REVIEW PHYSICAL B 68, 035404 2003
Enhanced radiative transition in Sin Gem nanoclusters
Ming Yu,1 C. S. Jayanthi,1 David A. Drabold,2 and S. Y. Wu1
2
Department of Physics, University of Louisville, Louisville, Kentucky 40292, USA Department of Physics and Astronomy, Condensed Matter and Surface Sciences Program, Ohio University, Athens, Ohio 45701-2979, USA Received 21 January 2003; revised manuscript received 31 March 2003; published 1 July 2003
1
Using an ab initio molecular-dynamics scheme the reball scheme , we determined the equilibrium structure of intermediate-size Sin Gem (n m 71) nanoclusters with and without hydrogen passivation on the surface. Due to the strong surface distortion, defect states are found to permeate the energy gap of Sin Gem clusters. However, the defect states are removed by adding H atoms on the surface of Sin Gem clusters, and the gap opens up to a few eV, indicating a blueshift for photoluminescence. It is also found that the radiative transition between the highest occupied molecular orbital HOMO and the lowest unoccupied molecular orbital LUMO states is enhanced by one to two orders of magnitude for Sin Gem nanoclusters with respect to the corresponding pure Si clusters. This signicant increase of the emission probability is attributed to the strong overlap of HOMO and LUMO wave functions centered mostly on the Ge atoms. DOI: 10.1103/PhysRevB.68.035404 PACS number s : 73.22. f, 61.46. w, 71.15.Pd, 78.67. n
I. INTRODUCTION
The optical properties of bulk Si and Ge are rather mediocre because the light-emission in the bulk Si and Ge is a phonon-assisted indirect process. Therefore, to improve on the light-emission feature in Si-based materials is a challenge for both technological and fundamental research. Luminescence is a result of a signicant overlap in electron and hole wave functions since the strength of the luminescence i.e., the emission rate and quantum efciency depends on the extent of this overlap and the transition probability. A possible means for increasing this overlap for the Si-based materials may be accomplished through, for example, alloying to change the band structure, introducing impurities to produce the intermediate state through which the electron can recombine with the hole, or zone folding to yield the desired quasidirect transition.1 However, the most important breakthrough in this topic is the observation of visible photoluminescence PL from porous Si Refs. 2 4 and Si quantum dots5 which opens possibilities for fabricating visible light-emitting devices from Si-based materials. The structural analysis of porous Si is quite difcult. But several measurements have conrmed that the principal feature of porous Si consists of extremely ne structures which are small enough to exhibit quantum connement effects.2,4,5 Various theoretical works have been conducted on Si nanowires6 10 and Si clusters.1115 They claried that the quantum connement effects give rise to a change in the electronic structure and the optical properties and they are the principle mechanism of the blueshift PL in porous Si and Si quantum dots. Experimental reports indicated that Ge quantum dots embedded in SiO2 glassy matrices16 or in porous Si Ref. 17 show a strong room-temperature luminescence. Theoretical studies on the structure and stability of Ge clusters,18 the polarizabilities of small Ge clusters,19 and the quantum connement effect on excitons in Ge quantum dots20 have also
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been reported. Since Ge has smaller electron and hole effective masses and a larger dielectric constant than the corresponding quantities for Si, the effective Bohr radius of the exciton in Ge is larger than that in Si, and the quantum connement effect appears more pronounced in Ge than in Si.16,17,20,21 These results suggest that Sin Gem nanoclusters could be possible candidates for components of nanoscale functional optical devices. In order to understand the physics of Sin Gem clusters, we performed an ab initio moleculardynamics simulation for Sin Gem clusters of an intermediate size, and systematically studied their electronic and optical properties. There is no doubt that the mismatch effect dominating the electronic and optical properties in Si1 x Gex alloys may introduce interesting optical features in Sin Gem clusters. But one has to keep in mind that the surface distortion associated with stabilizing the Sin Gem clusters will also play an important role. Therefore the competition between the lattice mismatch and the surface distortion is the basic issue in our investigation. For this purpose, we studied Sin Gem clusters of an intermediate size with and without the hydrogen passivation. By comparing our results between the two sets of Sin Gem clusters, we found that the surface distortion plays an important role in the intermediate size of Sin Gem clusters but the mismatch effect dominates when Sin Gem clusters of the intermediate size are passivated by hydrogen atoms to eliminate the dangling bonds so as to lessen the effect of the surface distortion. We also found that the latter shows an enhancement of radiative transition and a blueshift in PL. It should be noted that our simulations of the Sin Gem clusters without hydrogen passivation lead to only one of the more stable congurations among a large number of structural isomers. Hence its resulting structural and electronic properties may not exactly represent the corresponding properties of the true ground-state conguration of the Sin Gem clusters. However this caveat does not affect the main conclusions of our study, including the role played by the lattice
2003 The American Physical Society
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mismatch and the surface relaxation in Sin Gem clusters with and without hydrogen passivation, and the enhanced radiative transition in Sin Gem clusters.
II. METHOD
We considered 71-atom Sin Gem clusters, with m taking values 0, 18, 35, 53, and 71, respectively. The initial congurations for the ve clusters were generated randomly on a regular tetrahedral network. They were then relaxed by the ab initio molecular- dynamics scheme developed by Sankey and co-workers the reball scheme .22 This scheme is based on density-functional theory DFT in the local-density approximation LDA , where a local basis set is used to construct the Kohn-Sham orbitals. The basis functions are slightly excited pseudoatomic orbitals PAO . The KohnSham orbitals are calculated self-consistently using the and the Hamann-Schluter-Chiang pseudopotential23 Ceperley-Alder form of the exchange-correlation potential as parametrized by Perdew and Zunger.24 We have tested this method in the study of the strain relaxation of Si1 x Gex alloys and the results are in good agreement with the experimental observations.25 In our simulation, s p 3 -type PAO were used with connement radii of 5.0a B , 5.2a B , and 3.6a B for Si, Ge, and H atoms, respectively. A cubic cell with a lattice constant 50 was chosen as the unit cell. This cell size is sufciently large to ensure that spurious interactions between clusters will vanish. To obtain the stable conguration of a Sin Gem or a Sin Gem H84 cluster, molecular-dynamics simulations were performed on the clusters, starting from the initial random conguration, at a temperature of 103 K for about 20 ps until the network was equilibrated. The network was then slowly cooled to 300 K for 5 ps, and the dynamical quenching was nally performed to fully relax the system to 0 K. Charge transfer was calculated self-consistently in the simulations, which is important in modeling clusters. After performing the molecular-dynamical simulations, we obtained stable congurations, ve each for Sin Gem and Sin Gem H84 clusters, with the ratio m/(n m) 0.0, 0.25, 0.51, 0.75, and 1.0, respectively. As an example, Fig. 1 a shows the stabilized conguration of the Si36Ge35 cluster. Apparently, there is strong surface distortion in this case and the stabilized structure is a compact network in a more oblate structure. This kind of structure has also been found to be more stable for the intermediate size of Si clusters from other theoretical studies.26 It is found that the surface distortion leads to local bonding congurations with more than four bonds, in particular, in the vicinity of the surface. Further structural analysis shows that the average bond lengths of Si-Si (b SiSi), Si-Ge (b SiGe), and Ge-Ge (b GeGe) of the Sin Gem clusters are expanded relative to the bulk, as shown in Fig. 2 a . But they still maintain the relab SiGe b GeGe and are almost indepentionship of b SiSi dent of the ratio of m/(n m). In addition, as shown in Fig. 2 b , the average angles , which show the angle beand atoms, are less than the tetrahedral angle, tween except for Si Si Si in the Si18Ge53 cluster. The large uctuaas a function of the ratio of tion of the average angles
FIG. 1. The stabilized structure of the Si36Ge35 cluster shows a compact shape a and that of the Si36Ge35H84 cluster shows a spherical-like shape with tetrahedral symmetry in the interior part b . The Si atoms are marked by the grey color, Ge by the black, and H by the light gray.
m/(n m) indicates that the clusters have lost their initial tetrahedral symmetry. This means that the surface distortion strongly dominates in the intermediate-size Sin Gem clusters, whatever the ratio of Si/Ge. When the initial congurations were passivated with 84 H atoms to terminate the dangling bonds on the surface atoms, all the relaxed Sin Gem H84 clusters still maintain a tetrahedral symmetry in the interior, with only a minor distortion on the surface. Such features can be seen, for example, from the relaxed structure of Si36Ge35H84 shown in Fig. 1 b . The average bond lengths b SiSi , b SiGe , and b GeGe of the cluster see see Fig. 2 d are Fig. 2 c and the average angles quite close to the corresponding features in Si1 x Gex alloys comparing Figs. 2 c and 2 d to Figs. 3 and 7 in Ref. 25 . This is an indication that the surface distortion in hydrogenpassivated clusters is weak even for Sin Gem Hl clusters of an intermediate size. On the other hand, the mismatch effect
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PHYSICAL REVIEW B 68, 035404 2003
FIG. 2. The average bond lengths of b SiSi , b SiGe , and b GeGe in Sin Gem clusters a and in Sin Gem H84 clusters c as a function of the ratio of m/(n m). The average angles and and in the angle between two bonds Sin Gem clusters b and in Sin Gem H84 clusters d as a function of the ratio m/(n m) are shown in the right side of the panel. The dotted lines in b and d represent the tetrahedral angle of 109.47 in bulk Si and Ge.
becomes the dominating factor in hydrogenated Sin Gem clusters, similar to the situation exhibited in Si1 x Gex alloys.25
III. ELECTRONIC STRUCTURE
The LDA calculation is known to underestimate the energy gap in the single-particle energy spectrum of semiconductor clusters. Corrections to the underestimated energy gap can be calculated using the GW approach,14,27,28 where G refers to the Greens function and W refers to the screened interaction. However, GW calculations are expensive, even for clusters of an intermediate size considered in this work i.e., Sin Gem H84 clusters with n m 71). Our main interest in this study is to provide an understanding of the trend of change in the electronic structure and optical properties of Sin Gem clusters of a given size as the concentration of the Ge component varies. The LDA approach is expected to be sufciently accurate to provide a qualitative description of this trend of change. Therefore, we have calculated the electronic density of states EDOS of the relaxed Sin Gem clusters, using the same DFT/LDA scheme as that in the structural determination. The electronic density of states of Sin Gem and Sin Gem H84 clusters at the ve ratios of m/(n m) described previously are shown in Fig. 3. From the left panel of Fig. 3, many defect states associated with the surface distortion were seen in the highest occupied molecular orbital/lowest unoccupied molecular orbital HOMO-LUMO energy-gap region of Sin Gem clusters. These defect states were cleared up when the surface was passivated by H atoms see the right panel of Fig. 3 because H atoms terminate the dangling bonds of the surface atoms, resulting in the clusters keeping the basic tetrahedral symmetry. Even though the energy spectra of Sin Gem and Sin Gem H84 clusters are quite different, the gen-
eral feature of the energy spectrum in both cases was found to be rather insensitive to the ratio of m/(n m). Figures 4 a and 4 b illustrate the HOMO-LUMO energy gap of Sin Gem and Sin Gem H84 clusters as a function of the ratio of m/(n m), respectively. Because of the existence of the defect states in the energy-gap region, the HOMOLUMO gap of Sin Gem clusters is small only 0.4 0.5 eV and does not show any dependence on the ratio of Si to Ge. But the HOMO-LUMO gap of the hydrogenated Sin Gem clusters is opened up to several eV 3.5 4.0 eV and shows
FIG. 3. The electronic densities of states of Sin Gem clusters left column and Sin Gem H84 clusters right column at various ratios of m/(n m). The Fermi levels are denoted by the vertical bars and the broadening is 0.05 eV.
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at present, the application of the method is still limited to clusters of radii less than 1 nm. The study of the dielectric function via the optical transition matrix elements allows us to obtain information directly on absorption and photoluminescence spectra. It can also allow us to obtain information indirectly on the relevant radiative PL processes. The imaginary part of the dielectric function can be calculated by32 Im 16 e 2
2 S
e 0 v S
2
S
,
1
where e is the polarization vector of the light, v i/ H,r , 0 refers to the ground state, S the excited state, and S the excitation angular frequency. The excited state can be expanded in electron-hole pair conguration such that
hole elec
FIG. 4. The HOMO-LUMO energy gap of Sin Gem clusters a and Sin Gem H84 clusters b as a function of the ratio of m/(n m).
S
v
c
A S vc . vc
2
a linear dependence on the ratio of Si to Ge which reects the mismatch effect similar to that for the corresponding alloys.25 Specically, experimental observation29 and theoretical calculations30,31 indicate that the lattice mismatch associated with alloying in Si1 x Gex alloys induces a change of the conduction band, in particular, the lowest energy in the conduction band LUMO changes from the point near X point for Si-rich alloys to the L point for Ge-rich alloys,31 leading to a nearly linear decrease of the indirect band gap. In the case of Sin Gem clusters passivated by hydrogen, the dominating factor is again the lattice mismatch. This similar effect is observed from the HOMO-LUMO gap as shown in Fig. 4 b . The existence of such a large energy gap for Sin Gem clusters of this size with n m 71) can be attributed to the effect of quantum connement and is consistent with the scenario of a blueshift in PL.
IV. OPTICAL PROPERTIES
The coupling coefcient A S can be calculated within the vc framework of the two-particle Greens function by solving the corresponding Bethe-Salpeter equation. A reasonable approximation leads to the determination of A S as the positive vc solutions to the eigenvalue equation,32 Ec Ev AS vc
v ,c
K vc,v
AA
c
S
Av
S
c
S S A vc ,
3
where E c is the single-electron energy in the conduction band, E v the single-electron energy in the valence band, and AA K vc,v c the electron-hole interaction kernel. The most computationally intensive part in the calculation is the evaluation AA of the electron-hole interaction kernel K vc,v c . This bottleneck is one of the main culprits that limits the application of the GW approach to only very small clusters and that of the TDLAD to clusters with radii less than 1 nm. Using Eq. 2 , the optical transition matrix elements can be written as
hole elec
It is well known that an accurate calculation of the band gap is important for the determination of the optical polarizability of semiconductors and insulators. Many efforts have focused on the correction to the too-small band gap obtained by DFT/LDA calculations for the purpose of accurately predicting the optical response of these systems. Another key factor for determining the optical properties of these systems is a proper description of interacting electronhole pairs excitons . While calculations using the Greens function based on the GW approach have led to quite accurate predictions of the energy of low-lying excitations and the oscillator strength for small clusters,32 the application of this method to clusters of intermediate sizes is still computationally too expensive. Most recently, a method based on linear-response theory within the framework of the timedependent DFT/LDA TDLDA had been applied to calculate the optical spectra of clusters13 with results in general agreement with experimental results as well as more complicated theoretical methods e.g., the GW approach . However,
0vS
v
c
AS v v c . vc
4
The substitution of Eq. 4 into Eq. 1 shows that the imaginary part of the dielectric function depends on the transition matrix elements of electron-hole pair conguration v v c as well as the coupling coefcient A S . For clusters Sin Gem vc of a given size (n m const), the changing composition is expected to cause more effects on the transition matrix elements v v c than on the coupling coefcient A S because vc of the insensitivity of the electronic densities of state to the change in conguration see Eq. 3 and Fig. 3 . Since the main interest of this work is to understand the effect on the optical properties of Sin Gem clusters of a given intermediate size (n m const) by incorporating Ge atoms into the cluster, we therefore focus our attention on the effect on v v c due to the change in the composition in Sin Gem clusters, particularly on how the change in v v c affects the radiative transition probability of the clusters as their composition varies.
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PHYSICAL REVIEW B 68, 035404 2003 TABLE I. HOMO-LUMO spontaneous emission probability W HL and radiative lifetime HL of Sin Gem and Sin Gem H84 clusters. System Without hydrogenation Si53Ge18 Si71 Si36Ge35 Si18Ge53 Ge71 With hydrogenation Si71H84 Si53Ge18H84 Si36Ge35H84 Si18Ge53H84 Ge71H84 W HL (ns 0.164 2.049 1.415 4.078 0.824 0.0123 0.845 1.673 1.423 1.013 10 10 10 10 10 10 10 10 10 10
1
)
5 5 5 5 5
HL
(ns) 105 105 105 105 105 102 102 102 102 102
6.082 0.488 0.706 0.245 1.214 81.362 1.183 0.598 0.703 0.987
2 2 2 2 2
FIG. 5. The averaged imaginary part of the dielectric functions calculated from Eq. 5 of Sin Gem clusters left column and Sin Gem H84 clusters right column at various ratio of m/(n m). The broadening is 0.05 eV.
To set up a benchmark for the analysis of the trend of the optical features of Sin Gem clusters of a given size but with varying compositions within the framework considered in this work, we have calculated the imaginary part of the dielectric function without the electron-hole interaction. In this situation, Eq. 1 reduces to 16
22
edge of the optical spectra. Such peaks near the edge reect the high structural symmetry the tetrahedral symmetry in the present study of the Sin Gem H84 clusters. The large optical gaps in Sin Gem H84 clusters indicate the possibility of optical applications. Since the luminescence is a result of signicant overlap in electron and hole wave functions, and the strength of the luminescence depends on the extent of this overlap and the transition probability, we investigated the radiative transition probabilities of Sin Gem and Sin Gem H84 clusters. The state-tostate spontaneous transition probability (W i j ) the inverse of the radiative lifetime i j ) of the rst-order radiative process between states i and j is dened according to Fermis golden rule and is given by33 1
ij
Im
e
V
v re c
v
2
c
Ec Ev
. 5 Wij 4e 2
3 i jn
Here the sums are over all the eigenstates v and c . r is the position operator, and V the volume of the cluster. Specically, we calculated the average Im ( ) (16 2 e 2 /V) v c 1 ( v x c 2 vyc 2 v z c 2) 3 (E c E v )] since the shapes of the relaxed Sin Gem clusters are compact and suggest a more-or-less isotropic behavior. Figure 5 presents the calculated Im ( ) at various ratios of Si to Ge in the cases with and without H passivation. It is found that the optical gaps, dened by the onset energy of the spectral edge in Sin Gem clusters, are in the range 0.4 0.5 eV and are almost independent of the ratio of Si to Ge. The optical spectral peaks are smooth and broad. The spectra display low-energy transitions and show a tail near the optical edge. The tail corresponds to the defect states due to the surface distortion as shown in the EDOS of Sin Gem clusters Fig. 3 . Therefore, such optical properties are not suitable for application in optical devices. The optical gaps of H-terminated Sin Gem clusters, however, are in the range 3.3 4.1 eV and have a linear dependence on the ratio of Si to Ge. Unlike optical spectra of Sin Gem clusters, the spectral peaks of hydrogenated Sin Gem clusters are sharp with no tail at the
3hc 3
i re j
2
,
6
where e and m are the electron charge and mass, respectively. c is the speed of light, i j the energy difference divided by h) between states i and j , and n the refractive index of Sin Gem clusters. From ellipsometry34 and optical-absorption3 experiments for porous Si, it seems that the refractive index decreases with increasing porosity. Since there is no measurement of the refractive index for Sin Gem clusters, we choose n to be 1 in the present calculation. It is apparently from the formula of W i j that the energy difference 3j as well as the dipole matrix elements i r j i dominate the spontaneous transition probability. Table I lists the HOMO-LUMO spontaneous emission probabilities 1 3 HyL 2 W HL (4e 2 HL n/3hc 3 ) 3 ( H x L 2 2 H z L ) of Sin Gem and Sin Gem H84 clusters. It can be seen that W HL s for Sin Gem clusters are very small ( 10 5 ns 1 ). The corresponding radiative lifetimes HL are quite large, i.e., 608 s for Si71 , 49 s for Si53Ge18 , 71 s for Si36Ge35 , 25 s for Si18Ge53 , and 121 s for Ge71 , respectively. But the HOMO-LUMO spontaneous transition probabilities W HL of hydrogenated Sin Gem clusters are about two to three orders-of-magnitude larger
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PHYSICAL REVIEW B 68, 035404 2003 TABLE II. Dipole matrices i r j 2 3 ( i x j 2 iyj 2 2 2 i z j ) between HOMO-LUMO states ( ) of Sin Gem and Sin Gem H84 clusters. System Without hydrogenation Si71 Si53Ge18 Si36Ge35 Si18Ge53 Ge71 With hydrogenation Si71H84 Si53Ge18H84 Si36Ge35H84 Si18Ge53H84 Ge71H84 0.00294 0.292 0.623 0.600 0.464 0.0329 0.322 0.238 0.791 0.181 HrL
2
1
than the corresponding ones of Sin Gem clusters. Their corresponding radiative lifetimes HL are therefore shortened by about two to three orders, i.e., 8.1 s for Si71H84 , 0.12 s for Si53Ge18H84 , 0.059 s for Si36Ge35H84 , 0.07 s for Si18Ge53H84 , and 0.098 s for Ge71H84 , respectively. The large differences in W HL and HL between Sin Gem and Sin Gem H84 clusters are mainly due to the energy differences 3 of HL between HOMO-LUMO states in Sin Gem and Sin Gem H84 clusters. As seen in Fig. 3, the HOMO-LUMO energy differences HL of Sin Gem H84 clusters are about ten times larger than those of Sin Gem clusters. It is noted that the radiative lifetime HL calculated for the Si71H84 cluster (8.1 s is comparable to that of the Si66H64 cluster of a similar size (6 s obtained by Hirao et al.33 The spontaneous transition probabilities W HL of the Si71H84 cluster (0.012 104 ms 1 ) are also consistent with the recombination rate of an excited electron-hole pair in Si crystallites about 104 ms 1 ) for the photon energy of 4.0 eV at 5 K calculated by Delerue et al.6,10 The spontaneous transition probabilities W HL are found in our calculation to be higher in the pure Ge71H84 cluster (1.01 10 2 ns 1 ) than in the pure Si71H84 cluster (0.012 10 2 ns 1 ). This result is qualitatively consistent with the results of the radiative decay rate of excitons in Ge quantum dots about 0.2 10 2 ns 1 ) and in Si quantum dots about 0.05 10 2 ns 1 ) with the dot radius of 10 obtained by Takagahara and Takeda.20 Our analysis of the radiative transition in Sin Gem clusters indicates that the HOMO-LUMO spontaneous emission probability W HL radiative life-time HL ) of Sin Gem and Sin Gem H84 clusters is very sensitive to the Ge content m/(n m) in the cluster. Table I shows a sudden and substantial increase in W HL once Ge atoms are incorporated into the Si clusters. In particular, for hydrogen-passivated Sin Gem H84 clusters, the incorporation of Ge atoms into the Si cluster can bring about a dramatic increase in W HL of up to two orders of magnitude as compared to the pure hydrogenpassivated Si cluster. An increase in W HL of this magnitude for Sin Gem H84 with n m 71 suggests that the incorporation of Ge atoms into hydrogen-passivated Si clusters may pave the way to dramatically enhance the optical properties of pure Si clusters. To shed light on the underlying physics of this dramatic change in the HOMO-LUMO spontaneous emission probability W HL , we carried out a detailed analysis of factors that might affect W HL . From Eq. 6 , it can be seen that W HL is controlled by the interplay between the HOMO-LUMO gap HL and the dipole matrix element H r L . As shown in Fig. 4, HL for Sin Gem clusters of a given size does not exhibit any sensitive dependence on the Ge content. For Sin Gem H84 clusters, HL decreases with increasing Ge content m/(n m) and shows a linear dependence on the Ge content. However, the range of variation for HL is only by a factor of 1.22. On the other hand, the dependence of the dipole matrix element on m/(n m) shows a more complicated pattern see Table II . For example, H r L 2 1 2 HyL 2 H z L 2 ) for Sin Gem H84 3( H x L clusters shows a dramatic increase of two orders of magnitude once Ge atoms are incorporated into the Si cluster. It
peaks at the equal composition of Si and Ge atoms, and then reduces to a smaller but equal order of magnitude for the pure Ge cluster. Since the range of variation for H r L 2 3 over two orders of magnitude far surpasses that for HL less than one order of magnitude , it must be the behavior of H r L 2 that determines the pattern of behavior for W HL . As shown in Table I, W HL for Sin Gem H84 clusters indeed exhibits a similar pattern to that of H r L 2 , namely, a drastic, almost two orders-of- magnitude, increase for a small Ge content, peaking at an equal composition of Si and Ge atoms, reducing to a somewhat smaller value for the pure hydrogen-passivated Ge cluster. To shed light on the mechanism responsible for the dramatic increase in H r L 2 /W HL once Ge atoms are incorporated into the hydrogen-passivated Si clusters, we express the dipole matrix element in terms of the pseudoatomic orbitals within the reball scheme,22 namely,
HrL
i ,j
c iH *c L i j
rj
,
7
where c i denotes the coefcient of expansion of the wave function in the pseudoatomic orbital at the site i. The matrix element i r j depends only on the structural conguration of the cluster. Specically
i
rj
* r Ri r *r r
Ri r
r R j dr Ri j dr r Ri j dr , 8
*r
where (r Ri ) is the pseudoatomic orbital centered at atomic site i, Ri is the position vector of atomic site i, Ri j R j Ri , and r r Ri , respectively.
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FIG. 6. The absolute value of the coefcient c i vs the pseudoatomic orbital centered at atomic site i for the HOMO left column and LUMO states right column of Sin Gem H84 clusters. The numbers labeled at x axis denote the number of atomic site. At each atomic site, there are four histograms corresponding to the pseudoatomic orbital . The order of pseudo-atomic orbital is s, p x , p y , and p z , from left to right, respectively.
Since there is no substantial structural change for hydrogen-passivated Sin Gem Hl clusters of a given size (n m const), Eqs. 7 and 8 then indicate that the coefcients of expansion of the HOMO and LUMO states play the most signicant role in determining H r L . In Fig. 6, we plot the absolute value of the coefcient c i for the pseudoatomic orbital (r Ri ) vs atomic site i for the HOMO and LUMO states of hydrogen-passivated Sin Gem H84 (n m 71) clusters with various Ge contents. When the patterns of behavior for the hydrogen-passivated pure Si cluster (Si71Ge0 H84) are compared to those of the hydrogenpassivated pure Ge cluster (Si0 Ge71H84), it can be seen that the HOMO state for the two clusters shows similar distribution. However, there is a striking difference between the patterns exhibited by the LUMO states of the two clusters. In the case of the pure Si cluster, the expansion coefcients of the p x orbital give the major contribution to the LUMO states. On the other hand, two outstanding features differentiate the pattern of the LUMO state of the pure Ge cluster from that of the pure Si cluster: i The expansion coefcients of the s orbital give the major contribution to the LUMO state. ii The magnitude of the coefcients of the s orbital is gr...