Structural and Electronic Properties of Small Silicon Nanoclusters

Structural and Electronic Properties of Small Silicon Nanoclusters Prabodh Sahai Saxena a* and Amerendra Singh Sanger b a Physics Department, Lakshmi Narain College of Technology Excellence, Bhopal, India. b Gwalior Engineering College, Gwalior, India. LAKSHMI NRAIN COLLEGE OF TECHNOLOGY-EXCELLENCELAKSHMI NARAIN COLLEGE OF CHNOLOGY- Abstract We have investigated the lowest-energy structures, structural and electronic properties of small Si n (n=2-7) nanoclusters based on density functional theory (DFT) with local density approximation. The total energy, binding energy, HOMO- LUMO gap, bond length and Ionization Potential have been calculated. The results compared with the other theoretical calculations and found in good agreement. Keywords Nanocluster, total energy, binding energy, HOMO-LUMO gap, Ionization Potential. 1 Introduction Because of its great technological importance, silicon clusters are comprehensively studied. Various experimental and theoretical investigations proved that small silicon clusters are not the pieces of silicon crystal and it is also unrelated to bulk silicon in their structure and properties. Bachels and Schafer 1 produced the neutral silicon clusters by a laser vaporization source and measured binding energy of clusters having size from 65 to 890 atoms. Raghavachari and Logovinsky 2 studied the structure and bonding in small silicon clusters by using all electron ab-initio molecular-orbital techniques. Liu et al 3 reported the theoretical study of electronic properties of silicon nanoclusters. Mahtout and Belkhir 4 investigated the structure and relative stability of silicon clusters for 10-16 atoms by employing ab-initio molecular dynamics simulated annealing technique coupled with density functional theory in local density approximation. Pouchan et al 5 derived the relationship between the polarizability, stability and the geometry of small size silicon clusters by density functional theory methods. The effect of oxygen on structure and electronic properties of silicon nanoclusters for 5, 6, 10 and 18 atoms by employing electron-density functional theory and pseudopotentials was studied by Gnidenko and Zavodinsky 6. The structural transition to bulk diamond structure in nanosized silicon clusters was investigated by Yu et al 7. Bao-Xing Li 8 studied the stability for the neutral and charged silicon clusters for 26 to 30 atoms. The global geometry optimization of small silicon clusters with empirical potentials and at the DFT level has been reported by Tekin and Hartke 9. Ground state geometry of silicon clusters by employing artificial neural networks have been investigated by Lemes et al 10. Gillespie et al 11 studied the bond-order potential for silicon. In this study the tight-binding description of covalent bonding is used to propose a four-level bond order potential for elemental silicon. An Ab-initio molecular-dynamics simulation of Si clusters using the higher-order finite-difference-pseudopotential method has been performed by Jing et al 12. Simulation of silicon clusters by langevin molecular dynamics with quantum forces was performed by Binggeli et al 13. Zhu and Zeng 14 studied the structures and stabilities of small silicon clusters Si n (n=7-11) by using MP2/6-31 G(d) basis set. In another study, Jo and Lee 15 investigated the ionization potentials of small silicon and germanium clusters of 2 to 12 atoms from perameterized tight-binding method. Zhoa et al 16 calculated the ionization potential of small silicon clusters by using a localized orbital theory on the basis of the tight-binding approximation. Recently, Kostko et al 17 determined the ionization energies of small silicon clusters by using vacuum UV radiation. Despite many studies of small silicon nanoclusters, there are controversies with regard to the ground-state configurations. In this paper, we explore the lowest-energy configurations and investigate the structural and electronic properties of small Si nanoclusters. The extensive investigation of structural and electronic properties of small Si clusters is needed because of their important role in developing semiconducting nanodevices and to know the complex optical properties. 122 Prabodh Sahai Saxena and Amerendra Singh Sanger

123 Prabodh Sahai Saxena and Amerendra Singh Sanger International Journal of Engineering Technology Science and Research 2 Computational details The calculations have been performed using ABINIT code 18 which uses pseudopotential and the plane waves in local density approximation (LDA) of the density functional theory. An efficient fast Fourier transform algorithm 19 is used for the conversion of the wave functions between real and reciprocal lattices. We consider the exchange correlation potential in a local density approximation and the pseudopotential of Troullier and Martin 20 (TM). We find that the ground state total energy from other potentials is higher in comparison to TM potentials. These pseudopotentials are extremely efficient for the cases where the plane-wave expansion has a slow convergence. Thus, we believe that TM potentials and numerical basis set used in present calculation is reasonably good to explain various forms of nanoclusters. All the calculation has been performed in a selfconsistent manner. The studied structures have been optimized for Hellmann-Feynman forces as small as 10-2 ev/å to 10-3 ev/å on each atom. The energy cut-off of the plane wave basis is choosen as 40 Ryd (544.22 ev), which was sufficient for conversion test. We choose a simple cubic supercell having each side equal to 20 a.u. and relax the atomic position to achieve a minimum energy configuration. All the isomeric structures have been optimized to achieve minimum energy by relaxing both the lattice constants and the atomic position in the unit cell. 3 Results and Discussion 3.1 Stability of Structures The total energy is calculated to explore most stable structure. After achieving stable structures, the cluster binding energy per atom is calculated by using the following equation, b/n = [n E(Si) E(Si n )] /n where, n = 2,3,4,5,6,7. Where E(Si) represents the atomic energy of Si, and E(Si n ) is the energy of Si n nanocluster. The results for the total energy, binding energy per atom, HOMO-LUMO gap (i.e. energy difference of highest occupied molecular orbital and lowest unoccupied molecular orbital) and average bond length with other theoretical calculations are presented in Table-1.The most stable structures are the bold faced ones. All the considered structures are shown in fig.1. For isolated Si atom total energy is -115.84 ev. The total energy for silicon dimer is calculated to be -234.11 ev. For silicon dimer, our calculated binding energy is 1.21 ev, which is slightly lower than other calculated 2 value of 1.58 ev. However, calculated bond length 2.18 Å is in good agreement with other theoretical 13 calculation. For silicon dimer our calculated HOMO-LUMO gap is 1.82 ev. For three atom clusters we have considered linear, triangular and linear irregular structures. Among the considered structures the triangular morphology is most stable than others with high binding energy 2.58 ev and bond length of 2.26 Å. Here, our calculated binding energy is slightly higher than other reported 11 value 1.63 ev whereas average bond length is slightly lower than other calculation 11. For triangular structure HOMO-LUMO gap is 0.13 ev. The square, rhombus and tetrahedron structures are taken for four atom clusters. Here, square shape structure is found to be stable with binding energy of 3.13 ev and HOMO LUMO gap is 0.92 ev. The average bond length for the same is 2.30 Å. For five atoms size, tetrahedral structure has minimum energy configuration with highest binding energy value of 3.29 ev and bond length 2.40 Å. This calculated value of binding energy is in good agreement with other theoretical 2 calculation. For tetrahedral structure our calculated HOMO-LUMO gap is 0.50 ev. Other investigated structures in the same group like pentagonal, pyramidal and trigonal bipyramidal have the binding energy 3.04 ev, 3.06 ev and 3.18 ev respectively. For these structures, average bond length comes out to be 2.31 Å, 2.43 Å and 2.47 Å. Total energy of six atoms planar trapezoidal is found to be lower in comparison to the hexagonal structure. So, planar trapezoidal structure is more stable. For the stable structure our calculated binding energy and bond length is 3.40 ev and 2.32 Å respectively. Similarly for Si 7 cluster, it is found that pentagonal bipyramidal structure has more stability in comparison to planar hexagonal. Here, the bond length and binding energy of stable structure is 2.50 Å and 3.43 ev respectively and HOMO-LUMO gap is 0.15 ev. Table-1 reveals that our calculated binding energies are higher than other calculations which reflects the strong stability of the structures and thereby the accuracy of the presently employed method.

Fig. 2(a), 2(b) and 2(c) depicts the variation of binding energy per atom, HOMO-LUMO gap and average bond length with number of atoms for most stable silicon clusters. A perusal of fig.2 (a) reveals that the cluster binding energy increases rapidly with cluster size, which agrees well with the size-dependent properties of clusters. Fig. 2(b) shows that the two atoms silicon dimer has highest HOMO-LUMO gap. One observed that after four atom same is continuously decreases with number of atoms. The average bond length increases gradually with cluster size, but decreases for n=6 as is evident from the fig. 2(c). 3.2 Ionization Potential The Ionization Potential (IP) is defined as the amount of energy required to remove an electron from cluster. This is calculated as the energy difference between the cationic and neutral clusters. Our calculated IP is compared with other theoretical calculations in Table-2. The variation of IP with cluster size has been presented in fig.3. For the Si 2 dimer, the calculated IP is 8.03 ev. This calculated value of IP is in excellent agreement with other theoretical 15 value of 8.09 ev. Our calculated IP for three atoms triangular structure is 8.05 ev, while for linear and linear irregular structures IP is found to be 7.96 and 8.04 ev respectively. Four atoms square structure has IP of 8.10 ev. For other structures like, rhombus and tetrahedron IP is calculated to be 8.01 ev and 8.09 ev respectively. For rhombus structure, our calculated value is in good agreement with 8.02 ev reported by Jo and Lee 15 but for tetrahedron same is lower than other 15. For five atoms tetrahedral structure calculated IP is 8.08 ev which is higher than the other calculation 2. For the same atom size our calculated value of IP for pentagonal, pyramidal and trigonal bipyramidal structures is 8.07, 8.09 and 8.11 ev respectively which almost is in good agreement with other theoretical calculation 15. For six atoms silicon the ionization potentials for planar trapezoidal and hexagonal structure are 7.90 ev and 7.88 ev respectively. Our calculated IP for seven atoms pentagonal bipyramidal structure is in good agreement with other theoretical 15,16 calculations, this is 7.95eV. Ionization potential of stable silicon nanoclusters gradually increases up to four atoms, then decreases up to six atoms thereafter again increases for seven atoms cluster as is evident from fig. 3. 4 Conclusions The structural and electronic properties of different isomeric structures of small silicon nanoclusters Si n (n=2-7) have been investigated. We have determined ground state energy, binding energy per atom, HOMO-LUMO gap, bond length and ionization potential. We investigated that among the considered structure the most stable structures for Si 3, Si 4, Si 5, Si 6 and Si 7 are triangular, square, tetrahedral, planar trapezoidal and pentagonal bipyramidal respectively. At the nanoscale the total energy of silicon nanoclusters decreases with the number of atoms, indicating that the clusters become increasingly stabilized. It is revealed that binding energy per atom increases with cluster size as observed in other nanoclusters but our calculated binding energy is slightly higher than other calculations. Abrupt increment in binding energy per atom is observed from two to three atoms stable silicon cluster. It is also concluded that two atoms silicon dimer has highest HOMO-LUMO gap and even numbered stable clusters have higher HOMO-LUMO gap w.r.t. the neighbouring odd number clusters except in case of six atoms cluster. On the other hand, the even numbered Si n clusters show relatively higher stability. Average bond length is generally increases with cluster size with some distortions and is in good agreement with other investigations. According to this finding the ionization potential (IP) of small silicon nanoclusters increases up to four atoms continuously but thereafter it decreases up to six atoms. Then again increases for seven atoms. IP is maximum for four atoms stable silicon cluster. The findings of the present work can be extended in various directions. It would be interesting to extend the calculations towards larger cluster sizes and perform more comparisons with the results following from this method and other theory, based either on pseudopotentials or effective interatomic potentials. Acknowledgement The authors are thankful to Computational NanoScience & Technology Lab (CNTL) of ABV-Indian Institute of Information Technology & Management, Gwalior for providing the infrastructural facilities for computational work. 124 Prabodh Sahai Saxena and Amerendra Singh Sanger

References 1. Bachels T & Schafer R, Chem Phys Lett, 324 (2000) 365. 2. Raghavachari K & Logovinsky V, Phys Rev Lett, 55 (1985) 2853. 3. Liu L, Jayanthi C S & Wu S Y, J Appl Phys, 90 (2001) 4143. 4. Mahtout S & Belkhir M A, Acta Physica Polonica A, 109 (2006) 685. 5. Pouchan C, Begue D & Zhang DY, J Chem Phys, 121 (2004) 4628. 6. Gnidenko A A & Zavodinsky V G, Semiconductors, 42, (2008) 800. 7. Yu D K, Zhang R Q & Lee S T, Phys Rev, B65 (2002) 245417. 8. Bao-xing Li, Phys. Rev. B71 (2005) 235311. 9. Tekin A & Hartke B, Phys Chem Chem Phys, 6 (2004) 503. 10. Lemes M R, Marim L R, Jr A D P, Mat Res, 5 (2002) 281. 11. Gillespie B A, Zhou X W, Murdick D A, Wadley H N G, Drautz R & Pettifor D G, Phys Rev, B75 (2007) 155207. 12. Jing X, Troullier N, Dean D, Binggeli N, Chelikowsky J R, Wu K & Saad Y, Phys Rev, B50 (1994) 12234. 13. Binggeli N, Martins J L & Chelikowsky J R, Phys Rev Lett, 68 (1992) 2956. 14. Zhu X & Zeng X C, J Chem Phys, 118 (2003) 3558. 15. Jo C & Lee K, J Korean Phys Soc., 32 (1998) 182. 16. Zhao J, Chen X, Sun Q, Liu F & Wang G, Phys Lett, A198 (1995) 243. 17. Kostko O, Leone S R, Duncan M A, Ahmed M, J Phys Chem, A114 (2010) 3176. 18. Goedecker S, SIAM J Sci Comput 18 (1997) 1605. 19. Payne M C, Teter M P, Allan D C, Arias T A and Joannopoulos J D, Rev Mod Phys, 64 (1992) 1045. 20. Troullier N, Martin J L, Phys Rev, B43 (1991) 1993. n Table 1- The calculated total energy(ev), binding energy per atom E b /n (ev), HOMO-LUMO gap ΔE HL (ev), average bond length d (Å) and other theoretical calculations with number of atoms (n). Geometrical Structure Total energy (ev) Present cal. E b/n(ev) Othe r cal. E b/n( ev) Present cal. ΔE HL (ev) Present cal. bond length d(å) Other cal. bond length d(å) 1 One atom -115.84 2 2(a) Dimer -234.11 1.21 1.58 2 1.82 2.18 2.23 2, 2.20 13 3 3(a) Linear -354.03 2.17 1.61 11 1.26 2.19 2.37 11 3(b) Triangular -355.26 2.58 1.63 11 0.13 2.26 2.47 11 3(c) Linear irregular -354.07 2.18 2.56 2 1.10 2.20 2.17 2, 2.16 12, 2.17 13 4 4(a) Square -475.88 3.13 2.21 11 0.92 2.30 4(b) Rhombus -474.42 2.76 3.21 2 0.35 2.29 4(c) Tetrahedron -475.03 2.91 0.41 2.40 2.30 & 2.40 2, 2.31 & 2.40 12 5(a) Pentagonal -594.42 3.04 2.32 11 0.25 2.31 2.38 11 5 6 7 5(b) Pyramidal -594.50 3.06 2.25 11 0.12 2.43 2.61 & 2.50 11 5(c) Tetrahedral -595.65 3.29 3.34 2, 0.50 2.40 2.27 11 5(d) Trigonal bipyramidal -595.10 3.18 0.11 2.47 6(a) Planar trapezoidal -715.47 3.40 0.26 2.32 6(b) Hexagonal -714.27 3.20 2.50 11 0.24 2.52 2.55 11 7(a) Pentagonal bipyramidal -834.95 3.43 0.15 2.50 7(b) Planar hexagonal -834.03 3.30 0.38 2.32 125 Prabodh Sahai Saxena and Amerendra Singh Sanger

n Geometrical Structure Table 2- The calculated IP (ev) with other theoretical calculations. Present calculation IP (ev) Other calculations IP (ev) 2 2(a) Dimer 8.03 7.5 2, 8.09 15 3 4 3(a) Linear 7.96 3(b) Triangular 8.05 8.25 15 3(c) Linear irregular 8.04 7.9 2 4(a) Square 8.10 4(b) Rhombus 8.01 8.02 15, 7.95 16 4(c) Tetrahedron 8.09 8.35 15 5 6 7 5(a) Pentagonal 8.07 7.72 15 5(b) Pyramidal 8.09 7.99 15 5(c) Tetrahedral 8.08 7.8 2 5(d) Trigonal bipyramidal 8.11 8.07 15 6(a) Planar trapezoidal 7.90 6(b) Hexagonal 7.88 7(a) Pentagonal bipyramidal 7.95 7.73 15, 8.08 16 7(b) Planar hexagonal 7.87 Fig. 1 Isomeric structures of the Si n nanoclusters for n=2-7. 2(a) 3(a) 3(b) 3(c) 4(a) 4(b) 4(c) 5(a) 5(b) 5(c) 5(d) 6(a) 126 Prabodh Sahai Saxena and Amerendra Singh Sanger

Binding energy per atom (ev) HOMO-LUMO gap (ev) Average bond length (A0) International Journal of Engineering Technology Science and Research 6(b) 7(a) 7(b) Fig.2 Variation of (a) binding energy per atom (ev), (b) HOMO-LUMO gap (ev) and (c) average bond length (Å) for the most stable nanoclusters with number of atoms (n). 2.55 (c) 2.50 2.45 2.40 2.35 2.30 2.25 2.20 2.15 1.75 1.50 1.25 1.00 0.75 0.50 0.25 0.00 (b) 3.50 (a) 3.25 3.00 2.75 2.50 2.25 2.00 1.75 1.50 1.25 1.00 2 4 6 8 Number of atoms (n) Fig. 3 Variation of calculated Ionization Potential (ev) with number of atoms (n) for most stable nanoclusters. 127 Prabodh Sahai Saxena and Amerendra Singh Sanger

Ionization Potential (ev) International Journal of Engineering Technology Science and Research 8.10 8.05 8.00 7.95 7.90 2 4 6 Number of atoms (n) 128 Prabodh Sahai Saxena and Amerendra Singh Sanger