Theoretical study of the structural evolution of small hydrogenated silicon clusters: Si6Hx

18 October 1996

CHEMICAL PHYSICS LETTERS ELSEVIER

Chemical Physics Letters 261 (1996) 346-352

Theoretical study of the structural evolution of small hydrogenated silicon clusters: Si6Hx T a k e h i d e M i y a z a k i a,c, T s u y o s h i U d a b, I v a n ~ t i c h b, K i y o y u k i T e r a k u r a a a JRCAT, National Institute for Advanced Interdisciplinary Research, 1-1-4 Higashi, Tsukuba 305, Japan b JRCAT, Angstrom Technology Partnership, 1-1-4 Higashi, Tsukuba 305, Japan c Electrotechnical Laboratory, 1-1-4 Umezono, Tsukuba 305, Japan

Received 28 May 1996; in final form 27 August 1996

Abstract

Density functional calculations were performed for the structural properties and energetics of small hydrogenated silicon clusters: Si6Hx (0 ~< x ~< 14). We find that the structures of Si6Hx can be classified into several distinct families in terms of the arrangement of silicon atoms. In particular, we find a series of structures which are intermediate between compact and tetrahedral atomic arrangements. Based on calculated formation energies we address the relative stability of the Si6Hx clusters.

1. Introduction

Clusters represent a unique combination of molecular and condensed matter properties. Among numerous systems, pure silicon clusters (Sin) as a representative of covalent-bonded systems have been studied theoretically by a variety of techniques, including quantum chemistry methods [ 1-4], tight-binding schemes [5], density functional techniques [6], and quantum Monte Carlo methods [7]. The studies have shown that, in order to minimize the number of dangling bonds, small silicon clusters with n ~< 10 favor compact structures distinctly different from fragments of the diamond structures [ 1,3-7]. Larger clusters with 24 ~< n ~< 27 were found to exhibit a prolate-oblate shape transition [ 8]. Despite all these efforts, a number of fundamental questions, such as the one about the general rules governing the evolution of the cluster geometry with increasing n until the completion of a fully sp3-hybridized structure, have not yet been

answered. The case of hydrogenated silicon clusters (SinHx) is even more complicated than that of their bare counterparts. One may intuitively expect that hydrogenation would help the recovery of the tetrahedral bonding network by saturating the dangling bonds, However, the understanding of the structural evolution from a pure Sin to a crystal fragment as a function of x is incomplete. It is the purpose of this Letter to shed light on this problem. Hydrogenated silicon clusters have been known as byproducts during chemical vapor deposition of silicon in a silane plasma. Detailed experimental [9] as well as theoretical [ 10] investigations of the reaction of SinHx+ clusters (n up to ~ 4 ) with neutral silane molecules have clarified that they are grown incrementally by the addition of Sill2 accompanied by detachment of H2. Larger SinHx+ clusters (n up to 10) have been produced in the atmosphere of hydrosilicon radicals [ 11 ] and it was demonstrated that the stability

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T. Miyazaki et al. / Chemical Physics Letters 261 (1996) 346-352

of those clusters was sensitive to the coverage of the hydrogen atoms. Hydrogenated silicon clusters have also been regarded as a model of porous silicon [ 12]. However, most theoretical studies along this line have been limited to fragments of a bulk silicon crystal with dangling bonds fully saturated with H [ 13,14]. In this Letter we study the effect of hydrogenation on the structure of small silicon clusters, i.e. the structure evolution of a "host" silicon cluster as a function of the coverage of hydrogen atoms. We limit ourselves to the case of neutral SinHx clusters with n = 6 but allow x to vary in a wide range, x = 2 , 4 , 6 , 8 , 10, 12 and 14. The main results of our study are as follows: (a) the structures of Si6Hx can be classified into at least three distinct families in terms of the arrangement of silicon atoms: two kinds of compact structure and a bulk-like (tetrahedral) structure; (b) considering the sequential hydrogenation (Si6Hx-2 + H2 Si6Hx), attachment of hydrogen atoms to the cluster is systematic.

2. Method of calculation We have used two different methods based on density functional theory (DFT) [15]. Most of the structural optimization was performed in its planewave pseudopotential formulation [16] and generalized gradient approximation (GGA-PW'91) [ 17]. The search for the potential energy minimum was started either by a simulated annealing structure optimization (SASO) or simply guided by intuition or, where available, by structures previously proposed. The SASO was performed at a lower accuracy by taking the plane-wave cut-off of Epw = 10 Ry. A bare potential for H and an s-nonlocal only Kerker-type pseudopotential [ 18] for Si were used. The SASO optimized structures were then further refined with a higher accuracy. In this optimization, where quenched molecular dynamics was applied, we adopted the Vanderbilt ultrasofl pseudopotential [ 19] for the ls state of H and the Troullier-Martin-type of pseudopotential [20] tbr Si. The reason for choosing the pseudopotential for H is that if one used the bare Coulomb potential then a much larger cut-off energy would be required to achieve a similar accuracy because of the cusp condition at the origin. The wavefunctions were expanded with a cut off of Epw = 12.25 Ry.

347

In all the optimizations the clusters were placed in a cubic cell with edge length of 16/~ and the structure optimization was symmetry unrestricted. The optimization was terminated when the forces on all ions were smaller than ,~0.03 eV/,~. With these parameters, the differences in the total energies among different configurations were calculated with numerical uncertainty being ~0.02 eV/cluster 1. Further, the zero-point corrected 2 formation energy of a disilane molecule, defined here as the energy gain by hydrogenation of a silicon dimer, Si2 + 3H2 ~ Si2H6, was found to be 5.05 eV, close to the value obtained by GAUSSIAN-2 theory, 5.04 eV [ 23]. In order to analyze the shape of the molecular orbitals (MOs) of the clusters obtained, we also performed quantum chemistry calculations using the DMol program 3 . We note that both the optimized structures and the formation energies of the clusters obtained by the DMol program agreed well with those from the plane-wave calculation. This can be seen from the tests we performed for the H2, Sill4 and Si2H6 molecules (Table 1 ). The calculated structure parameters also agree well with the experimental values.

3. Structure of the Si6 cluster Before discussing hydrogenation of the Si6 cluster, we compare the structure of this cluster obtained in the present study with the results of structural optimizations performed previously [ 1,2] (see Fig. la and Table 2). Our optimized structure of the Si6 cluster has Cs symmetry. The one obtained by Raghavachari at the Hartree-Fock (HF) level of theory with the 631G* basis set [ 1 ] has C2v symmetry. The overall asI The artificial dispersion of the HOMO and LUMO due to periodic boundaryconditions (PBC) were found to be no greater than ,-~0.02eV, which is less than 1% of the HOMO-LUMOgaps of the obtained clusters (2-5 eV). The validity of PBC in cluster calculations is discussed in detail in Ref. [21 ]. 2We obtained 0.134 eV per H atom for Ez(H-H) from the calculated vibrational frequencyof H2 with a cut-off energy of 12.25 Ry. This value changedby no greater than 0.001 eV with cut-offenergyup to 16 Ry. As for Ez(H-Si), we assumed a value of 0.21 eV per H atom based on the result in Ref. [22]. 3We adopted the combination of the "Becke'88" [25] and the "PW'91" [17] for the exchange-correlation energy as well as potential.

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T. Miyazaki et al./Chemical Physics Letters 261 (1996) 346-352

Table 1 Calculated bond distances (/~) and bond angles (deg) of H2, Sill4 and Si2H6 Species

Structure parameters

Theory (plane wave)

Theory (DMol)

Expt.

H2

dH-H

0.773

0.750

0.741

SiH4

dsi-H

1.492

1.495

1.48

Si2H6

dsi_si

2,341 1,495 109.0 110.0

2.342 1.500 108.5 110.4

2.331 1.492 108.6 110.3

dsi-n /H-Si-H LSi-Si-H Table 2 Structure parameters (in/~) of Si6 in Fig. la. Atom pair

DFT (plane wave) a

DFF (DMol) a

HF/6_31G*b

MP2/6_31G*C

1-2 1-5 2-3 2-4 2-5 3-5 4-5 5-6

2.63 2.38 2.52 3.92(3.49) 2.42 2.41 2.36 2.70(3.49)

2.69 2.42 2.57 4.00 2.46 2.45 2.39 2.72

2.364 2.442 2.364 3.951 2.435 2.442 2.323 2.651

2.73 2.36 2.73 3.87 2.36 2.36 2.36 2.69

a This work. The numbers in parentheses are the corresponding distances in the regular octahedron. b Ref. [1]. c These bond lengths were reproduced by using the GAUSSIAN 92 program [34] in this study, because they are not shown in Ref. [2]. Note that the MP2/6-31G* optimized structure has D4h symmetry about the axis connecting atoms 5 and 6. pects o f these two structures are similar in the sense that both are edge-capped trigonal bipyramids. A difference is that the bond distances o f atom pairs ( 12) and ( 1 - 5 ) are 2.63 and 2.38 /~, in the structure in Fig. l a while they are 2.364 and 2 . 4 4 2 / ~ in the H F / 6 - 3 1 G * optimized counterpart, respectively. This means that atom 1 mainly bonds to atoms 5 and 6 in the former but to atom 2 in the latter. Honea et al. [2] found that the inclusion o f electron correlation to second order in M011er-Plesset perturbation theory [ 26] ( M P 2 / 6 - 3 1 G * ) leads to a geometry with Dah symmetry where the bond distances o f atom pairs ( 1 - 2 ) and ( 1 - 5 ) are 2.73 and 2.36/~, respectively, meaning that atom 1 (and equivalently, atoms 2, 3 and 4) bonds to atoms 5 and 6. This b o n d - l e n g t h ordering is the same as that in the structure calculated by DFT, although the symmetry is different. In this respect, the bonding character in the D F T optimized configuration o f the Si6 cluster is similar to that in the M P 2 / 6 - 3 1 G * rather

than to the HF/6-31G* ones. We would like to make a brief comment on the structure o f Fig. l a in relation to the structure o f the regular octahedron. The Si6 cluster with Oh symmetry has an open shell with four electrons in a triply degenerate HOMO. The geometry in Fig. l a is stabilized because Jahn-Teller distortion o f the compressive Tlu mode produces a H O M O - L U M O gap o f 2.2 eV. This mechanism and thus the structure o f Si6 in Fig. l a are both clearly different from the "triangle contraction" o f the chair-shaped six-membered ring and the resultant cluster with D3d symmetry which Saito et al. have proposed [ 27 ].

4. Hydrogenation-induced structure transformation of Si6 We start the discussion with the cases for x ~< 6 and consider the structures o f Si6Hx in Fig. 1, in which the

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T. Miyazaki et al./Chemical Physics Letters 261 (1996) 346-352

1

4

©

"5

3 (a) Si 6

(b) Si6H 2

(c) Si6H 4

(d) Si6H 6

Fig. 1. Ball-stick models of the structures of Si6Hx clusters with 0 ~< x x< 6. Open and closed circles represent Si and H atoms, respectively. Sticks connectingeach sphere are drawn if the distances of chosen pairs of Si atoms are less than 2.7 ~. The same convention applies to Figs. 2-4. arrangement of Si atoms remains close to that of the bare Si6 of Fig. la. We found that for the sequence of Si6Hx-2 + HE --~ Si6Hx, the attachment of H occurs at the site where the LUMO of Si6Hx-2 has a large amplitude for x = 2 and 6. The bonding interaction of the ls orbitals of the hydrogens with the LUMO of Si6Hx_2 should be the major cause of the stabilization of the clusters 4 . This explains the reason for the stability of Si6H2 of Fig. lb. A naive expectation for the attachment of H2 to the Si6 cluster of Fig. la may be at site 4, which is more convenient to form an spa-like configuration. However, this structure has a higher total energy than the structure of Fig. lb by ,-~0.1 eV. Furthermore, the structure with hydrogen atoms attached to atoms 1 and 3 is higher in energy by 0.55 eV than the one of Fig. lb. This configuration corresponds to the termination of the fully occupied HOMO of Si6, gaining no energy. In the case of x = 4, however, the 4The antibonding states formed from the occupied orbitals of Si6Hx-2 and the hydrogensare pushedup and will not be occupied, while the bondingstates between the low-lyingunoccupiedstates of Si6Hx-2 and the hydrogensmay become occupied.

(a)

(b) Fig. 2. Structures previously suggested for Si6H6, (a) hexsasilaprismane and (b) hexasilabenzene. next lowest unoccupied MO of Si6H2 participates in the stabilization of Si6H4, because the lobes of the LUMO of Si6H2 have nodes along the mirror plane spanned by atoms 1-2-3-4, whose termination with H gives rise to an unfavorable configuration about the Si atoms. It should be kept in mind that the arrangement of silicon atoms remains close to that of the bare Si6 in Fig. la. This suggests that the structures in Fig. 1 belong to a certain category which is characterized by the compact arrangement of the silicon atoms. Several quantum chemical calculations have been done on the low-energy structures of the Si6H6 isomers. Nagase et al. [28] and Sax and Janoschek [29] have suggested that hexasilaprismane (Fig. 2a) may be energetically favorable. We found that the structure in Fig. ld is 0.34 eV lower in energy than hexasilaprismane. We also performed MP2/6-31 G* optimizations of these clusters and found that this new structure is 0.47 eV lower in energy than hexasilaprismane. Hexasilabenzene (Fig. 2b), a silicon analogue of benzene (C6H6) [ 30 ], has, in our calculation, an energy higher by 0.67 eV than that of hexasilaprismane, in reasonable correspondence to the value 0.412 eV [28] or 0.581 eV [ 31 ], obtained in quantum chemical calculations. To our knowledge, no report has been available

350

T. Miyazaki et al./Chemical Physics Letters 261 (1996) 346-352

(a) Si s (a) Si6H 8

(c) Si6H12

(b) Si6Hlo

(b) Si6H 2

:
(d) Si6H 6

(e) SisH 8

(f) Si6Hlo

(d) Si6H14

Fig. 3. Structures of Si6Hx clusters with 8 ~< x ~< 14. tO date on the structure of Si6H2 and Si6H4 clusters (for a recent review, see Ref. [32] ). For x /> 8 (Fig. 3), we found another series of structures of Si6Hx to be energetically favorable. They are different from those for x ~< 6 in the sense that all the silicon atoms in the clusters are fourfold coordinated and terminated with either one or two hydrogen atoms. In this regime, the sequence Si6Hx-2 ÷ H2 ~ Si6Hx proceeds by breaking a bond of a pair of mono-hydrogenated silicon atoms and then saturating the broken bond with an additional two hydrogen atoms. The structure of Si6Hl0 in Fig. 3b agrees well with that calculated by Nagase et al. [33]. For x = 12, cyclohexasilane (Fig. 3c) has the lowest energy. We obtained three local minima of Si6H12. The one with a five-membered ring is 0.10 eV higher while the other two are 0.51 and 1.09 eV higher in energy than cyclohexasilane, respectively. The energy difference and the structure with the five-membered ring agree well with those obtained by Onida and Andreoni [ 13]. We further calculated the structure of the Si6HI4 clusters 5. We found that several structural isomers exist within a narrow energy range of ,--0.01 eV. An example of such geometries is illustrated in Fig. 3d.

5We used a supercell with the size of 16]~ x 16/~ x 24/k for this calculation.

Fig. 4. Structures of Si6Hx clusters with 0 ~< x ~< 10. Note that these structures are different from those illustrated in both Figs. 1 and 3. In addition to the compact and tetrahedral structures discussed above, we found yet another series of the Si6Hx clusters for 0 ~< x ~< 10 sharing some features of both compact and tetrahedral structures, illustrated in Fig. 4. First of all, the arrangement of silicon atoms of Si6 in Fig. 4a is distinct from that in Fig. la. Patterson and Messmer [4] classified these structures of the Sin clusters into two categories: the TBN ("tetrahedral-bond-network") and PBN ("polyhedralbonding-network") clusters. They identified the structures of Si6 in Fig. la and Fig. 4a to belong to the PBN and TBN, respectively. It is clear from Fig. 4 that the arrangement of silicon atoms is almost unchanged upon hydrogenation. Therefore the structures in Fig. 4 belong to a different structural category from those in Fig. 1 The structure of Si6H6 in Fig. 4d has a surprisingly low energy, 1 and 1.4 eV lower than those in Fig. ld and Fig. 2a, respectively. We found that the same trend of sequential hydrogenation (Si6Hx-2 ÷ H2 ~ Si6Hx) as in PBN Si6 (Fig. 1) applies here: stable structures of Si6Hx are generated from Si6Hx-2 by placing two H atoms at the apexes where the LUMO has a large amplitude. An exception occurs for x = 10,

T. Miyazaki et al./Chemical Physics Letters 261 (1996) 346-352

\

.

.

.

.

'oI\

SioHx

351

coverage (x ~> 10) regimes, while the "TBN" series is stable at the intermediate-coverage (4 ~< x <~ 10) regime, although this argument applies to the situation in which the neutral Si6Hx clusters are in equilibrium with the atmosphere of the H2 molecules. Consideration of the ionization of the clusters as well as clarification of the transition states during the attachment of Silly radicals to the clusters would be important for a deeper understanding of recent experimental results [ll].

6. Conclusions "

/

i

0

i

i

i

I

81'0121'4 Number of H atoms 4

;

Fig. 5. Zero-point corrected formation energy of Si6Hx clusters as a function of x. We include hexasilaprismane (Fig. 2a) in the "bulk" series.

where, because of saturation of all apexes in Si6H8 (Fig. 4e), a Si-Si bond is broken and the broken bond is terminated with an additional two H atoms. This construction of Si6Ht0 is the same as that of Si6Hx in Fig. 3, suggesting the onset of the "bulk" nature in Si6H10, whose total energy is indeed close to that of the counterpart in Fig. 3b.

5. Energetics of formation of the Si6Hx clusters Fig. 5 shows the formation energy of the optimized Si6Hx clusters, which we define as Eform(X) = E~tolusterfr] J r - x E H-si t ~

xE~-H). Here

(b-'cluster(r -- 0 ) -'[-xEtHo~ + ~tot ~"

E~ol~Ster(x) is the total energy of the optimized Si6Hx cluster, E~Ho~is the total energy of an H2 molecule per H atom, and EzH-si and EzH-H are the zero-point energies of a hydrogen atom in a cluster and of a hydrogen molecule, respectively (see footnote 2). We took a mixture of the Si6 cluster shown in Fig. la and H2 molecules as a reference for the formation energy. As discussed in the previous section, we have obtained three structure series characterized by the arrangement of silicon atoms: "PBN" (Fig. 1), "bulk" (Fig. 3) and "TBN" (Fig. 4). As for the relative stability of these, we observe that the "PBN" and "bulk" series are stable at the low-coverage (x ~< 2) and high-

In conclusion, we performed first-principles calculations on stable structures and energetics for hydrogenated silicon clusters: Si6Hx. The hydrogenation of silicon clusters categorizes their structures into at least three distinct families. We found a series of structures which are intermediate between compact and tetrahedral atomic arrangements. Calculated formation energies demonstrate that the Si6Hx clusters belonging to either of the three structure families are stabilized depending on x in the atmosphere of H2.

Acknowledgement We thank T. Kanayama and H. Murakami for stimulating discussions. TM is grateful to K. Tanaka and M. Fujita for suggesting this subject. TM also thanks H. Katagiri and Zhi-Hua Liu for useful comments. The plane-wave calculations have been performed on Fujitsu VPP500 at the JRCAT Supercomputer Laboratory. The DMol calculations have been done on a Cray C90 at the Research Information Processing Station (RIPS) of the Agency of Industrial Science and Technology (AIST). This work has been partly supported by the New Energy and Industrial Technology Development Organization (NEDO).

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