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Tight-binding calculation of ionization potentials of small silicon clusters
27 February1995 PHYSICS LETTERS A
ELSEVIER
Physics Letters A 198 (1995) 243-247
Tight-binding calculation of ionization potentials of small silicon clusters Jijun Zhao, Xiaoshuang Chen, Qiang Sun, Fengqi Liu, Guanghou Wang * Department q[ Physics and National Laboratory t~"Solid State Microstructures, Nanjing University, Nanjing 210093, China Center j'br Advanced Studies in Science and Technology of Microstructures, Nanjing 210093, China Received 14 November 1994; accepted for publication 15 December 1994 Communicated by J. Flouquet
Abstract The size dependence of ionization potentials (lPs) of silicon clusters is studied using a localized orbital theory on the basis of the tight-binding approximation. The geometric structures of the silicon clusters are from the theoretical results of the force field model, and the hopping integral and overlap integral in the tight-binding Hamiltonian matrix are obtained from the best fitted formula for the Slater-Koster parameters. It has been found that our results conform to experiments much better than the previous LDA calculation and the estimation of the spherical droplet model. The local maximum of the IPs for Sito is obtained, corresponding to the magic number behaviour observed by other experimental means. The structural isomer effect and relaxation effect on the IPs of the silicon clusters are also discussed.
1. Introduction
In recent years, small silicon clusters have been studied extensively both because of their intrinsic interest from the point of view of chemical bonding [ 1,2] and because of the potential technological applications of cluster-assembled materials [3,4]. Up to now, the structure and properties of Si,, clusters have been investigated by various experimental methods such as photofragmentation [5], collision induced dissociation [6], photoelectron spectroscopy [7], photoionization [8,9], ion-molecular reactions [ 10,11 ], and Raman spectra [ 12] etc. On the other hand, ab initio quantumchemical calculations [ 1,2,13,14] for clusters of up to ten atoms have been performed while semi-empirical methods such as tight-binding models [15,16] and force-field theory [17,18] have been introduced to * Corresponding author. 0375-9601/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD10375-9601 ( 9 4 ) 0 1 0 0 5 -6
study the larger systems. Most of these theoretical studies have focused on the geometric structures and the relative stability of the clusters, much less on the electronic structures and some other related properties. It is known that the ionization potential (IP) of a microcluster can provide a fundamental insight into the size dependent electronic structure of the cluster [ 19,20]. Very recently, the IPs of Sin clusters up to 40 atoms have been obtained by Fuke et al. [9]. However, the measured IP data are much higher than the adiabatic IPs for Si2_ 6 predicted by Raghavachari and Logovinsky [ 1 ] in their LDA calculation. So it is informative to have theoretical values of vertical IPs in comparison with the experiments. In this Letter, we shall carry out a tight-binding calculation to investigate the electronic structure and ionization potential of the silicon clusters. The nonorthogonality of the basis orbital is included in the frame of the localized orbital theory, which describes the size dependent IPs and the metal-non-
244
J. Z h a o et al. / P h y s i c s Letters A 198 (1995) 2 4 3 - 2 4 7
metal transition of the cadium clusters and mercury clusters successfully [21 ].
2. Theory In present work, we use a parameterized LCAO Hamiltonian constructed from the localized orbital theory [21-23]. The atomic-like orbital [ ~hit) are defined by the set of localized pseudopotential equations ( T + V i ) ]dp,t) - ~
(V}-
c~jt,)(dpj t, IVj)[ q5,~)
jl t
= Eil [ 4~/t> •
(1)
Here i and l label the atomic site and angular momentum of the orbital respectively. V~is the potential of the isolated atom i and V~ is the difference of the crystal potential from Vg, caused by the presence of the atom j. T is the kinetic energy operator. By introducing the coefficient matrix D, we can write Eq. ( 1) in an alternative way, jl r
= E
[~by)nj¢,,t.
(2)
jl t
On the basis of the linear independence of localized orbitals, it has been proved that the eigenenergy E can be obtained by solving the secular equation of the pseudo-Hamiltonian matrix D [ 22],
ID-EII = 0 ,
(3)
whose diagonal elements and off-diagonal elements are given by
By diagonalizing the pseudo-Hamiltonian matrix D, we can obtain the eigenenergy levels of the system. The vertical ionization potential can be derived approximately from the highest occupied molecular orbital (HOMO) of the cluster.
3. Results and discussions Although a considerable number of investigations have been carried out for the atomic structure of the silicon clusters, there is no common view of the structure and interatomic distance of these clusters. Phillips et al. have studied the atomic structures of the Si clusters by using the force field theory [ 17, l 8 ], taking into account the three-body effect and dangling-bond effect at the surface. The resolved equilibrium geometries of small clusters (i.e., less than seven atoms) are confirmed by a recent Raman spectroscopy experiment [ 12], while those for the larger clusters agreed with the ab initio calculations [ 1,13,141 and computer simulations [26]. Therefore, in this work, we take the stable atomic structures and bond lengths predicted by their force field model in Ref. [ 18]. The vertical IPs for these Si clusters are obtained by diagonalizing the pseudo-Hamiltonian matrix D constructed from the localized orbital theory. The calculated IPs and the experimental IPs are compared in Table 1, indicating that our model calculation agrees with the experiment nicely. The adopted equilibrium geometries and bond lengths for those clusters are also given in Table 1. Besides the precise electronic structure calculation, the IP of a dielectric sphere with radius R can be estimated by the spherical droplet model [ 19] as follows, E--1
o
D~t.it=en=ei, - ~ (4~gtl~hjt,)(qSjz,[V}l~b~z),
(4)
jl t
n,.j,, = ( 4),, I V} 14)j,, ).
(5)
Here e ° is the atomic Hartree-Fock energy [24]. In Eq. (4), we have neglected the crystal field integral because it is usually much smaller than the hopping integral. The latter term in the diagonal element corresponds to the correction for the nonorthogonality of the basis orbitals. In practice, the distance dependent hopping integral and overlap integral between different atomic orbitals in the matrix D are obtained from the best fitted Slater-Koster parameters given in Ref. [ 25 ].
IP(R) =WF+
e 2
---e 2R
(6)
Here WF is the work function of the bulk and E is the corresponding dielectronic constant. In Fig. 1, the theoretical predictions of the spherical droplet model are compared with our tight-binding calculation, previous LDA calculations [ 1] as well as the experimental values. One finds that the present tight-binding model conforms to the experiment better than the other calculations. It is worth noting that the local maximum of IPs for Silo corresponds to the magic number behaviour observed in other experiments [ 5-7,10,11]. Furthermore, for Si2 6 clusters, the values in our calculation
J. Zhao et a l . / Physics Letters A 198 (1995) 243-247
245
Table l Ionization potentials (eV) of Si,, clusters (Experiment refers to the experimental values taken from Ref. 19] ) Cluster
Geometry
Bond length (/~)
Theory
Experiment
Si2 Si~ Si4 Si~ Si~, SiT Sis Si9 Sire Si,, Si~_,
line triangle planar rhombus trigonal bipyramid octahedron pentagonal bipyramid bicapped octahedron tricapped trigonal prism bicapped tetragonal antiprism uncomplete icosahedral packing uncomplete icosahedral packing
2.73 2.69 2.65 2.66 2.71 2.73 2.74 2.71 2.70 2.66 2.61
8.24 8.49 7.95 8.53 8.42 8.08 7.80 7.72 7.81 7.41 6.86
> 8.49 > 8.49 7.97-8.49 7.79-8.49 7.87-7.97 7.87-7.97 7.46-7.87 7.46-7.87 7.87-7.97 7.46-7.87 7.17-7.46
are closer to the experimental ones while the LDA calculations give results generally 1 eV lower, though the trends of the two calculations ate more or less simular. Since the LDA theory has provided the adiabatic IPs for a cluster and our tight-binding gives the vertical ones, it seems the H O M O is more bonding in the present investigation than the previous LDA calculations indicate [8]. In our previous work [20], we found that the ionization potential of a transition-metal cluster depends sensitively on the cluster geometry. Both chemical
~
reactivity experiments [ 10,l 1 ] and theoretical calculations [ 13,26] revealed the possible existence of structural isomers in Sin clusters, especially Si 9 and Silo. On the other hand, it is expected that the relaxation effect may influence the electronic structures of a microcluster. To investigate both the structural and relaxation effects, we calculated the ionization potentials of three structural isomers of Silo clusters with the interatomic distance varying from 2.35 to 2.80.A. The atomic structures for these three possible isomers are bicapped tetragonal antiprism, tetracapped trigonal prism, and
10.0
.~,~1
"',., g.o
o ~
8.0
0 7.0
N
• ~,-,,I
0 6.0
I
I
I
I
I
I
I
I
2
3
,
s
8
7
8
9
I
I
I ;2
,3
C l u s t e r Size (n) Fig. 1. Size dependence of the ionization potentials (eV) of Si. clusters. The full line connecting the squares refers to the calculation results of the localized orbital theory; the dashed line connecting the crosses refers to the theoretical results of the LDA calculation; the dotted line refers to the prediction of the spherical droplet model; the black dots refer to the experimental results taken from Ref. [ 91 while the vertical bars denote their upper and lower limits.
246
J. Zhao et al. / Physics Letters A 198 (1995) 243-247 0,0
bioapped tetragonal antiprism
....... t e t r a e a p p e d trigonal prism tetracapped o c t a h e d r o n m ""~ 8.0
o
o
7.0
c~ N
o~
o 6,o
'
3.35
I
'
I
,..~5 8.~5 Interatomic
'
I
8.65 Distance
i
I
8.75
(~,)
Fig. 2. Dependence of the ionization potentials (eV) on the geometric structures and interatomic distance of Si~o clusters.
tetracapped octahdron, according to Refs. [ 13,26]. The IPs for these isomers of Silo as a function of bond length are shown in Fig. 2. We find that these structural isomers possess different IPs and the relations between the IP values and the interatomic distance for the isomers are different. For example, one finds that the IPs for Silo with the bicapped tetragonal antiprism structure oscillate with increasing bond length while those for the other ones only change slightly with varying interatomic distance. Furthermore, several studies have found structural distortions in the atomic structures of the small silicon clusters [13,26,27]. For example, in Ref. [26], the equilibrium structure for Si 6 is a bipyramid, consisting of a square with side length 2.71 A capped on top and bottom, with the distance between a vertex of the square and a cap being 2,36 ,&. Notice that the side length 2.71 ,& is just the bond length of the Si6 cluster with the octahedral structure predicted by the force-field theory [ 18] (see Table 1). One can take this bipyramid structure to be distorted from the octahedron in Ref. [ 18]. We have calculated the IP for the Si 6 cluster of this distorted structure and found the IP to be 8.40 eV, which is much closer to 8.42 eV for a perfect octahedron. Therefore, we conclude that the effect of structural distortion may not change the electronic structures and the ionization potentials of the silicon clusters significantly. Although the geometric structures used in our model calculation have not included structural dis-
tortion, we think the present results will not change largely upon including the structural distortion. In the normal tight-binding model, the basis functions for constructing the Hamiltonian matrix are presumed to be orthogonal and the overlap effects of the atomic orbitals are neglected. However, in this work, we find the overlap effect is important for determining the IPs of the silicon clusters. For instance, the IPs for S i 7 and Sire are 8.08 and 7.81 eV in our localized orbital model, while they will decrease to 7.67 and 7.32 eV if we neglect the overlap effect. In our other works, we have investigated the ionization potentials and metalinsulator transition of the divalent metal clusters Cd,, [21] and Hg~ [28] by means of the localized orbital theory and the theoretical results agree with the experiments quite well. Therefore, we think the localized orbital theory used in this paper is more suitable for describing the electronic structures and ionization potentials of the microclusters with covalent or semimetallic bonding, such as Si,, Cdn, Hgn. Further study on Gen and A1, clusters by our group is underway.
Acknowledgement This work is financially supported by the National Natural Science Foundation (China).
J. Zhao et al. / Physics Letters A 198 (1995) 243-247
References
I 1 ] K. Raghavachari and V. Logovinsky, Phys. Rev. Lett. 55 (1985) 2853. 121C.H. Patterson and R.P. Messmer, Phys. Rev. B 42 (1990) 7520. 13 ] M.F. Jarrold, Science 252 ( 1991 ) 1085. 141R.P. Andres, R.S. Averback, W.L. Brown, L.E. Brus, W.A. Goddard Ill, A. Kaldor, S.G. Louie, M. Moscovits, P.S. Peercy, S.J. Riley, R.W. Siegel, F. Spaepen and Y. Wang, J. Mater. Res. 4 (1989) 704. 151 L.A. Bloomfield, R.R. Freeman and W.L. Brown, Phys. Rev. Lett. 54 (1985) 2246; W. Begemann, K.H. Meiwes-Broer and R.O. Lutz, Phys. Rev. Lett. 56 (1986) 2248; Q.L. Zhang, Y. Liu, R.F. Curl, F.K. Tittel and R.E. Smalley, J. Chem. Phys. 88 (1988) 1670. 161 M.F. Jarrold and J.E. Bower, J. Phys. Chem. 92 (1988) 5702; M.F. Jarrold and E.C. Honea, J. Phys. Chem. 95 ( 1991 ) 9181. 17] T.N. Kitsoupoulos, C.J. Chick, A. Weaver and D.M. Neumark, J. Chem. Phys. 93 (1990) 6908; 95 ( 1991 ) 1441. 18] D.J. Trevor, D.M. Cox, K.C. Reichmann, R.O. Brichman and A. Kaldor, J. Phys. Chem. 91 (1987) 2598. 191 K. Fuke, K. Tsukamoto and F. Misaizu, Z. Phys. D 26 (1993) 204. I I0l M.F. Jarrold, J.E. Bower and K. Creegan, J. Chem. Phys. 90 (1989) 3615. I 11 ] J.M. Alfold, R.T. Laaksonen and R.E. Smalley, J. Chem. Phys. 94 (1991) 2618.
247
[ 12] E.C. Honea, A. Ogura, C.A. Murray, K. Raghavachari, W.O. Sprenger, M.F. Jarrold and W.L. Brown, Nature 366 (1993) 42. [13] K. Raghavachari, J. Chem. Phys. 84 (1986) 5672; K. Raghavachari and CM. Rohlfing, J. Chem. Phys. 89 ( 1988 ) 2219. [ 14] P. Ballone, W. Andreoni, R. Car and M. Parrinello, Phys. Rev. Lett. 60 (1988) 271. [ 15 ] O.F. Sankey, D.J. Nikewshi, D.A. Drabold and J.D. Dow, Phys. Rev. B 41 (1990) 12750. [ 16] I. Kwon, R. Biswas, C.Z. Wang, K.M. Ho and C.M. Soukoulis, Phys. Rev. B 49 (1994) 7242. [ 17] J.R. Chelikowsky, J.C. Phillips, M. Mamal and M. Strauss, Phys. Rev. Lett. 62 (1989) 292. [ 18] J.R. Chelikowsky and J.C. Phillips, Phys. Rev. B 41 (1990) 5735. [ 191 G. Makov, A. Nitzan and L.E. Brus, J. Chem. Phys. 88 (1988) 5076. [201 J.J. Zhao, M. Han and G.H. Wang, Phys. Rev. B 48 (1993) 15297. [ 21 ] J.J. Zfiao, X.S. Cheng and G.H. Wang, Europhys. Lett., in press. [22] LD. Weeks, P.W. Anderson and A.G.H. Davidson, J. Chem. Phys. 58 (1973) 1388. [23] V. Heine, in: Solid State Physics, Vol. 35 (Academic Press, New York, 1980). [24] C.F. Fischer, At. Data4 (1972) 301. [25] P.B. Alien, J.Q. Broughton and A.K. McMahan, Phys. Rev. B 34 (1986) 859. [26] X.G. Gong, Phys. Rev. B 47 (1993) 2329. [27] O.F. Sankey, D.J. Niklewski, D.A. Drabold and J.D. Dow, Phys. Rev. B 41 (1990) 12750.
ELSEVIER
Physics Letters A 198 (1995) 243-247
Tight-binding calculation of ionization potentials of small silicon clusters Jijun Zhao, Xiaoshuang Chen, Qiang Sun, Fengqi Liu, Guanghou Wang * Department q[ Physics and National Laboratory t~"Solid State Microstructures, Nanjing University, Nanjing 210093, China Center j'br Advanced Studies in Science and Technology of Microstructures, Nanjing 210093, China Received 14 November 1994; accepted for publication 15 December 1994 Communicated by J. Flouquet
Abstract The size dependence of ionization potentials (lPs) of silicon clusters is studied using a localized orbital theory on the basis of the tight-binding approximation. The geometric structures of the silicon clusters are from the theoretical results of the force field model, and the hopping integral and overlap integral in the tight-binding Hamiltonian matrix are obtained from the best fitted formula for the Slater-Koster parameters. It has been found that our results conform to experiments much better than the previous LDA calculation and the estimation of the spherical droplet model. The local maximum of the IPs for Sito is obtained, corresponding to the magic number behaviour observed by other experimental means. The structural isomer effect and relaxation effect on the IPs of the silicon clusters are also discussed.
1. Introduction
In recent years, small silicon clusters have been studied extensively both because of their intrinsic interest from the point of view of chemical bonding [ 1,2] and because of the potential technological applications of cluster-assembled materials [3,4]. Up to now, the structure and properties of Si,, clusters have been investigated by various experimental methods such as photofragmentation [5], collision induced dissociation [6], photoelectron spectroscopy [7], photoionization [8,9], ion-molecular reactions [ 10,11 ], and Raman spectra [ 12] etc. On the other hand, ab initio quantumchemical calculations [ 1,2,13,14] for clusters of up to ten atoms have been performed while semi-empirical methods such as tight-binding models [15,16] and force-field theory [17,18] have been introduced to * Corresponding author. 0375-9601/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD10375-9601 ( 9 4 ) 0 1 0 0 5 -6
study the larger systems. Most of these theoretical studies have focused on the geometric structures and the relative stability of the clusters, much less on the electronic structures and some other related properties. It is known that the ionization potential (IP) of a microcluster can provide a fundamental insight into the size dependent electronic structure of the cluster [ 19,20]. Very recently, the IPs of Sin clusters up to 40 atoms have been obtained by Fuke et al. [9]. However, the measured IP data are much higher than the adiabatic IPs for Si2_ 6 predicted by Raghavachari and Logovinsky [ 1 ] in their LDA calculation. So it is informative to have theoretical values of vertical IPs in comparison with the experiments. In this Letter, we shall carry out a tight-binding calculation to investigate the electronic structure and ionization potential of the silicon clusters. The nonorthogonality of the basis orbital is included in the frame of the localized orbital theory, which describes the size dependent IPs and the metal-non-
244
J. Z h a o et al. / P h y s i c s Letters A 198 (1995) 2 4 3 - 2 4 7
metal transition of the cadium clusters and mercury clusters successfully [21 ].
2. Theory In present work, we use a parameterized LCAO Hamiltonian constructed from the localized orbital theory [21-23]. The atomic-like orbital [ ~hit) are defined by the set of localized pseudopotential equations ( T + V i ) ]dp,t) - ~
(V}-
c~jt,)(dpj t, IVj)[ q5,~)
jl t
= Eil [ 4~/t> •
(1)
Here i and l label the atomic site and angular momentum of the orbital respectively. V~is the potential of the isolated atom i and V~ is the difference of the crystal potential from Vg, caused by the presence of the atom j. T is the kinetic energy operator. By introducing the coefficient matrix D, we can write Eq. ( 1) in an alternative way, jl r
= E
[~by)nj¢,,t.
(2)
jl t
On the basis of the linear independence of localized orbitals, it has been proved that the eigenenergy E can be obtained by solving the secular equation of the pseudo-Hamiltonian matrix D [ 22],
ID-EII = 0 ,
(3)
whose diagonal elements and off-diagonal elements are given by
By diagonalizing the pseudo-Hamiltonian matrix D, we can obtain the eigenenergy levels of the system. The vertical ionization potential can be derived approximately from the highest occupied molecular orbital (HOMO) of the cluster.
3. Results and discussions Although a considerable number of investigations have been carried out for the atomic structure of the silicon clusters, there is no common view of the structure and interatomic distance of these clusters. Phillips et al. have studied the atomic structures of the Si clusters by using the force field theory [ 17, l 8 ], taking into account the three-body effect and dangling-bond effect at the surface. The resolved equilibrium geometries of small clusters (i.e., less than seven atoms) are confirmed by a recent Raman spectroscopy experiment [ 12], while those for the larger clusters agreed with the ab initio calculations [ 1,13,141 and computer simulations [26]. Therefore, in this work, we take the stable atomic structures and bond lengths predicted by their force field model in Ref. [ 18]. The vertical IPs for these Si clusters are obtained by diagonalizing the pseudo-Hamiltonian matrix D constructed from the localized orbital theory. The calculated IPs and the experimental IPs are compared in Table 1, indicating that our model calculation agrees with the experiment nicely. The adopted equilibrium geometries and bond lengths for those clusters are also given in Table 1. Besides the precise electronic structure calculation, the IP of a dielectric sphere with radius R can be estimated by the spherical droplet model [ 19] as follows, E--1
o
D~t.it=en=ei, - ~ (4~gtl~hjt,)(qSjz,[V}l~b~z),
(4)
jl t
n,.j,, = ( 4),, I V} 14)j,, ).
(5)
Here e ° is the atomic Hartree-Fock energy [24]. In Eq. (4), we have neglected the crystal field integral because it is usually much smaller than the hopping integral. The latter term in the diagonal element corresponds to the correction for the nonorthogonality of the basis orbitals. In practice, the distance dependent hopping integral and overlap integral between different atomic orbitals in the matrix D are obtained from the best fitted Slater-Koster parameters given in Ref. [ 25 ].
IP(R) =WF+
e 2
---e 2R
(6)
Here WF is the work function of the bulk and E is the corresponding dielectronic constant. In Fig. 1, the theoretical predictions of the spherical droplet model are compared with our tight-binding calculation, previous LDA calculations [ 1] as well as the experimental values. One finds that the present tight-binding model conforms to the experiment better than the other calculations. It is worth noting that the local maximum of IPs for Silo corresponds to the magic number behaviour observed in other experiments [ 5-7,10,11]. Furthermore, for Si2 6 clusters, the values in our calculation
J. Zhao et a l . / Physics Letters A 198 (1995) 243-247
245
Table l Ionization potentials (eV) of Si,, clusters (Experiment refers to the experimental values taken from Ref. 19] ) Cluster
Geometry
Bond length (/~)
Theory
Experiment
Si2 Si~ Si4 Si~ Si~, SiT Sis Si9 Sire Si,, Si~_,
line triangle planar rhombus trigonal bipyramid octahedron pentagonal bipyramid bicapped octahedron tricapped trigonal prism bicapped tetragonal antiprism uncomplete icosahedral packing uncomplete icosahedral packing
2.73 2.69 2.65 2.66 2.71 2.73 2.74 2.71 2.70 2.66 2.61
8.24 8.49 7.95 8.53 8.42 8.08 7.80 7.72 7.81 7.41 6.86
> 8.49 > 8.49 7.97-8.49 7.79-8.49 7.87-7.97 7.87-7.97 7.46-7.87 7.46-7.87 7.87-7.97 7.46-7.87 7.17-7.46
are closer to the experimental ones while the LDA calculations give results generally 1 eV lower, though the trends of the two calculations ate more or less simular. Since the LDA theory has provided the adiabatic IPs for a cluster and our tight-binding gives the vertical ones, it seems the H O M O is more bonding in the present investigation than the previous LDA calculations indicate [8]. In our previous work [20], we found that the ionization potential of a transition-metal cluster depends sensitively on the cluster geometry. Both chemical
~
reactivity experiments [ 10,l 1 ] and theoretical calculations [ 13,26] revealed the possible existence of structural isomers in Sin clusters, especially Si 9 and Silo. On the other hand, it is expected that the relaxation effect may influence the electronic structures of a microcluster. To investigate both the structural and relaxation effects, we calculated the ionization potentials of three structural isomers of Silo clusters with the interatomic distance varying from 2.35 to 2.80.A. The atomic structures for these three possible isomers are bicapped tetragonal antiprism, tetracapped trigonal prism, and
10.0
.~,~1
"',., g.o
o ~
8.0
0 7.0
N
• ~,-,,I
0 6.0
I
I
I
I
I
I
I
I
2
3
,
s
8
7
8
9
I
I
I ;2
,3
C l u s t e r Size (n) Fig. 1. Size dependence of the ionization potentials (eV) of Si. clusters. The full line connecting the squares refers to the calculation results of the localized orbital theory; the dashed line connecting the crosses refers to the theoretical results of the LDA calculation; the dotted line refers to the prediction of the spherical droplet model; the black dots refer to the experimental results taken from Ref. [ 91 while the vertical bars denote their upper and lower limits.
246
J. Zhao et al. / Physics Letters A 198 (1995) 243-247 0,0
bioapped tetragonal antiprism
....... t e t r a e a p p e d trigonal prism tetracapped o c t a h e d r o n m ""~ 8.0
o
o
7.0
c~ N
o~
o 6,o
'
3.35
I
'
I
,..~5 8.~5 Interatomic
'
I
8.65 Distance
i
I
8.75
(~,)
Fig. 2. Dependence of the ionization potentials (eV) on the geometric structures and interatomic distance of Si~o clusters.
tetracapped octahdron, according to Refs. [ 13,26]. The IPs for these isomers of Silo as a function of bond length are shown in Fig. 2. We find that these structural isomers possess different IPs and the relations between the IP values and the interatomic distance for the isomers are different. For example, one finds that the IPs for Silo with the bicapped tetragonal antiprism structure oscillate with increasing bond length while those for the other ones only change slightly with varying interatomic distance. Furthermore, several studies have found structural distortions in the atomic structures of the small silicon clusters [13,26,27]. For example, in Ref. [26], the equilibrium structure for Si 6 is a bipyramid, consisting of a square with side length 2.71 A capped on top and bottom, with the distance between a vertex of the square and a cap being 2,36 ,&. Notice that the side length 2.71 ,& is just the bond length of the Si6 cluster with the octahedral structure predicted by the force-field theory [ 18] (see Table 1). One can take this bipyramid structure to be distorted from the octahedron in Ref. [ 18]. We have calculated the IP for the Si 6 cluster of this distorted structure and found the IP to be 8.40 eV, which is much closer to 8.42 eV for a perfect octahedron. Therefore, we conclude that the effect of structural distortion may not change the electronic structures and the ionization potentials of the silicon clusters significantly. Although the geometric structures used in our model calculation have not included structural dis-
tortion, we think the present results will not change largely upon including the structural distortion. In the normal tight-binding model, the basis functions for constructing the Hamiltonian matrix are presumed to be orthogonal and the overlap effects of the atomic orbitals are neglected. However, in this work, we find the overlap effect is important for determining the IPs of the silicon clusters. For instance, the IPs for S i 7 and Sire are 8.08 and 7.81 eV in our localized orbital model, while they will decrease to 7.67 and 7.32 eV if we neglect the overlap effect. In our other works, we have investigated the ionization potentials and metalinsulator transition of the divalent metal clusters Cd,, [21] and Hg~ [28] by means of the localized orbital theory and the theoretical results agree with the experiments quite well. Therefore, we think the localized orbital theory used in this paper is more suitable for describing the electronic structures and ionization potentials of the microclusters with covalent or semimetallic bonding, such as Si,, Cdn, Hgn. Further study on Gen and A1, clusters by our group is underway.
Acknowledgement This work is financially supported by the National Natural Science Foundation (China).
J. Zhao et al. / Physics Letters A 198 (1995) 243-247
References
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