Photoabsorption Spectra of Si n and Si n O (n 5)

Transcription

1 Commun. Theor. Phys. (Beijing, China) 51 (2009) pp c Chinese Physical Society and IOP Publishing Ltd Vol. 51, No. 4, April 15, 2009 Photoabsorption Spectra of Si n and Si n O (n 5) AN Fang-Fang, 1 ZHANG Hong, 2, and CHENG Xin-Lu 1 1 Institute of Atomic and Molecular Physics, Sichuan University, Chengdu , China 2 College of Physical Science and Technology, Sichuan University, Chengdu , China (Received June 2, 2008; Revised September 26, 2008) Abstract The photoabsorption spectra have been calculated for Si n and Si n O (n 5) clusters using time-dependent density-function theory. Our studies suggest that Si n 1 O clusters are relatively stable than those of corresponding Si n clusters. Moreover, substantial differences are observed among the absorption spectra of different molecules in the energy region (0 8 ev). Comparing two different exchange-correlation potentials, local-density and generalized-gradient approximations, both calculated optical spectra present the same spectral feature. PACS numbers: f, Mb, Cc Key words: Si n clusters, Si n O clusters, TDDFT, photoabsorption spectra 1 Introduction Silicon and silicon oxide clusters have attracted a lot of interest both experimentally and theoretically due to their applications in materials science (such as glass, optical fiber communication and so on). Especially, the electronic device industry still extensively relies on silicon oxide clusters. Therefore, a fundamental understanding of their stability, optical and electronic property is important. Many studies, which are concentrated on the structures, binding energies and polarizability of silicon clusters, have been performed. [1 4] However, the optical properties of Si n O clusters are studied rarely. And these properties are ultimately determined by their geometrical structures. So a better understanding for the structure of the ground-state structure is important, especially for the understanding of other properties. As we all know, the number of isomers increases exponentially with the number of atoms, and the energy differences between competing isomers are often quite small. Therefore one should employ various methods such as density-functional theoryor more exact one such as quantum Monte Carlo etc. to search the ground state structures. In view of this, it is better to resort to several spectroscopic tools available to both computational and experimental physics to characterize a system instead of relying on the ground state total energy. One of the experimental methods which can be used is photoabsorption spectra. A comparison between the calculated absorption spectra and experimental data can provide a useful tool to confirm the ground state structure. The absorption spectra of small Si n clusters were firstly studied by Angel Rubio et al. [5] then by Lgor Vasiliev et al. [6] And little study about the absorption spectra of Si n O clusters was presented. For this reason, in this work, we focus our attention on the optical properties especially the photoabsorption spectra of Si n and Si n O clusters calculated from first principles. Our procedure relies on the time-dependent densityfunctional theory (TDDFT) and it has been successfully used in our previous studies. [7,8] For the photoabsorption spectral calculations, there are two methods that can be implemented in the TDDFT, linear response methods and the methods of propagation the time-dependent Kohn Sham (KS) equations in real space. However, due to the fact that it needs a fair amount of unoccupied states, which are difficult to obtain in the linear-response theory, and with the increase of the size of cluster the linearresponse theory is much worse scaling with it, the timepropagation became the best choice to calculate the photoabsorption spectra. It is quite successful in describing many systems, which include molecules and clusters. [9,10] According to our knowledge, this method has not been used to describe the photoabsorption spectra of Si n and Si n O clusters. We expect these studies may help to highlight the further experimental study. 2 Method We take the initial structures from Refs. [1] and [2]. Then the geometry optimization was carried out by Gaussian03 package. [11] No significant variations are noticed with respect to the initial structures. The most stable geometries of Si n O are yielded by O substitution of the Si atom in the optimized Si n clusters or by capping the extra Si atom on the different positions of the Si n 1 O clusters. Based on these optimized ground-state structures, we perform the calculations of photoabsorption spectra. The main program used is OCTOPUS code, a real-space, real-time implementation of TDDFT. [12,13] Let us describe briefly the methodology. We start with the ground state calculation. Then the system is excited from its ground The project supported by the National Natural Science Foundation of China and China Academy of Engineering Physics under Grant No (NSAF) Corresponding author, address: hongzhang@scu.edu.cn

2 752 AN Fang-Fang, ZHANG Hong, and CHENG Xin-Lu Vol. 51 state with a delta electric filed. That is to say the ground state is instantaneously perturbed by the filed and then we propagate the time-dependent Kohn-Sham equations in real-time to get the dipole strength function. The dipole strength function is related to the imaginary part of the dynamical polarizability S(ω) = 4πm e h 2 ωiα(ω), (1) where h is Planck s constant, m e is the electron s mass, and α(ω) is the dynamical polarizability, which can be described by the following equation: α(ω) = 2 drzδn(r, ω), (2) E 0 where δn(r, ω) is the Fourier transform of the deviation of electron density (n(r, t) n (r, t = 0)). The optical absorption spectrum is obtained by averaging over all three orientations of the system. The electron-ion interaction is described by the normconserving Troullier Martins pseudopotentials. For the exchange-correlation functions, we use the local-density approximation (LDA) [14] and Perdew Burke Ernzerh (PBE) of generalized gradient approximation (GGA). [15] The main parameters used in spectral calculations were the following: mesh spacing of 0.25 Å; the wave function domain, a sphere of radius 7 Å; time step of h/ev; total propagation time of 20 h/ev. With these settings, a convergence of better than 0.1 ev was reached for the total energy. In addition, we have neglected the temperature effects and taken all the calculations in the zero temperature. 3 Results and Discussion In Table 1, we show the symmetry, average binding energies, and ionization potentials (I.P.) of Si n and Si n O clusters. And their optimized ground state structures and photoabsorption spectra are shown in Figs. [1 3]. The average binding energy for Si n and Si n O clusters can be obtained from the following expression, E b (Si n ) = 1 n [ne(si) E(Si n)], (3) E b (Si n O) = 1 n + 1 [ne(si) + E(O) E(Si no)], (4) where E(Si), E(O), E(Si n ), and E(Si n O) are the total energies for free atoms and clusters respectively. As shown in Table 1, the average binding energies of Si n O generally decrease with the increase of the size of cluster, and they are larger than those of the corresponding pure Si n clusters. In other words, the doping of O atom improves the stability of silicon clusters. The total binding energies are also calculated, which compare well with the experimental values. Fig. 1 Photoabsorption spectra of Si 2. Two exchangecorrelation approximations LDA (solid line) and PBE (dashed line) have been used and the spectrum calculated with linear-response method (LR) within TDDFT (dotted line) is plotted in arbitrary units for the sake of comparison. Table 1 The symmetry, binding energy and ionization potential (I.P.) of the lowest energy structures for Si n and Si n O clusters. All values in ev. Cluster Symmetry Per atom Binding energy Total (exp.) I.P. (exp.) Si 2 D h (3.17 ± 0.1 a ) 7.32(7.4 ± 0.3 b ) Si 3 C 2v (7.3 ± 0.4 a ) 7.59 Si 4 D 2h Si 5 D 3h SiO C v Si 2 O C 2v Si 3 O C 2v Si 4 O C 2V a Ref. [16], b Ref. [17] The I.P. obtained from the highest occupied KS eigenvalue of SiO cluster is bigger than those of other oxide clusters, which means that this cluster have weaker chemical reactivity. And as seen from Table 1, the I.P. of Si 2 O, Si 3 O, and Si 4 O increase with the increase of the size of cluster, and they are mostly bigger than the corresponding Si n clusters. That means the doping O atom in Si n clusters decreases their chemical reactivity.

3 No. 4 Photoabsorption Spectra of Si n and Si n O (n 5) 753 The results of our calculations of photoabsorption spectra for these clusters are plotted in Figs. 1 3, together with their optimized structures. Besides the absorption spectra, which averaged over the three polarization directions, we also plot their three components along the Cartesian axes. And in Fig. 2 we give the definition of the axes. For Si 2 cluster, the bond length we have optimized is 2.28 Å, which is bigger slightly than the experimental value. [18] However, the spectrum is less sensitive to this small structure differences. Both of the spectra obtained with LDA and PBE are quite similar to each other. We can see from Fig. 1 that they have a shape excitation at about 5.5 ev and two minor peaks in the 3 5 ev range. The Si 3 cluster in Fig. 2 exhibits a trend of increasing absorption spectra starting at the energy of 3 ev. And in the range of 3 8 ev, the spectrum consists with two minor peaks and four shape peaks. In the case of Si 4, the ground state geometry we have obtained is a planar rhombus (D 2h ). And the absorption spectra also starts at about 3 ev, which agrees well with the features seen for larger clusters. [18] It consists of four minor peaks in the range of ev, and the first shape excitation peak is divided into two peaks. Due to the high geometric symmetry of the D 3h the spectra of Si 5 exhibits shape excitation peak at about 7.5 ev. Fig. 2 Photoabsorption spectra of Si n (n = 2 5). The corresponding geometries are also shown in each panel together with their optimized bond lengths. LDA approximation has been used. The spectra of the three polarization directions are also shown. In comparison with the absorption spectra of Si 2 5 from Vasiliev et al. [6] which was obtained by using the linear-response theory (here we only give the Si 2 spectrum in Fig. 1), one can notice that their spectra start at about 5 ev except Si 5 cluster, and our calculated spectra all start at about 3 ev, which are in accordance with the feature found in larger silicon clusters. [18] Also we can notice that there are some common characteristics both in their spectra and ours, i.e., the two calculations (time-propagation and linear-response) have same peaks at about 5 and 5.5 ev for Si 2 cluster (Fig. 1), and the spectra of Si 3 5 clusters also has same excitation peaks at certain energies (see Ref. [6]). However, it can be easily found that with the energy increasing, their spectra and ours may differ a lot. The reason may be that one needs to take into account more unoccupied states in the linear-response calculation, and the implementation of the time-propagation has a much better scaling with the size of the system. The bond length of SiO is 1.52 Å, which is smaller than 2.32 Å in the Si 2 cluster. The ground-state geometries of Si 2 O, Si 3 O and Si 4 O are planar rhombus (C 2v )

4 754 AN Fang-Fang, ZHANG Hong, and CHENG Xin-Lu Vol. 51 structures. From Fig. 3, we can see the spectra of SiO exhibits two separate peaks in the range of ev. And the absorption spectra start at about 4 ev in the SiO and Si 2 O clusters, and at 2 ev in the Si 3 O and Si 4 O clusters. Except SiO cluster, the others all exhibit a series of close peaks. However, the differences between them are still sufficient to distinguish. Moreover, we noticed that between the three polarization directions the spectrum is stronger along the direction in which the length of cluster is more extended. The reason may be that with the length extended, the delocalization of the valence electrons increases. As seen from Si 4, the polarization in the x and y directions give a little contributions to the spectrum. Fig. 3 Photoabsorption spectra of Si n O (n = 1 4). In the insets we show the ground state structures together with their optimized bond lengths. LDA approximation has been used. The spectra of the three polarization directions are also shown. 4 Conclusion In this work, we have calculated the photoabsorption spectra of Si n and Si n O (n = 2 5) clusters using the TDDFT formalism. The spectra of Si 2 cluster exhibits a large peak at 5.5 ev. The first shape peak in Si 4 spectra is divided in two at about 6.5 ev. And the main peak in Si 5 spectrum is around 7.5 ev. In Si n O clusters, each exhibits a series of close absorption peaks in range of ev except SiO cluster which consists with two separate peaks. Furthermore, the average binding energies and I.P. of these clusters all exhibit sized dependencies. As compared to Si n clusters, the doping O atom significantly increases the chemical stability. Acknowledgments H. Zhang thanks Yoshiyuki Miyamoto for fruitful discussions. Also the authors thank Alberto Castro, M.A.L. Marques and Laura Koponen for their suggestions and comments on using OCTOPOUS code. References [1] Krishnan Raghavachari, J. Chem. Phys. 84 (1986) [2] Rene Fournier, Susan B. Sinnott, and Andrew E. De- Pristo, J. Chem. Phys. 97 (1992) [3] K. Fuke, K. Tsukamoto, F. Misaizu, and M. Sanekata, J. Chem. Phys. 99 (1993) [4] George Aroulis, Didier Begue, and Claude Pouchan, J. Chem. Phys. 119 (2003) 794.

5 No. 4 Photoabsorption Spectra of Si n and Si n O (n 5) 755 [5] A. Rubio, J. A. Alonso, X. Blase, L.C. Balbás, and Steven G. Louie, Phys. Rev. Lett.77 (1996) 247. [6] I. Vasiliev, Serdar Öǧüt, and J.R. Chelikowsky, Phys. Rev. B 65 (2002) [7] Y. Miyamoto and H. Zhang, Phys. Rev. B 77 (2008) [8] Y. Miyamoto and H. Zhang, Phys. Rev. B 77 (2008) R. [9] J.I. Martinez, A. Castro, A. Rubio, and J.A. Alonso, J. Chem. Phys. 115 (2006) [10] L. Koponen, L. Tunturivuori, M.J. Puska, and R.M. Nieminen, J. Chem. Phys. 126 (2007) [11] M.J. Frisch, G.W. Trucks, H.B. Schlegel, et al., GAUS- SIAN03, Pittsburgh, PA (2003). [12] A. Castro, H. Appel, Micael Oliveira, et al., Phys. Stat. Sol. B 243 (2006) [13] M.A.L. Marques, A. Castro, G.F. Bertsch, and A. Rubio, Comput. Phys. Commun. 151 (2003) 60. [14] J.P. Perdew and A. Zunger, Phys. Rev. B 23 (1981) [15] J.P. Perdew, K. Burke, and M. Ernzerhof, Phys. Rev. Lett. 77 (1996) [16] M.W. Chase Jr, C.A. Davies, J.R. Downey, Jr., D.J. Frurip, R.A. McDonald, and A.N. Syverud, J. Phys. Chem. Ref. Data 14 Suppl. No. 1 (1985). [17] J. Drowart, G. De Maria, and M.K. Inghram, J. Chem. Phys. 29 (1958) [18] K.D. Rinnen and M.L. Mandich, Phys. Rev. Lett. 69 (1992) 1823.