si120 usually contain not only pentagonal and hexagonal rings which occupy the majority (see Figure 5). For instance, Si46, Si60, Si72, Si84 and Si106 cages contains one, two, three, three, three seven-membered rings, respectively. Moreover, the outer Si120 cage contains ten seven-membered rings and one eight-membered ring. Based on the above analysis, the embryos of large-sized Si clusters are stuffed-cage structures filled with distorted bulk diamond fragments. To see the size trend of structural evolution, we have computed the average coordination number (CN) of these lowest-energy Si clusters using a cutoff 2.65 Å, as given in Table 1. With increasing cluster size, the average CN of core atoms and cage atoms gradually approach four and three, respectively. Again, this result indicates that the structures of core atoms for stuffed-cage clusters can be viewed as fragments of diamond lattice of silicon. In bulk Si with diamond lattice, each Si atom form four covalent bonds with the neighboring atoms via sp3 hybridization, leading to a tetrahedral bond angle of 109.47°. The bond angle distribution functions of the core atoms for lowest-energy structures for SiN clusters are plotted in Figure 6. The core atoms of Si60 form some triangles, thus the angle distribution functions show a few peaks near 60°. Meanwhile, the core atoms of Si80, Si100 and Si120 form some quadrangles; thus the angle distribution functions show some peaks around 90°. With increasing cluster size, the peaks of bond angle distribution of the cores become more and more concentrated. Especially, the characteristic peaks of core atoms for Si150 and Si170 obviously approach the tetrahedral bond angle of 109.47° in bulk lattice. The analysis of bond angles, along with the above discussions on average coordination numbers and core structures, clearly indicate that the core regions of these SiN clusters gradually become fragments of diamond lattice of Si, while there are still large number of surface atoms that substantially deviate from bulk structure.



From the trajectories of BH search, we also found two bilayer onion-like structures for Si80 and Si100, as shown in Figure 7, respectively. Different from the above stuffed cage structures with relatively compact core atoms, the onion-like structure can be characterized as an interior cage fitting inside an outer cage, i.e., Si16@Si64 for Si80 and Si20@Si80 for Si100, respectively, which have been mentioned by Yang et al.44 The core of Si16@Si64 is a highly symmetrical cage on which a square is opposite to a rhombus and surrounded by eight pentagons. The dodecahedron core of Si20@Si80 is also a cage composed of twelve pentagonal rings. The HOMO-LUMO gaps of Si16@Si64 and Si20@Si80 are rather small, i.e., 0.20 eV and 0.02 eV, respectively. However, from our GGA-PBE calculations, the energies of Si16@Si64 and Si20@Si80 are higher than the lowest-energy stuffed-cage Si20@Si60 and Si28@Si72 structures in Figure 1 by 0.15 eV and 0.11 eV, respectively. This demonstrates that the favorable motif of large Si clusters is stuffed-cage structure rather than onion-like structure suggested by Yang et al.44 However, we also do not deny the possibility that the onion structures may be the degenerate states of Si80 and Si100. Inspired by previous study by Yang et al44, we have also constructed two diamond-like (i.e., bulk-like) structures (as shown in Figure S3) for Si150 and Si170 clusters. These structures are initially taken from bulk diamond lattice and then fully relaxed by DFT optimization. The total energies of the low-energy diamond-like structures of Si150 and Si170 clusters are higher than those of triple-layered stuffed cage structures by 5.10 eV and 5.12 eV respectively. It is difficult to guarantee the best diamond-like structures of such large-sized clusters. Nevertheless, the significant energy difference over 5 eV is sufficient to ensure that the triple-layered stuffed cage structures are much more stable than the diamond-like counterparts. 3.2. Electronic properties Based on the lowest-energy structures determined from above section, we now discuss the electronic properties of Si clusters using the available experimental data for comparison. Table 2 compares the binding energy (Eb) of Si clusters and bulk Si in diamond phase with experimental values.9 One can see that deviations between our DFT calculations and experimental data are in


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the range of 0.16−0.2 eV. Eb increases with the cluster size and thus the larger Si cluster are relatively more stable. Using the lowest-energy structures of neutral SiN clusters as initial configurations, we further optimize the structures of negative and positive charged SiN clusters. The adiabatic ionization potentials (IPs) of the SiN clusters (N=60, 80, 100, 120, 150, 170) from our calculations are also summarized in Table 2. Compared to the experimental data by Fuke et al.7, our theoretical values reproduce the general size-dependent trend well, but systematically overestimate by 0.4−0.6 eV. With increasing cluster size, the IP gradually decreases and gradually approaches the work function of bulk Si. We further computed adiabatic detachment energy (ADE) of these SiN clusters and compared with experimental data6 in Table 2. Obviously, the ADE values from our simulation are consistent with the measured values by Hoffmann et al.6 with average deviation of 0.1 eV while both ADE values of onion-like structures Si16@Si64 and Si20@Si80 for Si80 and Si100 are 0.15 eV higher than experimental data. The general agreements with various experiments further support that the present structures obtained for Si clusters are reliable. Table 2 also presents the energy gap (Eg) between the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) of large-sized SiN (N=60−170) clusters. Note that the theoretical gap of 0.39 eV for Si60 is in accord with a previous value of 0.38 eV, which was experimentally estimated from a supported Si cluster with the comparable size using scanning tunneling spectroscopy.54 For reference, the indirect band gap of Eg = 0.76 eV for bulk Si from our DFT-PBE calculation is smaller than the experiment value of 1.1 eV,54 which is a well-known deficiency of GGA method. In Table 2, it can be observed that Eg of SiN decreases with increasing cluster size, e.g., from 0.39 eV at Si60 to 0.10 eV at Si170, in line with the analysis of the trend of gaps of Si clusters from experiments of Marsen et al.54 In particular, the very small gap of large clusters like Si150 and Si170 can be ascribed to large number of surface Si atoms with dangling bonds, which result in surface state that can be visualized in the plots of HOMO/LUMO in Figure 8. In addition, the band gaps of onion-like structures are significantly lower than the band gaps of stuffed-cage structures.



Photoelectron spectrum (PES) has been employed as fingerprint of the electronic structures of clusters and molecules. The comparison between the simulated and measured PES enables us to distinguish isomer structures.16 Based on the relaxed structures of cluster anions, the PES are simulated using the “generalized Koopman’s theorem”

55

. Figure 9 displays the simulated PES

for the lowest-structures of SiN clusters (N=60−170), compared with the experimental spectra by Hoffmann et al.6 Overall speaking, one can see satisfactory agreement between simulated and measured spectra, demonstrating the validity of our theoretical approaches. The onset locations and detailed PES features from the simulated spectra agree reasonably well with experimental data.6 There are some differences in the number of the peaks of spectra between the simulation and the experiment, which might be partly ascribed to the dense energy levels (especially for the large silicon clusters) and different manners of broadening of these levels in theory and experiment. The PES of the onion-like structure of Si80 is similar to that of the stuffed-cage lowest-energy structure. However, it is different from the stuffed-cage structure of Si100 that the PES of the onion-like structure has some distinct feature differences compared with experimental data from Hoffmann et al.6 To further examine the size-dependent evolution of electronic states, in Figure 10 we compare the partial electron density of states (PDOS) from the contribution of s and p orbital components for Si170 cluster and bulk Si. In fact, a small gap of 0.10 eV exists at the vicinity of the Fermi level of Si170, but does not appear in Figure 10 due to the Gaussian broadening. One can see that the conduction bands of Si170 are mainly contributed by the Si-3p states, the valence bands of Si170 are dominated by the Si-3p states between 0 and −4.5 eV and the Si-3s states below −4.5 eV. Interestingly, the essential feature of PDOS of Si170 is similar to that of Si bulk with a diamond lattice. We have also computed the PDOS of Si60 cluster and found some qualitative similarities as well. As representatives, the spatial distribution for HOMO and LUMO wave functions of SiN (N=60, 120, 170) clusters are plotted in Figure 8. First, all these molecular orbitals are delocalized and distributed in asymmetric manners. For Si60, the orbital distributions on outer


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cages are a little more than core filling atoms. On the other hand, the distributions of HOMO and LUMO almost locate around the outer cage for Si170. As the cluster size increases, the wave functions of molecular orbital distributions drift outwards of the clusters.

4. CONCLUSIONS Our unbiased global search shows that the most stable structures for the large Si clusters with 60−170 atoms are stuffed cages, which is a continuation of the structural motif for medium-sized SiN clusters. The ratio of the numbers of atoms between in outer shell and interior core obey an empirical rule derived from space-filling model. The outer cages contain not only the majority of the pentagonal and hexagonal rings but also small amount of seven-membered rings or eight-membered rings. The size-dependent evolution of structural and electronic properties of Si clusters has been discussed in comparison with bulk Si of diamond lattice. Our theoretical results for binding energy, ionization potential, adiabatic detachment energy, and photoelectron spectrum agree well with the available experimental data and their size dependency. The Si clusters with a diameter up to 18 Å possess small non-zero gaps, while Eg decreases with increasing cluster size partly owing to quantum confinement effect. Due to the surface electronic state, the wave functions of HOMO and LUMO orbitals gradually move to the outer cages of Si clusters as the cluster size increases. The electronic states of the largest Si170 cluster already resemble that of bulk Si. The present investigation of large-sized Si clusters provides a valuable reference for understanding the atomic structures and electronic properties of Si nanostructures.

ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China (41641038, 11604039, 11704057) and the Fundamental Research Funds for the Central Universities of China (DUT15RC(3)099, DUT16RC(3)089).



SUPPORTING INFORMATION Supporting Information. The relative energies from NTB model versus those from DFT single-point energies for 25 selected isomers of Si60. Structures and relative energies for isomers of Si60 from five independent BH search. The low-energy diamond-like structures of Si150 and Si170 clusters.


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Figure 1. Lowest-energy stuffed-cage structures for SiN clusters (N=60, 80, 100, 120, 150, 170). The core atoms are highlighted in pink and the atoms on outer cage are in yellow.


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Figure 2. The number m of interior atoms versus the number n of atoms on outer cage of Si clusters. The red curve is the previous function Eq. (1) by Zhao et al.32 Here the circles indicate the data of this work, the triangles indicate the data from Zhao et al.32



Figure 3. (a) Structures of the core-filling atoms within SiN (N=60, 80, 100, 120, 150, 170). The most inner cores of Si150 and Si170 are highlighted in blue; (b) structures of Si diamond fragments viewed from different lattice orientations.


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(a)

(b) Figure 4. Si150 and Si170 can be characterized as Si2@Si42@Si106 of (a) and Si2@Si48@Si120 of (b) respectively which are the triple-layered stuffed cage structures. From the inner core to the outer cage, the three layers are highlighted in blue, pink and yellow, respectively.



Figure 5. Here, Si46, Si60, Si72, Si84, Si106 and Si120 are outer cages of Si60, Si80, Si100, Si120, Si150 and Si170 clusters, respectively. Seven-membered and eight-membered rings are labeled by highlighted red and blue, respectively.


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Figure 6. The bond angle distribution functions of the core region for large-sized stuffed-cage Si clusters calculated with cutoff 2.65 Å. The red short dash line indicates the tetrahedron angle of 109.47°.



Figure 7. The onion-like structures Si16@Si64 and Si20@Si80, with the core atoms and outer shell atoms highlighted by pink and yellow, respectively.


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Figure 8. The isosurfaces of HOMO and LUMO orbital for SiN (N=60, 80, 100, 120, 150, 170) clusters (isovalue: 0.024 e/Å3). Blue and green denote different wavefunction phases of frontier orbitals. The endohedral atoms are highlighted by pink and the atoms on the outer cage are marked by yellow.



Figure 9. Simulated photoelectron spectra (PES) of the lowest-energy structures of SiN (N=60, 80, 100, 120, 150, 170) clusters (with 0.1 eV Gaussian broadening) are shown as red curves. The experimental data taken from Hoffmann et al6 are also shown as black curves in insets for comparison. The simulated PES of “onion-like” structures for Si80 and Si100 are shown as blue curves.


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Figure 10. The partial electron density of states (PDOS) for s and p orbitals of Si60, Si170 and bulk Si. The blue short dashed line denotes the Fermi level.



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Table 1. Average coordination numbers (CN) of cores and cages for stuffed-cage and onion-like structures of Si clusters calculated by cutoff distance of 2.65 Å.

Structure type

Stuffed-cage

Onion-like

Cluster Si60 Si80 Si100 Si120 Si150 Si170 Si80 Si100

CN of core 5.86 4.95 4.46 4.75 4.09 4.04 4.50

4.00


CN of cage 3.91 3.75 3.68 3.68 3.38 3.38 3.38 3.25

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Table 2. Ionization potential (IP), binding energy per atom (Eb), adiabatic detachment energy (ADE) and HOMO-LUMO gap (Eg) for low-energy structures of SiN (N=60−170) and bulk Si in diamond phase from our DFT-PBE calculation, as compared to experimental data (Expt.). Here superscript a, b and c are for Ref. 7, Ref. 9 and Ref. 6, respectively. The properties of onion-like structures Si16@Si64 and Si20@Si80 are also exhibited simultaneously.

Si60 Si80 Si80 Si100 Si100 Si120 Si150 Si170 Bulk

Structure Si14@Si46 Si20@Si60 Si16@Si64 Si28@Si72 Si20@Si80 Si36@Si84 Si44@Si106 Si50@Si120

IP (eV) Theo. Expt.a 5.98 5.40 5.86 5.25 5.83 5.25 5.70 5.20 5.70 5.20 5.69 5.10 5.70 5.10 5.54 5.10 5.15 4.87

Eb (eV) Theo. Expt.b 3.90 4.07 3.94 4.10 3.94 4.10 3.95 4.14 3.95 4.14 3.96 4.17 4.01 4.20 4.02 4.22 4.50 4.66


ADE (eV) Theo. Expt.c 3.46 3.44 3.63 3.56 3.71 3.56 3.61 3.72 3.87 3.72 3.80 3.74 4.10 3.83 3.95 3.88

Eg (eV) Theo. 0.39 0.33 0.20 0.28 0.02 0.24 0.11 0.10 0.76


TOC Graphic


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Low-energy Structures and Electronic Properties of Large-Sized Si

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